Indian Regional Math Olympiad (RMO) 2016 - All Sets
Regional Math Olympiad (RMO) 2016 Telengana Region
Let $ABC$ be a right angled triangle with $\angle B=90^{\circ}$. Let $I$ be the incentre of $\triangle ABC$. Suppose $AI$ is extended to meet $BC$ at $F$ . The perpendicular on $AI$ at $I$ is extended to meet $AC$ at $E$ . Prove that $IE = IF$. Let $ABC$ be a right angled triangle with $\angle B=90^{\circ}$. Let $I$ be the incentre of $\triangle ABC$. Suppose $AI$ is extended to meet $BC$ at $F$ . The perpendicular on $AI$ at $I$ is extended to meet $AC$ at $E$ . Prove that $IE = IF$.
Let $a,b,c$ be positive real numbers such that $\frac{a}{1+a}+\frac{b}{1+b}+\frac{c}{1+c}=1$.Prove that $abc\leq\frac{1}{8}$.
For any natural number $n$, expressed in base $10$, let $S(n)E$ denote the sum of all digits of $n$. Find all positive integers $n$ such that $n^3$ = $8Sn^3$+$6Sn(n+1)$.
Find all $6$ digit natural numbers, which consist of only the digits $1,2$ and $3$, in which $3$ occurs exactly twice and the number is divisible by $9$.
Let $ABC$ be a right angled triangle with $\angle B=90^{\circ}$. Let $AD$ be the bisector of angle $A$ with $D$ on $BC$ . Let the circumcircle of $\triangle ACD$ intersect $AB$ again at $E$; and let the circumcircle of $\triangle ABD$ intersect $AC$ again at $F$ . Let $K$ be the reflection of $E$ in the line $BC$ . Prove that $FK = BC$.
Show that the infinite arithmetic progression {$1,4,7,10 \cdots$} has infinitely many 3 -term sub sequences in harmonic progression such that for any two such triples {$a_1, a_2 , a_3$ } and {$b_1, b_2 ,b_3$} in harmonic progression , one has$$\frac{a_1} {b_1} \neq \frac {a_2}{b_2}$$
Let $ABC$ be a triangle and $D$ be the mid-point of $BC$. Suppose the angle bisector of $\angle ADC$ is tangent to the circumcircle of triangle $ABD$ at $D$. Prove that $\angle A=90^{\circ}$. Let $ABC$ be a triangle and $D$ be the mid-point of $BC$. Suppose the angle bisector of $\angle ADC$ is tangent to the circumcircle of $\triangle ABD$ at $D$. Prove that $\angle A=90^{\circ}$.
Let $a,b,c$ be three distinct positive real numbers such that $abc=1$. Prove that $$\frac{a^3}{(a-b)(a-c)}+\frac{b^3}{(b-c)(b-a)}+\frac{c^3}{(c-a)(c-b)} \geq 3$$
Let $a,b,c,d,e,d,e,f$ be positive integers such that $\frac a b <$; $\frac c d <$; $\frac e f$. Suppose $af-be=-1$. Show that $d \geq b+f$.
There are $100$ countries participating in an olympiad. Suppose $n$ is a positive integers such that each of the $100$ countries is willing to communicate in exactly $n$ languages. If each set of $20$ countries can communicate in exactly one common language, and no language is common to all $100$ countries, what is the minimum possible value of $n$?
Let $ABC$ be a right-angled triangle with $\angle B=90^{\circ}$. Let $I$ be the incentre if $ABC$. Extend $AI$ and $CI$; let them intersect $BC$ in $D$ and $AB$ in $E$ respectively. Draw a line perpendicular to $AI$ at $I$ to meet $AC$ in $J$, draw a line perpendicular to $CI$ at $I$ to meet $AC$ at $K$. Suppose $DJ=EK$. Prove that $BA=BC$.
(a). Given any natural number $N$, prove that there exists a strictly increasing sequence of $N$ positive integers in harmonic progression. (b). Prove that there cannot exist a strictly increasing infinite sequence of positive integers which is in harmonic progression.
Regional Math Olympiad (RMO) 2016 Maharashtra Region
Find distinct positive integers $n_1<n_2<\cdots<n_7$ with the least possible sum, such that their product $n_1 \times n_2 \times \cdots \times n_7$ is divisible by $2016$. Find distinct positive integers $n_1<n_2<\cdots<n_7$ with the least possible sum, such that their product $n_1 \times n_2 \times \cdots \times n_7$ is divisible by $2016$.
At an international event there are $100$ countries participating, each with its own flag. There are $10$ distinct flagpoles at the stadium, labelled $1,2,...,10$ in a row. In how many ways can all the $100$ flags be hoisted on these $10$ flagpoles, such that for each $i$ from $1$ to $10$, the flagpole $i$ has at least $i$ flags? (Note that the vertical order of the flagpoles on each flag is important)
Find all integers $k$ such that all roots of the following polynomial are also integers:$$f(x)=x^3-(k-3)x^2-11x+(4k-8)$$.
Let $\triangle ABC$ be scalene, with $BC$ as the largest side. Let $D$ be the foot of the perpendicular from $A$ on side $BC$. Let points (K,L) be chosen on the lines $AB$ and $AC$ respectively, such that $D$ is the midpoint of segment $KL$. Prove that the points $B,K,C,L$ are concyclic if and only if $\angle BAC=90^{\circ}$.
Let $x,y,z$ be non-negative real numbers such that $xyz=1$. Prove that$$(x^3+2y)(y^3+2z)(z^3+2x) \geq 27.$$
$ABC$ is an equilateral triangle with side length $11$ units. Consider the points $P_1,P_2, \cdots, P_10$ dividing segment $BC$ into $11$ parts of unit length. Similarly, define $Q_1, Q_2, \cdots, Q_10$ for the side $CA$ and $R_1,R_2,\cdots, R_10$ for the side $AB$. Find the number of triples ($i,j,k$) with $i,j,k$ in {$1,2,\cdots,10$} such that the centroids of $\triangle ABC$ and $P_iQ_jR_k$ coincide.
Let $ABC$ be a right-angled triangle with $\angle B=90^{\circ}$. Let $I$ be the incenter of $ABC$. Draw a line perpendicular to $AI$ at $I$. Let it intersect the line $CB$ at $D$. Prove that $CI$ is perpendicular to $AD$ and prove that $ID=\sqrt{b(b-a)}$ where $BC=a$ and $CA=b$.
Let $a,b,c$ be positive real numbers such that$$\frac{a}{1+a}+\frac{b}{1+b}+\frac{c}{1+c}=1.$$Prove that $abc \leq \frac{1}{8}$.
For any natural number $n$, expressed in base $10$, let $S(n)$ denote the sum of all digits of $n$. Find all natural numbers $n$ such that $n=2S(n)^2$.
Find the number of all 6-digits numbers having exactly three odd and three even digits.
Let $ABC$ be a triangle with centroid $G$. Let the circumcircle of $\triangle AGB$ intersect the line $BC$ in $X$ different from (B); and the circucircle of triangle $AGC$ intersect the line $BC$ in $Y$ different from $C$. Prove that $G$ is the centroid of $\triangle AXY.
Let ($a_1,a_2,\cdots$) be a strictly increasing sequence of positive integers in arithmetic progression. Prove that there is an infinite sub-sequence of the given sequence whose terms are in a geometric progression.
Given are two circles $\omega_1,\omega_2$ which intersect at points $X,Y$. Let $P$ be an arbitrary point on $\omega_1$. Suppose that the lines $PX,PY$ meet $\omega_2$ again at points $A,B$ respectively. Prove that the circumcircles of all $\triangle PAB$ have the same radius.
Consider a sequence $(a_k)_{k \geq 1}$ of natural numbers defined as follows: $a_1=a$ and $a_2=b$ with $a,b>1$ and $gcd(a,b)=1$ and for all $k>0$, $a_{k+2}=a_{k+1}+a_k$. Prove that for all natural numbers $n$ and $k$, $gcd(a_n,a_{n+k})$ <$\frac{a_k}{2}$.
Two circles $C_1$ and $C_2$ intersect each other at points $A$ and $B$. Their external common tangent (closer to $B$) touches $C_1$ at $P$ and $C_2$ at $Q$. Let $C$ be the reflection of $B$ in line $PQ$. Prove that $\angle CAP=\angle BAQ$.
Let $a,b,c$ be positive real numbers such that $a+b+c=3$. Determine, with certainty, the largest possible value of the expression $$ \frac{a}{a^3+b^2+c}+\frac{b}{b^3+c^2+a}+\frac{c}{c^3+a^2+b}$$
a.) A 7-tuple $a_1,a_2,a_3,a_4,b_1,b_2,b_3$ of pairwise distinct positive integers with no common factor is called a shy tuple if $$ a_1^2+a_2^2+a_3^2+a_4^2=b_1^2+b_2^2+b_3^2$$and for all $1 \leq i<j \leq 4$ and $1 \leq k \leq 3$, $a_i^2+a_j^2 \neq b_k^2$. Prove that there exists infinitely many shy tuples. b.) Show that $2016$ can be written as a sum of squares of four distinct natural numbers.
