Find all real values of $x$ such that $(\frac{1}{5}(x^2-10x+26))^{x^2-6x+5}=1$

If $\frac{y}{x - z} = \frac{x + y}{z} = \frac{x}{y}$ for three positive numbers $x$, $y$ and $z$, all different, then $\frac{x}{y} =$ $ \textbf{(A)}\ \frac{1}{2}\qquad\textbf{(B)}\ \frac{3}{5}\qquad\textbf{(C)}\ \frac{2}{3}\qquad\textbf{(D)}\ \frac{5}{3}\qquad\textbf{(E)}\ 2 $

The circle $\omega_1$ passes through the vertex $A$ of the parallelogram $ABCD$ and touches the rays $CB, CD$. The circle $\omega_2$ touches the rays $AB, AD$ and touches $\omega_1$ externally at point $T$. Prove that $T$ lies on the diagonal $AC$

If $a$ and $b$ are digits for which \begin{tabular}{ccc} & 2 & a\\ $\times$ & b & 3\\ \hline & 6 & 9\\ 9 & 2\\ \hline 9 & 8 & 9\end{tabular} Then $a+b =$ A. 3 B. 4 C. 7 D. 9 E. 12

Prove that for any tetrahedron the radius of the inscribed sphere $r <\frac{ ab}{ 2(a + b)}$, where $a$ and $b$ are the lengths of any pair of opposite edges.

Let $\omega$ be a circle with centre $O$. Let $\gamma$ be another circle passing through $O$ and intersecting $\omega$ at points $A$ and $B$. $A$ diameter $CD$ of $\omega$ intersects $\gamma$ at a point $P$ different from $O$. Prove that $\angle APC= \angle BPD$

Let $D$ be an infinite in both sides sequence of $0$s and $1$s. For each positive integer $n$ we denote with $a_n$ the number of different subsequences of $0$s and $1$s in $D$ of length $n$. Does there exist a sequence $D$ for which for each $n\geq 22$ the number $a_n$ is equal to the $n$-th prime number?

Let $\vartriangle ABC$ be a triangle with $\angle BAC = 90^o$, $\angle ABC = 60^o$, and $\angle BCA = 30^o$ and $BC = 4$. Let the incircle of $\vartriangle ABC$ meet sides $BC$, $CA$, $AB$ at points $A_0$, $B_0$, $C_0$, respectively. Let $\omega_A$, $\omega_B$, $\omega_C$ denote the circumcircles of triangles $\vartriangle B_0IC_0$ , $\vartriangle C_0IA_0$ , $\vartriangle A_0IB_0$, respectively. We construct triangle $T_A$ as follows: let $A_0B_0$ meet $\omega_B$ for the second time at $A_1\ne A_0$, let $A_0C_0$ meet $\omega_C$ for the second time at $A_2\ne A_0$, and let $T_A$ denote the triangle $\vartriangle A_0A_1A_2$. Construct triangles $T_B$, $T_C$ similarly. If the sum of the areas of triangles $T_A$, $T_B$, $T_C$ equals $\sqrt{m} - n$ for positive integers $m$, $n$, find $m + n$.

Given $n$ points $A_1, A_2, \ldots, A_n,$ no three collinear, show that the $n$- gon $A_1 A_2 \ldots A_n,$ is inscribed in a circle if and only if $A_1 A_2 \cdot A_3 A_n \cdot \ldots \cdot A_{n-1} A_n + A_2 A_3 \cdot A_4 A_n \cdot \ldots A_{n-1} A_n \cdot A_1 A_n + \ldots$ $+ A_{n-1} A_{n-2} \cdot A_1 A_n \cdot \ldots \cdot A_{n-3} A_n$ $= A_1 A_{n-1} \cdot A_2 A_n \cdot \ldots \cdot A_{n-2} A_n$, where $XY$ denotes the length of the segment $XY.$

Given a positive integer $x \le 233$, let $a$ be the remainder when $x^{1943}$ is divided by $233$. Find the sum of all possible values of $a$.

Find the greatest positive integer $m$, such that one of the $4$ letters $C,G,M,O$ can be placed in each cell of a table with $m$ rows and $8$ columns, and has the following property: For any two distinct rows in the table, there exists at most one column, such that the entries of these two rows in such a column are the same letter.

Find all pairs of solutions $(x,y)$: \[ x^3 + x^2y + xy^2 + y^3 = 8(x^2 + xy + y^2 + 1). \]

Find all functions $f: \mathbb{N} \to \mathbb{N}$ such that $$m+f(n) \mid f(m)^2 - nf(n)$$ for all positive integers $m$ and $n$. (Here, $f(m)^2$ denotes $\left(f(m)\right)^2$.)

