Let $k$ be a positive integer. Prove that there exists a positive integer $\ell$ with the following property: if $m$ and $n$ are positive integers relatively prime to $\ell$ such that $m^m\equiv n^n \pmod{\ell}$, then $m\equiv n \pmod k$.

Let $f: \{1, 2, 3, \ldots, 9\} \to \{1, 2, 3, \ldots, 9\}$ be a permutation chosen uniformly at random from the $9!$ possible permutations. Compute the expected value of $\underbrace{f(f(\cdots f(f(}_{2025 \ f\text{'s}}1))\cdots )).$

Find all integers $n$ such that $\frac{n+4}{3n-2}$ is integer

Consider a triangle $ABC$, inscribed in $O$ and $\angle A < \angle B$. Some point $P$ outside the circle satisfies $$\angle A=\angle PBA =180^{\circ}- \angle PCB$$ Let $D$ be the intersection of line $PB$ and $O$(different from $B$), and $Q$ the intersection of the tangent line of $O$ passing through $A$ and line $CD$. Show that $CQ : AB=AQ^2:AD^2$.

Find all integer solutions of the equation \[x^2+y^2=(x-y)^3.\]

There are $100$ positive numbers $a_1$, $a_2$, $...$, $a_{100}$ such that $$\frac{1}{a_1+1}+\frac{1}{a_2+1}+...+\frac{1}{a_{100}+1} \le 1.$$ Prove that $$a_1 \cdot a_2\cdot ... \cdot a_{100} \ge 99^{100}.$$

Let $n \ge 2$ be an integer. There are $n$ houses in a town. All distances between pairs of houses are different. Every house sends a visitor to the house closest to it. Find all possible values of $n$ (with full justification) for which we can design a town with $n$ houses where every house is visited.

After moving his sticky toy spider one morning, Tony heads outside to play "pirates" with his pal Nick, who lives a few doors down the street from the Kubiks. Tony and Nick imagine themselves as pirates in a rough skirmish over a chest of gold. Victorious over their foes, Tony and Nick claim the prize. However, they must split some of the gold with their crew, which they imagine consists of eight other bloodthirsty pirates. Each of the pirates receives at least one gold coin, but none receive the same number of coins, then Tony and Nick split the remainder equally. If there are $2000$ gold coins in the chest, what is the greatest number of gold coins Tony could take as his share? (Assume each gold coin is equally valuable.)

The $A$-excircle of a triangle $ABC$ touches the side $BC$ at the point $K$ and the extended side $AB$ at the point $L$. The $B$-excircle touches the lines $BA$ and $BC$ at the points $M$ and $N$, respectively. The lines $KL$ and $MN$ meet at the point $X$. Show that the line $CX$ bisects the angle $ACN$.

Let $x$, $y$, $z$ be arbitrary positive numbers such that $xy+yz+zx=x+y+z$. Prove that $$\frac{1}{x^2+y+1} + \frac{1}{y^2+z+1} + \frac{1}{z^2+x+1} \leq 1$$. When does equality occur? [i]Proposed by Marko Radovanovic[/i]

A book that is to be recorded onto compact discs takes $ 412$ minutes to read aloud. Each disc can hold up to $ 56$ minutes of reading. Assume that the smallest possible number of discs is used and that each disc contains the same length of reading. How many minutes of reading will each disc contain? $ \textbf{(A)}\ 50.2 \qquad \textbf{(B)}\ 51.5 \qquad \textbf{(C)}\ 52.4 \qquad \textbf{(D)}\ 53.8 \qquad \textbf{(E)}\ 55.2$

Let $n$ and $k$ be two positive integers. Consider a group of $k$ people such that, for each group of $n$ people, there is a $(n+1)$-th person that knows them all (if $A$ knows $B$ then $B$ knows $A$). 1) If $k=2n+1$, prove that there exists a person who knows all others. 2) If $k=2n+2$, give an example of such a group in which no-one knows all others.

In the quadrilateral $ABCD$, angles $B$ and $C$ are equal to $120^o$, $AB = CD = 1$, $CB = 4$. Find the length $AD$.

