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The third step in Math Olympiad in India. Past papers, sequential hints and training resources.

TRY CHEENTA INMO COURSEPast Papers of INMO (Indian National Math Olympiad India)

- Let D, E, F be points on the sides BC, CA, AB respectively of a triangle ABC such that BD = CE = AF and ∠BDF = ∠CED = ∠AFE. Prove that ABC is equilateral.
- Call a natural number n faithful, if there exist natural numbers a < b < c such that a divides b, b divides c and n = a + b + c. (i) Show that all but a finite number of natural numbers are faithful. (ii) Find the sum of all natural numbers which are not faithful.
- Consider two polynomials $ P(x)=a_nx^n+a_{n-1}x^{n-1}+......+a_1x+a_0=0 $ and $ Q(x)=b_nx^n+b_{n-1}x^{n-1}+......+b_1x+b_0=0 $ with integer coefficients such that $ a_n-b_n $ is a prime, $ a_{n-1}=b_{n-1} $ and $ a_nb_0-a_0b_n \neq 0 $. Suppose there exists a rational number r such that P(r) = Q(r) = 0. Prove that r is an integer.
- Suppose five of the nine vertices of a regular nine-sided polygon are arbitrarily chosen. Show that one can select four among these five such that they are the vertices of a trapezium.
- Let ABCD be a quadrilateral inscribed in a circle $ \Gamma $. Let E, F, G, H be the midpoints of the arcs AB, BC, CD, DA of the circle $ \Gamma $. Suppose AC.BD = EG .FH. Prove that AC, BD, EG, FH are concurrent.
- Find all functions $ f : R \mapsto R $ such that $ f(x+y)f(x-y)=(f(x)+f(y))^2-4x^2f(y) $, for all x, y ∈ R, where R denotes the set of all real numbers.

- Let ABC be a triangle with circum-circle $ \Gamma $.Let M be a point in the interior of the triangle ABC which is also on the bisector of $ \angle A $. Let AM, BM, CM meets $ \Gamma $ in $ A_1,B_1,C_1 $ respectively. Suppose P is the point of intersection of $ A_1,B_1 $ with AC. Prove that PQ is parallel to BC.
- Find all natural numbers n>1 such that $ n^2 $ does not divide (n-2)!.
- Find all non-zero real numbers $ x,y,z $which satisfy the system of equations: $ (x^2+xy+y^2)(y^2+yz+z^2)(z^2+zx+x^2)=xyz $, $ (x^4+x^2y^2+y^4)(y^4+y^2z^2+z^4)(z^4+z^2x^2+x^4)=x^3y^3z^3 $
- How many 6-tuples $ (a_1,a_2,a_3,a_4,a_5,a_6) $are there such that each of $ a_1,a_2,a_3,a_4,a_5,a_6 $ is from the set {1,2,3,4} and the six expressions $ a_j^2-a_ja_{j+1}+a_{j+1}^2 $ for $ j=1,2,3,4,5,6 $ (where $ a_7 $ is to be taken as $ a_1 $ ) are all equal to one another?
- Let ABC be an acute-angled triangle with altitude AK. Let H be its orthocentre and O be its circumcentre. Suppose KOH is an acute-angled triangle and P its circumcircle. Let Q be the reflection of P in the line HO. Show that Q lies on the line joining the mid-points of AB and AC.
- Define a sequence $ (a_n)_{n \ge 0} $ by $ a_0=0 $ , $ a_1=1 $ and $ a_{n}=2a_{n-1}+a_{n-2} $, for $ n \ge 2 $. (a) For every $ m>0 $ and $ 0 \le j \le m $,prove that $ 2a_m $ divides $ a_{m+j}+(-1)^{j}a_{m-j} $.(b) Suppose $ 2^{k} $ divides n for some natural numbers n and k. Prove that $ 2^k $ divides a_n.

- Let ABC be a triangle and P be a interior point such that $ \angle BPC $=$ 90^0 $, $ \angle BAP $ = $ \angle BCP $ . Let M,N be the mid-points of AC, BC respectively. Suppose BP=2PM. Prove A,P,N are collinear.
- Define a sequence $\left\langle a_{n}\right\rangle_{n=1}^{\infty}$ as follows
$a_{n}=\left\{\begin{array}{ll}0, & \text { if the number of positive divisors of } n \text { is odd } \\ 1, & \text { if the number of positive divisors of } n \text { is even. }\end{array}\right.$

(The positive divisors of $n$ include 1 as well as $n .$ ) Let $x=0 . a_{1} a_{2} a_{3} \ldots$ be the real number whose decimal expansion contains $a_{n}$ in the $n$ -th place, $n \geq 1$. Determine, with proof. whether $x$ is rational or irrational. - Find all real number x such that $ [x^2+2x]=[x]^2+2[x] $. (Here [] denotes the largest integer not exceeding x.)
- All the points in the plane are coloured using three colours. Prove that there exists a triangle with vertices having the same colour such that either it is isosceles or its angles are in geometric progression
- Let ABC be an acute-angled triangle and let H be its orthocentre. Let $ h_{\max} $ denotes the largest altitude of the triangle ABC. Prove that $ AH+BH+CH \le 2h_{\max} $.
- Let a,b,c be positive real numbers such that. Prove that $ a^2+b^2-c^2 > 6(c-a)(c-b) $.

