Found problems: 85335
Suppose a grid with $2m$ rows and $2n$ columns is given, where $m$ and $n$ are positive integers. You may place one pawn on any square of this grid, except the bottom left one or the top right one. After placing the pawn, a snail wants to undertake a journey on the grid. Starting from the bottom left square, it wants to visit every square exactly once, except the one with the pawn on it, which the snail wants to avoid. Moreover, it wants to fi nish in the top right square. It can only move horizontally or vertically on the grid.
On which squares can you put the pawn for the snail to be able to finish its journey?
Let $x, y, p, n$, and $k$ be positive integers such that $x^n + y^n = p^k$. Prove that if $n > 1$ is odd, and $p$ is an odd prime, then $n$ is a power of $p$.
[i]A. Kovaldji, V. Senderov[/i]
Suppose from $(m+2)\times(n+2)$ rectangle we cut $4$, $1\times1$ corners. Now on first and last row first and last columns we write $2(m+n)$ real numbers. Prove we can fill the interior $m\times n$ rectangle with real numbers that every number is average of it's $4$ neighbors.
Find all real numbers $x$ that satisfy the equation $$\frac{x-2020}{1}+\frac{x-2019}{2}+\cdots+\frac{x-2000}{21}=\frac{x-1}{2020}+\frac{x-2}{2019}+\cdots+\frac{x-21}{2000},$$ and simplify your answer(s) as much as possible. Justify your solution.
An immortal flea jumps on whole points of the number line, beginning with $0$. The length of the first jump is $3$, the second $5$, the third $9$, and so on. The length of $k^{\text{th}}$ jump is equal to $2^k + 1$. The flea decides whether to jump left or right on its own. Is it possible that sooner or later the flee will have been on every natural point, perhaps having visited some of the points more than once?
Let $ a,b,c$ be 3 complex numbers such that \[ a|bc| \plus{} b|ca| \plus{} c|ab| \equal{} 0.\] Prove that \[ |(a\minus{}b)(b\minus{}c)(c\minus{}a)| \geq 3\sqrt 3 |abc|.\]
Let $ABCD$ be a convex quadrilateral with $AB$ is not parallel to $CD$. Circle $\omega_1$ with center $O_1$ passes through $A$ and $B$, and touches segment $CD$ at $P$. Circle $\omega_2$ with center $O_2$ passes through $C$ and $D$, and touches segment $AB$ at $Q$. Let $E$ and $F$ be the intersection of circles $\omega_1$ and $\omega_2$. Prove that $EF$ bisects segment $PQ$ if and only if $BC$ is parallel to $AD$.
For every positive integer $n$ let $\tau (n)$ denote the number of its positive factors. Determine all $n \in N$ that satisfy the equality $\tau (n) = \frac{n}{3}$
An acute triangle $ABC$ is given. Let $D$ be the foot of altitude dropped for $A$. Tangents from $D$ to circles with diameters $AB$ and $AC$ intersects with the said circles at $K$ and $L$, in respective. Point $S$ in the plane is given so that $\angle ABC + \angle ABS = \angle ACB + \angle ACS = 180^\circ$. Prove that $A, K, L$ and $S$ lie on a circle.
Let $ \Psi\equal{}\langle A;...\rangle$ be an arbitrary, countable algebraic structure (that is, $ \Psi$ can have an arbitrary number of finitary operations and relations). Prove that $ \Psi$ has as many as continuum automorphisms if and only if for any finite subset $ A'$ of $ A$ there is an automorphism $ \pi_{A'}$ of $ \Psi$ different from the identity automorphism and such that \[ (x) \pi_{A'}\equal{}x\] for every $ x \in A'$.
[i]M. Makkai[/i]
In the Triangle ABC AB>AC, the extension of the altitude AD with D lying inside BC intersects the circum-circle of the Triangle ABC at P. The circle through P and tangent to BC at D intersects the circum-circle of Triangle ABC at Q distinct from P with PQ=DQ. Prove that AD=BD-DC
The sequence $\{x_n\}$ is given by
$$x_n=\frac14\left(\left(2+\sqrt3\right)^{2n-1}+\left(2-\sqrt3\right)^{2n-1}\right),\qquad n\in\mathbb N.$$Prove that each $x_n$ is equal to the sum of squares of two consecutive integers.
