Let $\omega$ be the circumcircle of $\Delta ABC$, where $|AB|=|AC|$. Let $D$ be any point on the minor arc $AC$. Let $E$ be the reflection of point $B$ in line $AD$. Let $F$ be the intersection of $\omega$ and line $BE$ and Let $K$ be the intersection of line $AC$ and the tangent at $F$. If line $AB$ intersects line $FD$ at $L$, Show that $K,L,E$ are collinear points

Let $n$ be a fixed positive integer. Initially, $n$ 1's are written on a blackboard. Every minute, David picks two numbers $x$ and $y$ written on the blackboard, erases them, and writes the number $(x+y)^4$ on the blackboard. Show that after $n-1$ minutes, the number written on the blackboard is at least $2^{\frac{4n^2-4}{3}}$. [i]Proposed by Calvin Deng[/i]

Let $ABC$ be a triangle with $AC > BC,$ let $\omega$ be the circumcircle of $\triangle ABC,$ and let $r$ be its radius. Point $P$ is chosen on $\overline{AC}$ such taht $BC=CP,$ and point $S$ is the foot of the perpendicular from $P$ to $\overline{AB}$. Ray $BP$ mets $\omega$ again at $D$. Point $Q$ is chosen on line $SP$ such that $PQ = r$ and $S,P,Q$ lie on a line in that order. Finally, let $E$ be a point satisfying $\overline{AE} \perp \overline{CQ}$ and $\overline{BE} \perp \overline{DQ}$. Prove that $E$ lies on $\omega$.

For any odd prime $p$ and any integer $n,$ let $d_p (n) \in \{ 0,1, \dots, p-1 \}$ denote the remainder when $n$ is divided by $p.$ We say that $(a_0, a_1, a_2, \dots)$ is a [i]p-sequence[/i], if $a_0$ is a positive integer coprime to $p,$ and $a_{n+1} =a_n + d_p (a_n)$ for $n \geqslant 0.$ (a) Do there exist infinitely many primes $p$ for which there exist $p$-sequences $(a_0, a_1, a_2, \dots)$ and $(b_0, b_1, b_2, \dots)$ such that $a_n >b_n$ for infinitely many $n,$ and $b_n > a_n$ for infinitely many $n?$ (b) Do there exist infinitely many primes $p$ for which there exist $p$-sequences $(a_0, a_1, a_2, \dots)$ and $(b_0, b_1, b_2, \dots)$ such that $a_0 <b_0,$ but $a_n >b_n$ for all $n \geqslant 1?$ [I]United Kingdom[/i]

Let $ABC$ be a non-obtuse triangle satisfying $AB>AC$ and $\angle B=45^{\circ}$. The circumcentre $O$ and incentre $I$ of triangle $ABC$ are such that $\sqrt{2}\ OI=AB-AC$. Find the value of $\sin A$.

There exist complex numbers $z_1,z_2,\dots,z_{10}$ which satisfy$$|z_ki^k+ z_{k+1}i^{k+1}| = |z_{k+1}i^k+ z_ki^{k+1}|$$for all integers $1 \leq k \leq 9$, where $i = \sqrt{-1}$. If $|z_1|=9$, $|z_2|=29$, and for all integers $3 \leq n \leq 10$, $|z_n|=|z_{n-1} + z_{n-2}|$, find the minimum value of $|z_1|+|z_2|+\cdots+|z_{10}|$. [i]Proposed by DeToasty3[/i]

A student puts $2010$ red balls and $1957$ blue balls into a box. Weiqing draws randomly from the box one ball at a time without replacement. She wins if, at anytime, the total number of blue balls drawn is more than the total number of red balls drawn. Assuming Weiqing keeps drawing balls until she either wins or runs out, ompute the probability that she eventually wins.

Find all quadruple $ (m,n,p,q) \in \mathbb{Z}^4$ such that \[ p^m q^n \equal{} (p\plus{}q)^2 \plus{} 1.\]

For a positive integer $n$, we define the function $f_n(x)=\sum_{k=1}^n |x-k|$ for all real numbers $x$. For any two-digit number $n$ (in decimal representation), determine the set of solutions $\mathbb{L}_n$ of the inequality $f_n(x)<41$. [i](41st Austrian Mathematical Olympiad, National Competition, part 1, Problem 2)[/i]

The temperatures $f^o F$ and $c^o C$ are equal when $f = \frac95 c + 32$. What temperature is the same in both $^o F$ and $^o C$?

