Define a sequence ${(n_k)_{k\ge 0}}$ by ${n_{0 }= n_{1} = 1}$, and ${n_{2k} = n_k + n_{k-1} }$ and ${n_{2k+1} = n_k}$ for ${k \ge 1}$. Let further ${q_k = n_k }$ / ${ n_{k-1} }$ for each ${k \ge 1}$. Show that every positive rational number is present exactly once in the sequence ${(q_k)_{k\ge 1}}$

In the square $ABCD$, points $E$ and $F$ were chosen on the sides $AB$ and $BC$ respectively, such that $BE=BF$. Let $L$ be midpoint of $EF$, $N$ be midpoint of $DF$, $O$ be the center of the square and $K=AL \cap DF$ (look at picture). Prove that points $C, K, L, O, N$ are lies on one circle. [i]A. Paleev[/i]

Determine all rational numbers $a$ for which the matrix $$\begin{pmatrix} a & -a & -1 & 0 \\ a & -a & 0 & -1 \\ 1 & 0 & a & -a\\ 0 & 1 & a & -a \end{pmatrix}$$ is the square of a matrix with all rational entries. [i]Proposed by Daniël Kroes, University of California, San Diego[/i]

Suppose that for a prime number $p$ and integers $a,b,c$ the following holds: \[6\mid p+1,\quad p\mid a+b+c,\quad p\mid a^4+b^4+c^4.\] Prove that $p\mid a,b,c$.

[b]p1.[/b] The length of the first side of a triangle is $1$, the length of the second side is $11$, and the length of the third side is an integer. Find that integer. [b]p2.[/b] Suppose $a, b$, and $c$ are positive numbers such that $a + b + c = 1$. Prove that $ab + ac + bc \le \frac13$. [b]p3.[/b] Prove that $1 +\frac12+\frac13+\frac14+ ... +\frac{1}{100}$ is not an integer. [b]p4.[/b] Find all of the four-digit numbers n such that the last four digits of $n^2$ coincide with the digits of $n$. [b]p5.[/b] (Bonus) Several ants are crawling along a circle with equal constant velocities (not necessarily in the same direction). If two ants collide, both immediately reverse direction and crawl with the same velocity. Prove that, no matter how many ants and what their initial positions are, they will, at some time, all simultaneously return to the initial positions. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Prove that for every positive integer $n$, there exist $n$ positive integers such that the sum of them is a perfect square and the product of them is a perfect cube.

If the tangents of all interior angles of a triangle are integers, what is the sum of these integers? $ \textbf{(A)}\ 4 \qquad\textbf{(B)}\ 5 \qquad\textbf{(C)}\ 6 \qquad\textbf{(D)}\ 9 \qquad\textbf{(E)}\ \text{None of above} $

Let ${ABC}$ be a triangle, ${k}$ its incircle and ${k_a,k_b,k_c}$ three circles orthogonal to ${k}$ passing through ${B}$ and ${C, A}$ and ${C}$ , and ${A}$ and ${B}$ respectively. The circles ${k_a,k_b}$ meet again in ${C'}$ ; in the same way we obtain the points ${B'}$ and ${A'}$ . Prove that the radius of the circumcircle of ${A'B'C'}$ is half the radius of ${k}$ .

Find all pairs $(x, y)$ of integers satisfying the following condition: each of the numbers $x^3 + y$ and $x + y^3$ is divisible by $x^2 + y^2$ . (S Zlobin)

Joe makes two cubes of sidelengths $9$ and $10$ from $1729$ randomly oriented and randomly arranged unit cubes, which are initially unpainted. These cubes are dipped into white paint. Then two cubes of sidelengths $1$ and $12$ are formed from the same unit cubes, again randomly oriented and randomly arranged, and these cubes are dipped into paint remover. Joe continues to alternately dip cubes of sides $9$ and $10$ into paint and cubes of sides $1$ and $12$ into paint remover ad nauseam. What is the limit of the expected number of painted unit cube faces immediately after dipping in paint remover?