- Let $A B C$ be a triangle, $I$ its in-centre; $A_{1}, B_{1}, C_{1}$ be the reflections of $I$ in $B C, C A, A B$ respectively. Suppose the circum-circle of triangle $A_{1} B_{1} C_{1}$ passes through $A$. Prove that $B_{1}, C_{1}$, $I, I_{1}$ are concyclic, where $I_{1}$ is the in-centre of triangle $A_{1} B_{1} C_{1}$
- Find all triples $(p, x, y)$ such that $p^{x}-y^{4}+4,$ where $p$ is a prime and $x, y$ are natural numbers.
- Let $A$ be a set of real numbers such that $A$ has at least four elements. Suppose $A$ has the property that $a^{2}+b c$ is a rational number for all distinct numbers $a, b, c$ in $A$. Prove that there exists a positive integer $M$ such that $a \sqrt{M}$ is a rational number for crery $a$ in $A$.
- All the points with integer coordinates in the $x y$ -plane are coloured using three colours, red, blue and green, each colour being used at least once. It is known that the point (0,0) is coloured red and the point (0,1) is coloured blue. Prove that there exist three points with integer coordinates of distinct colours which form the vertices of a right-angled triangle.
- Let $A B C$ be a triangle; $\Gamma_{A}, \Gamma_{B}, \Gamma_{C}$ be three equal, disjoint circles inside $A B C$ such that $\Gamma_{A}$ touches $A B$ and $A C ; \Gamma_{B}$ touches $A B ;$ and $B C,$ and $\Gamma_{C}$ touches $B C$ and $C A .$ Let $\Gamma$ be a circle touching circles $\Gamma_{A}, \Gamma_{B}, \Gamma_{C}$ externally. Prove that the line joining the circum-centre $O$ and the in-centre $I$ of triangle $A B C$ passes through the centre of $\Gamma$
- Let $P(x)$ be a given polynomial with integer coefficients. Prove that there exist two polynomials $Q(x)$ and $R(x),$ again with integer coefficients, such that (i) $P(x) Q(x)$ is a polynomial in $x^{2} ;$ and (ii) $P(x) R(x)$ is a polynomial in $x^{3}$.

- In a $\triangle ABC$ right-angled at $C$, the median through $B$ bisects the angle between $BA$ and the bisector of ∠B. Prove that \( \frac{5}{2} < \frac{AB}{BC} < 3 \).
- . Let $n$ be a natural number such that \( n = a^2 +b^2 +c^2 \) , for some natural numbers $ a, b, c $. Prove that $$ 9n = \\ (p_1a + q_1b + r_1c)^2 \\+ (p_2a + q_2b + r_2c)^2 \\+ (p_3a + q_3b + r_3c)^2 $$,

where \( p_j ’s \), \( q_j ’s\) , \( r_j ’s \) are all nonzero integers.

Further, of 3 does not divide at least one of ''a, b, c,'' prove that 9n can be expressed in the form \( x^2 + y^2 + z^2 \), where ''x, y, z'' are natural numbers none of which is divisible by 3. - Let $m$ and $n$ be positive integers such that the equation \( x^2−mx+n = 0 \) has real roots \( \alpha \) and \( \beta \).

Prove that \( \alpha \) and \( \beta \) are integers if and only if [\( m\alpha \)] + [\( m\beta \)] is the square of an integer. (Here [x] denotes the largest integer not exceeding x.) - Let \( \sigma = (a_1, a_2, a_3, . . . , a_n) \) be a permutation of \( (1, 2, 3, . . . , n) \). A pair \( (a_i, a_j ) \) is said to correspond to an inversion of \( \sigma \), if \( i < j \) but \( ai > aj \) . (Example: In the permutation $ (2, 4, 5, 3, 1) $, there are 6 inversions corresponding to the pairs \( (2, 1), (4, 3), (4, 1), (5, 3), (5, 1), (3, 1) \)

How many permutations of \( (1, 2, 3, . . . , n), (n \ge 3) \), have exactly two inversions. - Let $ABC$ be a triangle in which AB = AC. Let D be the midpoint of BC and P be a point on AD. Suppose E is the foot of the perpendicular from P on AC. If \( \frac{AP}{PD}=\frac{BP}{PE}= λ,\frac{BD}{AD}= m \) and \(z = m^2(1 + λ) \),

prove that \( z^2− (λ^3− λ^2− 2)z + 1 = 0 \). Hence show that \( λ ≥ 2 \) and \( λ = 2 \) if and only if ABC is equilateral. - If $x, y, z$ are positive real numbers, prove that $$ (x+y+z)^2(yz+zx+xy)^2 \\ ≤ \\ 3(y^2+yz+z^2) \\ \times (z^2+zx+x^2) \\ \times (x^2+xy+y^2) $$.
- Let \( f :Z \mapsto Z \) be a function satisfying $f(0) \neq 0 $ , $f(1)=0 $ and

1. \( f(xy) + f(x)f(y) = f(x) + f(y) \),

2. \( (f(x-y) – f(0))f(x)f(y) = 0 \) for all x , y in Z simultaneously.

1. Find the set of all possible values of the function f.

2. If \( f(10) \neq 0 \) and \( f(2) = 0 \), find the set of all integers n such that \( f(n) \neq 0 \) .

- In a non equilateral triangle $ ABC $, the sides $ a, b, c $ form an arithmetic progression. Let $ I $ and $ O $ denote the incentre and circumcentre of the triangle respectively.

(i) Prove that $ IO $ is perpendicular to $ BI $.

