A team of geologists on a field expedition have taken with them $80$ tin cans of provisions. The $80$ cans have different weights, which are known (there is a list). After a while the names of the contents of the cans have become illegible. The cook knows what is in each can and claims that he can prove it without opening any can and only using the list and a balance which indicates the difference of weight of the objects placed on its two pans. Show that in order to do so, (a) four weight measurements will be enough, (b) three will not (AK Tolpygo)

Find all quadruples $(a, b, c, d)$ of non-negative integers such that $ab =2(1 + cd)$ and there exists a non-degenerate triangle with sides of length $a - c$, $b - d$, and $c + d$.

Given a set $ M$ of points $ (x,y)$ with integral coordinates satisfying $ x^2 + y^2\leq 10^{10}$. Two players play a game. One of them marks a point on his first move. After this, on each move the moving player marks a point, which is not yet marked and joins it with the previous marked point. Players are not allowed to mark a point symmetrical to the one just chosen. So, they draw a broken line. The requirement is that lengths of edges of this broken line must strictly increase. The player, which can not make a move, loses. Who have a winning strategy?

Let $r > 1$ be a real number, and let $n$ be the largest integer smaller than $r$. Consider an arbitrary real number $x$ with $0 \leq x \leq \frac{n}{r-1}.$ By a [i]base-$r$ expansion[/i] of $x$ we mean a representation of $x$ in the form \[x=\frac{a_1}{r} + \frac{a_2}{r^2}+\frac{a_3}{r^3}+\cdots\] where the $a_i$ are integers with $0 \leq a_i < r.$ You may assume without proof that every number $x$ with $0 \leq x \leq \frac{n}{r-1}$ has at least one [i]base-$r$ expansion[/i]. Prove that if $r$ is not an integer, then there exists a number $p$, $0 \leq p \leq \frac{n}{r-1}$, which has infinitely many distinct [i]base-$r$ expansions[/i].

Suppose that $\cos\left(3x\right)+3\cos\left(x\right)=-2$. What is the value of $\cos\left(2x\right)$? $\text{(A) }-\frac{1}{2}\qquad\text{(B) }-\frac{1}{\sqrt[3]{2}}\qquad\text{(C) }\frac{1}{\sqrt[3]{2}}\qquad\text{(D) }\sqrt[3]{2}-1\qquad\text{(E) }\frac{1}{2}$

Let $AB,AC$ and $AD$ be the edges of a cube, $AB=\alpha$. Point $E$ was marked on the ray $AC$ so that $AE=\lambda \alpha$, and point $F$ was marked on the ray $AD$ so that $AF=\mu \alpha$ ($\mu> 0, \lambda >0$). Find (characterize) pairs of numbers $\lambda$ and $\mu$ such that the cross-sectional area of ​​a cube by any plane parallel to the plane $BCD$ is equal to the cross-sectional area of ​​the tetrahedron $ABEF$ by the same plane.

Given is the equation $(m, n) +[m, n] =m+n$ where $m, n$ are positive integers and m>n. a) Prove that n divides m. b) If $m-n=10$, solve the equation.

Let $m,n\geq2$ be positive integers, and let $a_1,a_2,\ldots ,a_n$ be integers, none of which is a multiple of $m^{n-1}$. Show that there exist integers $e_1,e_2,\ldots,e_n$, not all zero, with $\left|{\,e}_i\,\right|<m$ for all $i$, such that $e_1a_1+e_2a_2+\,\ldots\,+e_na_n$ is a multiple of $m^n$.

Let $d_i$ be the first decimal digit of $2^i$ for every non-negative integer $i$. Prove that for each positive integer $n$ there exists a decimal digit other than $0$ which can be found in the sequence $d_0, d_1, \dots, d_{n-1}$ strictly less than $\frac{n}{17}$ times.

Consider a rhombus $ABCD$ with center $O$. A point $P$ is given inside the rhombus, but not situated on the diagonals. Let $M,N,Q,R$ be the projections of $P$ on the sides $(AB), (BC), (CD), (DA)$, respectively. The perpendicular bisectors of the segments $MN$ and $RQ$ meet at $S$ and the perpendicular bisectors of the segments $NQ$ and $MR$ meet at $T$. Prove that $P, S, T$ and $O$ are the vertices of a rectangle.

See all the problems from 5-th Kyiv math festival [url=http://www.mathlinks.ro/Forum/viewtopic.php?p=506789#p506789]here[/url] Let $a, b, c, d$ be positive integers and $p$ be prime number such that $a^2+b^2=p$ and $c^2+d^2$ is divisible by $p.$ Prove that there exist positive integers $e$ and $f$ such that $e^2+f^2=\frac{c^2+d^2}{p}.$

Let $ U\subset\mathbb R^n$ be an open set, and let $ L: U\times\mathbb R^n\to\mathbb R$ be a continuous, in its second variable first order positive homogeneous, positive over $ U\times (\mathbb R^n\setminus\{0\})$ and of $ C^2$-class Langrange function, such that for all $ p\in U$ the Gauss-curvature of the hyper surface \[ \{ v\in\mathbb R^n \mid L(p,v) \equal{} 1 \}\] is nowhere zero. Determine the extremals of $ L$ if it satisfies the following system \[ \sum_{k \equal{} 1}^n y^k\partial_k\partial_{n \plus{} i}L \equal{} \sum_{k \equal{} 1}^n y^k\partial_i\partial_{n \plus{} k} L \qquad (i\in\{1,\dots,n\})\] of partial differetial equations, where $ y^k(u,v) : \equal{} v^k$ for $ (u,v)\in U\times\mathbb R^k$, $ v \equal{} (v^1,\dots,v^k)$.

