Cheenta hosted the final round of prestigious Sharygin Geometry Olympiad in India. The olympiad is intended for high-school students of four eldest grades. This post contains the problems from this contest.
Sharygin Olympiad is conducted by organisers from esteemed institutions in Russia.
Steklov Mathematical Institute RAS
Department of Education, Moscow City
Moscow Center for Pedagogical Mastery
Moscow Center for Continuous Mathematical Education
Journal of Classical Geometry
Moscow Institute of Physics and Technology
Awardees from India - 2024
Rishav Dutta (3rd prize)
Krishiv Khandelwal (3rd Prize)
Sayantan Mazumdar (3rd Prize)
Debarchan Neogi (3rd Prize)
Aratrik Pal (2nd Prize)
Kanav Talwar (1st Prize)
First Day Problems - Grade 8
A circle $\boldsymbol{\omega}$ centered at $O$ and a point $P$ inside it are given. Let $X$ be an arbitrary point of $\omega$, the line $X P$ and the circle $X O P$ meet $\omega$ for a second time at points $X_1, X_2$ respectively. Prove that all lines $X_1 X_2$ are parallel.
Let $C M$ be the median of an acute-angled triangle $A B C$, and $P$ be the projection of the orthocenter $H$ to the bisector of angle $C$. Prove that $M P$ bisects the segment $C H$.
Let $A D$ be the altitude of an acute-angled triangle $A B C$, and $A^{\prime}$ be the point of its circumcircle opposite to $A$. A point $P$ lies on the segment $A D$, and points $X, Y$ lie on the segments $A B, A C$ respectively in such a way that $\angle C B P=\angle A D Y$, $\angle B C P=\angle A D X$. Let $P A^{\prime}$ meet $B C$ at point $T$. Prove that $D, X, Y, T$ are concyclic.
A square with sidelength 1 is cut from the paper. Construct a segment with length $1 / 2024$ using at most 20 folds. No instruments are available, it is allowed only to fold the paper and to mark the common points of folding lines.
First Day Problems - Grade 9
Let $H$ be the orthocenter of an acute-angled triangle $A B C ; A_1, B_1, C_1$ be the touching points of the incircle with $B C, C A, A B$ respectively; $E_A, E_B$, $E_C$ be the midpoints of $\mathrm{AH}, \mathrm{BH}, \mathrm{CH}$ respectively. The circle centered at $E_A$ and passing through $A$ meets for the second time the bisector of angle $A$ at $A_2$; points $B_2, C_2$ are defined similarly. Prove that the triangles $A_1 B_1 C_1$ and $A_2 B_2 C_2$ are similar.
Points $A, B, C, D$ on the plane do not form a rectangle. Let the sidelengths of triangle $T$ equal $A B+C D, A C+B D, A D+B C$. Prove that the triangle $T$ is acute-angled.
Let $\left(P, P^{\prime}\right)$ and $\left(Q, Q^{\prime}\right)$ be two pairs of points isogonally conjugated with respect to a triangle $A B C$, and $R$ be the common point of lines $P Q$ and $P^{\prime} Q^{\prime}$. Prove that the pedal circles of points $P, Q$, and $R$ are coaxial.
For which $n>0$ it is possible to mark several different points and several different circles on the plane in such a way that:
exactly $n$ marked circles pass through each marked point;
exactly $n$ marked points lie on each marked circle;
the center of each marked circle is marked?
First Day Problems - Grade 10
The diagonals of a cyclic quadrilateral $A B C D$ meet at point $P$. The bisector of angle $A B D$ meets $A C$ at point $E$, and the bisector of angle $A C D$ meets $B D$ at point $F$. Prove that the lines $A F$ and $D E$ meet on the median of triangle $A P D$.
For which greatest $n$ there exists a convex polyhedron with $n$ faces having the following property: for each face there exists a point outside the polyhedron such that the remaining $n-1$ faces are seen from this point?
Let $B E$ and $C F$ be the bisectors of a triangle $A B C$. Prove that $2 E F \leq B F+C E$.
Let $I$ be the incenter of a triangle $A B C$. The lines passing through $A$ and parallel to $B I, C I$ meet the perpendicular bisector to $A I$ at points $S$, $T$ respectively. Let $Y$ be the common point of $B T$ and $C S$, and $A^*$ be a point such that $BICA^*$ is a parallelogram. Prove that the midpoint of segment $Y A^*$ lies on the excircle of the triangle touching the side $B C$.
Second Day Problems - Grade 8
The vertices $M, N, K$ of rectangle $K L M N$ lie on the sides $A B, B C, C A$ respectively of a regular triangle $A B C$ in such a way that $A M=2, K C=1$, the vertex $L$ lies outside the triangle. Find the value of angle $K M N$.
A circle $\boldsymbol{\omega}$ touches lines $a$ and $b$ at points $A$ and $B$ respectively. An arbitrary tangent to the circle meets $a$ and $b$ at $X$ and $Y$ respectively. Points $X^{\prime}$ and $Y^{\prime}$ are the reflections of $X$ and $Y$ about $A$ and $B$ respectively. Find the locus of projections of the center of the circle to the lines $X^{\prime} Y^{\prime}$.
A convex quadrilateral $A B C D$ is given. A line $l | A C$ meets the lines $A D, B C, A B, C D$ at points $X, Y, Z, T$ respectively. The circumcircles of triangles $X Y B$ and $Z T B$ meet for the second time at point $R$. Prove that $R$ lies on $B D$.
Two polygons are cut from the cartboard. Is it possible that for any disposition of these polygons on the plane they have common inner point or have only finite number of common points?
Second Day Problems - Grade 9
Let $A B C$ be an isosceles triangle $(A C=B C)$, $O$ be its circumcenter, $H$ be the orthocenter, and $P$ be a point inside the triangle such that $\angle A P H=$ $=\angle B P O=\pi / 2$. Prove that $\angle P A C=\angle P B A=\angle P C B$.