A deck of $52$ cards is given. There are four suites each having cards numbered $1,2,\cdots, 13$. The audience chooses some five cards with distinct numbers written on them. The assistant of the magician comes by, looks at the five cards and turns exactly one of them face down and arranges all five cards in some order. Then the magician enters and with an agreement made beforehand with the assistant, he has to determine the face down card (both suite and number). Explain how the trick can be completed.
Indian Regional Math Olympiad (RMO) 2015 - All Sets
Regional Math Olympiad (RMO) 2015 - Paper 1
In a cyclic quadrilateral $A B C D,$ let the diagonals $A C$ and $B D$ intersect at $X$. Let the circumcircles of triangles $A X D$ and $B X C$ intersect again at $Y$. If $X$ is the incentre of triangle $A B Y,$ show that $\angle C A D=90^{\circ}$.
Let $P_{1}(x)=x^{2}+a_{1} x+b_{1}$ and $P_{2}(x)=x^{2}+a_{2} x+b_{2}$ be two quadratic polynomials with integer coefficients. Suppose $a_{1} \neq a_{2}$ and there exist integers $m \neq n$ such that $P_{1}(m)=P_{2}(n), P_{2}(m)=P_{1}(n) .$ Prove that $a_{1}-a_{2}$ is even.
Find all fractions which can be written simultaneously in the forms $\frac{7 k-5}{5 k-3}$ and $\frac{6 l-1}{4 l-3},$ for some integers $k, l$.
Suppose 28 objects are placed along a circle at equal distances. In how many ways can 3 objects be chosen from among them so that no two of the three chosen objects are adjacent nor diametrically opposite?
Let $A B C$ be a right triangle with $\angle B=90^{\circ} .$ Let $E$ and $F$ be respectively the mid-points of $A B$ and $A C$. Suppose the incentre $I$ of triangle $A B C$ lies on the circumcircle of triangle $A E F$. Find the ratio $B C / A B$.
Find all real numbers $a$ such that $3 \langle a \langle4$ and $a(a-3\{a\})$ is an integer. (Here $\{a\}$ denotes the fractional part of $a$. For example $\{1.5\}=0.5 ;\{-3.4\}$ $=0.6 .)$
Let $A B C$ be a triangle. Let $B^{\prime}$ and $C^{\prime}$ denote respectively the reflection of $B$ and $C$ in the internal angle bisector of $\angle A$. Show that the triangles $A B C$ and $A B^{\prime} C^{\prime}$ have the same incentre.
Let $P(x)=x^{2}+a x+b$ be a quadratic polynomial with real coefficients. Suppose there are real numbers $s \neq t$ such that $P(s)=t$ and $P(t)=s$. Prove that $b-s t$ is a root of the equation $x^{2}+a x+b-s t=0$
Find all integers $a, b, c$ such that
$$
a^{2}=b c+1, \quad b^{2}=c a+1
$$
Suppose 32 objects are placed along a circle at equal distances. In how many ways can 3 objects be chosen from among them so that no two of the three chosen objects are adjacent nor diametrically opposite?
Two circles $\Gamma$ and $\Sigma$ in the plane intersect at two distinct points $A$ and $B$, and the centre of $\Sigma$ lies on $\Gamma$. Let points $C$ and $D$ be on $\Gamma$ and $\Sigma$, respectively, such that $C, B$ and $D$ are collinear. Let point $E$ on $\Sigma$ be such that $D E$ is parallel to $A C .$ Show that $A E=A B$
Find all real numbers $a$ such that $4\langle a\langle 5$ and $a(a-3\{a\})$ is an integer. (Here $\{a\}$ denotes the fractional part of $a$. For example $\{1.5\}=0.5 ;\{-3.4\}$ $=0.6 .)$
Two circles $\Gamma$ and $\Sigma,$ with centres $O$ and $O^{\prime},$ respectively, are such that $O^{\prime}$ lies on $\Gamma$. Let $A$ be a point on $\Sigma$ and $M$ the midpoint of the segment $A O^{\prime}$. If $B$ is a point on $\Sigma$ different from $A$ such that $A B$ is parallel to $O M,$ show that the midpoint of $A B$ lies on $\Gamma$.
Let $P(x)=x^{2}+a x+b$ be a quadratic polynomial where $a$ and $b$ are real numbers. Suppose $\left\langle P(-1)^{2}, P(0)^{2}, P(1)^{2}\right\rangle$ is an arithmetic progression of integers. Prove that $a$ and $b$ are integers.
Show that there are infinitely many triples $(x, y, z)$ of integers such that $x^{3}+y^{4}=z^{31}$.
Suppose 36 objects are placed along a circle at equal distances. In how many ways can 3 objects be chosen from among them so that no two of the three chosen objects are adjacent nor diametrically opposite?
Let $A B C$ be a triangle with circumcircle $\Gamma$ and incentre $I .$ Let the internal angle bisectors of $\angle A, \angle B$ and $\angle C$ meet $\Gamma$ in $A^{\prime}, B^{\prime}$ and $C^{\prime}$ respectively. Let $B^{\prime} C^{\prime}$ intersect $A A^{\prime}$ in $P$ and $A C$ in $Q,$ and let $B B^{\prime}$ intersect $A C$ in $R$. Suppose the quadrilateral $PIRQ$ is a kite; that is, $I P=I R$ and $Q P=Q R$ Prove that $A B C$ is an equilateral triangle.
Show that there are infinitely many positive real numbers $a$ which are not integers such that $a(a-3\{a\})$ is an integer. (Here $\{a\}$ denotes the fractional part of $a$. For example $\{1.5\}=0.5 ;\{-3.4\}=0.6 .)$
The length of each side of a convex quadrilateral $A B C D$ is a positive integer. If the sum of the lengths of any three sides is divisible by the length of the remaining side then prove that some two sides of the quadrilateral have the same length.
Let $P(x)=x^{2}+a x+b$ be a quadratic polynomial where $a$ is real and $b$ is rational. Suppose $P(0)^{2}, P(1)^{2}, P(2)^{2}$ are integers. Prove that $a$ and $b$ are integers.
Find all integers $a, b, c$ such that
$$
a^{2}=b c+4, \quad b^{2}=c a+4
$$
Suppose 40 objects are placed along a circle at equal distances. In how many ways can 3 objects be chosen from among them so that no two of the three chosen objects are adjacent nor diametrically opposite?
Two circles $\Gamma$ and $\Sigma$ intersect at two distinct points $A$ and $B$. A line through $B$ intersects $\Gamma$ and $\Sigma$ again at $C$ and $D,$ respectively. Suppose that $C A=$ $C D$. Show that the centre of $\Sigma$ lies on $\Gamma$.
How many integers $m$ satisfy both the following properties:
(i) $1 \leq m \leq 5000$
(ii) $[\sqrt{m}]=[\sqrt{m+125}] ?$
(Here $[x]$ denotes the largest integer not exceeding $x,$ for any real number
x.)
Let $A B C D$ be a convex quadrilateral with $A B=a, B C=b, C D=c$ and $D A=d$. Suppose
$$
a^{2}+b^{2}+c^{2}+d^{2}=a b+b c+c d+d a
$$
and the area of $A B C D$ is 60 square units. If the length of one of the diagonals is 30 units, determine the length of the other diagonal.
Determine the number of $3$-digit numbers in base $1$0 having at least one $5$ and at most one $3$.
Let $P(x)$ be a non-constant polynomial whose coefficients are positive integers. If $P(n)$ divides $P(P(n)-2015)$ for every natural number $n,$ prove that $P(-2015)=0 .$
Find all three digit natural numbers of the form $(a b c)_{10}$ such that $(a b c)_{10},(b c a)_{10}$ and $(c a b)_{10}$ are in geometric progression. (Here $(a b c)_{10}$ is representation in base $\left.10 .\right)$
Let $A B C$ be a right-angled triangle with $\angle B=90^{\circ}$ and let $B D$ be the altitude from $B$ on to $A C .$ Draw $D E \perp A B$ and $D F \perp B C .$ Let $P, Q, R$ and $S$ be respectively the incentres of triangle $D F C, D B F, D E B$ and $D A E .$ Suppose $S, R, Q$ are collinear. Prove that $P, Q, R$, $D$ lie on a circle.
Let $S=\{1,2, \ldots, n\}$ and let $T$ be the set of all ordered triples of subsets of $S,$ say $\left(A_{1}, A_{2}, A_{3}\right)$ such that $A_{1} \cup A_{2} \cup A_{3}=S .$ Determine, in terms of $n$,
$$
\sum_{\left(A_{1}, A_{2}, A_{3}\right) \in T}\left|A_{1} \cap A_{2} \cap A_{3}\right|
$$
where $|X|$ denotes the number of elements in the set $X .$ (For example, if $S=\{1,2,3\}$ and $A_{1}=\{1,2\}, A_{2}=\{2,3\}, A_{3}=\{3\}$ then one of the elements of $T$ is $\left.(\{1,2\},\{2,3\},\{3\}) .\right)$
Let $x, y, z$ be real numbers such that $x^{2}+y^{2}+z^{2}-2 x y z=1$. Prove that
$$
(1+x)(1+y)(1+z) \leq 4+4 x y z
$$
The length of each side of a convex quadrilateral $A B C D$ is a positive integer. If the sum of the lengths of any three sides is divisible by the length of the remaining side then prove that some two sides of the quadrilateral have the same length.