The average value of all the pennies, nickels, dimes, and quarters in Paula's purse is $ 20$ cents. If she had one more quarter, the average value would be $ 21$ cents. How many dimes does she have in her purse? $ \textbf{(A)}\ 0\qquad \textbf{(B)}\ 1\qquad \textbf{(C)}\ 2\qquad \textbf{(D)}\ 3\qquad \textbf{(E)}\ 4$

We have $2022$ $1s$ written on a board in a line. We randomly choose a strictly increasing sequence from ${1, 2, . . . , 2022}$ such that the last term is $2022$. If the chosen sequence is $a_1, a_2, ..., a_k$ ($k$ is not fixed), then at the $i^{th}$ step, we choose the first a$_i$ numbers on the line and change the 1s to 0s and 0s to 1s. After $k$ steps are over, we calculate the sum of the numbers on the board, say $S$. The expected value of $S$ is $\frac{a}{b}$ where $a, b$ are relatively prime positive integers. Find $a + b.$

A convex polygon of $n$ sides is considered. All its diagonals are drawn and we suppose that any three of them can only intersect on a vertex and that there is no pair of parallel diagonals. Under these conditions, we wish to compute a) The total number of intersection points of these diagonals, excluding the vertices. b) How many points, of these intersections, lie inside the polygon and how many lie outside.

Find the maximal value of \[S = \sqrt[3]{\frac{a}{b+7}} + \sqrt[3]{\frac{b}{c+7}} + \sqrt[3]{\frac{c}{d+7}} + \sqrt[3]{\frac{d}{a+7}},\] where $a$, $b$, $c$, $d$ are nonnegative real numbers which satisfy $a+b+c+d = 100$. [i]Proposed by Evan Chen, Taiwan[/i]

Compute the probability that a random permutation of the letters in BERKELEY does not have the three E’s all on the same side of the Y.

Let $p=2017$ be a prime. Find the remainder when \[\left\lfloor\dfrac{1^p}p\right\rfloor + \left\lfloor\dfrac{2^p}p\right\rfloor+\left\lfloor\dfrac{3^p}p\right\rfloor+\cdots+\left\lfloor\dfrac{2015^p}p\right\rfloor \] is divided by $p$. Here $\lfloor\cdot\rfloor$ denotes the greatest integer function. [i]Proposed by David Altizio[/i]

A staircase-brick with 3 steps of width 2 is made of 12 unit cubes. Determine all integers $ n$ for which it is possible to build a cube of side $ n$ using such bricks.

There are $101$ positive integers $a_1, a_2, \ldots, a_{101}$ such that for every index $i$, with $1 \leqslant i \leqslant 101$, $a_i+1$ is a multiple of $a_{i+1}$. Determine the greatest possible value of the largest of the $101$ numbers.

Let the line $L$ be perpendicular to the plane $P$. Three spheres touch each other in pairs so that each sphere touches the plane $P$ and the line $L$. The radius of the larger sphere is $1$. Find the minimum radius of the smallest sphere.

Fix a point $A$, a circle $\Omega$ centered at $O$, and reals $r$ and $\theta$. Let $X$ and $Y$ be variable points on $\Omega$ so that $\measuredangle XOY = \theta$. The tangents to $\Omega$ at $X$ and $Y$ meet at $T$, and a dilation at $T$ with scale factor $r$ sends $A$ to $A'$. Let $P$ be the foot from $A'$ to $TX$. $ $ $ $ $ $ $ $ $ $ Suppose that some point $P^*$ is the same for two different $X$. Show that $\measuredangle TXY = \measuredangle AP^\ast O$. (All angles are directed.) Proposed by [i]Karn Chutinan[/i]

A warehouse contains $175$ boots of size $8$, $175$ boots of size $9$ and $200$ boots of size $10$. Of these $550$ boots, $250$ are for the left foot and $300$ for the right foot. Let $n$ denote the total number of usable pairs of boots in the warehouse. (A usable pair consists of a left and a right boot of the same size.) (a) Is $n=50$ possible? (b) Is $n=51$ possible?

Let $ x,y,z$ be real numbers greater than or equal to $ 1.$ Prove that \[ \prod(x^{2} \minus{} 2x \plus{} 2)\le (xyz)^{2} \minus{} 2xyz \plus{} 2.\]

Let $(G, \cdot )$ be a group and $e$ an identity element. Prove that if all elements $x$ of $G$ satisfy $x\cdot x = e$ then $(G, \cdot)$ is abelian (that is, commutative).

Let $(a_1,a_2,\ldots, a_{13})$ be a permutation of $(1, 2, \ldots, 13)$. Ayvak takes this permutation and makes a series of [i]moves[/i], each of which consists of choosing an integer $i$ from $1$ to $12$, inclusive, and swapping the positions of $a_i$ and $a_{i+1}$. Define the [i]weight[/i] of a permutation to be the minimum number of moves Ayvak needs to turn it into $(1, 2, \ldots, 13)$. The arithmetic mean of the weights of all permutations $(a_1, \ldots, a_{13})$ of $(1, 2, \ldots, 13)$ for which $a_5 = 9$ is $\frac{m}{n}$, for coprime positive integers $m$ and $n$. Find $100m+n$. [i]Proposed by Alex Gu[/i]

Does there exist an positive integer $n$, so that for any positive integer $m<1002$, there exists an integer $k$ so that \[\displaystyle \frac{m}{1002} < \frac{k}{n} < \frac {m+1}{1003}\] holds? If $n$ does not exist, prove it; if $n$ exists, determine the minimum value of it. I know this problem was easy, but it still appeared on our TST, and so I posted it here.