The incircle of a triangle $A B C$ centered at $I$ touches the sides $B C, C A$, and $A B$ at points $A_1$, $B_1$, and $C_1$ respectively. The excircle centered at $J$ touches the side $A C$ at point $B_2$ and touches the extensions of $A B, B C$ at points $C_2, A_2$ respectively. Let the lines $I B_2$ and $J B_1$ meet at point $X$, the lines $I C_2$ and $J C_1$ meet at point $Y$, the lines $I A_2$ and $J A_1$ meet at point $Z$. Prove that if one of points $X, Y$, $Z$ lies on the incircle then two remaining points also lie on it.
Let $P$ and $Q$ be arbitrary points on the side $B C$ of triangle $A B C$ such that $B P=C Q$. The common points of segments $A P$ and $A Q$ with the incircle form a quadrilateral $X Y Z T$. Find the locus of common points of diagonals of such quadrilaterals.
Let points $P$ and $Q$ be isogonally conjugated with respect to a triangle $A B C$. The line $P Q$ meets the circumcircle of $A B C$ at point $X$. The reflection of $B C$ about $P Q$ meets $A X$ at point $E$. Prove that $A$, $P, Q, E$ are concyclic.
Second Day Problems - Grade 10
The incircle of a right-angled triangle $A B C$ touches the hypothenuse $A B$ at point $T$. The squares $A T M P$ and $B T N Q$ lie outside the triangle. Prove that the areas of triangles $A B C$ and $T P Q$ are equal.
A point $P$ lies on one of medians of triangle $A B C$ in such a way that $\angle P A B=\angle P B C=\angle P C A$. Prove that there exists a point $Q$ on another median such that $\angle Q B A=\angle Q C B=\angle Q A C$.
Let $A B C$ be a triangle with $\angle A=60^{\circ} ; A D$, $B E$, and $C F$ be its bisectors; $P, Q$ be the projections of $A$ to $E F$ and $B C$ respectively; and $R$ be the second common point of the circle $D E F$ with $A D$. Prove that $P, Q, R$ are collinear.
The common tangents to the circumcircle and an excircle of triangle $A B C$ meet $B C, C A, A B$ at points $A_1, B_1, C_1$ and $A_2, B_2, C_2$ respectively. The triangle $\Delta_1$ is formed by the lines $A A_1, B B_1$, and $C C_1$, the triangle $\Delta_2$ is formed by the lines $A A_2$, $B B_2$, and $C C_2$. Prove that the circumradii of these triangles are equal.
IOQM is the first level of (real) math olympiad in India. More than a 100,000 students take it every year. Success in IOQM leads to the second level which is RMO (Regional Math Olympiad) and third level INMO (Indian National Math Olympiad).
In this post we have added the problems and solutions from the IOQM 2024.
The smallest positive integer that does not divide $ \times 2 \times 3 \times 4 \times 5 \times 6 \times 7 \times 8 \times 9$ is:
Solution
Since this is 9 ! it will be divisible by all the numbers from $1-9$. Also, 9 ! contains $2 * 5=10$, it is divisible by 10 . Lets check for 11 . Since 11 is a prime number : 9! will not be divisible by 11 . So, the smallest positive number that does not divides $1 \times 2 \times \times 3 \times 4 \times 5 \times 6 \times$ $7 \times 8 \times 9$ is 11 .
Problem 2
The number of four-digit odd numbers having digits $1,2,3,4$, each occuring exactly once, is:
Solution
Number of odd numbers that we can form using $1,2,3,4$ without repeating implies that ones digit is either 1 or 3 . Rest of the three places can be occupied any of the remaining three digits. So the number of total ways $=3 \times 2 \times 1 \times 2=12$.
Problem 3
The number obtained by taking the last two digits of $5^{2024}$ in the same order is:
Solution
Last two digits of $5^n=25$
Problem 4
Let $A B C D$ be a quadriateral with $\angle A D C=70^{\circ}, \angle A C D=70^{\circ}, \angle A C B=10^{\circ}$ and $\angle B A D=110^{\circ}$. The measure of $\angle C A B$ (in degrees) is:
Problem 5
Let $a=\frac{x}{y}+\frac{y}{z}+\frac{z}{x}$, let $b=\frac{x}{z}+\frac{y}{x}+\frac{z}{y}$ and let $c=\left(\frac{x}{y}+\frac{y}{z}\right)\left(\frac{y}{z}+\frac{z}{x}\right)\left(\frac{z}{x}+\frac{x}{y}\right)$. The value of $|a b-c|$ is:
Solution
$a=\frac{x}{y}+\frac{y}{z}+\frac{z}{x}=\frac{x^2 z+y^2 x+z^2 y}{x y z}$
$b=\frac{x}{z}+\frac{y}{x}+\frac{z}{y}=\frac{x^2 y+y^2 z+z^2 x}{x y z}$
Find the number of triples of real numbers $(a, b, c)$ such that $a^{20}+b^{20}+c^{20}=a^{24}+b^{24}+c^{24}=1$
Solution
The triples $(a, b, c)$ is $(1,0,0),(0,1,0),(0,0,1),(-1,0,0),(0,-1,0),(0,0,-1)$
Problem 7
Determine the sum of all possible surface areas of a cube two of whose vertices are $(1,2,0)$ and $(3,3,2)$.
Solution
Notice that the distance between the points $(1,2,0)$ and $(3,3,2)=$ $\sqrt{2^2+1^2+2^2}=3$.
In the cube, we can have the vertices $(1,2,0)$ and $(3,3,2)$ as endpoints of a side of a cube (or) endpoints of a face diagonal (or) endpoints of a main diagonal and there is no other configuration. If $a$ denotes the side length of a cube, then $a=3, a \sqrt{2}=3, a \sqrt{3}=3$ are the only possibilities.