Segments $\overline{AB}, \overline{AC},$ and $\overline{AD}$ are edges of a cube and $\overline{AG}$ is a diagonal through the center of the cube. Point $P$ satisfies $BP=60\sqrt{10}$, $CP=60\sqrt{5}$, $DP=120\sqrt{2}$, and $GP=36\sqrt{7}$. Find $AP.$
Suppose $a$ and $b$ are two positive real numbers such that the roots of the cubic equation $x^3 - ax + b = 0$ are all real. If $\alpha$ is a root of this cubic with minimal absolute value, prove that \[ \dfrac{b}{a} < \alpha < \dfrac{3b}{2a}. \]
Given is a $n\times n$ grid with all squares on one diagonal being forbidden. You are allowed to start from any square, and move one step horizontally, vertically or diagonally. You are not allowed to visit a forbidden square or previously visited square. Your goal is to visit all non forbidden squares. Find, with proof, the minimum number of times you will have to move one step diagonally
Let $ A$ be a closed and bounded set in the plane, and let $ C$ denote the set of points at a unit distance from $ A$. Let $ p \in
C$, and assume that the intersection of $ A$ with the unit circle $ K$ centered at $ p$ can be covered by an arc shorter that a semicircle of $ K$. Prove that the intersection of $ C$ with a suitable neighborhood of $ p$ is a simple arc which $ p$ is not an endpoint.
[i]M. Bognar[/i]
Positive integers $x>1$ and $y$ satisfy an equation $2x^2-1=y^{15}$. Prove that 5 divides $x$.
Construct a tetromino by attaching two $2 \times 1$ dominoes along their longer sides such that the midpoint of the longer side of one domino is a corner of the other domino. This construction yields two kinds of tetrominoes with opposite orientations. Let us call them $S$- and $Z$-tetrominoes, respectively.
Assume that a lattice polygon $P$ can be tiled with $S$-tetrominoes. Prove that no matter how we tile $P$ using only $S$- and $Z$-tetrominoes, we always use an even number of $Z$-tetrominoes.
[i]Proposed by Tamas Fleiner and Peter Pal Pach, Hungary[/i]
Let triangle $ \vartriangle ABC$ have $AB = 17$, $BC = 14$, $CA = 12$. Let $M_A$, $M_B$, $M_C$ be midpoints of $\overline{BC}$, $\overline{AC}$, and $\overline{AB}$ respectively. Let the angle bisectors of $ A$, $ B$, and $C$ intersect $\overline{BC}$, $\overline{AC}$, and $\overline{AB}$ at $P$, $Q$, and $R$, respectively. Reflect $M_A$ about $\overline{AP}$, $M_B$ about $\overline{BQ}$, and $M_C$ about $\overline{CR}$ to obtain $M'_A$, $M'_B$, $M'_C$, respectively. The lines $AM'_A$, $BM'_B$, and $CM'_C$ will then intersect $\overline{BC}$, $\overline{AC}$, and $\overline{AB}$ at $D$, $E$, and $F$, respectively. Given that $\overline{AD}$, $\overline{BE}$, and $\overline{CF}$ concur at a point $K$ inside the triangle, in simplest form, the ratio $[KAB] : [KBC] : [KCA]$ can be written in the form $p : q : r$, where $p$, $q$ and $ r$ are relatively prime positive integers and $[XYZ]$ denotes the area of $\vartriangle XYZ$. Compute $p + q + r$.
A board of $6\times 6$ is totally covered by $18$ dominoes (of $2\times 1$), that is, there are no overlaps, gaps, and the tiles do not come off the board. Prove that, regardless of the arrangement of the tiles, there is always a line that divides the board into two non-empty parts, and without cutting tiles.
Prove that if the sides $ a $, $ b $, $ c $ of a triangle satisfy the inequality
$$a < b < c$$then the angle bisectors $ d_a $, $ d_b $, $ d_c $ of opposite angles satisfy the inequality
$$ d_a > d_b > d_c.$$
Prove that for every natural number $n > 1$ \[ \frac{1}{n+1} \left( 1 + \frac{1}{3} +\frac{1}{5} + \ldots + \frac{1}{2n-1} \right) > \frac{1}{n} \left( \frac{1}{2} + \frac{1}{4} + \ldots + \frac{1}{2n} \right) . \]
Evaluate $\int_{-\pi}^{\pi} \left(\sum_{k=1}^{2013} \sin kx\right)^2dx$.
Five positive consecutive integers starting with $a$ have average $b$. What is the average of $5$ consecutive integers that start with $b$?
${ \textbf{(A)}\ a+3\qquad\textbf{(B)}\ a+4\qquad\textbf{(C)}\ a+5\qquad\textbf{(D)}}\ a+6\qquad\textbf{(E)}\ a+7$
Let $n$ be a positive integer that is not divisible by either $2$ or $5$.
In the decimal expansion of $\frac{1}{n}= 0.a_1a_2a_3\cdots$ a finite number of digits after the decimal point are chosen arbitrarily to be deleted.
Clearly the decimal number obtained by this procedure is also rational, so it's equal to $\frac{a}{b}$ for some integers $a,b$. Prove that $b$ is divisible by $n$.