Given is a isosceles triangle ABC so that AB=BC. Point K is in ABC, so that CK=AB=BC and <KAC=30°.Find <AKB=?

There are twenty-four 4-digit numbers that use each of the four digits 2, 5, 7, and 4exactly once. Listed in numerical order from smallest to largest, the number in the $17th$ position in the list is $ \text{(A)}\ 4527\qquad\text{(B)}\ 5724\qquad\text{(C)}\ 5742\qquad\text{(D)}\ 7245\qquad\text{(E)}\ 7524 $

Inside a rhombus $ABCD$ with $\angle BAD=60$, points $F,H,G$ are choosen on lines $AD,DC,AC$ respectivily such that $DFGH$ is a paralelogram. Show that $BFH$ is a equilateral triangle.

Let ${\mathcal A}$ be a ${C^{\ast}}$ algebra with a unit element and let ${\mathcal A_+}$ be the cone of the positive elements of ${\mathcal A}$ (this is the set of such self adjoint elements in ${\mathcal A}$ whose spectrum is in ${[0,\infty)}$. Consider the operation \[ \displaystyle x \circ y =\sqrt{x}y\sqrt{x},\ x,y \in \mathcal A_+\] Prove that if for all ${x,y \in \mathcal A_+}$ we have \[ \displaystyle (x\circ y)\circ y = x \circ (y \circ y), \] then ${\mathcal A}$ is commutative. [i]Proposed by Lajos Molnár[/i]

Figures $ 0$, $ 1$, $ 2$, and $ 3$ consist of $ 1$, $ 5$, $ 13$, and $ 25$ nonoverlapping squares, respectively. If the pattern were continued, how many nonoverlapping squares would there be in figure $ 100$? [asy] unitsize(8); draw((0,0)--(1,0)--(1,1)--(0,1)--cycle); draw((9,0)--(10,0)--(10,3)--(9,3)--cycle); draw((8,1)--(11,1)--(11,2)--(8,2)--cycle); draw((19,0)--(20,0)--(20,5)--(19,5)--cycle); draw((18,1)--(21,1)--(21,4)--(18,4)--cycle); draw((17,2)--(22,2)--(22,3)--(17,3)--cycle); draw((32,0)--(33,0)--(33,7)--(32,7)--cycle); draw((29,3)--(36,3)--(36,4)--(29,4)--cycle); draw((31,1)--(34,1)--(34,6)--(31,6)--cycle); draw((30,2)--(35,2)--(35,5)--(30,5)--cycle); label("Figure",(0.5,-1),S); label("$0$",(0.5,-2.5),S); label("Figure",(9.5,-1),S); label("$1$",(9.5,-2.5),S); label("Figure",(19.5,-1),S); label("$2$",(19.5,-2.5),S); label("Figure",(32.5,-1),S); label("$3$",(32.5,-2.5),S);[/asy]$ \textbf{(A)}\ 10401 \qquad \textbf{(B)}\ 19801 \qquad \textbf{(C)}\ 20201 \qquad \textbf{(D)}\ 39801 \qquad \textbf{(E)}\ 40801$

A test has twenty questions. Seven points are awarded for each correct answer, two points are deducted for each incorrect answer and no points are awarded or deducted for each unanswered question. Joana obtained $87$ points. How many questions did she not answer?

In a group of nine persons it is not possible to choose four persons such that every one knows the three others. Prove that this group of nine persons can be partitioned into four groups such that nobody knows anyone from his or her group.

Let $ \gamma$ be circumcircle of $ \triangle ABC$.Let $ R_a$ be radius of circle touching $ AB,AC$&$ \gamma$ internally.Define $ R_b,R_c$ similarly. Prove That $ \frac {1}{aR_a} \plus{} \frac {1}{bR_b} \plus{} \frac {1}{cR_c} \equal{} \frac {s^2}{rabc}$.

Consider the cube $ABCDA'B'C'D'$ (with face $ABCD$ directly above face $A'B'C'D'$). a) Find the locus of the midpoints of the segments $XY$, where $X$ is any point of $AC$ and $Y$ is any piont of $B'D'$; b) Find the locus of points $Z$ which lie on the segment $XY$ of part a) with $ZY=2XZ$.

Pictures are taken of $100$ adults and $100$ children, with one adult and one child in each, the adult being the taller of the two. Each picture is reduced to $\frac 1k$ of its original size, where $k$ is a positive integer which may vary from picture to picture. Prove that it is possible to have the reduced image of each adult taller than the reduced image of every child.