This implies $a=3, \frac{3}{\sqrt{2}}, \sqrt{3}$. We know that the surface area of cube $=$ $6 a^2=6 \times 9$ (or) $6 \times \frac{9}{2}$ (or) $6 \times 3$. Hence, the sum of all possible surface areas $=6 \times 9+6 \times \frac{9}{2}+6 \times 3=6\left(9+\frac{9}{2}+3\right)=99$.
Problem 8
Let $n$ be the smallest integer such that the sum of digits of $n$ is divisible by 5 as well as the sum of digits of $(n+1)$ is divisible by 5 . What are the first two digits of $n$ in the same order?
Solution
Let $S(m)$ denote the sum of digits of a natural number $m$. Notice that, $S(m+1)=S(m)+1$, if $m$ does not end with 9 .
Clearly, if the number $n$ does not end with 9 , then $S(n)$ and $S(n+1)$ would only differ by 1 and hence both of them cannot be divisible by 5 simultaneously. Hence, $n=\overline{x 9}$.
Now, similarly if $x$ does not end with 9 , then $S(n)=S(x)+9 ; S(n+1)=$ $S(x+1)=S(x)+1 \Rightarrow S(n+1)-S(n)=8$, contradiction. Hence, $n=\overline{y 99}$. And again $y$ must end with 9 as otherwise, $S(n+1)-S(n)=9+9-1=17$, contradiction.
If we proceed in this way, then we would get $n=\overline{z 9999}$. Now, for the smallest value of such $n$, we take $z$ to be a single digit number. Note that $9+9+9+9=36$ is 1 modulo 5 and hence we need $z$ to be 4 modulo 5 . Thus, $z=4$ is the least possible such number and hence $n=49999$ is the least possible value of $n$.
Problem 9
Consider the grid of points $X={(m, n) \mid 0 \leq m, n \leq 4}$. We say a pair of points ${(a, b),(c, d)}$ in $X$ is a knight-move pair if $(c=a \pm 2$ and $d=b \pm 1$ ) or ( $c=a \pm 1$ and $d=b \pm 2$ ). The number of knight-move pairs in $X$ is:
Solution
Solution: We will find the possible number of knight-move pairs corresponding to each point and add all of them. But in this counting each point would be over counted exactly twice (as pair consits of two distinct points) and we divide by 2 at last in order to account for it and to get the final answer.
Let us consider only the points in $Y={(m, n) \mid 0 \leq m, n \leq 2} \subset X$ because using symmetry one can extend the number of possible pairs containing that point to all the other points. Counting by this way explicitly, one would get the number of pairs as shown in the diagram.
Hence, the sum of all the numbers $=4 \times(2+3+4+3)+4 \times(4+6)+8=96$. So, final answer $=\frac{96}{2}=48$.
Problem 10
Determine the number of positive integral values of $p$ for which there exists a triangle with sides $a, b$, and $c$ which satisfy
$$ a^2+\left(p^2+9\right) b^2+9 c^2-6 a b-6 p b c=0 . $$
Solution
Notice that
$$ a^2+\left(p^2+9\right) b^2+9 c^2-6 a b-6 p b c=(p b-3 c)^2+(a-3 b)^2=0 $$
Hence, individually the bracket values must be $0 \Rightarrow p b-3 c=0 ; a-3 b=0 \Rightarrow$ $a=3 b ; p b=3 c$. Hence, $(a, b, c)=\left(3 b, b, \frac{p b}{3}\right)$. We know that the necessary and sufficient condition for existence of a triangle is that the largest side being less than sum of other two sides. So, we take two cases now,
If $p<9 \Rightarrow \frac{p b}{3}<3 b \Rightarrow 3 b$ is the largest side. Hence, we need $3b<b+\frac{pb}{3}\Rightarrow p>6$. Hence, the allowed values for this case are $p=7,8$.
If $p \geq 9 \Rightarrow \frac{p b}{3}\geq 3 b \Rightarrow \frac{p b}{3}$ is the largest side. Hence, we need $\frac{p b}{3}<3 b+b \Rightarrow$ $p<12$. Hence, the possible values for this case are $p=9,10,11$.
Hence, there five possible values for $p$ which are $7,8,9,10,11$.
What is the value of $\frac{1}{a}+\frac{1}{b}+\frac{1}{c} ?$
Solution
By adding up the two equations, we would get, $$\frac{a+1}{2b+1}+\frac{2b+1}{3c+1}+\frac{3c+1}{a+1}=3\rightarrow Eq 1$$ Notice that each of these terms are positive real numbers and their product $\frac{a+1}{2b+1}\times\frac{2b+1}{3c+1}\times\frac{3c+1}{a+1}=1$.
Let $\alpha=\frac{a+1}{2b+1},\ \beta=\frac{2b+1}{3c+1},\ \gamma=\frac{3c+1}{a+1}$, then by the AM-GM inequality, $$\frac{\alpha+\beta+\gamma}{3}\geq\left(\alpha\beta\gamma\right)^{\frac{1}{3}}=1$$ with the equality holding true when $\alpha=\beta=\gamma=1$. But from the equation $Eq 1$, we know that the equality holds true and hence $\frac{a+1}{2b+1}=\frac{2b+1}{3c+1}=\frac{3c+1}{a+1}=1\Rightarrow \boxed{a=2b=3c}$. Now, substituting it in the first equation in the question, we get, $a=\frac{1}{2}=2b=3c\Rightarrow\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=12$.
Problem 12
Consider a square $A B C D$ of side length 16. Let $E, F$ be points on $C D$ such that $C E=E F=F D$. Let the line $B F$ and $A E$ meet in $M$. The area of $\triangle M A B$ is:
Solution
Drop perpendiculars from $M$ to $AB, CD$ and let the foot of perpendiculars be $X,Y$ respectively as shown.
Let $a=16=side\ of\ square$. Note that $CE=EF=FD\Rightarrow EF=\frac{a}{3}$. Since, $AB||EF\Rightarrow\triangle ABM\sim\triangle EFM\Rightarrow\frac{MX}{MY}=\frac{AB}{EF}=3$. But, we know that $XY=a\Rightarrow MX=\frac{3a}{4},\ MY=\frac{a}{4}$. Hence, area of $\triangle MAB=\frac{1}{2}\times MX\times AB=\frac{3a^2}{8}=96$.
