IMC 2023 Grade 7 — Selection round

For real numbers $x,y$, define $x\heartsuit y=xy-3x-3y+12$. Evaluate $A=1\heartsuit(2\heartsuit(3\heartsuit\cdots(2022\heartsuit2023)\cdots))$.

Nonnegative integers $a,b,c,d$ satisfy $(a+b+c+d)(a^2+b^2+c^2+d^2)^2=2023$. Find $a^3+b^3+c^3+d^3$.

How many ways can three distinct integers be chosen from $\{2021,2022,\ldots,2030\}$ so that their arithmetic mean is an integer?

The numbers $1,2,3,4,5,6$ fill the six boxes exactly once. What is the smallest possible value of the displayed expression?

Phong counts upward by fives starting at $3$. Minh counts downward by fours starting at $2023$. How many numbers from $3$ to $2023$, inclusive, are counted by both?

A circle is divided into five sectors as shown. How many ways can the sectors be colored with three given colors so that adjacent sectors have different colors? The five positions in the diagram are fixed.

In isosceles triangle $ABC$, $AB=AC$. The bisectors of $\angle ABC$ and $\angle ACB$ meet at $I$. Given $\angle CIA=130^\circ$, find $\angle CAB$.

Bình colors each square of a $3\times3$ grid red or green. How many colorings have at most one green square in each row and each column?

A two-digit number counts the students in a hall. Reversing its two digits gives a perfect square. What is the largest possible number of students?

For a positive integer $n$, let $n!=1\cdot2\cdots n$. Evaluate $A=\frac{1!}{2!+3!}+\frac{3!}{4!+5!}+\cdots+\frac{2021!}{2022!+2023!}$.

An has four different blue cards and four different red cards. How many sets of four cards contain at least one card of each color?

How many ordered pairs of primes $(p,q)$ satisfy $5pq^2+p=15p^3+q^3+7$?

Pairwise relatively prime integers $a,b,c$ satisfy $\frac{a}{b+c}=2$ and $\frac{b}{c+a}=3$. Find $a^2+b^2+c^2$.

In the figure, $AEIG$, $EBHI$, and $IHCF$ are rectangles and $AHJF$ is a parallelogram. Their three rectangle areas are respectively $3$, $5$, and $12$ square centimeters. Find the area of $AHJF$, in square centimeters.

The diagram has eight circles joined by segments; two contain $2$ and $26$. Khải fills the other six so that each number he writes is the average of its neighboring numbers. What is the sum of all eight numbers? Two circles are neighbors exactly when joined by a segment; crossings without circles are not vertices.

A polynomial $P(x)$ has all coefficients in $\{0,1,2,3,4,5\}$ and satisfies $P(6)=2049$. Find its coefficient of $x$.

The numbers $1,2,\ldots,2023$ are written on a board. An and Bình, starting with An, alternately remove one number until two remain. An wins if their sum is divisible by $8$; otherwise Bình wins. Who can force a victory? Give a detailed strategy.

What is the maximum number of tiles of the shown shape that fit on an $8\times9$ square-grid board without overlap? Tiles may be rotated and reflected, and each tile covers whole grid squares. Give a detailed proof.