Find all triples $(x,y,z)$ of positive integers such that $x \leq y \leq z$ and \[x^3(y^3+z^3)=2012(xyz+2).\]

Alireza multiplied a billion consecutive natural numbers, and Matin multiplied two million consecutive natural numbers. Prove that these two got different results or one of them has made a mistake.

Points $P$ and $Q$ are located inside square $ABCD$ such that $DP$ is parallel to $QB$ and $DP = QB = PQ$. Determine the minimum possible value of $\angle ADP$. [img]https://cdn.artofproblemsolving.com/attachments/c/c/be341e829c7a2663ba2b2329036946012486d7.jpg[/img]

Let $ E$ be the intersection of the diagonals of the convex quadrilateral $ ABCD$. The perimeters of $ \triangle AEB$, $ \triangle BEC$, $ \triangle CED$, and $ \triangle DEA$ are all same. If inradii of $ \triangle AEB$, $ \triangle BEC$, $ \triangle CED$ are $ 3,4,6$, respectively, then inradius of $ \triangle DEA$ will be ? $\textbf{(A)}\ \frac {9}{2} \qquad\textbf{(B)}\ \frac {7}{2} \qquad\textbf{(C)}\ \frac {13}{3} \qquad\textbf{(D)}\ 5 \qquad\textbf{(E)}\ \text{None}$

Let $\{x_i\}^{\infty}_{i=1}$ and $\{y_i\}^{\infty}_{i=1}$ be sequences of real numbers such that $x_1=y_1=\sqrt{3}$, $$x_{n+1}=x_n+\sqrt{1+x_n^2}\quad\text{and}\quad y_{n+1}=\frac{y_n}{1+\sqrt{1+y_n^2}}$$ for all $n\geq 1$. Prove that $2<x_ny_n<3$ for all $n>1$.

Prove that for arbitrary positive numbers the following inequality holds \[\frac{1}{a} + \frac{1}{b} + \frac{1}{c} \leq \frac{a^8 + b^8 + c^8}{a^3b^3c^3}.\]

Find all $ f: \mathbb{N}\to\mathbb{N}$ so that : $ f(n) \plus{} f(n \plus{} 1) \equal{} f(n \plus{} 2)f(n \plus{} 3) \minus{} 1996$

What is the largest possible area of an isosceles trapezoid in which the largest side is $13$ and the perimeter is $28$? $ \textbf{(A)}\ 13 \qquad\textbf{(B)}\ 24 \qquad\textbf{(C)}\ 27 \qquad\textbf{(D)}\ 28 \qquad\textbf{(E)}\ 30 $

Prove the identity \[\sum\limits_{k=0}^n\binom{n}{k}\left(\tan\frac{x}{2}\right)^{2k}\left(1+\frac{2^k}{\left(1-\tan^2\frac{x}{2}\right)^k}\right)=\sec^{2n}\frac{x}{2}+\sec^n x\] for any natural number $n$ and any angle $x.$

Let $a_n$ be a sequence of natural numbers defined by $a_1 = m$ and for $n > 1$. We call apair$ (a_k, a_{\ell })$ [i]interesting [/i] if (i) $0 < \ell - k < 2016$, (ii) $a_k$ divides $a_{\ell }$. Show that there exists a $m$ such that the sequence $a_n$ contains no interesting pair.

. Find all triplets $(x, y, z)$, $x > y > z$ of positive integers such that $\frac{1}{x}+\frac{2}{y}+\frac{3}{z}= 1$

Adams plans a profit of $ 10\%$ on the selling price of an article and his expenses are $ 15\%$ of sales. The rate of markup on an article that sells for $ \$5.00$ is: $ \textbf{(A)}\ 20\% \qquad\textbf{(B)}\ 25\% \qquad\textbf{(C)}\ 30\% \qquad\textbf{(D)}\ 33\frac {1}{3}\% \qquad\textbf{(E)}\ 35\%$

To two circles $r_1$ and $r_2$, intersecting at points $A$ and $B$, their common tangent $CD$ is drawn ($C$ and $D$ are tangency points, respectively, point $B$ is closer to line $CB$ than $A$). Line passing through $A$ , intersects $r_1$ and $r_2$ for second time at points $K$ and $L$, respectively ($A$ lies between $K$ and $L$). Lines $KC$ and $LD$ intersect at point $P$. Prove that $PB$ is the symmedian of triangle $KPL$. (Yu. Blinkov)