Problem 13
Three positive integers $a, b, c$ with $a>c$ satisfy the folowing equations:
$$ a c+b+c=b c+a+66, \quad a+b+c=32 $$
Find the value of $a$.
Solution
Problem 14
Initially, there are $3^{80}$ particles at the origin $(0,0)$. At each step the particles are moved to points above the $x$-axis as follows: if there are $n$ particles at any point $(x, y)$, then $\left\lfloor\frac{n}{3}\right\rfloor$ of them are moved to $(x+1, y+1),\left\lfloor\frac{n}{3}\right\rfloor$ are moved to $(x, y+1)$ and the remaining to $(x-1, y+1)$. For example, after the first step, there are $3^{79}$ particles each at $(1,1),(0,1)$ and $(-1,1)$. After the second step, there are $3^{78}$ particles each at $(-2,2)$ and $(2,2), 2 \times 3^{78}$ particles each at $(-1,2)$ and $(1,2)$, and $3^{79}$ particles at $(0,2)$. After 80 steps, the number of particles at $(79,80)$ is:
Solution
Problem 15
Let $X$ be the set consisting of twenty positive integers $n, n+2, \ldots, n+38$. The smallest value of $n$ for which any three numbers $a, b, c \in X$, not necessarily distinct, form the sides of an acute-angled triangle is:
Problem 16
Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a function satisfying the relation $4 f(3-x)+3 f(x)=x^2$ for any real $x$. Find the value of $f(27)-f(25)$ to the nearest integer. (Here $\mathbb{R}$ denotes the set of real numbers.)
Solution
Problem 17
Consider an isosceles triangle $A B C$ with sides $B C=30, C A=A B=20$. Let $D$ be the foot of the perpendicular from $A$ to $B C$, and let $M$ be the midpoint of $A D$. Let $P Q$ be a chord of the circumcircle of triangle $A B C$, such that $M$ lies on $P Q$ and $P Q$ is parallel to $B C$. The length of $P Q$ is:
Solution
Problem 18
Let $p, q$ be two-digit numbers neither of which are divisible by 10 . Let $r$ be the four-digit number by putting the digits of $p$ followed by the digits of $q$ (in order). As $p, q$ vary, a computer prints $r$ on the screen if $\operatorname{gcd}(p, q)=1$ and $p+q$ divides $r$. Suppose that the largest number that is printed by the computer is $N$. Determine the number formed by the last two digits of $N$ (in the same order).
Problem 19
Consider five points in the plane, with no three of them collinear. Every pair of points among them is joined by a line. In how many ways can we color these lines by red or blue, so that no three of the points form a triangle with lines of the same color.
Solution
Problem 20
On a natural number $n$ you are allowed two operations: (1) multiply $n$ by 2 or (2) subtract 3 from n. For example starting with 8 you can reach 13 as follows: $8 \rightarrow 16 \rightarrow 13$. You need two steps and you cannot do in less than two steps. Starting from 11, what is the least number of steps required to reach 121 ?
Solution
Problem 21
An intenger $n$ is such that $\left\lfloor\frac{n}{9}\right\rfloor$ is a three digit number with equal digits, and $\left\lfloor\frac{n-172}{4}\right\rfloor$ is a 4 digit number with the digits $2,0,2,4$ in some order. What is the remainder when $n$ is divided by $100 ?$
Solution
Problem 22
In a triangle $A B C, \angle B A C=90^{\circ}$. Let $D$ be the point on $B C$ such that $A B+B D=A C+C D$. Suppose $B D: D C=2: 1$. If $\frac{A C}{A B}=\frac{m+\sqrt{p}}{n}$, where $m, n$ are relatively prime positive integers and $p$ is a prime number, determine the value of $m+n+p$.
Solution
Problem 23
Consider the fourteen numbers, $1^4, 2^4, \ldots, 14^4$. The smallest natural number $n$ such that they leave distinct remainders when divided by $n$ is:
using Fermat’s Little theorem we get $a \equiv b(\bmod 31)$ which is not possible hence 31 is the right answer
Problem 24
Consider the set $F$ of all polynomials whose coefficients are in the set of ${0,1}$. Let $q(x)=x^3+x+1$. The number of polynomials $p(x)$ in $F$ of degree 14 such that the product $p(x) q(x)$ is also in $F$ is:
Solution
Problem 25
A finite set $M$ of positive integers consists of distinct perfect squares and the number 92 . The average of the numbers in $M$ is 85 . If we remove 92 from $M$, the average drops to 84 . If $N^2$ is the largest possible square in $M$, what is the value of $N$ ?
$x=16$ satisfies the equation if $\left.\lfloor x\rfloor=17 \Rightarrow 16+15 x+15 x^2\right)=17^3$ has a root in $(17,18)$ But $\lfloor x\rfloor=18$ has no solution $\text { sum }=16+17=33$
Problem 27
In a triangle $A B C$, a point $P$ in the interior of $A B C$ is such that $\angle B P C-\angle B A C=\angle C P A-\angle C B A=\angle A P B-\angle A C B$. Suppose $\angle B A C=30^{\circ}$ and $A P=12$. Let $D, E, F$ be the feet of perpendiculars form $P$ on to $B C, C A, A B$ respectively. If $m \sqrt{n}$ is the area of the triangle $D E F$ where $m, n$ are integers with prime, then what is the value of the product $m n$ ?
Solution
let $\alpha, \beta, \gamma$ be $\angle B P C, \angle C P A, \angle A P C$ respectively.
Find the largest positive integer $n<30$ such that $\frac{1}{2}\left(n^8+3 n^4-4\right)$ is not divisible by the square of ann prime number.