For which $n{}$ is it possible that a product of $n{}$ consecutive positive integers is equal to a sum of $n{}$ consecutive (not necessarily the same) positive integers? [i]Boris Frenkin[/i]

Fix an odd integer $n > 1$. For a permutation $p$ of the set $\{1,2,...,n\}$, let S be the number of pairs of indices $(i, j)$, $1 \le i \le j \le n$, for which $p_i +p_{i+1} +...+p_j$ is divisible by $n$. Determine the maximum possible value of $S$. Croatia

Let $L$ be the common point of the symmedians of triangle $ABC$, and $BH$ be its altitude. It is known that $\angle ALH = 180^o -2\angle A$. Prove that $\angle CLH = 180^o - 2\angle C$.

Alexandre and Bernado are playing the following game. At the beginning, there are $n$ balls in a bag. At first turn, Alexandre can take one ball from the bag; at second turn, Bernado can take one or two balls from the bag, and so on. So they take turns and in $k$ turn, they can take a number of balls from $1$ to $k$. Wins the one who makes the bag empty. For each value of $n$, find who has the winning strategy.

Tessa the hyper-ant is at the origin of the four-dimensional Euclidean space $\mathbb R^4$. For each step she moves to another lattice point that is $2$ units away from the point she is currently on. How many ways can she return to the origin for the first time after exactly $6$ steps? [i]Proposed by Yannick Yao

Consider the continuous functions $ f:[0,\infty )\longrightarrow\mathbb{R}, g: [0,1]\longrightarrow\mathbb{R} , $ where $ f $ has a finite limit at $ \infty . $ Show that: $$ \lim_{n \to \infty} \frac{1}{n}\int_0^n f(x) g\left( \frac{x}{n} \right) dx =\int_0^1 g(x)dx\cdot\lim_{x\to\infty} f(x) . $$

Let $S$ be a set with six elements. In how many different ways can one select two not necessarily distinct subsets of $S$ so that the union of the two subsets is $S$? The order of selection does not matter; for example, the pair of subsets $\{a, c\}$, $\{b, c, d, e, f\}$ represents the same selection as the pair $\{b, c, d, e, f\}$, $\{a, c\}$.

Solve the equation in the set of natural numbers $1+x^z + y^z = LCM(x^z,y^z)$

A brick has the shape of a cube of size $2$ with one corner unit cube removed. Given a cube of side $2^{n}$ divided into unit cubes from which an arbitrary unit cube is removed, show that the remaining figure can be built using the described bricks.

Let $p_1, p_2, \ldots, p_{42}$ be $42$ pairwise distinct prime numbers. Show that the sum \[\sum_{j=1}^{42}\frac{1}{p_j^2+1}\] is not a unit fraction $\frac{1}{n^2}$ of some integer square number.

In the triangle $ABC$, variable points $D, E, F$ are on the sides[lines] $BC, CA, AB$ respectively so the triangle $DFE$ is similar to the triangle $ABC$ in this order. Circumcircles of $BDF$ and $CDE$ intersect respectively the circumcircle of $ABC$ at $P$ and $Q$ for the second time. Prove that the circumcircle of $DPQ$ passes through a fixed point.

Medians $AM$ and $BE$ of a triangle $ABC$ intersect at $O$. The points $O, M, E, C$ lie on one circle. Find the length of $AB$ if $BE = AM =3$.

Segment $AB$ is both a diameter of a circle of radius 1 and a side of an equilateral triangle $ABC$. The circle also intersects $AC$ and $BD$ at points $D$ and $E$, respectively. The length of $AE$ is $\displaystyle \text{(A)} \ \frac{3}{2} \qquad \text{(B)} \ \frac{5}{3} \qquad \text{(C)} \ \frac{\sqrt 3}{2} \qquad \text{(D)} \ \sqrt{3} \qquad \text{(E)} \ \frac{2 + \sqrt 3}{2}$

Let $ABC$ be an acute triangle with $AB<AC$. Let $D, E,$ and $F$ be the feet of the perpendiculars from $A, B,$ and $C$ to the opposite sides, respectively. Let $P$ be the foot of the perpendicular from $F$ to line $DE$. Line $FP$ and the circumcircle of triangle $BDF$ meet again at $Q$. Show that $\angle PBQ = \angle PAD$.

An segment $S$ of length $50$ is covered by several segments of length $1$ , all of them contained in $S$. If any of these unit segments were removed, $S$ would no longer be completely covered. Find the maximum number of unit segments with this property. Clarification: Assume that the segments include their endpoints.