(ii) Suppose $ BI $ extended meets $ AC $ in $ K $, and $ D, E $ are the midpoints of $ BC, BA $ respectively. Prove that $ I $ is the circumcentre of triangle $ DKE $. - Prove that for every positive integer $ n $ there exists a unique ordered pair $ (a, b) $ of positive integers such that \( n =\frac{1}{2}(a + b − 1)(a + b − 2) + a \).
- Let $ X $ denote the set of all triples $ (a, b, c) $ of integers. Deﬁne a function $ f : X \mapsto X $ by \( f(a, b, c) = (a + b + c, ab + bc + ca, abc) \). Find all triples $ (a, b, c) $ in $ X $ such that \( f(f(a, b, c)) = (a, b, c) \).
- Some $46$ squares are randomly chosen from a $9 × 9$ chess board and are coloured red. Show that there exists a $2 × 2$ block of 4 squares of which at least three are coloured red.
- In a cyclic quadrilateral \( ABCD, AB = a, BC = b, CD = c \), \( ∠ABC = 120^0 \) , and \( ∠ABD = 30^0 \). Prove that

(i) \( c \ge a + b \);

(ii) \( |\sqrt{c + a} −\sqrt{c + b}| = \sqrt{c − a − b} \). - (a) Prove that if $ n $ is a positive integer such that $ n \geq 4011^2 $, then there exists an integer $ l $ such that \( n < l^2 < (1 +\frac{1}{2005})n \).

(b) Find the smallest positive integer $ M $ for which whenever an integer $ n $ is such that $ n \geq M $, there exists an integer $ l $, such that \( n < l^2 < (1 +\frac{1}{2005})n \).

- Let $M$ be the midpoint of side $BC$ of a $\triangle ABC$. Let the median $AM$ intersect the incircle of $ABC$ at $K$ and $L$, $K$ being nearer to $A$ than $L$. If $AK = KL = LM$, prove that the sides of $\triangle ABC$ are in the ratio $5 : 10 : 13$ in some order.
- Let \( \alpha \) and \( \beta \) be positive integers such that \( \frac{43}{197} < \frac{\alpha}{\beta} < \frac{17}{77} \).Find the minimum possible value of β.
- Let \( p, q, r \) be positive real numbers, not all equal, such that some two of the equations \( px^2+ 2qx + r = 0, qx^2+ 2rx + p = 0, rx^2 + 2px + q = 0 \), have a common root, say \( \alpha \). Prove that

(a) \( \alpha \) is real and negative; and

(b) the remaining third equation has non-real roots. - All possible $6$-digit numbers, in each of which the digits occur in non-increasing order (from left to right, e.g.,$877550$) are written as a sequence in increasing order. Find the 2005-th number in this sequence.
- Let \( x_1 \) be a given positive integer. A sequence \( (x_n)_{n=1} ^{\infty} = (x_1, x_2, x_3, · · ·) \) of positive integers is such that \( x_n \), for \( n \ge 2 \), is obtained from \( {x_n−1} \) by adding some nonzero digit of \( x_{n−1} \).

Prove that (a) the sequence has an even number;

(b) the sequence has inﬁnitely many even numbers. - Find all functions \( f : R \mapsto R \) such that \( f(x^2 + yf(z)) = xf(x) + zf(y) \), for all \( x, y, z \) in \( R \). (Here \( R \) denotes the set of all real numbers.)

- Consider a convex quadrilateral \( ABCD \), in which \( K, L, M, N \) are the midpoints of the sides \( AB, BC, CD, DA \) respectively . Suppose (a) BD bisects \( KM \) at \( Q \) ; (b) \( QA = QB = QC = QD \) ; and (c) \( LK/LM = CD/CB \) . Prove that ABCD is a square .
- Suppose $p$ is a prime greater than $3$. Find all pairs of integers (a, b) satisfying the equation \( a^2+ 3ab + 2p(a + b) + p^2= 0 \).
- If \( \alpha \) is a real root of the equation \( x^5 − x^3 + x − 2 = 0 \) , prove that [\(\alpha_6\)] = $3$ . (For any real number a , we denote by [a] the greatest integer not exceeding a . )
- Let $R$ denote the circumradius of a triangle \( ABC \); \( a, b, c \) its sides \( BC, CA, AB \) ; and \( r_a, r_b, r_c \) its exradii opposite \( A, B, C \). If \( 2R ≤ r_a \) , prove that

(i) \( a > b \) and \( a > c \) ;

(ii) \( 2R > r_b \) and \( 2R > r_c \) . - Let $S$ denote the set of all $6$-tuples \( (a, b, c, d, e, f) \) of positive integers such that \( a^2 + b^2 + c^2 +d^2 + e^2 = f^2 \). Consider the set \( T = {abcdef : (a, b, c, d, e, f) ∈ S} \). Find the greatest common divisor of all the members of T.
- Prove that the number of $5$-tuples of positive integers \( (a,b,c,d,e) \) satisfying the equation \( abcde = 5(bcde + acde + abde + abce + abcd) \) is an odd integer .

- Consider an acute triangle $ABC$ and let $P$ be an interior point of ABC . Suppose the lines BP and CP , when produced , meet AC and AB in E and F respectively . Let D be the point where AP intersects the line segment EF and K be the foot of perpendicular from $D$ on to $BC$ . Show that DK bisects <EKF .
- Find all primes $p$ and $q$ ,and even numbers n > 2 , satisfying the equation $p^n+ p^{n−1}+ · · · + p + 1 = q^2+ q + 1$ .
- Show that for every real number a the equation \( 8x^4− 16x^3+ 16x^2− 8x + a = 0 \) has at least one non-real root and ﬁnd the sum of all the non-real roots of the equation .
- Find all $7$-digit numbers formed by using only the digits $5$ and $7$ , and divisible by both $5$ and $7$ .
- Let $ABC$ be a triangle with sides \( a, b, c \) . Consider a triangle \( A_1B_1C_1 \) with sides equal to \( a +\frac{b}{2}, b +\frac{c}{2}, c +\frac{a}{2} \). Show that [\( A_1B_1C_1 \)] ≥ \( \frac{9}{4}[ABC] \), where $[XY Z]$ denotes the area of the triangle $XY Z$ .
- In a lottery tickets are given nine-digit numbers using only the digits $1, 2, 3$ . They are also coloured red , blue or green in such a way that two tickets whose numbers diﬀer in all the nine places get diﬀerent colours . Suppose the ticket bearing the number $122222222$ is red and that bearing the number $222222222$ is green . Determine , with proof , the colour of the ticket bearing the number $123123123$ .

