For a positive integer $n$, let $s(n)$ be the sum of the digits of $n$. If $n$ is a two-digit positive integer such that $\frac{n}{s(n)}$ is a multiple of $3$, compute the sum of all possible values of $n$.

Given isosceles tetrahedron $PABC$ (faces are equal triangles). Let $A_0$, $B_0$ and $C_0$ be the touchpoints of the circle inscribed in the triangle $ABC$ with sides $BC$, $AC$ and $AB$ respectively, $A_1$, $B_1$ and $C_1$ are the touchpoints of the excircles of triangles $PCA$, $PAB$ and $PBC$ with extensions of sides $PA$, $PB$ and $PC$, respectively (beyond points $A$, $B$, $C$). Prove that the lines $A_0A_1$, $B_0B_1$ and $C_0C_1$ intersect at one point.

Find all positive integers $x , y$ which are roots of the equation $2 x^y-y= 2005$ [u] Babis[/u]

For each positive integer $n$, consider the highest common factor $h_n$ of the two numbers $n!+1$ and $(n+1)!$. For $n<100$, find the largest value of $h_n$.

Circles $\omega_1$ and $\omega_2$ intersect at points $A$ and $B$, and $M$ is the midpoint of $AB$. Points $S_1$ and $S_2$ lie on the line $AB$ (but not between $A$ and $B$). The tangents drawn from $S_1$ to $\omega_1$ touch it at $X_1$ and $Y_1$, and the tangents drawn from $S_2$ to $\omega_2$ touch it at $X_2$ and $Y_2$. Prove that if the line $X_1X_2$ passes through $M$, then line $Y_1Y_2$ also passes through $M$.

Let $r$ be a natural number greater than $1$. Then there exist positive irrational numbers $x, y$ such that $x^y = r$ . Prove it.

Compute \[\sum_{k=0}^{100}\left\lfloor\dfrac{2^{100}}{2^{50}+2^k}\right\rfloor.\] (Here, if $x$ is a real number, then $\lfloor x\rfloor$ denotes the largest integer less than or equal to $x$.)

For real numbers $ a$ and $ b$, define $ a\$b\equal{}(a\minus{}b)^2$. What is $ (x\minus{}y)^2\$(y\minus{}x)^2$? $ \textbf{(A)}\ 0 \qquad \textbf{(B)}\ x^2\plus{}y^2 \qquad \textbf{(C)}\ 2x^2 \qquad \textbf{(D)}\ 2y^2 \qquad \textbf{(E)}\ 4xy$

We say that a set $S$ of integers is [i]rootiful[/i] if, for any positive integer $n$ and any $a_0, a_1, \cdots, a_n \in S$, all integer roots of the polynomial $a_0+a_1x+\cdots+a_nx^n$ are also in $S$. Find all rootiful sets of integers that contain all numbers of the form $2^a - 2^b$ for positive integers $a$ and $b$.

Ellina has twelve blocks, two each of red $\left({\bf R}\right),$ blue $\left({\bf B}\right),$ yellow $\left({\bf Y}\right),$ green $\left({\bf G}\right),$ orange $\left({\bf O}\right),$ and purple $\left({\bf P}\right).$ Call an arrangement of blocks [i]even[/i] if there is an even number of blocks between each pair of blocks of the same color. For example, the arrangement $$ {\text {\bf R B B Y G G Y R O P P O}}$$is even. Ellina arranges her blocks in a row in random order. The probability that her arrangement is even is $\frac mn$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

A point $M$ lies on the side $AC$ of a triangle $ABC$. The circle $\gamma$ with the diameter $BM$ intersects the lines $AB$ and $BC$ at $P$ and $Q$, respectively. Find the locus of the intersection point of the tangents to $\gamma$ at $P$ and $Q$ when point $M$ varies.

Let $ABC$ be a triangle inscribed on a circle with center $O$. Let $D$ and $E$ be points on the sides $AB$ and $BC$,respectively, such that $AD = DE = EC$. Let $X$ be the intersection of the angle bisectors of $\angle ADE$ and $\angle DEC$. If $X \neq O$, show that, the lines $OX$ and $DE$ are perpendicular.

Determine the number of three digit integers that are equal to $19$ times the sum of its digits.

Consider any sequence of real numbers $a_0$, $a_1$, $\cdots$. If, for all pairs of nonnegative integers $(m, s)$, there exists some integer $n \in [m+1, m+2024(s+1)]$ satisfying $a_m+a_{m+1}+\cdots+a_{m+s}=a_n+a_{n+1}+\cdots+a_{n+s}$, say that this sequence has [i]repeating sums[/i]. Is a sequence with repeating sums always eventually periodic?

