For natural number $t$, the repeating base-$t$ representation of the (base-ten) rational number $\frac{7}{51}$ is $0.\overline{23}_t=0.232323..._t$. What is $t$ ?

Right isosceles triangles are constructed on the sides of a 3-4-5 right triangle, as shown. A capital letter represents the area of each triangle. Which one of the following is true? [asy]/* AMC8 2002 #16 Problem */ draw((0,0)--(4,0)--(4,3)--cycle); draw((4,3)--(-4,4)--(0,0)); draw((-0.15,0.1)--(0,0.25)--(.15,0.1)); draw((0,0)--(4,-4)--(4,0)); draw((4,0.2)--(3.8,0.2)--(3.8,-0.2)--(4,-0.2)); draw((4,0)--(7,3)--(4,3)); draw((4,2.8)--(4.2,2.8)--(4.2,3)); label(scale(0.8)*"$Z$", (0, 3), S); label(scale(0.8)*"$Y$", (3,-2)); label(scale(0.8)*"$X$", (5.5, 2.5)); label(scale(0.8)*"$W$", (2.6,1)); label(scale(0.65)*"5", (2,2)); label(scale(0.65)*"4", (2.3,-0.4)); label(scale(0.65)*"3", (4.3,1.5));[/asy] $ \textbf{(A)}\ X\plus{}Z\equal{}W\plus{}Y \qquad \textbf{(B)}\ W\plus{}X\equal{}Z \qquad\textbf{(C)}\ 3X\plus{}4Y\equal{}5Z \qquad $ $\textbf{(D)}\ X\plus{}W\equal{}\frac{1}{2}(Y\plus{}Z) \qquad\textbf{(E)}\ X\plus{}Y\equal{}Z$

What is $o-w$, if $gun^2 = wowgun$ where $g,n,o,u,w \in \{0,1,2,\dots, 9\}$? $ \textbf{(A)}\ 1 \qquad\textbf{(B)}\ 2 \qquad\textbf{(C)}\ 3 \qquad\textbf{(D)}\ 5 \qquad\textbf{(E)}\ \text{None of above} $

Let $ l,d,k$ be natural numbers. We want to prove that for large numbers $ n$, for each $ k$-coloring of the $ n$-dimensional cube with side length $ l$, there is a $ d$-dimensional subspace that all of its vertices have the same color. Let $ H(l,d,k)$ be the least number such that for $ n\geq H(l,d,k)$ the previus statement holds. a) Prove that: \[ H(l,d \plus{} 1,k)\leq H(l,1,k) \plus{} H(l,d,k^l)^{H(l,1,k)} \] b) Prove that \[ H(l \plus{} 1,1,k \plus{} 1)\leq H(l,1 \plus{} H(l \plus{} 1,1,k),k \plus{} 1) \] c) Prove the statement of problem. d) Prove Van der Waerden's Theorem.

A positive integer is called [i]shiny[/i] if it can be written as the sum of two not necessarily distinct integers $a$ and $b$ which have the same sum of their digits. For instance, $2012$ is [i]shiny[/i], because $2012 = 2005 + 7$, and both $2005$ and $7$ have the same sum of their digits. Find all positive integers which are [b]not[/b] [i]shiny[/i] (the dark integers).

Suppose that $f(x) = \frac{x}{x^2 - 2x + 2}$ and $g(x_1, x_2, ... , x_7) = f(x_1) + f(x_2) +... + f(x_7)$. If $x_1, x_2,..., x_7$ are non-negative real numbers with sum $5$, determine for how many tuples $(x_1, x_2, ... , x_7)$ does $g(x_1, x_2, ... , x_7)$ obtain its maximal value.

If $ x$,$ y$, and $ z$ are real numbers such that \[(x \minus{} 3)^2 \plus{} (y \minus{} 4)^2 \plus{} (z \minus{} 5)^2 \equal{} 0,\] then $ x \plus{} y \plus{} z \equal{}$ $ \textbf{(A)}\ \minus{}12\qquad \textbf{(B)}\ 0\qquad \textbf{(C)}\ 8\qquad \textbf{(D)}\ 12\qquad \textbf{(E)}\ 50$

Stoyan and Nikolai have two $100\times 100$ chess boards. Both of them number each cell with the numbers $1$ to $10000$ in some way. Is it possible that for every two numbers $a$ and $b$, which share a common side in Nikolai's board, these two numbers are at a knight's move distance in Stoyan's board (that is, a knight can move from one of the cells to the other one with a move)? [i]Nikolai Beluhov[/i]

It is known that $\odot O$ is the circumcircle of $\vartriangle ABC$ wwith diameter $AB$. The tangents of $\odot O$ at points $B$ and $C$ intersect at $P$ . The line perpendicular to $PA$ at point $A$ intersects the extension of $BC$ at point $D$. Extend $DP$ at length $PE = PB$. If $\angle ADP = 40^o$ , find the measure of $\angle E$.

From $2008 \times 2008$ square we remove one corner cell $1 \times 1$. Is number of ways to divide this figure to corners from $3$ cells odd or even ?

