MathNet 2024 — batch of 27 questions

Compute the sum of all integers $n$ such that $n^{2}-3000$ is a perfect square.

Determine, with proof, whether there exist positive integers $x$ and $y$ such that $x+y$, $x^{2}+y^{2}$, and $x^{3}+y^{3}$ are all perfect squares.

Albert writes down all of the multiples of $9$ between $9$ and $999$, inclusive. Compute the sum of the digits he wrote.

The number $17^{6}$ when written out in base 10 contains 8 distinct digits from $1,2, \ldots, 9$, with no repeated digits or zeroes. Compute the missing nonzero digit.

Suppose $r$, $s$, and $t$ are nonzero reals such that the polynomial $x^{2} + r x + s$ has $s$ and $t$ as roots, and the polynomial $x^{2} + t x + r$ has $5$ as a root. Compute $s$.

A circle of area $1$ is cut by two distinct chords. Compute the maximum possible area of the smallest resulting piece.

Compute the smallest integer $n \gt 72$ that has the same set of prime divisors as $72$.

Jerry and Neil have a 3-sided die that rolls the numbers $1$, $2$, and $3$, each with probability $\frac{1}{3}$. Jerry rolls first, then Neil rolls the die repeatedly until his number is at least as large as Jerry's. Compute the probability that Neil's final number is $3$.

Six consecutive positive integers are written on slips of paper. The slips are then handed out to Ethan, Jacob, and Karthik, such that each of them receives two slips. The product of Ethan's numbers is $20$, and the product of Jacob's numbers is $24$. Compute the product of Karthik's numbers.

Compute the number of even positive integers $n \leq 2024$ such that $1,2, \ldots, n$ can be split into $\frac{n}{2}$ pairs, and the sum of the numbers in each pair is a multiple of $3$.

Compute the sum of all two-digit positive integers $x$ such that for all three-digit (base 10) positive integers $\underline{a}\ \underline{b}\ \underline{c}$, if $\underline{a}\ \underline{b}\ \underline{c}$ is a multiple of $x$, then the three-digit (base 10) number $\underline{b}\ \underline{c}\ \underline{a}$ is also a multiple of $x$.

Equilateral triangles $ABF$ and $BCG$ are constructed outside regular pentagon $ABCDE$. Compute $\angle FEG$.

Let $f(x)$ be a quotient of two quadratic polynomials. Given that $f(n) = n^{3}$ for all $n \in \{1,2,3,4,5\}$, compute $f(0)$.

Compute
$$
\sum_{i=1}^{4} \sum_{t=1}^{4} \sum_{e=1}^{4}\left\lfloor\frac{i t e}{5}\right\rfloor
$$

Suppose for some positive integer $n$, the numbers $2^n$ and $5^n$ have equal first digit. What are the possible values of this first digit?

Suppose that $a$, $b$, and $c$ are distinct positive integers such that $a^{b} b^{c} = a^{c}$. Across all possible values of $a$, $b$, and $c$, compute the minimum value of $a + b + c$.

Suppose $p$ is a prime number and $x, y, z$ are integers satisfying $0 \lt x \lt y \lt z \lt p$. If $x^3, y^3, z^3$ have equal remainders when divided by $p$, prove that $x^2 + y^2 + z^2$ is divisible by $x + y + z$.

Let $a$, $b$, and $c$ be real numbers such that
$$
\begin{aligned}
a + b + c &= 100 \\
ab + bc + ca &= 20, \text{ and } \\
(a + b)(a + c) &= 24
\end{aligned}
$$
Compute all possible values of $bc$.

Rectangle $R$ with area $20$ and diagonal of length $7$ is translated $2$ units in some direction to form a new rectangle $R'$. The vertices of $R$ and $R'$ that are not contained in the other rectangle form a convex hexagon. Compute the maximum possible area of this hexagon.

The graphs of the lines
$$
y = x + 2, \quad y = 3x + 4, \quad y = 5x + 6, \quad y = 7x + 8, \quad y = 9x + 10, \quad y = 11x + 12
$$
are drawn. These six lines divide the plane into several regions. Compute the number of regions the plane is divided into.