IMC 2024 Grade 8 — Selection round

Nonzero integers $a,b$ satisfy $a^2b^2+a^2+b^2=628$. Find $|a|+|b|$.

The four central squares of an $8\times8$ chessboard are removed as shown. What is the largest number of queens that can be placed on the remaining $60$ squares so that no two attack each other? A queen attacks the nearest piece in any common row, column, or diagonal. For this question, removed squares cannot hold queens but do not block attack lines.

Đức writes a $100$-digit integer $n$ containing no zeros. Let $m$ be the number formed by its first $50$ digits. If $n$ is divisible by $m$, how many zero digits occur in the quotient?

A positive real number $x$ satisfies $\sqrt[3]{6+\sqrt{x}}+\sqrt[3]{6-\sqrt{x}}=\sqrt[3]{3}$. Find $x$. All cube roots are real cube roots.

An operation on positive integers satisfies $a\otimes a=a+2$, $a\otimes b=b\otimes a$, and $\frac{a\otimes(a+b)}{a\otimes b}=\frac{a+b}{b}$. Find $8\otimes5$.

Phong assigns seven numbers, not necessarily distinct, from $\{1,2,\ldots,269\}$ to the black dots. The product of the three numbers on each shown straight line or circle must have the same remainder modulo $269$. If he uses $269$, what is the minimum number of times he must use it?

$ABCD$ is a square, $E$ lies on segment $BC$, and $F$ lies on segment $ED$. If $DF=BF$ and $EF=BE$, find $\angle DFA$.

How many seven-digit numbers use each of $1,2,3,4,5,6,7$ exactly once and have every two consecutive digits relatively prime?

The polynomial $P(x)=x^2+5x-7$ has roots $x_1,x_2$. Find $A=x_1^3+5x_1^2-4x_1+x_1^2x_2-4x_2$.

Fifty distinct integers are selected from $\{1,2,\ldots,100\}$ with sum $2900$. What is the smallest possible number of even integers selected?

What is the maximum number of tiles of the shown shape that can fit on a $10\times10$ square-grid board without overlap? Tiles may be rotated and cover whole grid squares.

Find all pairs of positive integers $(x,y)$ such that $xy$ is a perfect square and $x^2+4y^2$ is prime. Enter the pair in the form $(x,y)$.

How many unordered pairs of parallel diagonals can be chosen from a regular decagon?

How many ordered triples of integers $(a,b,c)$ satisfy $a+b+c=70$ and $ab+bc+ca=1633$?

Luffy and five crewmen sit in a circle to divide $99$ gold coins. Luffy chooses his share and each crewman’s share. Each crewman votes yes exactly when his share is greater than both neighbors’ shares. A majority of the five crewmen must vote yes for Luffy to survive. What is the greatest share Luffy can take? Shares are nonnegative integers; a person may receive zero coins.

Positive real numbers $a,b,c,d$ satisfy $abcd=169$. Find the least value of $A=3(a^2+b^2+c^2)+a(b+c+d)+b(c+d)+cd$.

$ABCD$ is a trapezoid with parallel bases $AB,CD$, where $AB<CD$. Point $M$ is the midpoint of $CD$, and $N$ is inside the trapezoid with $AN=BC$ and $BN=AD$. If $\angle AMB=90^\circ$, prove that quadrilaterals $ANMD$ and $BNMC$ have equal areas.

The sum of $2023$ rational numbers is an integer, and the product of any two of them is an integer. Prove that all the numbers are integers.