Solution
$f(n)=\frac{n^8+3 n^4-4}{2}$
Note, by Fermat’s little theorem,
$ x^4=1(\bmod 5) $
$ x^8=1(\bmod 5)$
Also, powers of $n$ are even. $n=k(\bmod 25) \quad$ when $k=1,2,3,4$
then $25 / f(n) \Rightarrow n \neq 21,22,23,24,26,27,28,29$ Also note $4 / f(25)$ $\therefore$ largest such $n=20$
Problem 29
Let $n=2^{19} 3^{12}$. Let $M$ denote the number of positive divisors of $n^2$ which are less than $n$ but would not divide $n$. What is the number formed by taking the last two digits of $M$ (in the same order)?
Solution
$n=2^{19} \cdot 3^{12}$
$n^2=2^{19} \cdot 3^{24}$ Let $2^a 3^b$ be such a divisor of $n^2$ which does not divide $n$ but is less than $n$.
Case 1. $a>19, b \leq 12=118$ Case 2. $a \leq 19, b>12=110$
$\therefore 118+110=228 \rightarrow 28$
Problem 30
Let $A B C$ be a right-angled triangle with $\angle B=90^{\circ}$. Let the length of the altitude $B D$ be equal to 12 What is the minimum possible length of $A C$, given that $A C$ and the perimeter of triangle $A B C$ are integers?
There are no Pythagorean triples with hypotenuse = 24.
$\therefore$ least possible $A C=25$
(15, 20, 25 triangle)
How to Build a Career in Mathematical Science - Steps and Resources
We discuss scope of a career in mathematical science. This includes paths in academic career and industry oriented career. We talk about how to prepare for these in school days and in college.
How to Solve Quadratic Diophantine Equation | ISI Entrance 2015 Objective 12
Quadratic diophantine equations can be hard to solve. However in this particular discussion, we will use a technique from elementary number theory to solve it. This problem is from the entrance of Indian Statistical Institute.
We use two facts:
Any square number produces remainders 0 or 1 when divided by 4
Remainders can be added.
Watch the video and then try the quiz that follows.
Watch Video
Try Quiz
Three Resources for IOQM and Math Olympiads
IOQM and other Math Olympiads like American Math Competitions can be really challenging. It requires the student to go beyond the school curriculum and think about non-routine problems. In this post I will share three resources that can help you to systematically take up the preparation.
Books for IOQM and Math Olympiad
In this age of abundance, there is a lot of great books on Math Olympiads. The trick is to choose one relatively good book, and solve all problems from it, cover to cover. In fact we strongly suggest our students to focus on a single book before moving to other resources.
You may start with Challenge and Thrill of Pre-College Mathematics by V Krishnamurthy, C R Pranesachar, and K N Ranganathan. This is an excellent book that covers all the topics for IOQM standard. If you are in Grade 8 or below then use Mathematical Circles (Russian Experience) by Dmitry Fomin, Sergey Genkin, Ilia Itenberg.
Ofcourse there are plenty of other great books. Here are a few of them:
Principles and Techniques in Combinatorics by Chen Chuan-Chon
Number Theory: Structures, Examples, and Problem by Titu Andreescu, Dorin Andrica
Euclidean Geometry in Mathematical Olympiads (EGMO) by Evan Chen
Algebra
Secrets in Inequalities by Pham Kim Huang
Functional Equation by Venkatchala
An Introduction to Diophantine Equations: A Problem-Based Approach by Dorin Andrica, Ion Cucurezeanu, et al.
Complex Numbers from A to Z by by Titu Andreescu and Dorin Andrica
Software for IOQM and Math Olympiad
Cheenta has an excellent software for Math Olympiad, Physics Olympiad and ISI - CMI Entrance related problem solving. This tool provides students with sequence of problem. The software adapts with the child's performance and provides problems accordingly. It is available at https://panini8.com/
The software also lets students ask doubt problems and communicate with fellow learners using Latex. We have observed that children who use this problem solving software for 15 minutes every day experience remarkable improvement in their performance.
Start With a Problem
At Cheenta every learning begins with a
Non-Routine Problem
Try the problems with hints
but try them on your own!
Solve With Hints
Ask Doubts
Cheenta community has hundreds of teachers and students
Ask your doubts. Respond to others.
Learn about your strengths and weaknesses
Know your topicwise progress.
Check Topic Wise Progress
Live Classes for IOQM and Math Olympiad
Cheenta has outstanding program for IOQM and Math Olympiads. Typically the program includes three components.
Compulsory Classes - twice a week
Concept Class
Homework Tutorial
Optional Problem Solving Class - upto five times a week
Optional 1-on-1 Class - for personalisation
Cheenta programs has been extremely successful in mathematical olympiads since 2010. For example in 2024, ten out of 78 students who qualified for Indian National Math Olympiad (from lakhs of Students) were from Cheenta.
Research Training Program and Research Projects in Cheenta in 2024
Cheenta has outstanding research programs for school and university students.
These programs offer a unique learning experience. They are also great addition to the academic portfolio of a learner. University applications greatly value research projects.
There are two types of research programs available at Cheenta.
Research Training Program - Duration 6 months - Meets once weekly
A research training program at Cheenta helps school and college students to gain skills to conduct research. In particular there is a weekly workshop for 6 months where they learn theoretical and practical tools from a particular area of one of these subjects: Mathematics, Physics, Computer Science, Statistics, Machine Learning, Artificial Intelligence.
Research Project Program - Duration 8 months to 1 year - Meets once weekly
A research project program at Cheenta helps students to work closely with advisors. The goal is to perform and publish expository or original work in some area of mathematical science.
Research Mentors at Cheenta are from the leading universities in India and the United States.
