The central square of a $9 \times 9$ chessboard is white. How many white squares are there on the board? (The squares of the chessboard are coloured alternately black and white.)
An integer $M$ is divisible by 4 but not by 256 . What is the number of distinct possible remainders when $M$ is divided by 256 ?
If $x_{1}, x_{2}, \ldots, x_{49}$ are non-zero integers such that $\sum_{i=1}^{49} x_{i}=0$, then what is the minimum possible value of $\sum_{i=1}^{49} x_{i}^{2} ?$
All six digits of three 2 -digit numbers are different. If $N$ is the largest possible sum of three such numbers, what is the sum of digits of $N$ ?
In triangle $A B C$, we are given that $\angle C A B=80^{\circ}$. Let the perpendicular bisector of $B C$ meet the circumcircle of triangle $A B C$ in $N$, where we assume that $A$ and $N$ lie on the same side of the chord $B C$. Then what is the measure of $\angle N B C$ in degrees?
Find the number of 2-digit positive integers $n$ such that $n=26+(a \times b)$, where $a$ and $b$ are the two digits of $n$.
In trapezium $A B C D$, it is given that $A B$ is parallel to $C D$. Assume that $A B=3 C D, C D=D A$ and $\angle C D A=$ $120^{\circ}$. If the largest angle of $A B C D$ is $x^{\circ}$ and the smallest angle is $y^{\circ}$, what is the value of $x / y$ ?
Let $N$ be the smallest positive integer whose digits add up to 2026 . What is the leading digit of $N+1$ ?
What is the number of integers in the set $\{0, \ldots, 20\}$ which can be expressed as the sum of two square integers?
Find the number of positive integers $n$ satisfying all the following conditions:
A $7 \times 7$ board is divided into 49 unit squares. We place checkers on the board, at most one per square. Find the largest number of checkers that can be placed on the unit squares so that each row, as well as each column, contains an even number of checkers.
Find the number of non-constant polynomials $P(x)$, with real coefficients, such that $P\left(x^{2}\right)=P(P(x))$
Let A be a 3 -digit number with distinct nonzero digits and $B$ be the number obtained by reversing the digits of A. Determine the largest possible prime factor of $|A-B|$.
In an isosceles triangle $A B C$, with $\angle A C B=90^{\circ}$. The point $D$ is on the side $B C$ such that $\angle A D C=75^{\circ}$. If the area of triangle $A D C$ is 81 , what is the length of segment $B D$ ?
Let $a, b, c, d$ be positive integers such that $a^{2}+b^{2}-c d^{2}=2026$. Find the minimum possible value of $a+b+c+d$.
Let $P$ be a regular polygon with 8 vertices. By a labelling of $P$ we mean an assignment of integers $1,2, \ldots, 8$ to the vertices in some order. A labelling is considered good if the path consisting of line segments 1 to 2,2 to 3, ... and 7 to 8 does not self-intersect. If $N$ is the number of good labellings, what is the remainder when $N$ is divided by 100 ?
A sequence $a_{1}, a_{2}, a_{3}, \ldots$ of real numbers satisfies $$ \frac{a_{n+3}-a_{n+2}}{a_{n}-a_{n+1}}=\frac{a_{n+3}+a_{n+2}}{a_{n}+a_{n+1}} $$ for all $n \geq 1$. Suppose $a_{55}=6, a_{66}=2$ and $a_{77}=1$. Let $N$ denote the sum $a_{1}^{2}+a_{2}^{2}+\cdots+a_{2026}^{2}$. What is the sum of the digits of $N$ ?
Let $M$ be the smallest positive integer with the following two properties:
Find the number of ordered triples $(x, y, z)$ of positive integers such that $1 \leq x, y, z \leq 8$ and $$ |x-y|+|y-z|+|z-x|=8 $$
Four points $A, B, C$ and $D$ lie on a straight line, in this order. A point $E$, not on the line, satisfies $\angle A E B=$ $\angle B E C=\angle C E D=45^{\circ}$. Let $F$ and $G$ be the midpoints of $A C$ and $B D$, respectively. If $\angle F E G=x^{\circ}$, what is the value of $x$ ?