Let $S$ be a square with sides length $100$. Let $L$ be a path within $S$ which does not meet itself and which is composed of line segments $A_0A_1,A_1A_2,A_2A_3,\ldots,A_{n-1}A_n$ with $A_0=A_n$. Suppose that for every point $P$ on the boundary of $S$ there is a point of $L$ at a distance from $P$ no greater than $\frac {1} {2}$. Prove that there are two points $X$ and $Y$ of $L$ such that the distance between $X$ and $Y$ is not greater than $1$ and the length of the part of $L$ which lies between $X$ and $Y$ is not smaller than $198$.

Let $\Omega$ be the circumcircle of cyclic quadrilateral $ABCD$. Consider such pairs of points $P$, $Q$ of diagonal $AC$ that the rays $BP$ and $BQ$ are symmetric with respect the bisector of angle $B$. Find the locus of circumcenters of triangles $PDQ$.

Find all integers $\,a,b,c\,$ with $\,1<a<b<c\,$ such that \[ (a-1)(b-1)(c-1) \] is a divisor of $abc-1.$

A sequence of seven digits is randomly chosen in a weekly lottery. Every digit can be any of the digits $0, 1, 2, 3, 4, 5, 6, 7, 8, 9.$ What is the probability of having at most five different digits in the sequence?

Find integer solutions of $x^3+y^3-2xy+x+y+2=0$

Consider $\vartriangle ABC$ such that $CA + AB = 3BC$. Let the incircle $\omega$ touch segments $\overline{CA}$ and $\overline{AB}$ at $E$ and $F$, respectively, and define $P$ and $Q$ such that segments $\overline{P E}$ and $\overline{QF}$ are diameters of $\omega$. Define the function $D$ of a point $K$ to be the sum of the distances from $K$ to $P$ and $Q$ (i.e. $D(K) = KP + KQ$). Let $W, X, Y$ , and $Z$ be points chosen on lines $\overleftrightarrow {BC}$, $\overleftrightarrow {CE}$, $\overleftrightarrow {EF}$, and $\overleftrightarrow {F B}$, respectively. Given that $BC =\sqrt{133}$ and the inradius of $\vartriangle ABC$ is $\sqrt{14}$, compute the minimum value of $D(W) + D(X) + D(Y ) + D(Z)$.

Given the rays $ OP$ and $OQ$.Inside the smaller angle $POQ$ selected points $M$ and $N$, such that $\angle POM=\angle QON $ and $\angle POM<\angle PON $ The circle, which concern the rays $OP$ and $ON$, intersects the second circle, which concern the rays $OM$ and $OQ$ at the points $B$ and $C$. Prove that$\angle POC=\angle QOB $

A sequence of integers fsng is defined as follows: fix integers $a$, $b$, $c$, and $d$, then set $s_1 = a$, $s_2 = b$, and $$s_n = cs_{n-1} + ds_{n-2}$$ for all $n \ge 3$. Create a second sequence $\{t_n\}$ by defining each $t_n$ to be the remainder when $s_n$ is divided by $2018$ (so we always have $0 \le t_n \le 2017$). Let $N = (2018^2)!$. Prove that $t_N = t_{2N}$ regardless of the choices of $a$, $b$, $c$, and $d$.

The diagram below shows a $1\times2\times10$ duct with $2\times2\times2$ cubes attached to each end. The resulting object is empty, but the entire surface is solid sheet metal. A spider walks along the inside of the duct between the two marked corners. There are positive integers $m$ and $n$ so that the shortest path the spider could take has length $\sqrt{m}+\sqrt{n}$. Find $m + n$. [asy] size(150); defaultpen(linewidth(1)); draw(origin--(43,0)--(61,20)--(18,20)--cycle--(0,-43)--(43,-43)--(43,0)^^(43,-43)--(61,-23)--(61,20)); draw((43,-43)--(133,57)--(90,57)--extension((90,57),(0,-43),(61,20),(18,20))); draw((0,-43)--(0,-65)--(43,-65)--(43,-43)^^(43,-65)--(133,35)--(133,57)); draw((133,35)--(133,5)--(119.5,-10)--(119.5,20)^^(119.5,-10)--extension((119.5,-10),(100,-10),(43,-65),(133,35))); dot(origin^^(133,5)); [/asy]

We call a positive integer $\textit{cheesy}$ if we can obtain the average of the digits in its decimal representation by putting a decimal separator after the leftmost digit. Prove that there are only finitely many $\textit{cheesy}$ numbers.

Find all pairs of positive integers $(m,n)$ for which there exist relatively prime integers $a$ and $b$ greater than $1$ such that $$\frac{a^m+b^m}{a^n+b^n}$$ is an integer.