This website contains problems from math contests. Problems and corresponding tags were obtained from the Art of Problem Solving website.

Tags were heavily modified to better represent problems.

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Found problems: 11

2020-21 KVS IOQM India, 9
Tags: IOQM , KV , algebra , system of equations
find the number of ordered triples $(x,y,z)$ of real numbers that satisfy the system of equations: $x+y+z=7; x^2+y^2+z^2=27; xyz=5$.

2020-21 KVS IOQM India, 11
Tags: IOQM , KV , number theory , prime numbers
The prime numbers $a,b$ and $c$ are such that $a+b^2=4c^2$. Determine the sum of all possible values of $a+b+c$.

2020-21 KVS IOQM India, 7
Tags: IOQM , KV
$a,b,c$ are positive real numbers such that $a^2+b^2=c^2$ and $ab=c$. Determine the value of $\left\lvert{\frac{(a+b+c)(a+b-c)(b+c-a)(c+a-b)}{c^2}}\right\rvert$

2020-21 KVS IOQM India, 2
Tags: IOQM , KV , geometry , rectangle
If $ABCD$ is a rectangle and $P$ is a point inside it such that $AP=33, BP=16, DP=63$. Find $CP$.

2020-21 KVS IOQM India, 4
Tags: IOQM , KV , geometry , incenter
Let $ABC$ be an isosceles triangle with $AB=AC$ and incentre $I$. If $AI=3$ and the distance from $I$ to $BC$ is $2$, what is the square of length on $BC$?

2020-21 KVS IOQM India, 10
Tags: IOQM , KV
Let $A$ and $B$ be two finite sets such that there are exactly $144$ sets which are subsets of $A$ or subsets of $B$. Find the number of elements in $A \cup B$.

2020-21 KVS IOQM India, 6
Tags: IOQM , KV
Let $ABCD$ be a square with side length $100$. A circle with centre $C$ and radius $CD$ is drawn. Another circle of radius $r$, lying inside $ABCD$, is drawn to touch this circle externally and such that the circle also touches $AB$ and $AD$. If $r=m+n\sqrt{k}$, where $m,n$ are integers and $k$ is a prime number, find the value of $\frac{m+n}k$.

2020-21 KVS IOQM India, 8
Tags: IOQM , KV
Find the largest $2$-digit number $N$ which is divisible by $4$, such that all integral powers of $N$ end with $N$.

2020-21 KVS IOQM India, 3
Tags: IOQM , KV
Sita and Geeta are two sisters. If Sita's age is written after Geeta's age a four digit perfect square (number) is obtained. If the same exercise is repeated after 13 years another four digit perfect square (number) will be obtained. What is the sum of the present ages of Sita and Geeta?

2020-21 KVS IOQM India, 5
Tags: IOQM , KV
Find the number of positive integers $n$ such that the highest power of $7$ dividing $n!$ is $8$.

2020-21 KVS IOQM India, 1
Tags: IOQM , KV
If $a,b,c$ are real numbers and $(a+b-5)^2+(b+2c+3)^2+(c+3a-10)^2=0$ find the integer nearest to $a^3+b^3+c^3$.