Let $p$ be a prime number. Find all pairs $(x, y)$ of positive integers such that $x^3 + y^3 - 3xy = p -1$.

In a village, the maximum distance between two houses is $M$ and the minimum distance is $m$. Prove that if the village has $6$ houses, then $\frac{M}{m} \ge \sqrt3$.

In a new school $40$ percent of the students are freshmen, $30$ percent are sophomores, $20$ percent are juniors, and $10$ percent are seniors. All freshmen are required to take Latin, and $80$ percent of the sophomores, $50$ percent of the juniors, and $20$ percent of the seniors elect to take Latin. The probability that a randomly chosen Latin student is a sophomore is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

Find the number of functions $ A\stackrel{f}{\longrightarrow } A $ for which there exist two functions $ A\stackrel{g}{\longrightarrow } B\stackrel{h}{\longrightarrow } A $ having the properties that $ g\circ h =\text{id.} $ and $ h\circ g=f, $ where $ B $ and $ A $ are two finite sets.

On two intersecting lines $\ell_1$ and $\ell_2$, segments $AB$ and $CD$ of a given length are selected, respectively. Prove that the volume of the tetrahedron $ABCD$ does not depend on the position of the segments $AB$ and $CD$ on the lines $\ell_1$ and $\ell_2$.

Let $ABC$ be a triangle with side lengths $5$, $4\sqrt 2$, and $7$. What is the area of the triangle with side lengths $\sin A$, $\sin B$, and $\sin C$?

An urn contains $n$ balls labeled $1, 2, ... , n$. We draw the balls out one by one (without replacing them) until we obtain a ball whose number is divisible by $k$. Find all $k$ such that the expected number of balls removed is $k$.

Let $\mathbb N$ be the set of all positive integers. A subset $A$ of $\mathbb N$ is [i]sum-free[/i] if, whenever $x$ and $y$ are (not necessarily distinct) members of $A$, their sum $x+y$ does not belong to $A$. Determine all surjective functions $f:\mathbb N\to\mathbb N$ such that, for each sum-free subset $A$ of $\mathbb N$, the image $\{f(a):a\in A\}$ is also sum-free. [i]Note: a function $f:\mathbb N\to\mathbb N$ is surjective if, for every positive integer $n$, there exists a positive integer $m$ such that $f(m)=n$.[/i]

Consider a function $f(x)=\frac{\sin x}{9+16\sin ^ 2 x}\ \left(0\leq x\leq \frac{\pi}{2}\right).$ Let $a$ be the value of $x$ for which $f(x)$ is maximized. Evaluate $\int_a^{\frac{\pi}{2}} f(x)\ dx.$ [i]2010 Saitama University entrance exam/Mathematics[/i] Last Edited

Let $S_n=\sum_{k=1}^n (k^5+k^7).$ Let the prime factorization of $\text{gcd}(S_{2020},S_{6060})$ be $p_1^{k_1}\cdot p_2^{k_2}\cdots p_i^{k_i}$. Compute $p_1+p_2+\cdots +p_i+k_1+k_2+\cdots + k_i $.

Let $d_1,d_2,d_3,d_4$ be the four smallest divisors of a positive integer $n$ (having at least four divisors). Find all $n$ such that $d_1^2+d_2^2+d_3^2+d_4^2 = n$.

All the squares of a $2022 \times 2022$ board will be colored white or black. Chips will be placed in several of these boxes, at most one per box. We say that two tokens attack each other, when the following two conditions are met: a) There is a path of squares that joins the squares where the pieces were placed. This path can have a horizontal, vertical, or diagonal direction. b) All the squares in this path, including the squares where the pieces are, are of the same color. For example, the following figure shows a small example of a possible coloring of a $6 \times 6$ board with $A, B, C, D$, and $E$ tiles placed. The pairs of checkers that attack each other are $(D, E)$, $(C, D)$, and $(B, E)$. [img]https://cdn.artofproblemsolving.com/attachments/2/0/52ec7b7d1c02e266b666e4f8b25e87c58f0c89.png[/img] What is the maximum value of $k$ such that it is possible to color the board and place $k$ tiles without any two of them attacking each other?

