Is it true, that for all pairs of non-negative integers $a$ and $b$ , the system \begin{align*} \tan{13x} \tan{ay} =& 1 \\ \tan{21x} \tan{by}= & 1 \end{align*} has at least one solution?

Each square of the chessboard was painted in one of eight colors in such a way that the number of squares colored by all the colors are equal. Is it always possible to put $8$ rooks not threatening each other on multi-colored cells?

Given that $x, y,$ and $z$ are positive real numbers such that $$x^{\log_2(yz)}=2^8\cdot3^4, \quad y^{\log_2(zx)}=2^9\cdot3^6, \quad \text{and}\quad z^{\log_2(xy)}=2^5 \cdot 3^{10},$$ compute the smallest possible value of $xyz.$

Let $ABCD$ be a convex quadrilateral and let the diagonals $AC$ and $BD$ intersect at $O$. Let $I_1, I_2, I_3, I_4$ be respectively the incentres of triangles $AOB, BOC, COD, DOA$. Let $J_1, J_2, J_3, J_4$ be respectively the excentres of triangles $AOB, BOC, COD, DOA$ opposite $O$. Show that $I_1, I_2, I_3, I_4$ lie on a circle if and only if $J_1, J_2, J_3, J_4$ lie on a circle.

One side of a given triangle is $ 18$ inches. Inside the triangle a line segment is drawn parallel to this side forming a trapezoid whose area is one-third of that of the triangle. The length of this segment, in inches, is: $ \textbf{(A)}\ 6\sqrt {6} \qquad \textbf{(B)}\ 9\sqrt {2} \qquad \textbf{(C)}\ 12 \qquad \textbf{(D)}\ 6\sqrt {3} \qquad \textbf{(E)}\ 9$

There exists a unique pair of positive integers $k,n$ such that $k$ is divisible by $6$, and $\sum_{i=1}^ki^2=n^2$. Find $(k,n)$.

For a given natural number $k > 1$, find all functions $f:\mathbb{R} \to \mathbb{R}$ such that for all $x, y \in \mathbb{R}$, $f[x^k + f(y)] = y +[f(x)]^k$.

Let $w, x, y, z$ be integers from $0$ to $3$ inclusive. Find the number of ordered quadruples of $(w, x, y, z)$ such that $5x^2 + 5y^2 + 5z^2 - 6wx-6wy -6wz$ is divisible by $4$. [i]Proposed by Timothy Qian[/i]

[asy] //size(100);//local size(200); real r1=2; pair O=(0,0), D=(.5,.5*sqrt(3)), C=(D.x+.5*3,D.y), B, B_prime=endpoint(arc(D, 3, 0,-2)); B=B_prime; path c1=circle(O, r1); pair C=midpoint(D--B_prime); path arc2=arc(B_prime, 6/2, 158.25,250); draw(c1); draw(O--D); draw(D--C); draw(C--B_prime); pair A=beginpoint(arc2); draw(B_prime--A); //dot(O^^D^^C^^A); //dot(B_prime); label("\scriptsize{$O$}",O,.6dir(D--O)); label("\scriptsize{$C$}",C,.5dir(-55)); label("\scriptsize{$D$}", D,.2NW); //label("\scriptsize{$B$}",B,S); label("\scriptsize{$B$}", B_prime, .5*dir(D--B_prime)); label("\scriptsize{$A$}",A,.5dir(NE)); label("\tiny{2}", O--D, .45*LeftSide); label("\tiny{3}", D--C, .45*LeftSide); label("\tiny{6}", B_prime--A, .45*RightSide); label("\tiny{3}", waypoint(C--B_prime,.1), .45*N); //Credit to Klaus-Anton for the diagram[/asy] In the adjoining figure, $AB$ is tangent at $A$ to the circle with center $O$; point $D$ is interior to the circle; and $DB$ intersects the circle at $C$. If $BC=DC=3$, $OD=2$, and $AB=6$, then the radius of the circle is $\textbf{(A) }3+\sqrt{3}\qquad\textbf{(B) }15/\pi\qquad\textbf{(C) }9/2\qquad\textbf{(D) }2\sqrt{6}\qquad \textbf{(E) }\sqrt{22}$

Given two parallel lines $r$ and $r'$ and a point $P$ on the plane that contains them and that is not on them, determine an equilateral triangle whose vertex is point $P$ , and the other two, one on each of the two lines. [img]https://cdn.artofproblemsolving.com/attachments/9/3/1d475eb3e9a8a48f4a85a2a311e1bda978e740.png[/img]

Prove that if a lattice parallelogram contains at most three lattice points in addition to its vertices, then those are on one of the diagonals.

