Point $K$ is marked on the diagonal $AC$ in rectangle $ABCD$ so that $CK = BC$. On the side $BC$, point $M$ is marked so that $KM = CM$. Prove that $AK + BM = CM$.

Two pirates divide the loot, consisting of two bags of coins and a diamond, according to the following rules. First the first pirate takes take a few coins from any bag and transfer them from this bag in the other the same number of coins. Then the second pirate does the same (choosing the bag from which he takes the coins at his discretion) and etc. until you can take coins according to these rules. The pirate who takes the coins last gets the diamond. Who will get the diamond if is each of the pirates trying to get it? Give your answer depending on the initial number of coins in the bags.

Let $x,y,z,w$ be integers such that $2^x+2^y+2^z+2^w=24.375$. Find the value of $xyzw$.

Find the sum of all possible sums $a + b$ where $a$ and $b$ are nonnegative integers such that $4^a + 2^b + 5$ is a perfect square.

Suppose $\overline{a_1a_2...a_{2009}}$ is a $2009$-digit integer such that for each $i = 1,2,...,2007$, the $2$-digit integer $\overline{a_ia_{i+1}}$ contains $3$ distinct prime factors. Find $a_{2008}$ (Note: $\overline{xyz...}$ denotes an integer whose digits are $x, y,z,...$.)

Let $ABC$ be an acute-angled triangle with the property that the bisector of $\angle BAC$, the altitude through $B$ and the perpendicular bisector of $AB$ intersect in one point. Determine the angle $\alpha = \angle BAC$.

A teacher and her 30 students play a game on an infinite cell grid. The teacher starts first, then each of the 30 students makes a move, then the teacher and so on. On one move the person can color one unit segment on the grid. A segment cannot be colored twice. The teacher wins if, after the move of one of the 31 players, there is a $1\times 2$ or $2\times 1$ rectangle , such that each segment from it's border is colored, but the segment between the two adjacent squares isn't colored. Prove that the teacher can win.

On the coordinate plane, draw a set of points whose coordinates $(x, y)$ satisfy the inequality $$2 arc \cos x \ge arc \cos y$$

Given are 100 different positive integers. We call a pair of numbers [i]good[/i] if the ratio of these numbers is either 2 or 3. What is the maximum number of good pairs that these 100 numbers can form? (A number can be used in several pairs.) [i]Proposed by Alexander S. Golovanov, Russia[/i]

Determine the maximum number $ h$ satisfying the following condition: for every $ a\in [0,h]$ and every polynomial $ P(x)$ of degree 99 such that $ P(0)\equal{}P(1)\equal{}0$, there exist $ x_1,x_2\in [0,1]$ such that $ P(x_1)\equal{}P(x_2)$ and $ x_2\minus{}x_1\equal{}a$. [i]Proposed by F. Petrov, D. Rostovsky, A. Khrabrov[/i]

I have a very good solution of this but I want to see others. Let the midpoint$ M$ of the side$ AB$ of an inscribed quardiletar, $ABCD$.Let$ P $the point of intersection of $MC$ with $BD$. Let the parallel from the point $C$ to the$ AP$ which intersects the $BD$ at$ S$. If $CAD$ angle=$PAB$ angle= $\frac{BMC}{2}$ angle, prove that $BP=SD$.

We are given $30$ boots standing in a row, $15$ of which are for right feet and $15$ for the left. Prove that there are ten successive boots somewhere in this row with $5$ right and $5$ left boots among them. (D. Fomin, Leningrad)

The insphere and the exsphere opposite to the vertex $D$ of a (not necessarily regular) tetrahedron $ABCD$ touch the face $ABC$ in the points $X$ and $Y$, respectively. Show that $\measuredangle XAB=\measuredangle CAY$.

Find the number of ways to split the numbers from $1$ to $12$ into $4$ non-intersecting sets of size $3$ such that each set has sum divisible by $3$.

Prove that there exist infinitely many positive integers $ n$ such that $ n!$ is not divisible by $ n^2\plus{}1$.

Natural numbers $a, b, c, d$ are given such that $c>b$. Prove that if $a + b + c + d = ab-cd$, then $a + c$ is a composite number.

Let $V$ be a point in the exterior of a circle of center $O$, and let $T_1,T_2$ be the points where the tangents from $V$ touch the circle. Let $T$ be an arbitrary point on the small arc $T_1T_2$. The tangent in $T$ at the circle intersects the line $VT_1$ in $A$, and the lines $TT_1$ and $VT_2$ intersect in $B$. We denote by $M$ the intersection of the lines $TT_1$ and $AT_2$. Prove that the lines $OM$ and $AB$ are perpendicular.

Let $S$ be the set of the first $2018$ positive integers, and let $T$ be the set of all distinct numbers of the form $ab$, where $a$ and $b$ are distinct members of $S$. What is the $2018$th smallest member of $T$? [i]2018 CCA Math Bonanza Lightning Round #2.1[/i]

Show that it is possible to express $1$ as a sum of $6$, and as a sum of $9$ reciprocals of odd positive integers. Generalize the problem.