Right triangle $\triangle{DEF}$ with $\angle{D}=90^\circ$ and $\angle{F}=30^\circ$ is inscribed in equilateral triangle $\triangle{ABC}$ such that $D, E,$ and $F$ lie on segments $\overline{BC}, \overline{CA},$ and $\overline{AB},$ respectively. Given that $BD=7$ and $DC=4,$ compute $DE.$

At the start of a game there are three boxes with $2008, 2009$ and $2010$ game pieces Anja and Bernd play in turns according to the following rule: [i]When it is your turn, select two boxes, empty them and then distribute the pieces from the third box to the three boxes, such that no box may remain empty.If you can no longer complete a turn, you have lost. [/i] Who has a winning strategy when Anja starts?

A [i]slip[/i] on an integer $n\geq 2$ is an operation that consists in choosing a prime divisor $p$ of $n$ and replacing $n$ by $\frac{n+p^2}{p}.$ Starting with an arbitrary integer $n\geq 5$, we successively apply the slip operation on it. Show that one eventually reaches $5$, no matter the slips applied.

Evin’s calculator is broken and can only perform $3$ operations: Operation $1$: Given a number $x$, output $2x$. Operation $2$: Given a number $x$, output $4x +1$. Operation $3$: Given a number $x$, output $8x +3$. After initially given the number $0$, how many numbers at most $128$ can he make?

We are given a cyclic quadrilateral $ABCD$ with a point $E$ on the diagonal $AC$ such that $AD=AE$ and $CB=CE$. Let $M$ be the center of the circumcircle $k$ of the triangle $BDE$. The circle $k$ intersects the line $AC$ in the points $E$ and $F$. Prove that the lines $FM$, $AD$ and $BC$ meet at one point. [i](4th Middle European Mathematical Olympiad, Individual Competition, Problem 3)[/i]

Let $f_0(x)=x+|x-100|-|x+100|$, and for $n\geq 1$, let $f_n(x)=|f_{n-1}(x)|-1$. For how many values of $x$ is $f_{100}(x)=0$? $\textbf{(A) }299\qquad \textbf{(B) }300\qquad \textbf{(C) }301\qquad \textbf{(D) }302\qquad \textbf{(E) }303\qquad$

In the national math league, there are $7$ teams. Their season is a round robin format, where each team plays other. Find the number of ways the games could go such that they have equal number of wins. [i]Proposed by Ishin Shah[/i]

A deck of cards has only red cards and black cards. The probability of a randomly chosen card being red is $\frac13$. When $4$ black cards are added to the deck, the probability of choosing red becomes $\frac14$. How many cards were in the deck originally. $\textbf{(A) }6 \qquad \textbf{(B) }9 \qquad \textbf{(C) }12 \qquad \textbf{(D) }15 \qquad \textbf{(E) }18$

Let $\mathbb{Z} _{>0}$ be the set of positive integers. Find all functions $f: \mathbb{Z} _{>0}\rightarrow \mathbb{Z} _{>0}$ such that \[ m^2 + f(n) \mid mf(m) +n \] for all positive integers $m$ and $n$.

Let $P$ be the set of all $2012$ tuples $(x_1, x_2, \dots, x_{2012})$, where $x_i \in \{1,2,\dots 20\}$ for each $1\leq i \leq 2012$. The set $A \subset P$ is said to be decreasing if for each $(x_1,x_2,\dots ,x_{2012} ) \in A$ any $(y_1,y_2,\dots, y_{2012})$ satisfying $y_i \leq x_i (1\leq i \leq 2012)$ also belongs to $A$. The set $B \subset P$ is said to be increasing if for each $(x_1,x_2,\dots ,x_{2012} ) \in B$ any $(y_1,y_2,\dots, y_{2012})$ satisfying $y_i \geq x_i (1\leq i \leq 2012)$ also belongs to $B$. Find the maximum possible value of $f(A,B)= \dfrac {|A\cap B|}{|A|\cdot |B|}$, where $A$ and $B$ are nonempty decreasing and increasing sets ($\mid \cdot \mid$ denotes the number of elements of the set).

Four distinct circles of radius $r$ are on the surface of a unit sphere such that they are pairwise tangent. Find $r$. [i]Proposed by Thomas Lam[/i]

Find the last two digits of $2023+202^3+20^{23}$. [i]Proposed by Anthony Yang[/i]

There are $n \ge 2$ numbers on the blackboard: $1, 2,..., n$. It is permitted to erase two of those numbers $x,y$ and write $2x - y$ instead. Find all values of $n$ such that it is possible to leave number $0$ on the blackboard after $n - 1$ such procedures.

It is given that the number $4^{11}+1$ is divisible by some prime greater than $1000$. Determine this prime. [i] Proposed by David Altizio [/i]

A solid box is $ 15$ cm by $ 10$ cm by $ 8$ cm. A new solid is formed by removing a cube $ 3$ cm on a side from each corner of this box. What percent of the original volume is removed? $ \textbf{(A)}\ 4.5 \qquad \textbf{(B)}\ 9 \qquad \textbf{(C)}\ 12 \qquad \textbf{(D)}\ 18 \qquad \textbf{(E)}\ 24$