- For a convex hexagon $ABCDEF$ , consider the following six statements : $ (a_1) $ AB is parallel to DE : $ (a_2) $ AE = BD; $ (b_1) $ BC is parallel to EF : $ (b_2) $ BF = CE; $ (c_1) $ CD is parallel to $FA$ : $ (c_2) CA = DF$. (a) Show that if all the six statements are true , then the hexagon is cyclic ( i. e. , it can be inscribed in a circle ). (b) Prove that , in fact , any five of these six statements also imply that the hexagon is cyclic.
- Determine the least positive value taken by the expression $a^3 + b^3 + c^3 = 3abc$ as $a, b, c$ vary over all positive integers . Find also all triples $(a, b, c)$ for which this least value is attained .
- Let $x, y$ be positive reals such that $x + y = 2$. Prove that $ x^3y^3(x^3 + y^3) \leq 2 $.
- Do there exist 100 lines in the plane , no three of them concurrent , such that they intersect exactly in $2002$ points ?
- Do there exist three distinct positive real numbers $a, b, c$ such that the numbers $a, b, c, b + c −a, c + a − b, a + b − c$ and $a + b + c$ form a $7$-term arithmetic progression in some order ?
- Suppose the $ n^2 $ numbers $1, 2, 3, · · · , n^2 $ are arranged to form an n by n array consisting of $n$ rows and $n$ columns such that the numbers in each row ( from left to right ) and each column ( from top to bottom ) are in increasing order . Denote by $ a_{jk} $ the number in the $j$ − th row and $k$ − th column . Suppose $ b_j $ is the maximum possible number of entries that can occur as $ a_{jj} $ , $ 1 \leq j \leq n $ . Prove that $$ b_1 + b_2 + b_3 + · · · b_n \le \frac{n}{3}(n^2 − 3n + 5) $$.( Example : In the case n = 3 , the only numbers which can occur as $ a_{22} $ are 4, 5 or 6 so that $ b_2 $ = 3 . )

- Let ABC be a triangle in which no angle is $ 90^0 $. For any point P in the plane of the triangle, let $ A_1,B_1,C_1 $ denote the reflections of P in the sides BC,CA,AB respectively. Prove the following statements

(a) If P is the incentre or an excentre of ABC, then P is the circumcentre of $ A_1B_1C_1 $;

(b) If P is the circumcentre of ABC, then P is the orthocentre of $ A_1B_1C_1 $;

(c) If P is the orthocentre of ABC, then P is either the incentre or an excentre of $ A_1B_1C_1 $. - Show that the equation \( x^2 + y^2 + z^2 = (x − y)(y − z)(z − x) \) has infinitely many solutions in integers x, y, z.
- If a, b, c are positive real numbers such that abc = 1, prove that $$ a^{b+c} b^{c+a} c^{a+b} \le 1 $$.
- Given any nine integers show that it is possible to choose, from among them, four integers a, b, c, d such that a + b − c − d is divisible by 20. Further show that such a selection is not possible if we start with eight integers instead of nine.
- Let ABC be a triangle and D be the midpoint of side BC. Suppose <DAB = <BCA and <DAC = $15^{\circ} $. Show that ADC is obtuse. Further, if O is the circumcentre of ADC, prove that triangle AOD is equilateral.
- Let R denote the set of all real numbers. Find all functions $ f : R \mapsto R $ satisfying the condition f(x + y) = f(x)f(y)f(xy) for all x, y in R.

- The incircle of triangle ABC touches the sides BC,CA and AB in K,L and M respectively. The line through A and parallel to LK meets MK in P and the line through A and parallel to MK meets LK in Q. Show that the line PQ bisects the sides AB and AC of the triangle ABC.
- Solve for integers x, y, z: $ x+y=1-z,x^3+y^3=1-z^2 $.
- If a, b, c, x are real numbers such that $ abc \neq 0 $ and $ \frac{xb+(1-x)c}{a}=\frac{xc+(1-x)a}{b}=\frac{xa+(1-x)b}{c} $then prove that a=b=c.
- In a convex quadrilateral PQRS, PQ = RS, $ (\sqrt{3}-1) $QR = SP and < RSP − < SPQ = $ 30^0 $. Prove that <PQR - <QRS = $ 90^0 $
- Let a, b, c be three real numbers such that $ 1 \ge a \ge b \ge c \ge 0 $. Prove that if $ \lambda $ is a root of the cubic equation $ x^3+ax^2+bx+c=0 $ (real or complex), then $ |\lambda| \le 1 $.
- For any natural numbers n, $ (n \ge 3) $, let f(n) denote the number of non congruent integer-sided triangles with perimeter n (e. g., f(3) = 1, f(4) = 0, f(7) = 2). Show that (a) f(1999) > f(1996);(b) f(2000) = f(1997).