Indian Regional Math Olympiad (RMO) 2014 - All Sets
Regional Math Olympiad (RMO) 2014 - Paper 1
Let $A B C$ be a triangle and let $A D$ be the perpendicular from $A$ on to $B C .$ Let $K, L, M$ be points on $A D$ such that $A K=K L=L M=M D$. If the sum of the areas of the shaded regions is equal to the sum of the areas of the unshaded regions, prove that $B D=D C$
Let $a_{1}, a_{2}, \ldots, a_{2 n}$ be an arithmetic progression of positive real numbers with common difference d. Let
(i) $a_{1}^{2}+a_{3}^{2}+\cdots+a_{2 n-1}^{2}=x$
(ii) $a_{2}^{2}+a_{4}^{2}+\cdots+a_{2 n}^{2}=y,$ and
(iii) $a_{n}+a_{n+1}=z$.
Express $d$ in terms of $x, y, z, n .$
Suppose for some positive integers $r$ and $s$, the digits of $2^{r}$ is obtained by permuting the digits of $2^{x}$ in decimal expansion. Prove that $r=8$.
Is it possible to write the numbers $17,18,19, \ldots, 32$ in a $4 \times 4$ grid of unit squares, with one number in each square, such that the product of the numbers in each $2 \times 2$ sub-grids AMRG, GRND, MBHR and $R H C N$ is divisible by $16 ?$
Let $A B C$ be an acute-angled triangle and let $H$ be its ortho-centre. For any point $P$ on the circum-circle of triangle $A B C$, let $Q$ be the point of intersection of the line $B H$ with the line $A P$. Show that there is a unique point $X$ on the circum-circle of $A B C$ such that for every point $P \neq A, B$. the circum-circle of $H Q P$ pass through $X$.
Let $x_{1}, x_{2}, \ldots, x_{2014}$ be positive real numbers such that $\sum_{j=1}^{2014} x_{j}=1 .$ Determine with proof the smallest constant $K$ such that
$$
K \sum_{j=1}^{2014} \frac{x_{j}^{2}}{1-x_{j}} \geq 1
$$
In an acute-angled triangle $A B C, \angle A B C$ is the largest angle. The perpendicular bisectors of $B C$ and $B A$ intersect $A C$ at $X$ and $Y$ respectively. Prove that circumcentre of triangle $A B C$ is incentre of triangle $B X Y$.
Let $x, y, z$ be positive real numbers. Prove that
$$
\frac{y^{2}+z^{2}}{x}+\frac{z^{2}+x^{2}}{y}+\frac{x^{2}+y^{2}}{z} \geq 2(x+y+z)
$$
$\text { Find all pairs of }(x, y) \text { of positive integers such that } 2 x+7 y \text { divides } 7 x+2 y \text { . }$
For any positive integer $n>1,$ let $P(n)$ denote the largest prime not exceeding $n$. Let $N(n)$ denote the next prime larger than $P(n)$. (For example $P(10)=7$ and $N(10)=11,$ while $P(11)=11$ and $N(11)=13 .)$ If $n+1$ is a prime number, prove that the value of the sum
$$
\frac{1}{P(2) N(2)}+\frac{1}{P(3) N(3)}+\frac{1}{P(4) N(4)}+\cdots+\frac{1}{P(n) N(n)}=\frac{n-1}{2 n+2}
$$
Let $A B C$ be a triangle with $A B>A C$. Let $P$ be a point on the line $A B$ beyond $A$ such that $A P+P C=A B .$ Let $M$ be the mid-point of $B C$ and let $Q$ be the point on the side $A B$ such that $C Q \perp A M .$ Prove that $B Q=2 A P$.
Let $n$ be an odd positive integer and suppose that each square of an $n \times n$ grid is arbitrarily filled with either by 1 or by $-1 .$ Let $r_{j}$ and $c_{k}$ denote the product of all numbers in $j$ -th row and $k$ -th column respectively, $1 \leq j, k \leq n$. Prove that
$$
\sum_{j=1}^{n} r_{j}+\sum_{k=1}^{n} c_{k} \neq 0
$$
Let $A B C$ be an acute-angled triangle and suppose $\angle A B C$ is the largest angle of the triangle. Let $R$ be its circumcentre. Suppose the circumcircle of triangle $A R B$ cuts $A C$ again in $X .$ Prove that $R X$ is pependicular to $B C$
Find all real numbers $x$ and $y$ such that
$$
x^{2}+2 y^{2}+\frac{1}{2} \leq x(2 y+1)
$$
Prove that there does not exist any positive integer $n<2310$ such that $n(2310-n)$ is a multiple of 2310
Find all positive real numbers $x, y, z$ such that
$$
2 x-2 y+\frac{1}{z}=\frac{1}{2014}, \quad 2 y-2 z+\frac{1}{x}=\frac{1}{2014}, \quad 2 z-2 x+\frac{1}{y}=\frac{1}{2014}
$$
Let $A B C$ be a triangle. Let $X$ be on the segment $B C$ such that $A B=A X$. Let $A X$ meet the circumcircle $\Gamma$ of triangle $A B C$ again at $D .$ Show that the circumcentre of $\triangle B D X$ lies on $\Gamma$.
For any natural number $n$, let $S(n)$ denote the sum of the digits of $n$. Find the number of all 3 -digit numbers $n$ such that $S(S(n))=2$.
Let $ABCD$ be an isosceles trapezium having an incircle; let $AB$ and $CD$ be
the parallel sides and let $CE$ be the perpendicular from $C$ on to $AB$. Prove
that $CE$ is equal to the geometric mean of $AB$ and $CD$.
If $x$ and $y$ are positive real numbers, prove that
$$
4 x^{4}+4 y^{3}+5 x^{2}+y+1 \geq 12 x y
$$
Determine all pairs $m>n$ of positive integers such that
$$
1=\text{gcd}(n+1, m+1)=\text{gcd}(n+2, m+2)=\cdots=\text{gcd}(m, 2 m-n)
$$
$\text { What is the minimal area of a right-angled triangle whose inradius is } 1 \text { unit? } $
Let $A B C$ be an acute-angled triangle and let $I$ be its incentre. Let the incircle of triangle $A B C$ touch $B C$ in $D .$ The incircle of the triangle $A B D$ touches $A B$ in $E$; the incircle of the triangle $A C D$ touches $B C$ in $F$. Prove that $B, E, I, F$ are concyclic.
In the adjacent figure, can the numbers $1,2,3,4, \cdots, 18$ be placed, one on each line segment, such that the sum of the numbers on the three line segments meeting at each point is divisible by $3 ?$
Three positive real numbers $a, b, c$ are such that $a^{2}+5 b^{2}+4 c^{2}-4 a b-4 b c=0 .$ Can $a, b, c$ be the lengths of the sides of a triangle? Justify your answer.
The roots of the equation
$$
x^{3}-3 a x^{2}+b x+18 c=0
$$
form a non-constant arithmetic progression and the roots of the equation
$$
x^{3}+b x^{2}+x-c^{3}=0
$$
form a non-constant geometric progression. Given that $a, b, c$ are real numbers, find all positive integral values $a$ and $b$.
Let $A B C$ be an acute-angled triangle in which $\angle A B C$ is the largest angle. Let $O$ be its circumoentre. The perpendicular bisectors of $B C$ and $A B$ meet $A C$ at $X$ and $Y$ respectively. The internal bisectors of $\angle A X B$ and $\angle B Y C$ meet $A B$ and $B C$ at $D$ and $E$ respectively. Prove that $B O$ is perpendicular to $A C$ if $D E$ is parallel to $A C$.
A person moves in the $x-y$ plane moving along points with integer co-ordinates $x$ and $y$ only. When she is at point $(x, y),$ she takes a step based on the following rules:
(a) if $x+y$ is even she moves to either $(x+1, y)$ or $(x+1, y+1)$;
(b) if $x+y$ is odd she moves to either $(x, y+1)$ or $(x+1, y+1)$.
How many distinct paths can she take to go from (0,0) to (8,8) given that she took exactly three steps to the right $((x, y)$ to $(x+1, y)) ?$
Let $a, b, c$ be positive numbers such that
$$
\frac{1}{1+a}+\frac{1}{1+b}+\frac{1}{1+c} \leq 1
$$
Prove that $\left(1+a^{2}\right)\left(1+b^{2}\right)\left(1+c^{2}\right) \geq 125 .$ When does the equality hold?