Let $k$ be the smallest positive integer for which there exist distinct integers $m_{1}, m_{2}, m_{3}, m_{4}, m_{5}$ such that the polynomial $$p(x)=\left(x-m_{1}\right)\left(x-m_{2}\right)\left(x-m_{3}\right)\left(x-m_{4}\right)\left(x-m_{5}\right)$$ has exactly $k$ nonzero coefficients. Find, with proof, a set of integers $m_{1}, m_{2}, m_{3}, m_{4}, m_{5}$ for which this minimum $k$ is achieved.

Find all prime numbers $p$ such that the number $$3^p+4^p+5^p+9^p-98$$ has at most $6$ positive divisors.

In a company of people some pairs are enemies. A group of people is called [i]unsociable[/i] if the number of members in the group is odd and at least $3$, and it is possible to arrange all its members around a round table so that every two neighbors are enemies. Given that there are at most $2015$ unsociable groups, prove that it is possible to partition the company into $11$ parts so that no two enemies are in the same part. [i]Proposed by Russia[/i]

On the sides of the triangle $ABC$ externally constructed right triangles $ABC_1$, $BCA_1$, $CAB_1$. Prove that the points of intersection of the medians of the triangles $ABC$ and $A_1B_1C_1$ coincide.

In an acute triangle $ABC$, the incircle $\omega$ touches $BC$ at $D$. Let $I_a$ be the excenter of $ABC$ opposite to $A$, and let $M$ be the midpoint of $DI_a$. Prove that the circumcircle of triangle $BMC$ is tangent to $\omega$. [i]Patrik Bak (Slovakia)[/i]

Given is a triangle $ABC$ and a circle through $A, B$. The perpendicular bisector of $AB$ meets the circle at $P, Q$, such that $AP>AQ$. Let $M$ be a point on the segment $AB$. The lines through $M$, parallel to $QA, QB$ meet $PB, PA$ at $R, S$. Prove that $MQ$ bisects $RS$.

$ABCD$ is a quadrilateral on the complex plane whose four vertices satisfy $z^4+z^3+z^2+z+1=0$. Then $ABCD$ is a $\textbf{(A)}~\text{Rectangle}$ $\textbf{(B)}~\text{Rhombus}$ $\textbf{(C)}~\text{Isosceles Trapezium}$ $\textbf{(D)}~\text{Square}$

Let $A$ be a set of $8$ elements, and $B := (B_1,...,B_7)$ be an ordered $7$-tuple of subsets of $A$. Let $N$ be the number of such $7$-tuples $B$ such that there exists a unique $4$-element subset $I \subseteq \{1,2,...,7\}$ for which the intersection $\cap _{ i\in I} B_i$ is nonempty. Find the remainder when $N$ is divided by $67$.

Cut the $10$ cm $\times 20$ cm rectangle into two pieces with one straight cut so that they can fit inside the $19.5$ cm diameter circle without intersecting.

A sphere touches all edges of a tetrahedron. Let $a, b, c$ and d be the segments of the tangents to the sphere from the vertices of the tetrahedron. Is it true that that some of these segments necessarily form a triangle? (It is not obligatory to use all segments. The side of the triangle can be formed by two segments)

Find all functions $f : R \to R$ which satisfy $f(x)f(y) = f(x + y) + xy$ for all $x, y \in R$.

Let $b$ be the probability that the cards are from different suits. Compute $\lfloor1000b\rfloor$.

Non-negative real numbers $x, y, z$ satisfy the following relations: $3x + 5y + 7z = 10$ and $x + 2y + 5z = 6$. Find the minimum and maximum of $w = 2x - 3y + 4z$.