- Let ABC be an acute angled triangle in which D, E, F are points on BC, CA, AB respectively such that AD is perpendicular to BC; AE = EC; and CF bisects <C internally. Suppose CF meets AD and DE in M and N respectively. If FM = 2, MN = 1, NC = 3, find the perimeter of the triangle ABC.
- In a village 1998 persons volunteered to clean up, for a fair, a rectangular field with integer sides and perimeter equal to 3996 feet. For this purpose, the field was divided into 1998 equal parts. If each part had an integer area ( measured in Sq. ft.), find the length and breadth of the field.
- Show that there do not exist polynomials p(x) and q(x) each having integer coefficients and of degree greater than or equal to 1 such that $latex p(x)q(x)=x^5+2x+1 $.
- Let \( \Gamma \) and \( \Gamma^{'} \) be two concentric circles. Let ABC and \( A^{'}B^{'}C^{'} \) be any two equilateral triangles inscribed in \( \Gamma \) and \( \Gamma^{'} \) respectively. If $latex P $ and \( P^{'} \) are any two points on \( \Gamma \) and \( \Gamma^{'} \) respectively, show that \( {P^{'}A}^2+{P^{'}B}+{P^{'}C}={A^{'}P}^2+{B^{'}P}^2+{C^{'}P}^2 \)
- Given any four distinct real numbers, show that one can choose three numbers, say, A, B, C from among them such that all the three quadratic equations \( Bx^2+x+C=0, Cx^2+x+A=0, Ax^2+x+B=0 \) have only real roots or all the three equations have only imaginary roots.
- For which positive integer values of n can the set {1, 2, 3, 4, . . . ,4n} be split into n disjoint 4-element subsets {a, b, c, d} such that in each of these sets \( a=\frac{b+c+d}{3} \)?

- In a circle \( C_1 \) with centre O, let AB be a chord that is not a diameter. Let M be the midpoint of AB. Take a point T on the circle \( C_2 \) with OM as diameter. Let the tangent to \( C_2 \) at T meet \( C_1 \) in P. Show that \( PA^2+PB^2=4PT^2 \)
- Let a and b be two positive \( a^{\frac{1}{3}}+b^{\frac{1}{3}} \) rational numbers such that \( a^{\frac{1}{3}}+b^{\frac{1}{3}} \) is also a rational number. Prove that themselves are rational numbers.
- Let p, q, r, s be four integers such that s is not divisible by 5. If there is an integer a such that \( pa^3+qa^2+ra+s \) is divisible by 5, prove that there is an integer b such that \( sb^3+rb^2+qb+p \) is also divisible by 5.
- Suppose ABCD is a cyclic quadrilateral inscribed in a circle of radius one unit. If \( AB.BC.CD.DA \ge 4 \),prove that ABCD is a square.
- Suppose a, b, c are three real numbers such that the quadratic equation \( x^2-(a+b+c)x+(ab+bc+ca)=0 \) has roots of the form \( \alpha +i \beta \) where \( \alpha > 0 \)and \( \beta \neq 0 \) are real numbers [here \( i=\sqrt{-1} \)]. Show that (i) the numbers a, b, c are all positive; (ii) the numbers \( \sqrt{a},\sqrt{b},\sqrt{c} \) form the sides of a triangle.
- It is desired to choose n integers from the collection of 2n integers, namely, 0, 0, 1, 1, 2, 2, . . . , n−1, n−1 such that the average (that is, the arithmetic mean) of these n chosen integers is itself an integer and as minimum as possible. Show that this can be done for each positive integer n and find this minimum average for each n.

- Let ABCD be a parallelogram. Suppose a line passing through C and lying outside the parallelogram meets AB and AD produced at E and F respectively. Show that \( AC^2+CE.CF=AB.AE+AD.AF \)
- Show that there do not exist positive integers m and n such that \( \frac{m}{n}+\frac{m+1}{n}=4 \).
- If a, b, c are three distinct real numbers and \( (a+ \frac{1}{b})+(b+\frac{1}{c})+(c+\frac{1}{a})=t \) for some real number t, prove that abc + t = 0.
- In a unit square one hundred segments are drawn from the center to to the sides dividing the square into one hundred parts (triangles and possibly quadrilaterals). If all the parts have equal perimeter p show that 1 · 4 < p < 1 · 5.
- Find the number of 4 × 4 arrays whose entries are from the set {0, 1, 2, 3} and which are such that the sum of numbers in each of the four rows and each of the four columns is divisible by 4. (An m × n array is an arrangement of mn numbers in m rows and n columns.)
- Suppose a and b are two positive real numbers such that the roots of the cubic equation \( x^3-ax+b=0 \) are all real. If is a root of this cubic with minimum absolute value, prove that \( \frac{b}{a} \le \alpha \le \frac{3b}{2a} \)

- (a) Given any positive integer n, show that there exist distinct positive integers x and y such that x + j divides y + j for j = 1, 2, 3, · · · , n. (b) If for some positive integers x and y, x + j divides y + j for all positive integers j, prove that x = y.
- Let C1 and C2 be two concentric circles in the plane with radii R and 3R respectively. Show that the orthocentre of any triangle inscribed in circle C1 lies in the interior of circle C2. Conversely, show that also every point in the interior of C2 is the orthocentre of some triangle inscribed in C1.
- Solve the following system of equations for real numbers a, b, c, d, e. \( 3a=(b+c+d)^2 \), \( 3b=(c+d+e)^2 \) , \(3c=(d+e+a)^2 \), \(3d=(e+a+b)^2\), \( 3e=(a+b+c)^2 \).
- Let X be a set containing n elements. Find the number of all ordered triples (A,B,C) of subsets of X such that A is a subset of B and B is a proper subset of C.
- Define a sequence \( (a_n)_{n \ge 1} \) by \( a_1=1,a_2=2 \) and \( a_{n+2}=2a_{n+1}-a_{n}+2 \) for \( n \ge 1 \). Prove that for any m, \( a_{m}a_{m+1} \) is also a term in the sequence.
- There is a 2n × 2n array (matrix) consisting of 0’s and 1’s and there are exactly 3n zeros.Show that it is possible to remove all the zeros by deleting some n rows and some n columns. [Note: A m × n array is a rectangular arrangement of mn numbers in which there are m horizontal rows and n vertical columns.]