Prove that for any four positive real numbers $ a$, $ b$, $ c$, $ d$ the inequality \[ \frac {(a \minus{} b)(a \minus{} c)}{a \plus{} b \plus{} c} \plus{} \frac {(b \minus{} c)(b \minus{} d)}{b \plus{} c \plus{} d} \plus{} \frac {(c \minus{} d)(c \minus{} a)}{c \plus{} d \plus{} a} \plus{} \frac {(d \minus{} a)(d \minus{} b)}{d \plus{} a \plus{} b}\ge 0\] holds. Determine all cases of equality. [i]Author: Darij Grinberg (Problem Proposal), Christian Reiher (Solution), Germany[/i]

For a positive integer $m$ denote by $S(m)$ and $P(m)$ the sum and product, respectively, of the digits of $m$. Show that for each positive integer $n$, there exist positive integers $a_1, a_2, \ldots, a_n$ satisfying the following conditions: \[ S(a_1) < S(a_2) < \cdots < S(a_n) \text{ and } S(a_i) = P(a_{i+1}) \quad (i=1,2,\ldots,n). \] (We let $a_{n+1} = a_1$.) [i]Problem Committee of the Japan Mathematical Olympiad Foundation[/i]

For a ring $ A, $ and an element $ a $ of it, define $ s_a,d_a:A\longrightarrow A, s_a(x)=ax,d_a=xa.$ [b]a)[/b] Prove that if $ A $ is finite, then $ s_a $ is injective if and only if $ d_a $ is injective. [b]b)[/b] Give example of a ring which has an element $ b $ for which $ s_b $ is injective and $ d_b $ is not, or, conversely, $ s_b $ is not injective, but $ d_b $ is.

When $x$ is added to both the numerator and the denominator of the fraction $a/b, a\neq b, b\neq 0$, the value of the fraction is changed to $c/d$. Then $x$ equals: $\textbf{(A) }\dfrac1{c-d}\qquad \textbf{(B) }\dfrac{ad-bc}{c-d}\qquad \textbf{(C) }\dfrac{ad-bc}{c+d}\qquad \textbf{(D) }\dfrac{bc-ad}{c-d}\qquad \textbf{(E) }\dfrac{bc-ad}{c+d}$

Determine all positive integers $n$ such that $\frac{2^n}{n^2}$ is an integer.

Scalene triangle $ABC$ has incenter $I$ and circumcircle $\Omega$ with center $O$. $H$ is the orthocenter of triangle $BIC$, and $T$ is a point on $\Omega$ for which $\angle ATI=90^\circ$. Circle $(AIO)$ intersects line $IH$ again at $X$. Show that the lines $AX, HT$ intersect on $\Omega$.

Prove that for every non-negative integer $ n, $ there exists a non-negative integer $ m $ such that $$ \left( 1+\sqrt{2} \right)^n=\sqrt m +\sqrt{m+1} . $$

Find all differentiable functions $f: (0,\infty)\mapsto (0,\infty)$ for which there is a positive real number $a$ such that \[ f'\left(\frac ax\right)=\frac x{f(x)} \] for all $x>0.$

Triangle $ABC$ has $AB = 5$, $BC = 7$, and $CA = 8$. New lines not containing but parallel to $AB$, $BC$, and $CA$ are drawn tangent to the incircle of $ABC$. What is the area of the hexagon formed by the sides of the original triangle and the newly drawn lines?

$D_n$ is a set of domino pieces. For each pair of non-negative integers $(a, b)$ with $a \le b \le n$, there is one domino, denoted $[a, b]$ or $[b, a]$ in $D_n$. A [i]ring [/i] is a sequence of dominoes $[a_1, b_1], [a_2, b_2], ... , [a_k, b_k]$ such that $b_1 = a_2, b_2 = a_3, ... , b_{k-1} = a_k$ and $b_k = a_1$. Show that if $n$ is even there is a ring which uses all the pieces. Show that for n odd, at least $(n+1)/2$ pieces are not used in any ring. For $n$ odd, how many different sets of $(n+1)/2$ are there, such that the pieces not in the set can form a ring?

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

Find all primes $q_1, q_2, q_3, q_4, q_5$ such that $q_1^4+q_2^4+q_3^4+q_4^4+q_5^4$ is the product of two consecutive even integers.

Problem 8. For every non-negative integer n, define an n-variable function K_n (x_1,x_2,…,x_n ) as follows: K_0=1 K_1 (x_1 )=〖x_1〗^2 K_(n+2) (x_1,x_2,…,x_(n+2) )=〖x_(n+2)〗^2.K_(n+1) (x_1,x_2,…,x_(n+1) )+(x_(n+2)+x_(n+1))K_n (x_1,x_2,…,x_n ) Prove that: K_n (x_1,x_2,…,x_n )=K_n (x_n,…〖,x〗_2,x_1 )

Determine all functions $f: \mathbb{Q} \to \mathbb{Q}$ such that $$f(2xy + \frac{1}{2}) + f(x-y) = 4f(x)f(y) + \frac{1}{2}$$ for all $x,y \in \mathbb{Q}$.