Let $D, E, F$ be the points of contact of the incircle of an acute-angled triangle $A B C$ with $B C, C A, A B$ respectively. Let $I_{1}, I_{2}, I_{3}$ be the incentres of the triangles $A F E, B D F, C E D$ respectively. Prove that the lines $I_{1} D, I_{2} E, I_{3} F$ are concurrent.
Indian Regional Math Olympiad (RMO) 2013 - All Sets
CRMO 2013 - Paper 1
Let $A B C$ be an acute-angled triangle. The circle $\Gamma$ with $B C$ as diameter intersects $A B$ and $A C$ again at $P$ and $Q,$ respectively. Determine $\angle B A C$ given that the orthocentre of triangle $A P Q$ lies on $\Gamma$.
Let $f(x)=x^{3}+a x^{2}+b x+c$ and $g(x)=x^{3}+b x^{2}+c x+a,$ where $a, b, c$ are integers with $c \neq 0$. Suppose that the following conditions hold:
(a) $f(1)=0$;
(b) the roots of $g(x)=0$ are the squares of the roots of $f(x)=0$.
Find the value of $a^{2013}+b^{2013}+c^{2013}$.
Find all primes $p$ and $q$ such that $p$ divides $q^{2}-4$ and $q$ divides $p^{2}-1$.
Find the number of 10 -tuples $\left(a_{1}, a_{2}, \ldots, a_{10}\right)$ of integers such that $\left|a_{1}\right| \leq 1$ and
$$
a_{1}^{2}+a_{2}^{2}+a_{3}^{2}+\cdots+a_{10}^{2}-a_{1} a_{2}-a_{2} a_{3}-a_{3} a_{4}-\cdots-a_{9} a_{10}-a_{10} a_{1}=2
$$.
Let $A B C$ be a triangle with $\angle A=90^{\circ}$ and $A B=A C .$ Let $D$ and $E$ be points on the segment $B C$ such that $B D: D E: E C=3: 5: 4$. Prove that $\angle D A E=45^{\circ} .$
Suppose that $m$ and $n$ are integers such that both the quadratic equations $x^{2}+m x-n=0$ and $x^{2}-m x+n=0$ have integer roots. Prove that $n$ is divisible by 6.
Prove that there do not exist natural numbers $x$ and $y,$ with $x>1,$ such that
$$
\frac{x^{7}-1}{x-1}=y^{5}+1
$$.
In a triangle $A B C, A D$ is the altitude from $A,$ and $H$ is the orthocentre. Let $K$ be the centre of the circle passing through $D$ and tangent to $B H$ at $H .$ Prove that the line $D K$ bisects $A C .$
Consider the expression
$$
2013^{2}+2014^{2}+2015^{2}+\cdots+n^{2}
$$
Prove that there exists a natural number $n>2013$ for which one can change a suitable number of plus signs to minus signs in the above expression to make the resulting expression equal 9999.
Let $A B C$ be a triangle with $\angle A=90^{\circ}$ and $A B=A C .$ Let $D$ and $E$ be points on the segment $B C$ such that $B D: D E: E C=1: 2: \sqrt{3}$. Prove that $\angle D A E=45^{\circ} .$
For positive integers n, define A(n) to be \( \frac{(2n)!}{(n!)^2} \).Determine the sets of positive integers n for which (a) A(n) is an even number, (b) A(n) is a multiple of 4.
Let $n \geq 3$ be a natural number and let $P$ be a polygon with $n$ sides. Let $a_{1}, a_{2}, \ldots, a_{n}$ be the lengths of the sides of $P$ and let $p$ be its perimeter. Prove that
$$
\frac{a_{1}}{p-a_{1}}+\frac{a_{2}}{p-a_{2}}+\cdots+\frac{a_{n}}{p-a_{n}}<2.
$$
For a natural number $n$, let $T(n)$ denote the number of ways we can place $n$ objects of weights $1,2, \ldots, n$ on a balance such that the sum of the weights in each pan is the same. Prove that $T(100)>T(99)$
Prove that the polynomial $f(x)=x^{4}+26 x^{3}+56 x^{2}+78 x+1989$ cannot be expressed as a product $f(x)=p(x) q(x)$, where $p(x), q(x)$ are both polynomials with integral coefficients and with degree less than 4.
Find all 4 -tuples $(a, b, c, d)$ of natural numbers with $a \leq b \leq c$ and $a !+b !+c !=3^{d}$.
In an acute-angled triangle $A B C$ with $AB$ < $AC$, the circle $\Gamma$ touches $A B$ at $B$ and passes through $C$ intersecting $A C$ again at $D $. Prove that the orthocentre of triangle $A B D$ lies on $\Gamma$ if and only if it lies on the perpendicular bisector of $B C$.
A polynomial is called a Fermat polynomial if it can be written as the sum of the squares of two polynomials with integer coefficients. Suppose that $f(x)$ is a Fermat polynomial such that $f(0)=1000 .$ Prove that $f(x)+2 x$ is not a Fermat polynomial.
Let $A B C$ be a triangle which it not right-angled. Define a sequence of triangles $A_{i} B_{i} C_{i}$, with $i \geq 0$, as follows: $A_{0} B_{0} C_{0}$ is the triangle $A B C ;$ and, for $i \geq 0, A_{i+1}, B_{i+1}, C_{i+1}$ are the reflections of the orthocentre of triangle $A_{i} B_{i} C_{i}$ in the sides $B_{i} C_{i}, C_{i} A_{i}, A_{i} B_{i},$ respectively. Assume that $\angle A_{m}=\angle A_{n}$ for some distinct natural numbers $m, n .$ Prove that $\angle A=60^{\circ} .$
Let $n \geq 4$ be a natural number. Let $A_{1} A_{2} \cdots A_{n}$ be a regular polygon and $X=\{1,2, \ldots, n\}$. A subset $\left\{i_{1}, i_{2}, \ldots, i_{k}\right\}$ of $X,$ with $k \geq 3$ and $i_{1}$ < $i_{2}<\cdots$ < $i_{k},$ is called a good subset if the angles of the polygon $A_{i_{1}} A_{i_{2}} \cdots A_{i_{k}},$ when arranged in the increasing order, are in an arithmetic progression. If $n$ is a prime, show that a proper good subset of $X$ contains exactly four elements.
Let $\Gamma$ be a circle with centre $O .$ Let $\Lambda$ be another circle passing through $O$ and intersecting $\Gamma$ at points $A$ and $B$. A diameter $C D$ of $\Gamma$ intersects $\Lambda$ at a point $P$ different from $O .$ Prove that
$$
\angle A P C=\angle B P D
$$.
Determine the smallest prime that does not divide any five-digit number whose digits are in a strictly increasing order.
Given real numbers $a, b, c, d, e>1$ prove that
$$
\frac{a^{2}}{c-1}+\frac{b^{2}}{d-1}+\frac{c^{2}}{e-1}+\frac{d^{2}}{a-1}+\frac{e^{2}}{b-1} \geq 20
$$.
Let $x$ be a non-zero real number such that $x^{4}+\frac{1}{x^{4}}$ and $x^{5}+\frac{1}{x^{5}}$ are both rational numbers. Prove that $x+\frac{1}{x}$ is a rational number.
In a triangle $A B C,$ let $H$ denote its orthocentre. Let $P$ be the reflection of $A$ with respect to $B C .$ The circumcircle of triangle $A B P$ intersects the line $B H$ again at $Q,$ and the circumcircle of triangle $A C P$ intersects the line $C H$ again at $R$. Prove that $H$ is the incentre of triangle $P Q R$.
Suppose that the vertices of a regular polygon of 20 sides are coloured with three colours red, blue and green - such that there are exactly three red vertices. Prove that there are three vertices $A, B, C$ of the polygon having the same colour such that triangle $A B C$ is isosceles.
Let $A B C$ be an isosceles triangle with $A B=A C$ and let $\Gamma$ denote its circumcircle. A point $D$ is on arc $A B$ of $\Gamma$ not containing $C .$ A point $E$ is on arc $A C$ of $\Gamma$ not containing $B$. If $A D=C E$ prove that $B E$ is parallel to $A D .$
Find all triples $(p, q, r)$ of primes such that $p q=r+1$ and $2\left(p^{2}+q^{2}\right)=r^{2}+1 .$
A finite non-empty set of integers is called 3 -good if the sum of its elements is divisible by 3 . Find the number of non-empty 3-good subsets of $\{0,1,2, \ldots, 9\}$.
In a triangle $A B C,$ points $D$ and $E$ are on segments $B C$ and $A C$ such that $B D=3 D C$ and $A E=4 E C .$ Point $P$ is on line $E D$ such that $D$ is the midpoint of segment $E P .$ Lines $A P$ and $B C$ intersect at point $S .$ Find the ratio $B S / S D$.
Let $a_{1}, b_{1}, c_{1}$ be natural numbers. We define
$$
a_{2}={gcd}\left(b_{1}, c_{1}\right), \quad b_{2}=g c d\left(c_{1}, a_{1}\right), \quad c_{2}={gcd}\left(a_{1}, b_{1}\right)
$$
and
$$
a_{3}={lcm}\left(b_{2}, c_{2}\right), \quad b_{3}={lcm}\left(c_{2}, a_{2}\right), \quad c_{3}={lcm}\left(a_{2}, b_{2}\right)
$$
Show that $g c d\left(b_{3}, c_{3}\right)=a_{2}$.
