Let $n$ be a positive integer divisible by $4$. Find the number of permutations $\sigma$ of $(1,2,3,\cdots,n)$ which satisfy the condition $\sigma(j)+\sigma^{-1}(j)=n+1$ for all $j \in \{1,2,3,\cdots,n\}$.

Let $ABC$ be a triangle with $\angle BAC=60^\circ$. Let $D$ be a point on the line $AC$ such that $AB = AD$ and $A$ lies between $C$ and $D$. Suppose that there are two points $E \ne F$ on the circumcircle of the triangle $DBC$ such that $AE = AF = BC$. Prove that the line $EF$ passes through the circumcenter of $ABC$.

You are given a deck of $n \cdot m$ different cards where $n$ and $m$ are fixed numbers both between $10^{99}$ and $10^{100}$. You perform an [i]$m$-perfect shuffle[/i] for some times. In a single $m$-perfect shuffle, you divide the deck into $m$ piles with $n$ consecutive cards in each pile. You take one card from each pile, in order of the piles, for $n$ times to form the new deck. (The $m$-perfect shuffle is deterministic) For example, if the cards are labeled 12345678 where $n=4$ and $m=2$, you divide the deck into 1234 and 5678, and after one $2$-perfect shuffle you get 15263748. In another example, if the cards are labeled 123456789 where $n=3$ and $m=3$, you divide the deck into 123, 456, and 789, and after one $3$-perfect shuffle you get 147258369. Find an algorithm that, in at most $k$ steps, outputs the smallest positive number of $m$-perfect shuffle after which the deck is exactly the same as the original deck. In each step, you can do one arithmetic operation in $\{+, -, *, /, \bmod\}$, do one comparison, break out of a loop, or store one number to a specific location of an array. You can use the following precomputed numbers of steps in your solution: [list] [*] Checking if $a$ divides $b$ for any two integers $a$ and $b$ takes 2 steps because you need to compute $b \bmod a$ then compare with $0$. [*]A loop over $k$ iterations takes $2k$ steps because you need to increment the loop index by $1$ $k$ times and check the loop guard $k$ times. [*]Simulating one "$m$-perfect shuffle" takes $7nm$ steps because there is one loop index increment, four arithmetic operations, and one store in each iteration of the loop. [/list] [b]Scoring:[/b] An algorithm that completes in at most $k$ steps will be awarded: [list] [*] 1 pt for $k > 10^{10^{10^{10}}}$ [*] 10 pts for $k = 10^{10^{10^{10}}}$ [*] 20 pts for $k = 10^{420}$ [*] 30 pts for $k = 10^{360}$ [*] 50 pts for $k = 10^{240}$ [*] 70 pts for $k = 10^{202}$ [*] 80 pts for $k = 10^{201}$ [*] 95 pts for $k = 10^{120}$ [*] 98 pts for $k = 10^{102}$ [*] 100 pts for $k = 10^{101}$ [/list] [i]Proposed by Mingkuan Xu[/i]

Let \( P(x) \) be a polynomial with natural coefficients. We denote by \( d(n) \) the number of positive divisors of the natural number \( n \), and by \( \sigma(n) \), the sum of these divisors. The sequence \( a_n \) is defined as follows: \[ a_{n+1} \in \left\{ \begin{array}{ll} \sigma(P(d(a_n))) \\ d(P(\sigma(a_n))) \end{array} \right. \] That is, \( a_{n+1} \) is one of the two terms above. Show that there exists a constant \( C \), depending on \( a_1 \) and \( P(x) \), such that for all \( i \), \( a_i < C \); in other words, show that the sequence \( a_n \) is bounded.

How many ways are there to completely fill a $3 \times 3$ grid of unit squares with the letters $B, M$, and $T$, assigning exactly one of the three letters to each of the squares, such that no $2$ adjacent unit squares contain the same letter? Two unit squares are adjacent if they share a side.

Determine all prime numbers $p, q$ and $r$ with $p + q^2 = r^4$. [i](Karl Czakler)[/i]

Let $ ABCD$ be an isosceles trapezoid with $ \overline{AD}\parallel{}\overline{BC}$ whose angle at the longer base $ \overline{AD}$ is $ \dfrac{\pi}{3}$. The diagonals have length $ 10\sqrt {21}$, and point $ E$ is at distances $ 10\sqrt {7}$ and $ 30\sqrt {7}$ from vertices $ A$ and $ D$, respectively. Let $ F$ be the foot of the altitude from $ C$ to $ \overline{AD}$. The distance $ EF$ can be expressed in the form $ m\sqrt {n}$, where $ m$ and $ n$ are positive integers and $ n$ is not divisible by the square of any prime. Find $ m \plus{} n$.

Find all positive integers $n$ with $n \ge 2$ such that the polynomial \[ P(a_1, a_2, ..., a_n) = a_1^n+a_2^n + ... + a_n^n - n a_1 a_2 ... a_n \] in the $n$ variables $a_1$, $a_2$, $\dots$, $a_n$ is irreducible over the real numbers, i.e. it cannot be factored as the product of two nonconstant polynomials with real coefficients. [i]Proposed by Yang Liu[/i]

Consider fractions $\frac{a}{b}$ where $a$ and $b$ are positive integers. (a) Prove that for every positive integer $n$, there exists such a fraction $\frac{a}{b}$ such that $\sqrt{n} \le \frac{a}{b} \le \sqrt{n+1}$ and $b \le \sqrt{n}+1$. (b) Show that there are infinitely many positive integers $n$ such that no such fraction $\frac{a}{b}$ satisfies $\sqrt{n} \le \frac{a}{b} \le \sqrt{n+1}$ and $b \le \sqrt{n}$.