- In an acute-angled triangle ABC,<A=\( 30^0 \), H is the orthocenter and M is the mid-point of BC. On the line HM, take a point T such that HM = MT. Show that AT = 2BC.
- Show that there are infinitely many pairs (a, b) of relatively prime integers (not necessarily positive) such that both quadratic functions \( x^2+ax+b=0 \) and \( x^2+2ax+b=0 \) have integer roots.
- Show that the number of 3−element subset {a, b, c} of {1, 2, 3, . . . , 63} with a + b + c < 95 is less than the number of those with a + b + c > 95.
- Let ABC be triangle and a circle \( \Gamma^{'} \) be drawn inside the triangle, touching its incircle \( \Gamma \) externally and also touching the two sides AB and AC. Show the ratio of the radii of the circles \( \Gamma \) and \( \Gamma^{'} \) is equal to \( \tan^2{\frac{\pi-A}{4}} \).
- Let \( a_1 ,a_2 ,a_3 , . . . ,a_n \) be n real numbers all greater than 1 and such that \( 1 \le k \le {n-1} \) for. Show that \( \frac{a_1}{a_2}+ \frac{a_2}{a_3} + . . . . .+ \frac{a_{n-1}}{a_n}+ \frac{a_n}{a_{1}} < 2n-1 \)
- Find all primes p for which the quotient \( 2^{p-1}-1 \) is a square.

- Let G be the centroid of a triangle ABC in which the angle C is obtuse and AD and CF be the medians from A and C respectively onto the sides BC and AB. If the four points B, D, G and F are concyclic, show that \( \frac{AC}{BC} > \sqrt{2} \). If further P is a point on the line BG extended such that AGCP is a parallelogram, show that the triangle ABC and GAP are similar.
- If \( x^5-x^3+x=a \), prove that \( x^6 \ge 2a-1 \).
- In any set of 181 square integers, prove that one can always find a subset of 19 numbers, sum of whose elements is divisible by 19.
- Find the number of nondegenerate triangles whose vertices lie in the set of points (s, t) in the plane such that \( 0 \le s \le 4 \), \( 0 \le t \le 4 \) with s and t integers.
- A circle passes through a vertex C of a triangle ABCD and touches its sides AB and AD at M and N respectively. If the distance from C to the line segment MN is equal to 5 units, find the area of the rectangle ABCD.
- If \( f:R \mapsto R \) is a function satisfying the properties (a) f(−x) = −f(x), (b) f(x + 1) = f(x) + 1, (c) \( f(\frac{1}{x})=\frac{f(x)}{x^2} \), for \( x \neq 0) \), prove that\( f(x) = x \) for all real values of x. Here \( R \) denotes the set of all real numbers.

- The diagonals AC and BD of a cyclic quadrilateral ABCD intersect at P. Let O be the circumcenter of triangle APB and H be the orthocenter of triangle CPD. Show that the points H, P, O are collinear.
- If a, b, c, d are 4 non-negative real numbers and a + b + c + d = 1, show that \( ab+bc+cd \le \frac{1}{4} \).
- Let ABC be a triangle in a plane \( \sum \). Find the set of all points P (distinct from A, B, C) in the plane \( \sum \) such that the circumcircles of triangles ABP, BCP and CAP have the same radii.
- Let \( P(x)=x^2+ax+b \)= be a quadratic polynomial in which a and b are integers. Given any integer n, show that there is an integer M such that P(n) · P(n + 1) = P(M).
- Show that there is a natural number n such that n! when written in decimal notation (that is, in base 10) ends exactly in 1993 zeros.
- Let ABC be triangle right-angled at A and S be its circumcircle. Let \( S_1 \) be the circle touching the lines AB and AC and the circle S internally. Further let \( S_2 \) be the circle touching the lines AB and AC, and the circle S externally. If \( r_1 \) and \( r_2 \) be the radii of the circles \( S_1 \) and \( S_2 \) respectively, show that \( r_1·r_2=4(area \Delta ABC) \).
- Let A = {1, 2, 3, . . . , 100} and B be a subset of A having 53 elements. Show that B has two distinct elements x and y whose sum is divisible by 11.
- Let f be a bijective (1-1 and onto) function from A = {1, 2, 3 . . . , n} to itself. Show that there is positive number such that, for each i in A. denotes the composite function \( f \) m-times.
- Show that there exists a convex hexagon in the plane such that (a) all its interior angles are equal, (b) all its sides are 1, 2, 3, 4, 5, 6 in some order.