Srijit Mukherjee
(Pursuing) PhD from Pennsylvania State University. BStat - MStat from Indian Statistical Institute
Dr. Ashani Dasgupta
PhD from University of Wisconsin-Milwaukee
Dr. Sourayan Banerjee
PhD from Indian Institute of Science Education and Research
Swarnabja Bhowmick
Computer Scientist (University of Calcutta)
Scope of Research for School and College Students at Cheenta in 2024
In 2024, Cheenta is offering research opportunities in the following areas:
Mathematics (Pure) - Hyperbolic Geometry, Topology, Group Theory
Statistical Analysis
Machine Learning - Computer Vision
Artificial Intelligence
Mathematics (Pure) - Algebraic Geometry
Get Started with a Research Mentoring Session
Real Olympiads versus Fake Olympiads: Recommended by Cheenta
Mathematical Olympiads began with a noble objective: promote non-routine problem solving and a culture of research among school students. The real International Math Olympiad is a 9 - hour marathon examination consisting of only 6 problems. Here is an example of a problem from IMO 2023. Notice its difference from the 'multiple choice problems' found in hundreds of so called private olympiads that operate all around the world.
IMO 2023, Problem 1.
Determine all composite integers $n>1$ that satisfy the following property: if $d_1, d_2, \ldots, d_k$ are all the positive divisors of $n$ with $1=d_1<d_2<\cdots<d_k=n$, then $d_i$ divides $d_{i+1}+d_{i+2}$ for every $1 \leqslant i \leqslant k-2$.
How can I participate in the real IMO?
In order to qualify for IMO, a student goes through a rigorous, multi-step process. These steps are different in every country. For example, India has a 4 step process:
IOQM [Starts at 8th grade]
RMO [For those who qualify in IOQM]
INMO [For those who qualify in RMO]
IMO Training Camp
At the end of this 4-step process, only 6 students from India are sent to the actual International Math Olympiad.
Similarly, the United States has a 4 step process:
American Math Competition 10 or 12
AIME
USAMO / USAJMO
MOSP
What topics are tested in the real math olympiads?
Mathematical Olympiads test ingenuity and problem solving skill of young mathematicians. This is usually developed over several years, through consistent hard work. In particular, the problems are derived from the following four areas of pre-college mathematics.
Number Theory
Geometry
Algebra
Polynomials
Complex Numbers
Functional Equations
Inequalities
Combinatorics
What contests are recommended in elementary school or middle school if I am not ready for real IMO?
The rule of the thumb is: go for olympiads created by real mathematicians, who are usually part of Mathematical Teacher's Association of a particular country. Here are some of the recommended contests:
India (upto Grade 7 --> After that you should focus on IOQM)
NMTC Contests
Australian Math Competition
Math Kangaroo
American Math Competition 8
United States (upto Grade 7 --> After that you should focus on AMC 10)
Australian Math Competition
Math Kangaroo
American Math Competition 8
MOEMS
University of Waterloo Contests
United Kingdom
UKMT
American Math Competition 8
University of Waterloo Contests
Why should you avoid other private olympiads?
There are hundreds of private olympiads run by publishing houses and coaching centers who are interested in selling books and courses. They have created their own contests. They use the buzzword 'olympiad', 'imo' to lure students and parents. Unfortunately these private contests add little to no value to a student's academic development. In order to become popular they keep the difficulty level of their questions quite low. Students who do well in those poor-quality problems, are handed out with gold and silver medals (certainly if your child gets a 'gold' in so-and-so 'olympiad', chances are, you will spread the word).
In most cases the private olympiads play a negative role. Since most of the problems in a private contest requires the students to remember some extra formulae, it defeats the central philosophy of the real mathematical olympiads: to think out-of-the-box. We often hear a third grader's parent boasting, 'my child knows tenth grade math'. However when challenged with some non-routine problem from their own standard, these students fumble desperately. Indeed the private olympiads can change the focus of a student from ingenuity to rote learning.
Another issue that often comes up is this: students who received 'gold' and 'silver' in the private competitions, develop a false sense of complacency. They lose valuable time for preparation for the real contests. When faced with the real olympiads, they find themselves completely unprepared.
What books should you follow for real math olympiad preparation?
For classes 1 to 5
Math Circle by the Bay: Topics for Grades $1-5$ by Ilya Zakharevich, Laura Givental, and Maria Nemirovskaya
Math from Three to Seven: The Story of a Mathematical Circle for Preschoolers by Alexander K. Zvonkin Mathematics can be fun by Yakov Perelman
Math Circles for Elementary School Students by Natasha Rozhkovskaya
For classes 6 to 8
Mathematical Circles: (Russian Experience) by Dmitrii Vladimirovich Fomin, Ilia Itenberg, and Sergei Aleksandrovich Genkin
Algebra by Alexander Shen and Israel Gelfand
Kiselev's Geometry: Planimetry by Alexander Givental
For classes 9 to 12
Challenge and Thrill of Pre-College Mathematics by C. R. Pranesachar and V Krishnamurthy
An Excursion In Mathematics. by M. R. Modak
Trigonometry by Israel Gelfand and Mark Saul
Complex Numbers from A to …Z by Dorin Andrica and Titu Andreescu Euclidean Geometry in Mathematical Olympiads by Evan Chen Principles and Techniques in Combinatorics by Chuan Chong Chen and KOH KHEE MENG
Secrets in Inequalities by Pham Kim Hung
Functional Equations by Venkatachala Test of Mathematics at the 10+2 Level by East West Press (useful for ISI-CMI Entrances.Polynomials by Edward Barbeau
Start with these 7 steps for Math Olympiad, ISI-CMI Entrance, IOQM, AMC, INMO
What are the Real Benefits of Math Olympiad ?
AMC 8, 2024 Problems, Solutions and Concepts
Problem 1 What is the ones digit of $$ 222,222-22,222-2,222-222-22-2 $$ (A) 0 (B) 2 (C) 4 (D) 6 (E) 8
Problem 2 What is the value of this expression in decimal form? $$ \frac{44}{11}+\frac{110}{44}+\frac{44}{1100} $$ (A) 6.4 (B) 6.504 (C) 6.54 (D) 6.9 (E) 6.94
Problem 3 Four squares of side lengths $4,7,9$, and 10 units are arranged in increasing size order so that their left edges and bottom edges align. The squares alternate in the color pattern white-gray-white-gray, respectively, as shown in the figure. What is the area of the visible gray region in square units?