Find all positive integer $n$ and prime number $p$ such that $p^2+7^n$ is a perfect square

A rectangular yard contains two flower beds in the shape of congruent isosceles right triangles. THe remainder of the yard has a trapezoidal shape, as shown. The parallel sides of the trapezoid have lengths $ 15$ and $ 25$ meters. What fraction of the yard is occupied by the flower beds? [asy]unitsize(2mm); defaultpen(linewidth(.8pt)); fill((0,0)--(0,5)--(5,5)--cycle,gray); fill((25,0)--(25,5)--(20,5)--cycle,gray); draw((0,0)--(0,5)--(25,5)--(25,0)--cycle); draw((0,0)--(5,5)); draw((20,5)--(25,0));[/asy]$ \textbf{(A)}\ \frac18\qquad \textbf{(B)}\ \frac16\qquad \textbf{(C)}\ \frac15\qquad \textbf{(D)}\ \frac14\qquad \textbf{(E)}\ \frac13$

Let $n$ be a positive integer and $G$ be a finite group of order $n.$ A function $f:G \to G$ has the $(P)$ property if $f(xyz)=f(x)f(y)f(z)~\forall~x,y,z \in G.$ $\textbf{(a)}$ If $n$ is odd, prove that every function having the $(P)$ property is an endomorphism. $\textbf{(b)}$ If $n$ is even, is the conclusion from $\textbf{(a)}$ still true?

[list=a][*] Find infinitely many pairs of integers $a$ and $b$ with $1<a<b$, so that $ab$ exactly divides $a^{2}+b^{2}-1$. [*] With $a$ and $b$ as above, what are the possible values of \[\frac{a^{2}+b^{2}-1}{ab}?\] [/list]

New license plates consist of two letters, three digits, and two letters (from the English alphabet of$ 26$ letters). What is the largest possible number of such license plates if it is required that every two of them differ at no less than two positions?

Thelma writes a list of four digits consisting of $1$, $3$, $5$, and $7$, and each digit can appear one time, multiples time, or not at all. The list has a unique [i]mode[/i], or the number that appears the most. Thelma removes two numbers of that mode from the list; her list now has no unique mode! How many lists are possible? Suppose that all possible lists are unordered. $\textbf{(A) }18\qquad\textbf{(B) }24\qquad\textbf{(C) }30\qquad\textbf{(D) }36\qquad\textbf{(E) }48$

There's a group of $21$ people such that each person has no more than $7$ friends among the others and any two friends have a different number of total friends. Prove that there are $6$ people, none of which knows the others.

a) (version for grades 10-11) Let $P$ be a point lying in the interior of a triangle. Show that the product of the distances from $P$ to the sides of the triangle is at least $8$ times less than the product of the distances from $P$ to the tangents to the circumcircle at the vertices of the triangle. b) (version for grades 8-9) Is it true that for any triangle there exists a point $P$ for which equality in the inequality from a) holds?

Find the smallest positive integer $n$ that is divisible by $100$ and has exactly $100$ divisors.

For continuously differentiable function \(f : [0, 1] \to\mathbb{R}\) with \(f (1/2) = 0\), show that \[\left(\int_0^1 f(x)\mathrm{d}x\right)^2\leq \frac{1}{4}\int_0^1\left(f'(x)\right)^2\mathrm{d}x\]

From a bag containing 5 pairs of socks, each pair a different color, a random sample of 4 single socks is drawn. Any complete pairs in the sample are discarded and replaced by a new pair draw from the bag. The process continues until the bag is empty or there are 4 socks of different colors held outside the bag. What is the probability of the latter alternative?

Let $\triangle{ABC}$ be an acute, scalene triangle with orthocenter $H$, and let $AH$ meet the circumcircle of $\triangle{ABC}$ at a point $D \neq A$. Points $E$ and $F$ are chosen on $AC$ and $AB$ such that $DE \perp AC$ and $DF \perp AB$. Show that $BE$, $CF$, and the line through $H$ parallel to $EF$ concur.

In the Cartesian plane $XOY$, there is a rhombus $ABCD$ whose side lengths are all $4$ and $|OB|=|OD|=6$, where $O$ is the origin. [b](1)[/b] Prove that $|OA|\cdot |OB|$ is a constant. [b](2)[/b] Find the locus of $C$ if $A$ is a point on the semicircle \[(x-2)^2+y^2=4 \quad (2\le x\le 4).\]

If $r$ is a number such that $r^2-6r+5=0$, find $(r-3)^2$

Is it possible to cut a square into five squares?

Determine all $4$-tuples $(a,b, c, d)$ of positive real numbers satisfying $a + b +c + d = 1$ and $\max (\frac{a^2}{b},\frac{b^2}{a}) \cdot \max (\frac{c^2}{d},\frac{d^2}{c}) = (\min (a + b, c + d))^4$