Fill each cell with an integer from $1$-$7$ so each number appears exactly once in each row and column. In each ``cage" of three cells, the three numbers must be valid lengths for the sides of a non-degenerate triangle. Additionally, if a cage has an ``A", the triangle must be acute, and if the cage has an ``R", the triangle must be right. [asy] for(int i = 0; i < 8; ++i){ draw((0,i) -- (7,i)^^(i,0)--(i,7), gray(0.7)); } draw((2.1,6.1) -- (4.9, 6.1)--(4.9, 6.9) -- (2.1,6.9)--cycle); draw((5.1,6.1) -- (6.1, 6.1) -- (6.1, 5.1) -- (6.9, 5.1) -- (6.9, 6.9) --(5.1, 6.9) -- cycle); label(scale(0.5)*"R", (5.1, 6.9), SE); draw((1.1, 5.9) -- (1.1, 4.1) -- (2.9, 4.1)-- (2.9, 4.9) -- (1.9, 4.9) -- (1.9, 5.9) -- cycle); draw((3.1, 3.1) -- (3.9, 3.1) -- (3.9, 5.9) -- (3.1, 5.9) -- cycle); draw(shift((3,0))*((1.1, 5.9) -- (1.1, 4.1) -- (2.9, 4.1)-- (2.9, 4.9) -- (1.9, 4.9) -- (1.9, 5.9) -- cycle)); draw(shift((3,-1))*((3.1, 3.1) -- (3.9, 3.1) -- (3.9, 5.9) -- (3.1, 5.9) -- cycle)); label(scale(0.5)*"A", (6.1, 4.9), SE); draw(shift((2,-2))*((3.1, 3.1) -- (3.9, 3.1) -- (3.9, 5.9) -- (3.1, 5.9) -- cycle)); draw((3.1, 2.1) -- (4.9, 2.1) -- (4.9, 3.9) -- (4.1, 3.9) -- (4.1, 2.9) -- (3.1, 2.9) -- cycle); label(scale(0.5)*"R", (4.1, 3.9), SE); draw((0.1, 2.1) -- (0.1, 3.9) -- (1.9, 3.9) -- (1.9, 3.1) -- (0.9, 3.1) -- (0.9, 2.1) -- cycle); draw(shift((0, -3))*((1.1, 5.9) -- (1.1, 4.1) -- (2.9, 4.1)-- (2.9, 4.9) -- (1.9, 4.9) -- (1.9, 5.9) -- cycle)); label(scale(0.5)*"R", (1.1, 2.9), SE); draw(shift((-2, -6)) * ((2.1,6.1) -- (4.9, 6.1)--(4.9, 6.9) -- (2.1,6.9)--cycle)); label(scale(0.5)*"A", (0.1, 0.9), SE); draw(shift((0,-2))*((3.1, 2.1) -- (4.9, 2.1) -- (4.9, 3.9) -- (4.1, 3.9) -- (4.1, 2.9) -- (3.1, 2.9) -- cycle)); label(scale(0.5)*"A", (4.1, 1.9), SE); draw(shift((2,-2))*((3.1, 2.1) -- (4.9, 2.1) -- (4.9, 3.9) -- (4.1, 3.9) -- (4.1, 2.9) -- (3.1, 2.9) -- cycle)); [/asy]

Let $a,b,c,d$ be positive integers such that $ad \neq bc$ and $gcd(a,b,c,d)=1$. Let $S$ be the set of values attained by $\gcd(an+b,cn+d)$ as $n$ runs through the positive integers. Show that $S$ is the set of all positive divisors of some positive integer.

Prove the inequality $$ \frac{y^2-x^2}{2x^2+1}+\frac{z^2-y^2}{2y^2+1}+\frac{x^2-z^2}{2z^2+1} \geq 0$$ where $x$, $y$ and $z$ are real numbers

Consider an $n\times{n}$ grid formed by $n^2$ unit squares. We define the centre of a unit square as the intersection of its diagonals. Find the smallest integer $m$ such that, choosing any $m$ unit squares in the grid, we always get four unit squares among them whose centres are vertices of a parallelogram.

We have a half-circle with endpoints $A$ and $B$ and center $S$. The points $C$ and $D$ lie on the half-circle such that $ \angle BAC \equal{} 20^\circ$ and the lines $ AC$ and $ SD$ are perpendicular to each other. What is the angle between the lines $ AC$ and $ BD$? [asy] size(8cm); pair A = (-1, 0), B = (1, 0), S = (0, 0), C = (sqrt(3)/2, 1/2); path circ = arc(S, 1, 0, 180); pair P = foot(S, A, C); pair D = intersectionpoints(circ, S--(7*(P-S)+S))[0]; draw(circ); draw(A--C--B--cycle); draw(S--D--B); dot(A); dot(B); dot(S); dot(C); dot(D); label("$A$", A, SW); label("$B$", B, SE); label("$S$", S, SW); label("$D$", D, NW); label("$C$", C, NE); markscalefactor *= 0.5; draw(rightanglemark(A, P, D)); draw(anglemark(S, A, C)); label("$20^\circ$", A + (0.3, 0.05), E);[/asy] A. $ 45^\circ$ B. $ 55^\circ$ C. $ 60^\circ$ D. $ 67 \frac{1}{2}^\circ$ E. $ 72^\circ$

Equilateral triangle $ABC$ has side lengths of $24$. Points $D$, $E$, and $F$ lies on sides $BC$, $CA$, $AB$ such that ${AD}\perp{BC}$, ${DE}\perp{AC}$, and ${EF}\perp{AB}$. $G$ is the intersection of $AD$ and $EF$. Find the area of quadrilateral $BFGD$