Let $P(x)=x^{3}+a x^{2}+b$ and $Q(x)=x^{3}+b x+a,$ where $a, b$ are non-zero real numbers. Suppose that the roots of the equation $P(x)=0$ are the reciprocals of the roots of the equation $Q(x)=0$. Prove that $a$ and $b$ are integers. Find the greatest common divisor of $P(2013 !+1)$ and $Q(2013 !+1)$.
Indian Regional Math Olympiad (RMO) 2012 - All Sets
CRMO 2012 - Paper 1
Let $A B C$ be an acute-angled triangle. The circle $\Gamma$ with $B C$ as diameter intersects $A B$ and $A C$ again at $P$ and $Q,$ respectively. Determine $\angle B A C$ given that the orthocentre of triangle $A P Q$ lies on $\Gamma$
Let $f(x)=x^{3}+a x^{2}+b x+c$ and $g(x)=x^{3}+b x^{2}+c x+a,$ where $a, b, c$ are integers with $c \neq 0$. Suppose that the following conditions hold:
(a) $f(1)=0$;
(b) the roots of $g(x)=0$ are the squares of the roots of $f(x)=0$.
Find the value of $a^{2013}+b^{2013}+c^{2013}$
Find all primes $p$ and $q$ such that $p$ divides $q^{2}-4$ and $q$ divides $p^{2}-1$
Find the number of 10 -tuples $\left(a_{1}, a_{2}, \ldots, a_{10}\right)$ of integers such that $\left|a_{1}\right| \leq 1$ and
$$
a_{1}^{2}+a_{2}^{2}+a_{3}^{2}+\cdots+a_{10}^{2}-a_{1} a_{2}-a_{2} a_{3}-a_{3} a_{4}-\cdots-a_{9} a_{10}-a_{10} a_{1}=2
$$
Let $A B C$ be a triangle with $\angle A=90^{\circ}$ and $A B=A C .$ Let $D$ and $E$ be points on the segment $B C$ such that $B D: D E: E C=3: 5: 4$. Prove that $\angle D A E=45^{\circ} .$
Suppose that $m$ and $n$ are integers such that both the quadratic equations $x^{2}+m x-n=0$ and $x^{2}-m x+n=0$ have integer roots. Prove that $n$ is divisible by 6.
Let $A B C D$ be a unit square. Draw a quadrant of a circle with $A$ as centre and $B, D$ as end points of the arc. Similarly, draw a quadrant of a circle with $B$ as centre and $A, C$ as end points of the arc. Inscribe a circle $\Gamma$ touching the arc $A C$ internally, the arc $B D$ internally and also touching the side $A B .$ Find the radius of the circle $\Gamma$.
Let $a, b, c$ be positive integers such that $a$ divides $b^{4}, b$ divides $c^{4}$ and $c$ divides $a^{4}$. Prove that $a b c$ divides $(a+b+c)^{21}$
Let $a$ and $b$ be positive real numbers such that $a+b=1$. Prove that
$$
a^{a} b^{b}+a^{b} b^{a} \leq 1
$$
Let $X=\{1,2,3, \ldots, 12\} .$ Find the the number of pairs $\{A, B\}$ such that $A \subseteq X$, $B \subseteq X, A \neq B$ and $A \cap B=\{2,3,5,7,8\} .$
Let $A B C$ be a triangle. Let $D, E$ be a points on the segment $B C$ such that $B D=$ $D E=E C .$ Let $F$ be the mid-point of $A C .$ Let $B F$ intersect $A D$ in $P$ and $A E$ in $Q$ respectively. Determine $B P / P Q$.
Show that for all real numbers $x, y, z$ such that $x+y+z=0$ and $x y+y z+z x=-3$, the expression $x^{3} y+y^{3} z+z^{3} x$ is a constant.
Let $A B C D$ be a unit square. Draw a quadrant of a circle with $A$ as centre and $B, D$ as end points of the arc. Similarly, draw a quadrant of a circle with $B$ as centre and $A, C$ as end points of the arc. Inscribe a circle $\Gamma$ touching the arcs $A C$ and $B D$ both externally and also touching the side $C D .$ Find the radius of the circle $\Gamma$.
Let $a, b, c$ be positive integers such that $a$ divides $b^{5}, b$ divides $c^{5}$ and $c$ divides $a^{5}$. Prove that $a b c$ divides $(a+b+c)^{31}$.
Let $a$ and $b$ be positive real numbers such that $a+b=1$. Prove that
$$
a^{a} b^{b}+a^{b} b^{a} \leq 1
$$
Let $X=\{1,2,3, \ldots, 10\} .$ Find the the number of pairs $\{A, B\}$ such that $A \subseteq X$, $B \subseteq X, A \neq B$ and $A \cap B=\{5,7,8\}$
Let $A B C$ be a triangle. Let $D, E$ be a points on the segment $B C$ such that $B D=$ $D E=E C .$ Let $F$ be the mid-point of $A C .$ Let $B F$ intersect $A D$ in $P$ and $A E$ in $Q$ respectively. Determine the ratio of the area of the triangle $A P Q$ to that of the quadrilateral $P D E Q$.
Find all positive integers $n$ such that $3^{2 n}+3 n^{2}+7$ is a perfect square.
Let $A B C D$ be a unit square. Draw a quadrant of a circle with $A$ as centre and $B, D$ as end points of the arc. Similarly, draw a quadrant of a circle with $B$ as centre and $A, C$ as end points of the arc. Inscribe a circle $\Gamma$ touching the arc $A C$ externally, the arc $B D$ internally and also touching the side $A D .$ Find the radius of the circle $\Gamma$
Let $a, b, c$ be positive integers such that $a$ divides $b^{2}, b$ divides $c^{2}$ and $c$ divides $a^{2}$. Prove that $a b c$ divides $(a+b+c)^{7}$.
Let $a$ and $b$ be positive real numbers such that $a+b=1 .$ Prove that
$$
a^{a} b^{b}+a^{b} b^{a} \leq 1
$$
Let $X=\{1,2,3, \ldots, 11\} .$ Find the the number of pairs $\{A, B\}$ such that $A \subseteq X$, $B \subseteq X, A \neq B$ and $A \cap B=\{4,5,7,8,9,10\}$
Let $A B C$ be a triangle. Let $E$ be a point on the segment $B C$ such that $B E=2 E C .$ Let $F$ be the mid-point of $A C .$ Let $B F$ intersect $A E$ in $Q .$ Determine $B Q / Q F$.
Solve the system of equations for positive real numbers:
$$
\frac{1}{x y}=\frac{x}{z}+1, \quad \frac{1}{y z}=\frac{y}{x}+1, \quad \frac{1}{z x}=\frac{z}{y}+1
$$
Let $ABC$ be an acute-angled triangle; $AD$ be the bisector of $\angle BAC$ with $D$ on $BC$; and $BE$ be the altitude from $B$ on $AC$. Show that $\angle CED > 45^{\circ}$.
Let $a,b,c$ be three natural numbers such that ( $a\leq b\leq c$ ) and ( $gcd(c-a,c-b)=1$ ). Suppose that there exits an integer $d$ such that $a+d,b+d,c+d$ form the sides of a right-angled triangle. Prove that there exist integers $l,m$ such that $c+d=l^2+m^2$.
Find all pairs ($a,b$) of real numbers such that whenever $\alpha$ is a root $x^2+ax+b=0$, $\alpha^2-1$ is also a root of the equation.
How many $6$-digits numbers are there such that: (a) the digits of all the numbers are from the set { $1,2,3,.... $}; (b) any digits that appears in the number appears twice? (Example: $225252$ is an admissible number, while $222133$ is not.)
A trapezium $ABCD$, in which $AB$ is parallel to $CD$, is inscribed in a circle with centre $O$. Suppose the diagonal $AC$ and $BD$ of the trapezium intersect at $M$ and $OM=2$. (a) If $\angle AMB$ is determine with proof the difference between the length between the parallel sides. (b) If $\angle AMD$ is find the difference between the parallel sides.
Prove that: (a) $5 < 5^{\frac{1}{2}}+5^{\frac{1}{3}}+5^{\frac{1}{4}}$; (b) $8 < 8^{\frac{1}{2}}+8^{\frac{1}{3}}+8^{\frac{1}{4}}$; (c) $n < n^{\frac{1}{2}}+n^{\frac{1}{3}}+n^{\frac{1}{4}} $ for all integers greater than or equal to $9$.
Let $ABC$ be an acute-angled triangle and let $D, E, F$ be the feet of perpendiculars from $A,B,C$ respectively to $BC,CA,AB$. Let the perpendiculars from $F$ to $CB, CA, AD, BE$ meet them in $P, Q,M,N$ respectively. Prove that $P, Q,M,N$ are collinear.
