Determine all triples of integers $(x, y, z)$ that satisfy the equation \[x^z + y^z = z.\]

The bisectors of angles $A$ and $C$ of triangle $ABC$ intersect the circumcircle of this triangle at points $A_0$ and $C_0$, respectively. A straight line passing through the center of the inscribed circle of a triangle $ABC$ is parallel to side $AC$ and intersects line $A_0C_0$ at point $P$. Prove that line $PB$ is tangent to the circumcircle of the triangle $ABC$.

[b]p1.[/b] Two players play the following game. On the lowest left square of an $8\times 8$ chessboard there is a rook. The first player is allowed to move the rook up or to the right by an arbitrary number of squares. The second player is also allowed to move the rook up or to the right by an arbitrary number of squares. Then the first player is allowed to do this again, and so on. The one who moves the rook to the upper right square wins. Who has a winning strategy? [b]p2.[/b] In Crocodile Country there are banknotes of $1$ dollar, $10$ dollars, $100$ dollars, and $1,000$ dollars. Is it possible to get 1,000,000 dollars by using $250,000$ banknotes? [b]p3.[/b] Fifteen positive numbers (not necessarily whole numbers) are placed around the circle. It is known that the sum of every four consecutive numbers is $30$. Prove that each number is less than $15$. [b]p4.[/b] Donald Duck has $100$ sticks, each of which has length $1$ cm or $3$ cm. Prove that he can break into $2$ pieces no more than one stick, after which he can compose a rectangle using all sticks. [b]p5.[/b] Three consecutive $2$ digit numbers are written next to each other. It turns out that the resulting $6$ digit number is divisible by $17$. Find all such numbers. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Prove that $ \frac {( \minus{} 1)^n}{n!}\int_1^2 (\ln x)^n\ dx \equal{} 2\sum_{k \equal{} 1}^n \frac {( \minus{} \ln 2)^k}{k!} \plus{} 1$.

A student council must select a two-person welcoming committee and a three-person planning committee from among its members. There are exactly $10$ ways to select a two-person team for the welcoming committee. It is possible for students to serve on both committees. In how many different ways can a three-person planning committee be selected? $ \textbf{(A)}\ 10\qquad\textbf{(B)}\ 12\qquad\textbf{(C)}\ 15\qquad\textbf{(D)}\ 18\qquad\textbf{(E)}\ 25$

Find the least integer $n$ greater than $345$ such that $\frac{3n+4}{5}, \frac{4n+5}{3},$ and $\frac{5n+3}{4}$ are all integers.

Let $BD$ be an altitude of $\triangle ABC$ with $AB < BC$ and $\angle B > 90^\circ$. Let $M$ be the midpoint of $AC$, and point $K$ be symmetric to point $D$ with respect to point $M$. A perpendicular drawn from point $M$ to the line $BC$ intersects line $AB$ at point $L$. Prove that $\angle MBL = \angle MKL$. [i]Proposed by Oleksandra Yakovenko[/i]

Find all positive integers $r$ with the property that there exists positive prime numbers $p$ and $q$ so that $$p^2 + pq + q^2 = r^2 .$$

An excircle centered at $I_{A}$ touches the side $BC$ of a triangle $ABC$ at point $D$. Prove that the pedal circles of $D$ with respect to the triangles $ABI_{A}$ and $ACI_{A}$ are congruent. Proposed by:K.Belsky

Three different non-zero real numbers $a,b,c$ satisfy the equations $a+\frac{2}{b}=b+\frac{2}{c}=c+\frac{2}{a}=p $, where $p$ is a real number. Prove that $abc+2p=0.$

11.6 Construct for each vertex of the quadrilateral of area $S$ a symmetric point wrt to the diagonal, which doesn't contain this vertex. Let $S'$ be an area of the obtained quadrilateral. Prove that $\frac{S'}{S}<3$. ([i]L. Emel'yanov[/i])

A hunter and an invisible rabbit play a game on an infinite square grid. First the hunter fixes a colouring of the cells with finitely many colours. The rabbit then secretly chooses a cell to start in. Every minute, the rabbit reports the colour of its current cell to the hunter, and then secretly moves to an adjacent cell that it has not visited before (two cells are adjacent if they share an edge). The hunter wins if after some finite time either:[list][*]the rabbit cannot move; or [*]the hunter can determine the cell in which the rabbit started.[/list]Decide whether there exists a winning strategy for the hunter. [i]Proposed by Aron Thomas[/i]

A convex polyhedron has $n$ faces that are all congruent triangles with angles $36^{\circ}, 72^{\circ}$, and $72^{\circ}$. Determine, with proof, the maximum possible value of $n$.

i) If $b^2+c^2=a^2, \,\,\,\, b\ne \pm c$ , calculate the expression $\frac{b^3+c^3}{b+c}+\frac{b^3-c^3}{b-c}$. ii) If $a+\frac{1}{a}=k, a\ne 0$, find the expression $a^4+\frac{1}{a^4}$ in terms of $k$.