Rabbits Peter and Pauline have three offspring—Flopsie, Mopsie, and Cotton-tail. These five rabbits are to be distributed to four different pet stores so that no store gets both a parent and a child. It is not required that every store gets a rabbit. In how many different ways can this be done? $\textbf{(A)} \ 96 \qquad \textbf{(B)} \ 108 \qquad \textbf{(C)} \ 156 \qquad \textbf{(D)} \ 204 \qquad \textbf{(E)} \ 372 $

Let $n\ge 2$ be a natural number. Let $a_1\le a_2\le a_3\le \cdots \le a_n$ be real numbers such that $a_1+a_2+\cdots +a_n>0$ and $n(a_1^2+a_2^2+\cdots +a_n^2)=2(a_1+a_2+\cdots +a_n)^2.$ If $m=\lfloor n/2\rfloor+1$, the smallest integer larger than $n/2$, then show that $a_m>0.$

Let $M$ and $N$ be two palindrome numbers, each having $9$ digits and the palindromes don't start with $0$. If $N>M$ and between $N$ and $M$ there aren't any palindromes, find all values of $N-M$.

Let $ABC$ be a triangle. Let $C^{\prime}$ and $A^{\prime}$ be the reflections of its vertices $C$ and $A$, respectively, in the altitude of triangle $ABC$ issuing from $B$. The perpendicular to the line $BA^{\prime}$ through the point $C^{\prime}$ intersects the line $BC$ at $U$; the perpendicular to the line $BC^{\prime}$ through the point $A^{\prime}$ intersects the line $BA$ at $V$. Prove that $UV \parallel CA$. Darij

An ordered pair $(m,n)$ of non-negative integers is called "simple" if the addition $m+n$ in base $10$ requires no carrying. Find the number of simple ordered pairs of non-negative integers that sum to $1492$.

Sam writes three $3$-digit positive integers (that don't end in $0$) on the board and adds them together. Jessica reverses each of the integers, and adds the reversals together. (For example, $\overline{XYZ}$ becomes $\overline{ZYX}$.) What is the smallest possible positive three-digit difference between Sam's sum and Jessica's sum?

Point $S$ is the midpoint of arc $ACB$ of the circumscribed circle $k$ around triangle $ABC$ with $AC>BC$. Let $I$ be the incenter of triangle $ABC$. Line $SI$ intersects $k$ again at point $T$. Let $D$ be the reflection of $I$ across $T$ and $M$ be the midpoint of side $AB$. Line $IM$ intersects the line through $D$, parallel to $AB$, at point $E$. Prove that $AE=BD$.

In a quadrilateral $ABCD$ of area $1$, the parallel sides $BC$ and $AD$ are in the ratio $1 :2$ . $K$ is the midpoint of the diagonal $AC$ and $L$ is the point of intersection of the line $DK$ and the side $AB$. Determine the area of the quadrilateral $BCKL$ . (M G Sonkin)

Let $K $ be a convex set in the $xy$-plane, symmetric with respect to the origin and having area greater than $4 $. Prove that there exists a point $(m, n) \neq (0, 0)$ in $K$ such that $m$ and $n$ are integers.

The material of new ball corset of the princess is quadrilateral . The tailor must sew four decorative strips on it. Two of gold, two of silver. Two of the same color on two opposite sides and the other two on it to a midline not intersecting them. The tailor is not yet familiar with the dress final shape. However, you will definitely sew the dress to be the cheapest (i.e., the gold stripe should be shorter than the silver). For design, it would be important to know what color stripe is centered. Can you decide this without knowing the the exact shape of the dress? [img]https://cdn.artofproblemsolving.com/attachments/8/1/85d40e7a352e468d0c9da7530c6a0378575de0.png[/img]

Compute the number of nonempty subsets $S$ of $\{ 1,2,3,4,5,6,7,8,9,10 \}$ such that the median of $S$ is an element of $S$.

We place the numbers from $1$ to $81$ in a $9\times $ board. Prove that exist $k \in \{1,2,...,9\}$ so that the product of the numbers in the $k$-th column is diferent to the product of the numbers in the $k$-th row.