Find the least possible value of $a + b$, where $a, b$ are positive integers such that $11$ divides $a + 13b$ and $13$ divides $a + 11b$.
If $a, b, c$ are three positive real numbers, prove that $\frac{a^2+1}{b+c}+\frac{b^2+1}{c+a}+\frac{c^2+1}{a+b} \geq 3 $.
A $6×6$ square is dissected into $9$ rectangles by lines parallel to its sides such that all these rectangles have only integer sides. Prove that there are always two congruent rectangles.
Let $ABCD$ be a quadrilateral in which $AB$ is parallel to $CD$ and perpendicular to $AD$; $AB = 3CD$; and the area of the quadrilateral is $4$. If a circle can be drawn touching all the sides of the quadrilateral, find its radius.
Prove that there are infinitely many positive integers $n$ such that $n(n+ 1)$ can be expressed as a sum of two positive squares in at least two different ways. (Here $a^2+b^2$ and $b^2+a^2$ are considered as the same representation.)
Let $X$ be the set of all positive integers greater than or equal to $8$ and let $f:X \mapsto X $ be a function such that $f(x + y) = f(xy)$ for all $x \geq 4 $, $y \geq 4 $. If $f(8) = 9$, determine $f(9)$.
Let $ABCD$ be a convex quadrilateral; $P, Q,R, S$ be the midpoints of $AB,BC,CD,DA$ respectively such that $\triangle AQR$ and $\triangle CSP$ are equilateral. Prove that $ABCD$ is a rhombus. Determine its angles.
If $x, y$ are integers and $17$ divides both the expressions $x^2-2xy-y^2-5x+7y$ and $x^2-3xy+2y^2+x-y$,then prove that $17$ divides $xy − 12x + 15y$.
If $a, b, c$ are three real numbers such that $|a-b| \geq c,|b-c| \geq a,|c-a| \geq b $, then prove that one of $a, b, c$ is the sum of the other two.
Find the number of all $5$-digit numbers (in base $10$) each of which contains the block $15$ and is divisible by $15$. (For example, $31545$, $34155$ are two such numbers.)
In $\triangle ABC$, let $D$ be the midpoint of $BC$. If $\angle ADB = 45^{\circ} $ and $\angle ACD = 30^{\circ} $, determine $\angle BAD $.
Determine all triples ($a, b, c$) of positive integers such that $a \leq b \leq c $ and $a + b + c + ab + bc + ca = abc + 1$.
Let $a, b, c$ be three positive real numbers such that $a + b + c = 1$.Let $\gamma$=min{ $a^3+a^2bc,b^3+ab^2c,c^3+abc^2$}.Prove that the roots of the equation $x^2+x+4 \gamma$ are real.
Consider in the plane a circle $\Gamma$ with center $O$ and a line $l$ not intersecting circle $\Gamma$. Prove that there is a point $Q$ on the perpendicular drawn from $O$ to the line $l$, such that for any point $P$ on the line $l$, $PQ$ represents the length of the tangent from $P$ to the circle $\Gamma$.
Positive integers are written on all the faces of a cube, one on each. At each corner (vertex) of the cube, the product of the numbers on the faces that meet at the corner is written. The sum of the numbers written at all the corners is $2004$. If $T$ denotes the sum of the numbers on all the faces, find all the possible values of $T$.
Let $\alpha$ and $\beta$ be the roots of the quadratic equation $x^2+mx-1$.where $m$ is an odd integer. Let $\gamma_m = \alpha^n + \beta^n$,for $n \geq 0$. Prove that for $n \geq 0$, (a) is an integer and (b) $gcd(\gamma_n, \gamma_{n+1})=1$.
Prove that the number of triples ($A,B,C$) where $A,B,C$ are subsets of {$1, 2, · · · , n$} such that $A \cap B \cap C = \phi$, $A \cap B \neq \phi$, $B \cap C = \phi$ is $7^n+2.6^n+5^n$.
Let $ABCD$ be a quadrilateral ; $X$ and $Y$ be the midpoints of $AC$ and $BD$ respectively ; and the lines through $X$ and $Y$ respectively parallel to BD,AC meet in $O$. Let $P, Q,R, S$ be the midpoints of $AB,BC,CD,DA$ respectively. Prove that (a) quadrilaterals $APOS$ and $APXS$ have the same area ; (b) the areas of the quadrilaterals $APOS,BQOP,CROQ,DSOR$ are all equal .
Let $(p_1p_2p_3.....p_n....)$ be a sequence of primes defined by $p_1=2$ and for $n \geq 1$, $p_{n+1}$ is the largest prime factor of $p_1p_2p_3.....p_n+1$ (Thus $p_2=3,p_3=7$). Prove that $p_n \neq 5$ for any $n$.
Let $x$ and $y$ be positive real numbers such that $y^3+y \leq x-x^3$.Prove that (a) $y < x < 1$; and (b). $x^2+y^2 \leq 1$.
Let $ABC$ be a triangle in which $AB = AC$ and $\angle CAB = 90^{\circ}$. Suppose $M$ and $N$ are points on the hypotenuse $BC$ such that $BM^2 +CN^2= MN^2$. Prove that $\angle MAN = 45^{\circ}$.
If $n$ is an integer greater than $7$, prove that ( ${n} \choose {7}$ - $l$ floor $\frac{n}{7}$ $r$ floor ) is divisible by $7$. [ Here $\frac{n}{7}$ denotes the number of ways of choosing $7$ objects from among $n$ objects; also for any real number $x$, $[x]$ denotes the greatest integer not exceeding $x$.
Let $a, b, c$ be three positive real numbers such that $a + b + c = 1$. Prove that among the three numbers $a − ab, b − bc, c − ca$ there is one which is at most $\frac{1}{4}$ and there is one which is at least $\frac{2}{9}$.
Find the number of ordered triples ($x, y, z$) of nonnegative integers satisfying the conditions: (i) $x \leq y \leq z$, (ii) $x+y+z \leq 100$.
Suppose $P$ is an interior point of a $\triangle ABC$ such that the ratios $d(A,BC)$ $\frac{d(A,BC)}{d(P,BC)} , \frac{d(B,CA)}{d(P,CA)},\frac{d(C,AB)}{d(P,AB)}$ are all equal. Find the common value of these ratios. [ Here $d(X, Y Z)$ denotes the perpendicular distance from a point $X$ to the line $Y Z$.].
Find all real numbers a for which the equation $x^2+(a-1)x+1=3|x|$ has exactly three distinct real solutions in $x$.
Consider the set $X = {1, 2, 3, · · · , 9, 10}$. Find two disjoint nonempty subsets $A$ and $B$ of $X$ such that. (a) $A \cup B = X $; (b) $\prod(A)$ is divisible by $\prod(B)$, where for any finite set of numbers $C$, $\prod(C)$ denotes the product of all numbers in $C$; (c)the quotient $\prod(A)/ \prod(B)$ is as small as possible.
In an acute $\triangle ABC$, points $D;E; F$ are located on the sides $BC;CA;AB$ respectively such that $\frac{CD}{CE}=\frac{CA}{CB},\frac{AE}{AF}=\frac{AB}{AC},\frac{BF}{BD}=\frac{BC}{BA}$ Prove that $AD;BE;CF$ are the altitudes of $ABC$.
Solve the following equation for real $x$: $(x^2+x-2)^3+(2x^2-x-1)^3=27(x^2-1)^3$.
Let $a; b; c$ be positive integers such that a divides $b^2$, $b$ divides $c^2$ and $c$ divides $a^2$. Prove that abc divides $la(a+b+c) $.
Suppose the integers $1; 2; 3;...; 10$ are split into two disjoint collections $a_1a_2a_3a_4a_5$ and $b_1b_2b_3b_4b_5$ such that $a_1 < a_2 < a_3 < a_4, b_2 > b_3 > b_4 > b_5 $ (i) Show that the larger number in any pair { $a_i$, $b_j $}, $1 \leq j \leq 5$ is at least $6$. (ii) Show that $|a_1-b_1|+|a_2-b_2|+|a_3-b_3|+|a_4-b_4|+|a_5-b_5|=25$ for every such partition.
The circumference of a circle is divided into eight arcs by a convex quadrilateral $ABCD$, with four arcs lying inside the quadrilateral and the remaining four lying outside it. The lengths of the arcs lying inside the quadrilateral are denoted by $p, q, r, s$ in counter-clockwise direction starting from some arc. Suppose $p + r = q + s$. Prove that $ABCD$ is a cyclic quadrilateral.
For any natural number $n> 1$, prove the inequality: $\frac{1}{2} \leq \frac{1}{1+n^2} + \frac{2}{2+n^2} + \frac{3}{3+n^2} +.....+ \frac{n}{n+n^2} \leq \frac{1}{2} + \frac{1}{2n}$
Find all integers $a; b; c; d$ satisfying the following relations : (i) $1 \leq a \leq b \leq c \leq d$, (ii) $ab + cd = a + b + c + d + 3$.