There exists a fraction $x$ that satisfies $ \sqrt{x^2+5} - x = \tfrac{1}{3}$. What is the sum of the numerator and denominator of this fraction? $\textbf{(A) }8\qquad\textbf{(B) }21\qquad\textbf{(C) }25\qquad\textbf{(D) }32\qquad\textbf{(E) }34$

The graph relates the distance traveled [in miles] to the time elapsed [in hours] on a trip taken by an experimental airplane. During which hour was the average speed of this airplane the largest? [asy] unitsize(12); for(int a=1; a<13; ++a) { draw((2a,-1)--(2a,1)); } draw((-1,4)--(1,4)); draw((-1,8)--(1,8)); draw((-1,12)--(1,12)); draw((-1,16)--(1,16)); draw((0,0)--(0,17)); draw((-5,0)--(33,0)); label("$0$",(0,-1),S); label("$1$",(2,-1),S); label("$2$",(4,-1),S); label("$3$",(6,-1),S); label("$4$",(8,-1),S); label("$5$",(10,-1),S); label("$6$",(12,-1),S); label("$7$",(14,-1),S); label("$8$",(16,-1),S); label("$9$",(18,-1),S); label("$10$",(20,-1),S); label("$11$",(22,-1),S); label("$12$",(24,-1),S); label("Time in hours",(11,-2),S); label("$500$",(-1,4),W); label("$1000$",(-1,8),W); label("$1500$",(-1,12),W); label("$2000$",(-1,16),W); label(rotate(90)*"Distance traveled in miles",(-4,10),W); draw((0,0)--(2,3)--(4,7.2)--(6,8.5)); draw((6,8.5)--(16,12.5)--(18,14)--(24,15));[/asy] $ \text{(A)}\ \text{first (0-1)}\qquad\text{(B)}\ \text{second (1-2)}\qquad\text{(C)}\ \text{third (2-3)}\qquad\text{(D)}\ \text{ninth (8-9)}\qquad\text{(E)}\ \text{last (11-12)} $

Let $T$ be the set of all 2-digit numbers whose digits are in $\{1,2,3,4,5,6\}$ and the tens digit is strictly smaller than the units digit. Suppose $S$ is a subset of $T$ such that it contains all six digits and no three numbers in $S$ use all six digits. If the cardinality of $S$ is $n$, find all possible values of $n$.

Prove that $x = y = z = 1$ is the only positive solution of the system \[\left\{ \begin{array}{l} x+y^2 +z^3 = 3\\ y+z^2 +x^3 = 3\\ z+x^2 +y^3 = 3\\ \end{array} \right. \]

Alice creates the graphs $y=|x-a|$ and $y=c-|x-b|$ , where $a,b,c\in\mathbb{R^+}$. She observes that these two graphs and $x$ axis divides the positive side of the plane ($x,y>0$) into two triangles and a quadrilateral. Find the ratio of sums of two triangles' areas to the area of quadrilateral. [hide=There might be a translation error] In the original statement,it says $XOY$ plane,instead of positive side of the plane. I think these 2 are the same,but I might be wrong [/hide]

Two jokers are added to a $52$ card deck and the entire stack of $54$ cards is shuffled randomly. What is the expected number of cards that will be strictly between the two jokers?

Suppose we are given a $19\times19$ chessboard (a table with $19^2$ squares) and remove the central square. Is it possible to tile the remaining $19^2-1=360$ squares with $4\times1$ and $1\times4$ rectangles? (So that each of the $360$ squares is covered by exactly one rectangle.) Justify your answer.

In a room containing $N$ people, $N > 3$, at least one person has not shaken hands with everyone else in the room. What is the maximum number of people in the room that could have shaken hands with everyone else? $\textbf{(A) }0\qquad\textbf{(B) }1\qquad\textbf{(C) }N-1\qquad\textbf{(D) }N\qquad \textbf{(E) }\text{none of these}$

Let $a_1, a_2, ..., a_n, a_{n+1}$ be a finite sequence of real numbers satisfying $a_0 = a_{n+1} = 0$ and $|a_{k-1} - 2a_{k} + a_{k+1}| \leq 1$ for $k = 1, 2, ..., n$ Prove that for $k=0, 1, ..., n+1,$ $|a_k| \leq \frac{k(n+1-k)}{2}$

Let $a\in\mathbb N$ and $m=a^2+a+1$. Find the number of $0\leq x\leq m$ that:\[x^3\equiv1(\mbox{mod}\ m)\]

If $ a,b,n$ are positive integers, number of solutions of the equaition $ a^2 \plus{} b^4 \equal{} 5^n$ is $\textbf{(A)}\ 1 \qquad\textbf{(B)}\ 2 \qquad\textbf{(C)}\ 3 \qquad\textbf{(D)}\ 4 \qquad\textbf{(E)}\ \text{Infinitely many}$

Let $ABC$ be an acute triangle such that $\angle{B}=60$. The circle with diameter $AC$ intersects the internal angle bisectors of $A$ and $C$ at the points $M$ and $N$, respectively $(M\neq{A},$ $N\neq{C})$. The internal bisector of $\angle{B}$ intersects $MN$ and $AC$ at the points $R$ and $S$, respectively. Prove that $BR\leq{RS}$.