Found problems: 85335
2005 Sharygin Geometry Olympiad, 23
Envelop the cube in one layer with five convex pentagons of equal areas.
2020-2021 Winter SDPC, #3
Let $ABCD$ be a quadrilateral, let $P$ be the intersection of $AB$ and $CD$, and let $O$ be the intersection of the perpendicular bisectors of $AB$ and $CD$. Suppose that $O$ does not lie on line $AB$ and $O$ does not lie on line $CD$. Let $B'$ and $D'$ be the reflections of $B$ and $D$ across $OP$. Show that if $AB'$ and $CD'$ meet on $OP$, then $ABCD$ is cyclic.
1950 AMC 12/AHSME, 14
For the simultaneous equations
\[ 2x\minus{}3y\equal{}8\]
\[ 6y\minus{}4x\equal{}9\]
$\textbf{(A)}\ x=4,y=0 \qquad
\textbf{(B)}\ x=0,y=\dfrac{3}{2}\qquad
\textbf{(C)}\ x=0,y=0 \qquad\\
\textbf{(D)}\ \text{There is no solution} \qquad
\textbf{(E)}\ \text{There are an infinite number of solutions}$
JOM 2014, 2.
In ZS Chess, an Ivanight attacks like a knight, except that if the attacked square is out of range, it goes
through the edge and comes out from the other side of the board, and attacks that square instead. The
ZS chessboard is an $8 \times 8$ board, where cells are coloured with $n$ distinct colours, where $n$ is a natural
number, such that a Ivanight placed on any square attacks $ 8 $ squares that consist of all $n$ colours, and
the colours appear equally many times in those $ 8 $ squares. For which values of $n$ does such a ZS chess
board exist?
2017 CMIMC Combinatorics, 10
Ryan stands on the bottom-left square of a 2017 by 2017 grid of squares, where each square is colored either black, gray, or white according to the pattern as depicted to the right. Each second he moves either one square up, one square to the right, or both one up and to the right, selecting between these three options uniformly and independently. Noting that he begins on a black square, find the probability that Ryan is still on a black square after 2017 seconds.
[center][img]http://i.imgur.com/WNp59XW.png[/img][/center]
1994 Poland - First Round, 12
The sequence $(x_n)$ is given by
$x_1=\frac{1}{2},$ $x_n=\frac{2n-3}{2n} \cdot x_{n-1}$ for $n=2,3,... .$
Prove that for all natural numbers $n \geq 1$ the following inequality holds
$x_1+x_2+...+x_n < 1$.
2017-2018 SDPC, 7
Let $n > 1$ be a fixed integer. On an infinite row of squares, there are $n$ stones on square $1$ and no stones on squares $2$, $3$, $4$, $\ldots$. Curious George plays a game in which a [i]move[/i] consists of taking two adjacent piles of sizes $a$ and $b$, where $a-b$ is a nonzero even integer, and transferring stones to equalize the piles (so that both piles have $\frac{a+b}{2}$ stones). The game ends when no more moves can be made. George wants to analyze the number of moves it takes to end the game.
(a) Suppose George wants to end the game as quickly as possible. How many moves will it take him?
(b) Suppose George wants to end the game as slowly as possible. Show that for all $n > 2$, the game will end after at most $\frac{3}{16}n^2$ moves.
[i]Scoring note:[/i] For part (b), partial credit will be awarded for correct proofs of weaker bounds, eg. $\frac{1}{4}n^2$, $n^k$, or $k^n$ (for some $k \geq 2$).