- In a triangle ABC, angle A is twice angle B. Show that \( a^2=b.(b+c) \).
- If x, y and z are three real numbers such that x + y + z = 4 and \( x^2+y^2+z^2=6 \), then show that each of x, y and z lies in the closed interval [2/3, 2], that is \( \frac{2}{3} \le x \le 2 \)$, \( \frac{2}{3} \le y \le 2 \),and. Can x attain the extreme value 2/3 or 2 ?
- Find the remainder when \( 19^{92} \) is divided by 92.
- Find the number of permutations \( (P_1, P_2, P_3, P_4, P_5, P_6) \) of 1, 2, 3, 4, 5, 6 such that for any k, \( 1 \le k \le 5 \). \( (P_1, P_2, . . . P_k) \) does not form a permutation of {1, 2, . . . k}. That is \( P_1 \neq 0 \); \( (P_1, P_2) \) is not permutation of {1, 2}; \( (P_1, P_2, P_3) \) is not a permutation of {1, 2, 3}, etc.
- Two circles C1 and C2 intersect at two distinct points P and Q in a plane. Let a line passing through P meet the circles C1 and C2 in A and B respectively. Let Y be the mid-point of AB and QY meet the circles C1 and C2 in X and Z respectively. Show that Y is also the mid-point of XZ.
- Let \( f(x) \) be a polynomial in x with integer coefficients and suppose that for 5 distinct integers \( a_1,a_2,a_3,a_4 \) and \( a_5 \) one has \( f(a_1)=f(a_2)=f(a_3)=f(a_4)=f(a_5)=2 \). Show that there does not exist an integer b such that \( f(b) = 9 \).
- Find the number of ways in which one can place the numbers \( 1, 2, 3, . . ., n^2 \) on the \( n^2 \) squares of n × n chessboard, one on each, such that the numbers in each row and each column are in arithmetic progression. (Assume \( n \ge 3 ) \).
- Determine all pairs (m, n) of positive integers for which \( 2^n+3^n \) is a perfect square.
- Let \( A_1A_2A_3 . . .A_n \) be an n-sided regular polygon such that \( \frac{1}{A_1A_2}=\frac{A_1}{A_3}+\frac{A_1}{A_4} \). Determine n, the number of sides of the polynomial.
- Determine all functions \( f:R{0,1} \mapsto R \) satisfying the functional relation,where x is a real number different from 0 and 1.

- Find the number of positive integers n for which (a) \( n \le 1991 \) and (b) 6 is a factor of \( n^2+3n+2 \).
- Given any acute-angled triangle ABC, let points \( A',B',C' \) be located as follows : \( A^{'} \)is the point where altitude from A on BC meets the outwards facing semi-circle drawn on BC as diameter. Points \( B^{'},C^{'} \) are located similarly. Prove that \( |BCA^{'}|^2+|CAB^{'}|^2+|ABC^{'}|^2 \\ =|ABC|^2 \), where [ABC] denotes the area of triangle ABC, etc.
- Given a triangle ABC, define the quantities x, y, z as follows: \( x = \tan((B − C)/2) \tan(A/2) \) , \( y = \tan((C − A)/2) \tan(B/2) \), \( z = \tan((A − B)/2) \tan(C/2) \). Prove that : $$ x + y + z + xyz = 0. $$
- Let a, b, c be real numbers with 0 < a < 1, 0 < b < 1, 0 < c < 1 and a+b+c = 2. Prove that $$ \frac{a}{1-a}.\frac{b}{1-b}.\frac{c}{1-c} \ge 8 $$.
- Triangle ABC has incenter I. Let points X, Y be located on the line segment AB, AC respectively so that : \( BX.AB=IB^2 \) and \( CY.AC=IC^2 \) Given that the points X, I, Y lie on a straight line, find the possible values of the measure of angle A.
- (a) Determine the set of all positive integers n for which \( 3^{n+1} \) divides \( 2^{j+1} \). (b) Prove that \( 3^{n+2} \) does not divide \( 2^{3^{n}}+1 \) for any positive integer n.
- Solve the following system of equations for real x, y, z: \( x+y-z=4, x^2-y^2+z^2=4,xyz=6 \).
- There are 10 objects with total weight 20, each of the weights being a positive integer. Given that none of the weights exceeds 10, prove that the 10 objects can be divided into two groups that balance each other when placed on the two pans of a balance.

- Prove that the polynomial \( f(x)=x^4+26x^3+56x^2+78x+1989 \) 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.
- Let a, b, c and d be any four real numbers, not all equal to zero. Prove that the roots of the polynomial \( x^6+ax^3+bx^2+cx+d \) cannot all be real.
- Let A denote a subset of the set {1, 11, 21, 31, . . . 541, 551} having the property that no two elements of A add up to 552. Prove that A cannot have more than 28 elements.
- Determine with proof, all the positive integers n for which: (a) n is not the square of any integer; and (b) \( [\sqrt{n}] \) divides \( n^2 \). (Notation : [x] denotes the largest integer that is less than or equal to x).
- 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.
- Triangle ABC has incenter I and the incircle touches BC, CA at D, E respectively. Let BI meet DE at G. Show that AG is perpendicular to BC.
- Let A be one of the two points of intersection of two circles with centers X , Y respectively. The tangents at A to the two circles meet the circles again at B.C . Let a point P belocated so that PX AY is a parallelogram. Show that P is also the circum-center of triangle ABC .

- Prove that the polynomial \( f(x)=x^4+26x^3+56x^2+78x+1989 \) 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.
- Let a, b, c and d be any four real numbers, not all equal to zero. Prove that the roots of the polynomial \( x^6+ax^3+bx^2+cx+d \) cannot all be real.
- Let A denote a subset of the set {1, 11, 21, 31, . . . 541, 551} having the property that no two elements of A add up to 552. Prove that A cannot have more than 28 elements.
- Determine with proof, all the positive integers n for which: (a) n is not the square of any integer; and (b) \( [\sqrt{n}] \) divides \( n^2 \). (Notation : [x] denotes the largest integer that is less than or equal to x).
- 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.
- Triangle ABC has incenter I and the incircle touches BC, CA at D, E respectively. Let BI meet DE at G. Show that AG is perpendicular to BC.
- Let A be one of the two points of intersection of two circles with centers X , Y respectively. The tangents at A to the two circles meet the circles again at B.C . Let a point P belocated so that PX AY is a parallelogram. Show that P is also the circum-center of triangle ABC .