Determine all integers $n>1$ whose positive divisors add up to a power of $3.$ [i]Andrei Bâra[/i]

Let $f:\mathbb{R}\to (0;+\infty )$ be a continuous function such that $\underset{x\to -\infty }{\mathop{\lim }}\,f(x)=\underset{x\to +\infty }{\mathop{\lim }}\,f(x)=0.$ a) Prove that $f(x)$ has the maximum value on $\mathbb{R}.$ b) Prove that there exist two sequeneces $({{x}_{n}}),({{y}_{n}})$ with ${{x}_{n}}<{{y}_{n}},\forall n=1,2,3,...$ such that they have the same limit when $n$ tends to infinity and $f({{x}_{n}})=f({{y}_{n}})$ for all $n.$

Let $f$ be any function that maps the set of real numbers into the set of real numbers. Prove that there exist real numbers $x$ and $y$ such that \[f\left(x-f(y)\right)>yf(x)+x\] [i]Proposed by Igor Voronovich, Belarus[/i]

The figure $ABCD$ is bounded by a semicircle $CDA$ and a quarter circle $ABC$. Given that the distance from $A$ to $C$ is $18$, find the area of the figure. [asy] size(200); defaultpen(linewidth(0.8)); pair A=(-9,0),B=(0,9*sqrt(2)-9),C=(9,0),D=(0,9); dot(A^^B^^C^^D); draw(arc(origin,9,0,180)^^arc((0,-9),9*sqrt(2),45,135)); label("$A$",A,S); label("$B$",B,N); label("$C$",C,S); label("$D$",D,N); [/asy]

Initially, the numbers $1$ and $2$ are written on the board. A move consists of choosing a positive real number $x$ and replacing $(a,b)$ on the board by $\left(a+\frac{x}{b},b+\frac{x}{a}\right)$. Is it possible to create in finitely many moves a situation where the numbers on the board are $2$ and $3$?

Given a $2n \times 2m$ table $(m,n \in \mathbb{N})$ with one of two signs ”+” or ”-” in each of its cells. A union of all the cells of some row and some column is called a cross. The cell on the intersectin of this row and this column is called the center of the cross. The following procedure we call a transformation of the table: we mark all cells which contain ”−” and then, in turn, we replace the signs in all cells of the crosses which centers are marked by the opposite signs. (It is easy to see that the order of the choice of the crosses doesn’t matter.) We call a table attainable if it can be obtained from some table applying such transformations one time. Find the number of all attainable tables.

For every integer $ a>1$ an infinite list of integers is constructed $ L(a)$, as follows: [list] $ a$ is the first number in the list $ L(a)$.[/list] [list] Given a number $ b$ in $ L(a)$, the next number in the list is $ b\plus{}c$, where $ c$ is the largest integer that divides $ b$ and is smaller than $ b$.[/list] Find all the integers $ a>1$ such that $ 2002$ is in the list $ L(a)$.

The expression $\sin2^\circ\sin4^\circ\sin6^\circ\cdots\sin90^\circ$ is equal to $p\sqrt{5}/2^{50}$, where $p$ is an integer. Find $p$.

Jonathan and Justin each flip a coin eight times. Jonathan and Justin get $m, n$ heads respectively. What is the probability that the difference of that $|m-n| \equiv 0 $ mod $4$? [i]Lightning 3.4[/i]

Squares of an $n \times n$ table ($n \ge 3$) are painted black and white as in a chessboard. A move allows one to choose any $2 \times 2$ square and change all of its squares to the opposite color. Find all such n that there is a finite number of the moves described after which all squares are the same color.

Jamal wants to store $ 30$ computer files on floppy disks, each of which has a capacity of $ 1.44$ megabytes (MB). Three of his files require $ 0.8$ MB of memory each, $ 12$ more require $ 0.7$ MB each, and the remaining $ 15$ require $ 0.4$ MB each. No file can be split between floppy disks. What is the minimal number of floppy disks that will hold all the files? $ \textbf{(A)}\ 12 \qquad \textbf{(B)}\ 13 \qquad \textbf{(C)}\ 14 \qquad \textbf{(D)}\ 15 \qquad \textbf{(E)}\ 16$