2002 Miklós Schweitzer, 4
For a given natural number $n$, consider those sets $A\subseteq \mathbb{Z}_n$ for which the equation $xy=uv$ has no other solution in the residual classes $x,y,u,v\in A$ than the trivial solutions $x=u$, $y=v$ and $x=v$, $y=u$. Let $g(n)$ be the maximum of the size of such sets $A$. Prove that
$$\limsup_{n\to\infty}\frac{g(n)}{\sqrt{n}}=1$$
2000 Moldova National Olympiad, Problem 6
Let $(a_n)_{n\ge0}$ be a sequence of positive numbers that satisfy the relations $a_{i-1}a_{i+1}\le a_i^2$ for all $i\in\mathbb N$. For any integer $n>1$, prove the inequality
$$\frac{a_0+\ldots+a_n}{n+1}\cdot\frac{a_1+\ldots+a_{n-1}}{n-1}\ge\frac{a_0+\ldots+a_{n-1}}n\cdot\frac{a_1+\ldots+a_n}n.$$
2007 Junior Tuymaada Olympiad, 8
Several knights are arranged on an infinite chessboard. No square is attacked by more than one knight (in particular, a square occupied by a knight can be attacked by one knight but not by two). Sasha outlined a $ 14\times 16$ rectangle. What maximum number of knights can this rectangle contain?
2016 Iran Team Selection Test, 5
Let $P$ and $P '$ be two unequal regular $n-$gons and $A$ and $A'$two points inside $P$ and$ P '$, respectively.Suppose $\{ d_1 , d_2 , \cdots d_n \}$ are the distances from $A $ to the vertices of $P$ and $\{ d'_1 , d'_2 , \cdots d'_n \}$ are defines similarly for $P',A'$. Is it possible for $\{ d'_1 , d'_2 , \cdots d'_n \}$ to be a permutation of $\{ d_1 , d_2 , \cdots d_n \}$ ?
2022 Spain Mathematical Olympiad, 4
Let $P$ be a point in the plane. Prove that it is possible to draw three rays with origin in $P$ with the following property: for every circle with radius $r$ containing $P$ in its interior, if $P_1$, $P_2$ and $P_3$ are the intersection points of the three rays with the circle, then \[|PP_1|+|PP_2|+|PP_3|\leq 3r.\]
Kvant 2020, M2628
There are $m$ identical two-pan weighting scales. One of them is broken and it shows any outcome, at random. The other scales always show the correct outcome. Moreover, the weight of the broken scale differs from those of the other scales, which are all equal. At a move, we may choose a scale and place some of the other scales on its pans. Determine the greatest value of $m$ for which we may find the broken scale with no more than three moves.
[i]Proposed by A. Gribalko and O. Manzhina[/i]
II Soros Olympiad 1995 - 96 (Russia), 11.9
Solve the equation $$x(2^{1-2x}-1)=2^{x-2x^2}-1$$
2018 Baltic Way, 3
Let $a,b,c,d$ be positive real numbers such that $abcd=1$. Prove the inequality
\[\frac{1}{\sqrt{a+2b+3c+10}}+\frac{1}{\sqrt{b+2c+3d+10}}+\frac{1}{\sqrt{c+2d+3a+10}}+\frac{1}{\sqrt{d+2a+3b+10}} \le 1.\]
2004 Germany Team Selection Test, 3
Let $n \geq 2$ be a natural number, and let $\left( a_{1};\;a_{2};\;...;\;a_{n}\right)$ be a permutation of $\left(1;\;2;\;...;\;n\right)$. For any integer $k$ with $1 \leq k \leq n$, we place $a_k$ raisins on the position $k$ of the real number axis. [The real number axis is the $x$-axis of a Cartesian coordinate system.]
Now, we place three children A, B, C on the positions $x_A$, $x_B$, $x_C$, each of the numbers $x_A$, $x_B$, $x_C$ being an element of $\left\{1;\;2;\;...;\;n\right\}$. [It is not forbidden to place different children on the same place!]
For any $k$, the $a_k$ raisins placed on the position $k$ are equally handed out to those children whose positions are next to $k$. [So, if there is only one child lying next to $k$, then he gets the raisin. If there are two children lying next to $k$ (either both on the same position or symmetric with respect to $k$), then each of them gets one half of the raisin. Etc..]
After all raisins are distributed, a child is unhappy if he could have received more raisins than he actually has received if he had moved to another place (while the other children would rest on their places).
For which $n$ does there exist a configuration $\left( a_{1};\;a_{2};\;...;\;a_{n}\right)$ and numbers $x_A$, $x_B$, $x_C$, such that all three children are happy?
2014-2015 SDML (High School), 2
A circle of radius $5$ is inscribed in an isosceles right triangle, $ABC$. The length of the hypotenuse of $ABC$ can be expressed as $a+a\sqrt{2}$ for some $a$. What is $a$?