(A) 42 (B) 45 (C) 49 (D) 50 (E) 52
Problem 4 When Yunji added all the integers from 1 to 9 , she mistakenly left out a number. Her incorrect sum turned out to be a square number. What number did Yunji leave out? (A) 5 (B) 6 (C) 7 (D) 8 (E) 9
Problem 5 Aaliyah rolls two standard 6 -sided dice. She notices that the product of the two numbers rolled is a multiple of 6 . Which of the following integers cannot be the sum of the two numbers? (A) 5 (B) 6 (C) 7 (D) 8 (E) 9
Problem 6 Sergai skated around an ice rink, gliding along different paths. The gray lines in the figures below show foru of the paths labeled $P, Q, R$, and $S$. What is the sorted order of the four paths from shortest to longest?
Problem 7 A $3 \times 7$ rectangle is covered without overlap by 3 shapes of tiles: $2 \times 2,1 \times 4$, and $1 \times 1$, shown below. What is the minimum possible number of $1 \times 1$ tiles used? (A) 1 (B) 2 (C) 3 (D) 4 (E) 5
Problem 8 On Monday Taye has $\$ 2$. Every day, he either gains $\$ 3$ or doubles the amount of money he had on the previous day. How many different dollar amounts could Taye have on Thursday, 3 days later? (A) 3 (B) 4 (C) 5 (D) 6 (E) 7
Problem 9 All of the marbles in Maria's collection are red, green, or blue. Maria has half as many red marbles as green marbles and twice as many blue marbles as green marbles. Which of the following could be the total number of marbles in Maria's collection? (A) 24 (B) 25 (C) 26 (D) 27 (E) 28
Problem 10 In January 1980 the Mauna Loa Observatory recorded carbon dioxide (CO_2) levels of $338 \mathrm{ppm}$ (parts per million). Over the years the average CO_2 reading has increased by about 1.1515 ppm each year. What is the expected CO_2 level in ppm in January 2030? Round your answer to the nearest integer. (A) 399 (B) 414 (C) 420 (D) 444 (E) 459
Problem 11
The coordinates of $\triangle A B C$ are $A(5,7), B(11,7)$ and $C(3, y)$, with $y>7$. The area of $\triangle A B C$ is 12 . What is the value of $y$ ?
(A) 8 (B) 9 (C) 10 (D) 11 (E) 12
Problem 12 Rohan keeps a total of 90 guppies in 4 fish tanks. There is 1 more guppy in the 2 nd tank than the 1 st tank. There are 2 more guppies in the 3 rd tank than the 2nd tank. There are 3 more guppies in the 4 th tank than the 3rd tank. How many guppies are in the 4 th tank? (A) 20 (B) 21 (C) 23 (D) 24 (E) 26
Problem 13 Buzz Bunny is hopping up and down a set of stairs, one step at a time. In how many ways can Buzz start on the ground, make a sequence of 6 hops, and end up back on the ground? (For example, one sequence of hops is up-up-down-down-up-down.)
(A) 8 (B) 9 (C) 10 (D) 11 (E) 12
Problem 14
The one-way routes connecting towns $A, M, C, X, Y$, and $Z$ are shown in the figure below (not drawn to scale). The distances in kilometers along each route are marked. Traveling along these routes, what is the shortest distance from $A$ to $Z$ in kilometers?\
(A) 28 (B) 29 (C) 30 (D) 31 (E) 32
Problem 15 Let the letters $F, L, Y, B, U, G$ represent distinct digits. Suppose $\underline{F} \underline{L} \underline{Y} \underline{F} \underline{L} \underline{Y}$ is the greatest number that satisfies the equation $$ \text { 8. } \underline{F} \underline{L} \underline{Y} \underline{F} \underline{L} \underline{Y}=\underline{B} \underline{U} \underline{G} \underline{B} \underline{U} \underline{G} . $$
What is the value of $\underline{F} \underline{L} \underline{Y}+\underline{B} \underline{U} \underline{G}$ ? (A) 1089 (B) 1098 (C) 1107 (D) 1116 (E) 1125
Problem 16 Minh enters the numbers 1 through 81 into the cells of a $9 \times 9$ grid in some order. She calculates the product of the numbers in each row and column. What is the least number of rows and columns that could have a product divisible by 3 ? (A) 8 (B) 9 (C) 10 (D) 11 (E) 12
Problem 17
A chess king is said to attack all the squares one step away from it, horizontally, vertically, or diagonally. For instance, a king on the center square of a $3 \times 3$ grid attacks all 8 other squares, as shown below. Suppose a white king and a black king are placed on different squares of a $3 \times 3$ grid so that they do not attack each other. In how many ways can this be done?
(A) 20 (B) 24 (C) 27 (D) 28 (E) 32
Problem 18
Three concentric circles centered at $O$ have radii of 1,2 , and 3 . Points $B$ and $C$ lie on the largest circle. The region between the two smaller circles is shaded, as is the portion of the region between the two larger circles bounded by central angle $B O C$, as shown in the figure below. Suppose the shaded and unshaded regions are equal in area. What is the measure of $\angle B O C$ in degrees?
(A) 108 (B) 120 (C) 135 (D) 144 (E) 150
Problem 19 Jordan owns 15 pairs of sneakers. Three fifths of the pairs are red and the rest are white. Two thirds of the pairs are high-top and the rest are low-top. The red high-top sneakers make up a fraction of the collection. What is the least possible value of this fraction?
Any three vertices of the cube $P Q R S T U V W$, shown in the figure below, can be connected to form a triangle. (For example, vertices $P, Q$, and $R$ can be connected to form isosceles $\triangle P Q R$.) How many of these triangles are equilateral and contain $P$ as a vertex?