Let $BE$ and $CF$ be the altitudes of an acute $\triangle ABC$, with $E$ on $AC$ and $F$ on $AB$. Let $O$ be the point of intersection of $BE$ and $CF$. Take any line $KL$ through $O$ with $K$ on $AB$ and $L$ on $AC$. Suppose $M$ and $N$ are located on $BE$ and $CF$ respectively, such that $KM$ is perpendicular to $BE$ and $LN$ is perpendicular to $CF$.Prove that $FM$ is parallel to $EN$.
Find all primes $p$ and $q$ such that $p^2+7pq+q^2$ is the square of an integer.
Find the number of positive integers $x$ which satisfy the condition $[\frac{x}{99}]=[\frac{x}{101}]$. (Here $[z]$ denotes, for any real $z$, the largest integer not exceeding $z$ ; e. g. $[\frac{7}{4}]$.
Consider an $n × n$ array of numbers. Suppose each row consists of the $n$ numbers $1, 2, . . . , n$ in some order and $a_{ij}=a_{ji}$ for $i = 1, 2, . . . , n$ and $j = 1, 2, . . . , n$. If $n$ is odd, prove that the numbers $a_{11},a_{22},a_{33},....,a_{nn}$ are $1, 2, . . . , n$ in some order.
In a triangle ABC , D is a point on BC such that AD is the internal bisector of $\angle A$. Suppose $\angle B = 2 \angle C$ and $CD = AB$. Prove that $\angle A = 72^{\circ}$ .
If $x, y, z$ are the sides of a triangle, then prove that ( $|x^2(y-z)+y^2(z-x)+z^2(x-y)| \leq xyz$ ).
Prove that the product of the first $1000$ positive even integers differs from the product of the first $1000$ odd integers by a multiple of $2001$.
Let $AC$ be a line segment in the plane and $B$ a point between $A$ and $C$. Construct isosceles triangles $PAB$ and $QBC$ on one side of the segment $AC$ such that $\angle APB = \angle BQ= 120^{\circ}$ and an isosceles $\triangle RAC$ on the other side of $AC$ such that $\angle ARC= 120^{\circ}$. Show that $PQR$ is an equilateral triangle.
Solve the equation $y^3=x^3+8x^2-6x+8$,for positive integers $x$ and $y$.
Suppose $(x_1, x_2,....., x_n)$ is a sequence of positive real numbers such that ( $x_1 \geq x_2 \geq x_3 \geq ... \geq x_n... $) and for all $n$ $\frac{x_1}{1}+\frac{x_4}{2}+\frac{x_9}{3}+....+\frac{x_{n^2}}{n}\leq 1$ . Show that for all $k$ the following inequality is satisfied: $\frac{x_1}{1}+\frac{x_2}{2}+\frac{x_3}{3}+....+\frac{x_k}{k}\leq 3$.
All the $7$-digit numbers containing each of the digits 1, 2, 3, 4, 5, 6, 7 exactly once, and not divisible by $5$, are arranged in increasing order. Find the $2000$-th number in this list.
The internal bisector of $\angle A$ in a $\triangle ABC$ with $AC > AB$, meets the circumcircle ( $\Gamma$ ) of the triangle in $D$. Join $D$ to the centre $O$ of the circle ( $\Gamma$ ) and suppose $DO$ meets $AC$ in $E$, possibly when extended. Given that $BE$ is perpendicular to $AD$, show that $AO$ is parallel to $BD$.
(i) Consider two positive integers $a$ and $b$ which are such that $a^a b^b$ is divisible by $2000$. What is the least possible value of the product $ab$? (ii) Consider two positive integers $a$ and $b$ which are such that $a^a b^b$ is divisible by $2000$. What is the least possible value of the product $ab$?
Find all real values of $a$ for which the equation $x^4-2ax^2+x+a^2-a=0$ has all its root real.
Prove that the inradius of a right-angled triangle with integer sides is an integer.
Find the number of positive integers which divide $10^{999}$ but not $10^{998}$.
Let $ABCD$ be a square and $M,N$ points on sides $AB,BC$ respectably, such that $\angle MDN =45^{\circ}$. If $R$ is the midpoint of $MN$ show that $RP = RQ$ where $P,Q$ are the points of intersection of $AC$ with the lines $MD, ND$.
If $p, q, r$ are the roots of the cubic equation $x^3-3px^2+3q^2x-r^3=0$, show that $p=q=r$.
If $a, b, c$ are the sides of a triangle prove the following inequality: $\frac{a}{c+a-b}+\frac{b}{a+b-c}+\frac{c}{b+c-a}\geq 3 $.
Find all solutions in integers $m, n$ of the equation $(m-n)^2=\frac{4mn}{m+n-1}$.
Find the number of quadratic polynomials $ax^2+bx+c$, which satisfy the following conditions: (a) $a, b, c$ are distinct; (b) $a, b, c \in {1, 2, 3, . . . 1999}$ and (c) $x + 1$ divides $ax^2+bx+c$.
Let $ABCD$ be a convex quadrilateral in which $\angle BAC = 50^{\circ} ,\angle CAD = 60^{\circ} ,\angle CBD = 30^{\circ}$,and $\angle BDC = 25^{\circ}$. If $E$ is the point of intersection of $AC$ and $BD$, find $\angle AEB$.
Let $n$ be a positive integer and $p_1p_2......p_n$ be $n$ prime numbers all larger than $5$ such that $6$ divides $p_{1}^{2}+p_{2}^{2}+p_{n}^{2}$. Prove that $6$ divides $n$.
Prove the following inequality for every natural number $n$: $\frac{1}{n+1}(1+\frac{1}{3}+\frac{1}{5}+.....+\frac{1}{2n-1})\geq \frac{1}{n}(\frac{1}{2}+\frac{1}{4}+\frac{1}{6}+.........+\frac{1}{2n}) $.
Let $ABC$ be a triangle with $AB = BC$ and $\angle BAC = 30^{\circ}$. Let $A^{'}$ be the reflection of $A$ in the line $BC$; $B^{'}$ be the reflection of $B$ in the line $CA$; $C^{'}$ be the reflection of $C$ in the line $AB$. Show that $A^{'},B^{'},C^{'}$ form the vertices of an equilateral triangle.
Find the minimum possible least common multiple (lcm) of twenty (not necessarily distinct) natural numbers whose sum is $801$.
Given the $7$-element set $A = {a, b, c, d, e, f, g}$, find a collection $T$ of $3$-element subsets of $A$ such that each pair of elements from $A$ occurs exactly in one of the subsets of $T$.
Let $P$ be an interior point of a $\triangle ABC$ and let $BP$ and $CP$ meet $AC$ and $AB$ in $E$ and $F$ respectively. If [$BPF$] = $4$, [$BPC$] = $8$ and $[CPE] = 13$, find [$AFPE$]. (Here $[·]$ denotes the area of a triangle or a quadrilateral, as the case may be.)
For each positive integer $n$, define $a_n = 20 + n^2$, and $d_n=GCD(a_n,a_{n+1})$. Find the set of all values that are taken by $ d_n$ and show by examples that each of these values are attained.
Solve for real $x$: $\frac{1}{[x]}+\frac{1}{[2x]}=9(x)+\frac{1}{3}$,where $[x]$ is the greatest integer less than or equal to $x$ and $(x)$ = $x − [x]$, [e.g. $[3.4] = 3$ and $(3.4) = 0.4$].
In a quadrilateral $ABCD$, it is given that $AB$ is parallel to $CD$ and the diagonals $AC$ and $BD$ are perpendicular to each other. Show that
$(a) AD.BC \geq AB.CD$,
$(b) AD + BC \geq AB+CD$.
Let $x, y$ and $z$ be three distinct real positive numbers. Determine with proof whether or not the three real numbers $|\frac{x}{y}-\frac{y}{x}|,|\frac{y}{z}-\frac{z}{y}|,|\frac{z}{x}-\frac{x}{z}|$ can be the lengths of the sides of a triangle.
Find the number of unordered pairs ${A,B}$ (i.e., the pairs ${A,B}$ and ${B,A}$ are considered to be the same) of subsets of an $n$-element set $X$ which satisfy the conditions:
$(a) A \neq B$,
$(b) A \cup B =X$.
[e.g., if $X = \{a, b, c, d\}$, then $\{\{a, b\}, \{b, c, d\}\}, \{\{a\}, \{b, c, d\}\}, \{\phi,\{a, b, c, d\}\}$ are some of the admissible pairs.]
The sides of a triangle are three consecutive integers and its inradius is four units. Determine the circumradius.
Find all triples $(a, b, c)$ of positive integers such that $(1+ \frac{1}{a})(1+ \frac{1}{b})(1+ \frac{1}{c})=3$.
Solve for real number $x$ and $y$: $xy^2=15x^2+17xy+15y^2$, $x^2y=20x^2+3y^2$.
Suppose $N$ is an $n$-digit positive integer such that (a) all the $n$-digits are distinct; and (b) the sum of any three consecutive digits is divisible by $5$. Prove that $n$ is at most $6$. Further, show that starting with any digit one can find a six-digit number with these properties.