1996 Bundeswettbewerb Mathematik, 3
Let $ABC$ be a triangle, and erect three rectangles $ABB_1A_2$, $BCC_1B_2$, $CAA_1C_2$ externally on its sides $AB$, $BC$, $CA$, respectively. Prove that the perpendicular bisectors of the segments $A_1A_2$, $B_1B_2$, $C_1C_2$ are concurrent.
2014 NIMO Problems, 2
Let $ABC$ be an equilateral triangle. Denote by $D$ the midpoint of $\overline{BC}$, and denote the circle with diameter $\overline{AD}$ by $\Omega$. If the region inside $\Omega$ and outside $\triangle ABC$ has area $800\pi-600\sqrt3$, find the length of $AB$.
[i]Proposed by Eugene Chen[/i]
2012 Today's Calculation Of Integral, 856
On the coordinate plane, find the area of the part enclosed by the curve $C: (a+x)y^2=(a-x)x^2\ (x\geq 0)$ for $a>0$.
1994 AMC 12/AHSME, 26
A regular polygon of $m$ sides is exactly enclosed (no overlaps, no gaps) by $m$ regular polygons of $n$ sides each. (Shown here for $m=4, n=8$.) If $m=10$, what is the value of $n$?
[asy]
size(200);
defaultpen(linewidth(0.8));
draw(unitsquare);
path p=(0,1)--(1,1)--(1+sqrt(2)/2,1+sqrt(2)/2)--(1+sqrt(2)/2,2+sqrt(2)/2)--(1,2+sqrt(2))--(0,2+sqrt(2))--(-sqrt(2)/2,2+sqrt(2)/2)--(-sqrt(2)/2,1+sqrt(2)/2)--cycle;
draw(p);
draw(shift((1+sqrt(2)/2,-sqrt(2)/2-1))*p);
draw(shift((0,-2-sqrt(2)))*p);
draw(shift((-1-sqrt(2)/2,-sqrt(2)/2-1))*p);[/asy]
$ \textbf{(A)}\ 5 \qquad\textbf{(B)}\ 6 \qquad\textbf{(C)}\ 14 \qquad\textbf{(D)}\ 20 \qquad\textbf{(E)}\ 26 $
2012 Olympic Revenge, 4
Say that two sets of positive integers $S, T$ are $\emph{k-equivalent}$ if the sum of the $i$th powers of elements of $S$ equals the sum of the $i$th powers of elements of $T$, for each $i= 1, 2, \ldots, k$. Given $k$, prove that there are infinitely many numbers $N$ such that $\{1,2,\ldots,N^{k+1}\}$ can be divided into $N$ subsets, all of which are $k$-equivalent to each other.
2003 Cuba MO, 3
Let $ABC$ be an acute triangle and $T$ be a point interior to this triangle. that $\angle ATB = \angle BTC = \angle CTA$. Let $M,N$ and $P$ be the feet of the perpendiculars from $T$ to $BC$, $CA$ and $AB$ respectively. Prove that if the circle circumscribed around $\vartriangle MNP$ cuts again the sides $ BC$, $CA$ and $AB$ in $M_1$, $N_1$, $P_1$ respectively, then the $\vartriangle M_1N_1P_1$ It is equilateral.
2018 Switzerland - Final Round, 9
Let $n$ be a positive integer and let $G$ be the set of points $(x, y)$ in the plane such that $x$ and $y$ are integers with $1 \leq x, y \leq n$. A subset of $G$ is called [i]parallelogram-free[/i] if it does not contains four non-collinear points, which are the vertices of a parallelogram. What is the largest number of elements a parallelogram-free subset of $G$ can have?
May Olympiad L2 - geometry, 2007.5
In the triangle $ABC$ we have $\angle A = 2\angle C$ and $2\angle B = \angle A + \angle C$. The angle bisector of $\angle C$ intersects the segment $AB$ in $E$, let $F$ be the midpoint of $AE$, let $AD$ be the altitude of the triangle $ABC$. The perpendicular bisector of $DF$ intersects $AC$ in $M$.
Prove that $AM = CM$.