[b](a)[/b] Determine if exist an integer $n$ such that $n^2 -k$ has exactly $10$ positive divisors for each $k = 1, 2, 3.$ [b](b) [/b]Show that the number of positive divisors of $n^2 -4$ is not $10$ for any integer $n.$

$\triangle KWU$ is an equilateral triangle with side length $12$. Point $P$ lies on minor arc $\overarc{WU}$ of the circumcircle of $\triangle KWU$. If $\overline{KP} = 13$, find the length of the altitude from $P$ onto $\overline{WU}$. [i]Proposed by Bradley Guo[/i]

A simple calculator is given to you. (It contains 8 digits and only does the operations +,-,*,/,$ \sqrt{\mbox{}}$) How can you find $ 3^{\sqrt{2}}$ with accuracy of 6 digits.

Prove that there exist infinitely many positive integers $x,y,z$ for which the sum of the digits in the decimal representation of $~4x^4+y^4-z^2+4xyz$ $~$ is at most $2$. (Proposed by Gerhard Woeginger, Austria)

Consider the lattice of all algebraically closed subfields of the complex field $ \mathbb{C}$ whose transcendency degree (over $ \mathbb{Q}$) is finite. Prove that this lattice is not modular. [i]L. Babai[/i]

Consider the triangle $\triangle ABC$ with $AB \not = AC$. Let point $O$ be the circumcenter of $\triangle ABC$. Let the angle bisector of $\angle BAC$ intersect $BC$ at point $D$. Let $E$ be the reflection of point $D$ across the midpoint of the segment $BC$. The lines perpendicular to $BC$ in points $D,E$ intersect the lines $AO,AD$ at the points $X,Y$ respectively. Prove that the quadrilateral $B,X,C,Y$ is cyclic.

Jerry is bored one day, so he makes an array of Cocoa pebbles. He makes 8 equal rows with the pebbles remaining in a box. When Kramer drops by and eats one, Jerry yells at him until Kramer realizes he can make 9 equal rows with the remaining pebbles. After Kramer eats another, he finds he can make 10 equal rows with the remaining pebbles. Find the smallest number of pebbles that were in the box in the beginning.

Find the eighth term of the sequence $1440,$ $1716,$ $1848,\ldots,$ whose terms are formed by multiplying the corresponding terms of two arithmetic sequences.

In a triangle $ABC$, the angle bisector of $\widehat{ABC}$ meets $[AC]$ at $D$, and the angle bisector of $\widehat{BCA}$ meets $[AB]$ at $E$. Let $X$ be the intersection of the lines $BD$ and $CE$ where $|BX|=\sqrt 3|XD|$ ve $|XE|=(\sqrt 3 - 1)|XC|$. Find the angles of triangle $ABC$.

Let $n\ge 2$ be an integer. $a_1,a_2,\dotsc,a_n$ are arbitrarily chosen positive integers with $(a_1,a_2,\dotsc,a_n)=1$. Let $A=a_1+a_2+\dotsb+a_n$ and $(A,a_i)=d_i$. Let $(a_2,a_3,\dotsc,a_n)=D_1, (a_1,a_3,\dotsc,a_n)=D_2,\dotsc, (a_1,a_2,\dotsc,a_{n-1})=D_n$. Find the minimum of $\prod\limits_{i=1}^n\dfrac{A-a_i}{d_iD_i}$

Let $x_1,x_2,\ldots,x_n$ be positive real numbers such that \[ x_1+\frac{1}{x_2} = x_2+\frac{1}{x_3} = x_3+\frac{1}{x_4} = \dots = x_{n-1}+\frac{1}{x_n} = x_n+\frac{1}{x_1}. \] Prove that $x_1=x_2=x_3=\dots=x_n$.

Andy the Ant lives on a coordinate plane and is currently at $(-20, 20)$ facing east (that is, in the positive $x$-direction). Andy moves $1$ unit and then turns $90^{\circ}$ degrees left. From there, Andy moves $2$ units (north) and then turns $90^{\circ}$ degrees left. He then moves $3$ units (west) and again turns $90^{\circ}$ degrees left. Andy continues his progress, increasing his distance each time by $1$ unit and always turning left. What is the location of the point at which Andy makes the $2020$th left turn? $\textbf{(A)}\ (-1030, -994)\qquad\textbf{(B)}\ (-1030, -990)\qquad\textbf{(C)}\ (-1026, -994)\qquad\textbf{(D)}\ (-1026, -990)\qquad\textbf{(E)}\ (-1022, -994)$

Let the roots of \[x^{2022}-7x^{2021}+8x^2+4x+2\] be $r_1,r_2,\dots,r_{2022}$, the roots of \[x^{2022}-8x^{2021}+27x^2+9x+3\] be $s_1,s_2,\dots,s_{2022}$, and the roots of \[x^{2022}-9x^{2021}+64x^2+16x+4\] be $t_1,t_2,\dots,t_{2022}$. Compute the value of \[\sum_{1\le i,j\le2022}r_is_j+\sum_{1\le i,j\le2022}s_it_j+\sum_{1\le i,j\le2022}t_ir_j.\]

Tim is participating in the following three math contests. On each contest his score is the number of correct answers. $\bullet$ The Local Area Inspirational Math Exam consists of 15 problems. $\bullet$ The Further Away Regional Math League has 10 problems. $\bullet$ The Distance-Optimized Math Open has 50 problems. For every positive integer $n$, Tim knows the answer to the $n$th problems on each contest (which are pairwise distinct), if they exist; however, these answers have been randomly permuted so that he does not know which answer corresponds to which contest. Unaware of the shuffling, he competes with his modified answers. Compute the expected value of the sum of his scores on all three contests. [i]Proposed by Evan Chen[/i]

Given six consecutive positive integers, prove that there exists a prime such that one and only one of these six integers is divisible by this prime.