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 $BFGD$

Find all pairs $(a,b)$ of positive integers that satisfy the equation \[a^{a^{a}}= b^{b}.\]

If $ 10^{2y} \equal{} 25$, then $ 10^{ \minus{} y}$ equals: $ \textbf{(A)}\ \minus{} \frac {1}{5} \qquad\textbf{(B)}\ \frac {1}{625} \qquad\textbf{(C)}\ \frac {1}{50} \qquad\textbf{(D)}\ \frac {1}{25} \qquad\textbf{(E)}\ \frac {1}{5}$