- Let \( m_{1},m_{2},m_{3},\dots,m_{n} \) be a rearrangement of the numbers \( 1,2,\dots,n \). Suppose that \( n \) is odd. Prove that the product $$[\left(m_{1}-1\right)\left(m_{2}-2\right)\cdot\left(m_{n}-n\right)]$$ is an even integer.
- Prove that the product of 4 consecutive natural numbers cannot be a perfect cube.
- Five men \( A \), \( B \), \( C \), \( D \), \( E \) are wearing caps of black or white colour without each knowing the colour of his cap. It is known that a man wearing black cap always speaks the truth while the ones wearing white always tell lies. If they make the following statements, find the colour worn by each of them:

\( A \) : I see three black caps and one white cap.

\( B \) : I see four white caps

\( C \) : I see one black cap and three white caps

\( D \) : I see your four black caps. - If \( a \) and \( b \) are positive and \( a+b = 1 \), prove that $$[\left(a+\frac{1}{a}\right)^{2}+\left(b+\frac{1}{b}\right)^{2}\geq\frac{25}{2}]$$
- Show that there do not exist any distinct natural numbers \( a \), \( b \), \( c \), \( d \) such that \( a^{3}+b^{3}=c^{3}+d^{3} \) and \( a^{3}+b^{3}=c^{3}+d^{3} \).
- If \( a_{0},a_{1},\dots,a_{50} \) are the coefficients of the polynomial [\(\left(1+x+x^{2}\right)^{25}\)] show that \( a_{0}+a_{2}+a_{4}+\cdots+a_{50} \) is even.
- Given an angle \( \angle QBP \) and a point \( L \) outside the angle \( \angle QBP \). Draw a straight line through \( L \) meeting \( BQ \) in \( A \) and \( BP \) in \( C \) such that the triangle \( \triangle ABC \) has a given perimeter.
- A river flows between two houses \( A \) and \( B \), the houses standing some distances away from the banks. Where should a bridge be built on the river so that a person going from \( A \) to \( B \), using the bridge to cross the river may do so by the shortest path? Assume that the banks of the river are straight and parallel, and the bridge must be perpendicular to the banks.
- Show that for a triangle with radii of circumcircle and incircle equal to \( R \), \( r \) respectively, the inequality \( R\geq 2r \) holds.

- Given and as relatively prime positive integers greater than one, show that \([\frac{\log_{10}m}{\log_{10}n} ]\) is not a rational number.

Determine the largest number in the infinite sequence \([ 1,\sqrt[2]{2},\sqrt[3]{3},\sqrt[4]{4},dots,\sqrt[n]{n},dots ]\) - Let \( T \) be the set of all triplets \( (a,b,c) \) of integers such that \( 1\leq a < b < c\leq 6 \) For each triplet \( (a,b,c) \) in \( T \), take number \( ac.bc.c \). Add all these numbers corresponding to all the triplets in \( T \). Prove that the answer is divisible by 7.
- If \( x \), \( y \), \( z \), and \( n \) are natural numbers, and \( n\geq z \) then prove that the relation \( x^{n}+y^{n}= z^{n} \) does not hold.
- Find a finite sequence of 16 numbers such that: (a) it reads same from left to right as from right to left. (b) the sum of any 7 consecutive terms is \( -1 \), (c) the sum of any 11 consecutive terms is \( +1 \).
- Prove that if coefficients of the quadratic equation \( ax^{2}+bx+c=0 \) are odd integers, then the roots of the equation cannot be rational numbers.
- Construct the \( \triangle ABC \), given \( h_{a} \), \( h_{b} \) (the altitudes from \( A \) and \( B \)) and \( m_{a} \), the median from the vertex \( A \).
- Three congruent circles have a common point \( O \) and lie inside a given triangle. Each circle touches a pair of sides of the triangle. Prove that the incentre and the circumcentre of the triangle and the common point \( O \) are collinear.
- Prove that any triangle having two equal internal angle bisectors (each measured from a vertex to the opposite side) is isosceles.

- A person who left home between 4 p.m. and 5 p.m. returned between 5 p.m. and 6 p.m. and found that the hands of his watch had exactly exchanged place, when did he go out ?
- Solve $$ \log_{2}x+\log_{4}y+\log_{4}z \\ = 2 \log_{3}y+\log_{9}z+\log_{9}x \\ =2 \log_{4}z+\log_{16}x+\log_{16}y=2 $$
- Two circles with radii a and b respectively touch each other externally. Let c be the radius of a circle that touches these two circles as well as a common tangent to the two circles. Prove that $$ \frac{1}{\sqrt{c}}=\frac{1}{\sqrt{a}}+\frac{1}{\sqrt{b}} $$
- Find the least natural number whose last digit is 7 such that it becomes 5 times larger when this last digit is carried to the beginning of the number.
- If \( P(x) \) is a polynomial with integer coefficients and a, b, c , three distinct integers, then show that it is impossible to have $ P(a)=b$, $ P(b)=c$, $ P(c)=a$.
- Construct a quadrilateral which is not a parallelogram, in which a pair of opposite angles and a pair of opposite sides are equal.
- If \( a \), \( b \), \( x \), \( y \) are integers greater than 1 such that \( a \) and \( b \) have no common factor except 1 and \( x^{a}= y^{b} \) show that , for some integer greater than 1.
- Suppose \( A_{1},\cdots, A_{6}\) are six sets each with four elements and \( B_{1},\cdots,B_{n}\) are sets each with two elements, Let \( S = A_{1}\cup A_{2}\cup \cdots \cup A_{6} \\ = B_{1}\cup \cdots \cup B_{n} \). Given that each elements of belongs to exactly four of the \( A \)'s and to exactly three of the \( B \)'s, find \( n\).
- Show that among all quadrilaterals of a given perimeter the square has the largest area

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