(A) 0 (B) 1 (C) 2 (D) 3 (E) 6
Problem 21 A group of frogs (called an army) is living in a tree. A frog turns green when in the shade and turns yellow when in the sun. Initially, the ratio of green to yellow frogs was $3: 1$. Then 3 green frogs moved to the sunny side and 5 yellow frogs moved to the shady side. Now the ratio is $4: 1$. What is the difference between the number of green frogs and the number of yellow frogs now? (A) 10 (B) 12 (C) 16 (D) 20 (E) 24
Problem 22
A roll of tape is 4 inches in diameter and is wrapped around a ring that is 2 inches in diameter. A cross section of the tape is shown in the figure below. The tape is 0.015 inches thick. If the tape is completely unrolled, approximately how long would it be? Round your answer to the nearest 100 inches.
(A) 300 (B) 600 (C) 1200 (D) 1500 (E) 1800
Problem 23 Rodrigo is drawing lines on the coordinate plane, and counting how many unit squares they go through. He draws a line with endpoints $(2000,3000)$ and $(5000,8000)$. How many unit squares does this segment go through? (A) 6000 (B) 6500 (C) 7000 (D) 7500 (E) 8000
Problem 24 Jean has made a piece of stained glass art in the shape of two mountains, as shown in the figure below. One mountain peak is 8 feet high while the other peak is 12 feet high. Each peak forms a $90^{\circ}$ angle, and the straight sides form a $45^{\circ}$ angle with the ground. The artwork has an area of 183 square feet. The sides of the mountain meet at an intersection point near the center of the artwork, $h$ feet above the ground. What is the value of $h$ ? (A) 4 (B) 5 (C) $4 \sqrt{2}$ (D) 6 (E) $5 \sqrt{2}$
Problem 25 A small airplane has 4 rows of seats with 3 seats in each row. Eight passengers have boarded the plane and are distributed randomly among the seats. A married couple is next to board. What is the probability there will be 2 adjacent seats in the same row for the couple?
Given a triangle $A B C$ with $\angle A C B=120^{\circ}$. The point $L$ is marked on the side $A B$ so that $C L$ is the bisector of $\angle A C B$. The points $N$ and $K$ are marked on the sides $A C$ and $B C$, respectively, so that $C N+C K=C L$. Prove that the triangle $K L N$ is equilateral.
Problem 2
Given a prime number $p$ such that the number $2 p$ is equal to the sum of the squares of some four consecutive positive integers. Prove that $p-7$ is divisible by 36 .
Problem 3
Let $f(x)$ be a polynomial with real coefficients of degree 2. Suppose that for some pairwise distinct real numbers $a, b, c$ we have $$ f(a)=b c ; f(b)=c a ; f(c)=a b . $$
Determine $f(a+b+c)$ in terms of $a, b, c$.
Problem 4
The set $X$ of $N$ four-digit numbers formed from the digits $1,2,3,4,5,6,7,8$ satisfies the following condition: for any two different digits from 1, 2, 3, 4, 5, 6, 7, 8 there exists a number in $X$ which contains both of them.
Determine the smallest possible value of $N$.
Problem 5
The side-lengths $a, b, c$ of a triangle $A B C$ are positive integers. Let $$ T_n=(a+b+c)^{2 n}-(a-b+c)^{2 n}-(a+b-c)^{2 n}+(a-b-c)^{2 n} $$ for any positive integer $n$. If $\frac{T_2}{2 T_1}=2023$ and $a>b>c$, determine all possible perimeters of the triangle $A B C$.
Problem 6
The diagonals $A C$ and $B D$ of a cyclic quadrilateral $A B C D$ meet at $P$. The point $Q$ is chosen on the segment $B C$ so that $P Q$ is perpendicular to $A C$. Prove that the line joining the centres of the circumcircles of triangles $A P D$ and $B Q D$ is parallel to $A D$.
Version 2
Problem 1
Let $\mathbb{N}$ be the set of all positive integers and $S={(a, b, c, d) \in \mathbb{N}^4: a^2+b^2+c^2=d^2}$. Find the largest positive integer $m$ such that $m$ divides $a b c d$ for all $(a, b, c, d) \in S$.
Problem 2
Let $\omega$ be a semicircle with $A B$ as the bounding diameter and let $C D$ be a variable chord of the semicircle of constant length such that $C, D$ lie in the interior of the $\operatorname{arc} A B$. Let $E$ be a point on the diameter $A B$ such that $C E$ and $D E$ are equally inclined to the line $A B$. Prove that (a) the measure of $\angle C E D$ is a constant; (b) the circumcircle of triangle $C E D$ passes through a fixed point.
Problem 3
For any natural number $n$, expressed in base 10 , let $s(n)$ denote the sum of all its digits. Find all natural numbers $m$ and $n$ such that $m<n$ and $$ (s(n))^2=m \quad \text { and } \quad(s(m))^2=n . $$
Problem 4
Let $\Omega_1, \Omega_2$ be two intersecting circles with centres $O_1, O_2$ respectively. Let $l$ be a line that intersects $\Omega_1$ at points $A, C$ and $\Omega_2$ at points $B, D$ such that $A, B, C, D$ are collinear in that order. Let the perpendicular bisector of segment $A B$ intersect $\Omega_1$ at points $P, Q$; and the perpendicular bisector of segment $C D$ intersect $\Omega_2$ at points $R, S$ such that $P, R$ are on the same side of $l$. Prove that the midpoints of $P R, Q S$ and $O_1 O_2$ are collinear.
Problem 5
Let $n>k>1$ be positive integers. Determine all positive real numbers $a_1, a_2, \ldots, a_n$ which satisfy $$ \sum_{i=1}^n \sqrt{\frac{k a_i^k}{(k-1) a_i^k+1}}=\sum_{i=1}^n a_i=n $$
Problem 6
Consider a set of 16 points arranged in a $4 \times 4$ square grid formation. Prove that if any 7 of these points are coloured blue, then there exists an isosceles right-angled triangle whose vertices are all blue.