A $1 \times 5$ rectangle is divided into five $1 \times 1$ squares by drawing four line segments parallel to the shorter side of the rectangle. Each of the resulting sixteen unit-length line segments is coloured red, blue or green. A $1 \times 1$ square is called colourful if all the three colours are used in colouring its sides. If $N$ is the number of ways of colouring such that all the five $1 \times 1$ squares are colourful, find the remainder when $N$ is divided by 100 .
Let $E=\left\{p^{4}+p^{2}-2 \mid p\right.$ is a prime, $\left.p>3\right\}$. What is the largest positive integer that divides all the numbers in $E$ ?
Let $A B C D$ be a rectangle and let $E$ be a point on $B D$ such that $A E$ is perpendicular to $B D$. If $A E=12$ and $C E=\sqrt{193}$, compute the area of the rectangle $A B C D$.
Complex numbers $x, y, z$ satisfy the following system of equations:
$$ \begin{aligned} & x^{2}+y^{2}+z=x y\end{aligned} $$
$$ \begin{aligned} & x+y^{2}+z^{2}=y z\end{aligned} $$
$$ \begin{aligned} & x^{2}+y+z^{2}=x z \end{aligned} $$
Determine the sum of all distinct possible values of $\left|\left(x^{2}-y\right)\left(y^{2}-z\right)\left(z^{2}-x\right)\right|$.
Let $N$ be the number of distinct 8 -digit numbers obtained by arranging the six numbers $0,1,2,3,10,23$ where the first digit of the 8 digit number is not zero. Find the sum of the digits of $N$.
There are $n$ points in the plane, no three of which are collinear. Every pair of points is joined by a segment which is coloured red or blue such that the following conditions hold:
The lengths of the sides of a convex quadrilateral are $\sqrt{a}, \sqrt{a+3}, \sqrt{a+2}$ and $\sqrt{2 a+5}$, in this order. The length of each diagonal is $\sqrt{2 a+5}$. If $\theta^{\circ}$ is the difference between the largest angle and the second largest angle of the quadrilateral then determine the value of $\theta$.
Let $a_{1}, a_{2}, \ldots$ and $b_{1}, b_{2}, \ldots$ be strictly increasing sequences of positive integers such that
Find the sum of all possible values of $a_{1}+b_{1}$.
Let $n=\frac{4^{31}-1}{3}$. Find the remainder when $2^{n-1}$ is divided by $n$.
In triangle $A B C$, it is given that $\angle C A B=50^{\circ}$ and $\angle A B C=70^{\circ}$. Points $D$ and $E$ are chosen on sides $B C$ and $A C$ respectively such that $\angle A B E=\angle D A B=30^{\circ}$. If $\angle D E B=x^{\circ}$, what is the value of $r$ ?
In 2026, the following Cheenta students have been successful for Indian Statistical Institute's M.Stat Entrance. They ranked within the first 50 in the entire country in these entrances. I.S.I. M.Stat Entrance
In 2026, the following Cheenta students have been successful for Indian Statistical Institute's B.Stat Entrance and Chennai Mathematical Institute's B.Sc. Math Entrance. They ranked within the first 200 in the entire country in these entrances. Most of these students attended the problem solving workshops regularly, which happen 5 days every week. CMI B.Sc. Math Entrance […]
In 2025, 8 students from Cheenta Academy cracked the prestigious Regional Math Olympiad. In this post, we will share some of their success stories and learning strategies. The Regional Mathematics Olympiad (RMO) and the Indian National Mathematics Olympiad (INMO) are two most important mathematics contests in India.These two contests are for the students who are […]
Cheenta Academy proudly celebrates the success of 27 current and former students who qualified for the Indian Olympiad Qualifier in Mathematics (IOQM) 2025, advancing to the next stage — RMO. This accomplishment highlights their perseverance and Cheenta’s ongoing mission to nurture mathematical excellence and research-oriented learning.