Let $ABC$ be a triangle and $h_a$ the altitude through $A$. Prove that $(b+c)^2 \geq a^2 + 4h_{a}^{2}$.(As usual $a, b, c$ denote the sides $BC, CA, AB$ respectively.)
Given any positive integer $n$ show that there are two positive rational numbers $a$ and $b$,$a \neq b$, which are not integers and which are such that $a-b$, $a^2-b^2,a^3-b^3,.....,a^n-b^n$ are all integers.
If $A$ is a fifty-element subset of the set $\{1, 2, 3, . . . , 100\}$ such that no two numbers from $A$ add up to $100$ show that $A$ contains a square.
In triangle $ABC$, $K$ and $L$ are points on the side $BC$ ($K$ being closer to $B$ than $L$) such that $BC.KL = BK.CL$ and $AL$ bisects $\angle KAC $ . Show that $AL$ is perpendicular to $AB$.
Call a positive integer n good if there are n integers, positive or negative, and not necessarily distinct, such that their sum and product are both equal to $n$ (eg. 8 is good, since $8 = 4.2.1.1.1.1.(-1)(-1) = 4 + 2 + 1 + 1 + 1 + 1 + (-1) + (-1) )$. Show that integers of the form 4k + 1 and 4l are good.
Prove that among any $18$ consecutive three-digit numbers there is at least one number which is divisible by the sum of it's digits.
Show that the quadratic equation $ x^2+7x-14(q^2+1)=0 $ , where $q$ is an integer, has no integer root.
Show that for any triangle $ABC$, the following inequality is true: $ a^2+b^2+c^2 > \sqrt{3}$ max ${(|a^2-b^2|,|b^2-c^2|,|c^2-a^2|}) $ where $a, b, c$ are, as usual, the sides of the triangle.
Let $ A_1A_2A_3....A_{21} $ be a 21-sided reqular polygon inscribed in a circle with center O. How many triangles $ A_iA_jA_k $, contain the point O in their interior?
Show that for any real number $x$, $ x^2 \sin {x}+x \cos {x}+x^2+\frac{1}{2} > 0 $.
A leaf is torn from a paperback novel. The sum of the numbers on the remaining pages is $15000$. What are the page numbers on the torn leaf.
In the $\triangle ABC$, the incircle touches the sides $BC, CA$ and $AB$ respectively at $D, E$ and $F$. If the radius of the incircle is $4$ units and if $BD, CE$ and $AF$ are consecutive integers, find the sides of the $\triangle ABC$.
Find all $6$-digit natural numbers $ a_1a_2a_3a_4a_5a_6 $ formed by using the digits $1, 2, 3, 4, 5, 6$ once each such that the number $ a_1a_2a_3...a_k $ is divisible by $k$, for $1 \leq k \leq 6 $.
Solve the system of equations for real $x$ and $y$ : $ 5x (1+\frac{1}{x^2+y^2})=12 $ , $5y(1-\frac{1}{x^2+y^2})=12 $.
Let $A$ be a set of $16$ positive integers with the property that the product of any two distinct numbers of A will not exceed $1994$. Show that there are two numbers $a$ and $b$ in $A$ which are not relatively prime.
Let $AC$ and $BD$ be two chords of a circle with center O such that they intersect at right angles inside the circle at the point $M$. Suppose $K$ and $L$ are the mid-points of the chord $AB$ and $CD$ respectively. Prove that $OKML$ is a parallelogram.
Find the number of all rational numbers $m/n$ such that (a) $0 < m/n < 1$, (b) $m$ and $n$ are relatively prime, (c) $mn = 25!$.
If $a, b$ and $c$ are positive real numbers such that $a + b + c = 1$, prove that $(1+a)(1+b)(1+c) \geq 8(1-a)(1-b)(1-c) $.
Let $ABC$ be an acute-angled triangle and $CD$ be the altitude through $C$. If $AB = 8$ and $CD = 6$, find the distance between the mid-points of $AD$ and $BC$.
Prove that the ten’s digit of any power of 3 is even. [e.g. the ten’s digit of $3^{6} = 729$ is $2$]
Suppose $A_1A_2A_3.....A_n $ is a $20$-sided regular polygon. How many non-isosceles (scalene) triangles can be formed whose vertices are among the vertices of the polygon but whose sides are not the sides of the polygon?
Let $ABCD$ be a rectangle with $AB = a$ and $BC = b$. Suppose $ r_1 $ is the radius of the circle passing through $A$ and $B$ and touching $CD$; and similarly $r_2 $ is the radius of the circle passing through $B$ and $C$ and touching $AD$. Show that $r_1 +r_2 \geq \frac{5}{8}(a+b) $.
Show that $19^{93}+13^{99} $ is a positive integer divisible by $162$.
If $a, b, c, d$ are four positive real numbers such that $abcd = 1$, prove that $ (1+a)(1+b)(1+c)(1+d) \geq 16 $.
In a group of ten persons, each person is asked to write the sum of the ages of all the other $9$ persons. If all the ten sums form the $9$-element set $\{82, 83, 84, 85, 87, 89, 90, 91, 92\}$ find the individual ages of the persons (assuming them to be whole numbers of years).
I have $6$ friends and during a vacation I met them during several dinners. I found that I dined with all the $6$ exactly on $1$ day; with every $5$ of them on $2$ days; with every $4$ of them on $3$ days; with every $3$ of them on $4$ days; with every $2$ of them on $5$ days. Further every friend was present at $7$ dinners and every friend was absent at $7$ dinners. How many dinners did I have alone?
Determine the set of integers $n$ for which $n^2 +19n + 92 $ is a square of an integer.
If $\frac{1}{a}+\frac{1}{b}=\frac{1}{c} $ where $a, b, c$ are positive integers with no common factor, prove that (a + b) is the square of an integer.
Determine the largest $3$-digit prime factor of the integer $ {2000} \choose {1000} $ .
$ABCD$ is a cyclic quadrilateral with $AC$ perpendicular to $BD$; $AC$ meets $BD$ at $E$. Prove that $ R $ is the radius of the circumscribing circle.
$ABCD$ is a cyclic quadrilateral; $x, y, z$ are the distances of $A$ from the lines $BD, BC, CD$ respectively. Prove that $\frac{BD}{x}=\frac{BC}{y}+\frac{CD}{z} $
$ABCD$ is a quadrilateral and $P, Q$ are mid-points of $CD$, $AB$ respectively. Let $AP, DQ$ meet at $X$, and $BP$, $CQ$ meet at $Y$ . Prove that area of $ADX$ + area of $BCY$ = area of quadrilateral $PXQY$ .
Solve the system $(x + y)(x + y + z) = 18$, $(y + z)(x + y + z) = 30$, $(z + x)(x + y + z) = 2A$ in terms of the parameter $A$.
The cyclic octagon $ABCDEFGH$ has sides $a, a, a, a, b, b, b, b$ respectively. Find the radius of the circle that circumscribes $ABCDEFGH$ in terms of $a$ and $b$.
Let $P$ be an interior point of $\triangle ABC$ and $AP, BP, CP$ meet the sides $BC, CA, AB$ in $D, E, F$ respectively. Show that \( \frac{AP}{PD} = \frac{AF}{FB} +\frac{AE}{EC} \).
If $a, b, c$ and $d$ are any four positive real numbers, then prove that $\frac{a}{b}+\frac{b}{c}+\frac{c}{d}+\frac{d}{a} \ge 4 $.
A four-digit number has the following properties:
it is a perfect square;
its first two digits are equal to each other;
its last two digits are equal to each other; Find all such four digit numbers.
There are two Urns each containing an arbitrary number of balls (both are non empty to begin with). We are allowed two types of operations:
remove an equal number of balls simultaneously from both the urns and
double the number of balls in any one of them. Show that after performing these operations finitely many times , both the urns can be made empty.
Take any point $P_{1} $ on the side $BC$ of a $\triangle ABC$ and draw the following chain of lines: $P_{1}P_{2} $ parallel to $AC$, $P_{2}P_{3} $ parallel to $BC$, $P_{3}P_{4} $ parallel to $BC$, $ P_{4}P_{5} $ parallel to $CA$, and $ P_{4}P_{5} $ parallel to $BC$. Here lie on AB $P_{3}P_{4} $ on $CA$ and $ P_{4} $ on BC. Show that $ P_{6}P_{1} $ is parallel to $AB$.
Find all integer values of a such that the quadratic expression $(x+a)(x+1991) + 1$ can be factored as a product $(x+b)(x+c)$ where $b$ and $c$ are integers.
Prove that $n^{4}+4^{n} $ is composite for all integer values of $n > 1$.
The $64$ squares of a $8 \times 8 $ chessboard are filled with positive integers in such a way that each integer is the average of the integers on the neighbouring squares. (Two squares are neighbours if they share a common edge or a vertex. Thus a square can have $8, 5$ or $3$ neighbours depending on its position).Show that all the $64$ integers entries are in fact equal.
