Found problems: 85335
2022 Estonia Team Selection Test, 5
(a) Is it true that, for arbitrary integer $n{}$ greater than $1$ and distinct positive integers $i{}$ and $j$ not greater than $n{}$, the set of any $n{}$ consecutive integers contains distinct numbers $i^{'}$ and $j^{'}$ whose product $i^{'}j^{'}$ is divisible by the product $ij$?
(b) Is it true that, for arbitrary integer $n{}$ greater than $2$ and distinct positive integers $i, j, k$ not greater than $n{}$, the set of any $n{}$ consecutive integers contains distinct numbers $i^{'},j^{'},k^{'}$ whose product $i^{'}j^{'}k^{'}$ is divisible by the product $ijk$?
2002 Estonia Team Selection Test, 2
Consider an isosceles triangle $KL_1L_2$ with $|KL_1|=|KL_2|$ and let $KA, L_1B_1,L_2B_2$ be its angle bisectors. Prove that $\cos \angle B_1AB_2 < \frac35$
2020 Sharygin Geometry Olympiad, 16
Cevians $AP$ and $AQ$ of a triangle $ABC$ are symmetric with respect to its bisector. Let $X$, $Y$ be the projections of $B$ to $AP$ and $AQ$ respectively, and $N$, $M$ be the projections of $C$ to $AP$ and $AQ$ respectively. Prove that $XM$ and $NY$ meet on $BC$.
2006 Australia National Olympiad, 4
There are $n$ points on a circle, such that each line segment connecting two points is either red or blue.
$P_iP_j$ is red if and only if $P_{i+1} P_{j+1}$ is blue, for all distinct $i, j$ in $\left\{1, 2, ..., n\right\}$.
(a) For which values of $n$ is this possible?
(b) Show that one can get from any point on the circle to any other point, by doing a maximum of 3 steps, where one step is moving from a point to another point through a red segment connecting these points.
2015 Online Math Open Problems, 11
Let $S$ be a set. We say $S$ is $D^\ast$[i]-finite[/i] if there exists a function $f : S \to S$ such that for every nonempty proper subset $Y \subsetneq S$, there exists a $y \in Y$ such that $f(y) \notin Y$. The function $f$ is called a [i]witness[/i] of $S$. How many witnesses does $\{0,1,\cdots,5\}$ have?
[i]Proposed by Evan Chen[/i]
2010 Today's Calculation Of Integral, 594
In the $x$-$y$ plane, two variable points $P,\ Q$ stay in $P(2t,\ -2t^2+2t),\ Q(t+2,-3t+2)$ at the time $t$.
Let denote $t_0$ as the time such that $\overline{PQ}=0$. When $t$ varies in the range of $0\leq t\leq t_0$, find the area of the region swept by the line segment $PQ$ in the $x$-$y$ plane.
2013 NIMO Summer Contest, 10
Let $P(x)$ be the unique polynomial of degree four for which $P(165) = 20$, and \[ P(42) = P(69) = P(96) = P(123) = 13. \] Compute $P(1) - P(2) + P(3) - P(4) + \dots + P(165)$.
[i]Proposed by Evan Chen[/i]
2001 Romania Team Selection Test, 4
Three schools have $200$ students each. Every student has at least one friend in each school (if the student $a$ is a friend of the student $b$ then $b$ is a friend of $a$).
It is known that there exists a set $E$ of $300$ students (among the $600$) such that for any school $S$ and any two students $x,y\in E$ but not in $S$, the number of friends in $S$ of $x$ and $y$ are different.
Show that one can find a student in each school such that they are friends with each other.
2009 USA Team Selection Test, 1
Let $m$ and $n$ be positive integers. Mr. Fat has a set $S$ containing every rectangular tile with integer side lengths and area of a power of $2$. Mr. Fat also has a rectangle $R$ with dimensions $2^m \times 2^n$ and a $1 \times 1$ square removed from one of the corners. Mr. Fat wants to choose $m + n$ rectangles from $S$, with respective areas $2^0, 2^1, \ldots, 2^{m + n - 1}$, and then tile $R$ with the chosen rectangles. Prove that this can be done in at most $(m + n)!$ ways.
[i]Palmer Mebane.[/i]
Estonia Open Junior - geometry, 2007.1.4
Call a scalene triangle K [i]disguisable[/i] if there exists a triangle K′ similar to K with two shorter sides precisely as long as the two longer sides of K, respectively. Call a disguisable triangle [i]integral[/i] if the lengths of all its sides are integers.
(a) Find the side lengths of the integral disguisable triangle with the smallest possible perimeter.
(b) Let K be an arbitrary integral disguisable triangle for which no smaller integral
disguisable triangle similar to it exists. Prove that at least two side lengths of K are
perfect squares.
1996 Estonia National Olympiad, 4
Prove that for each prime number $p > 5$ there exists a positive integer n such that $p^n$ ends in $001$ in decimal representation.
1979 All Soviet Union Mathematical Olympiad, 282
The convex quadrangle is divided by its diagonals onto four triangles. The circles inscribed in those triangles are equal. Prove that the given quadrangle is a diamond.
1998 Iran MO (3rd Round), 5
In a triangle $ABC$, the bisector of angle $BAC$ intersects $BC$ at $D$. The circle $\Gamma$ through $A$ which is tangent to $BC$ at $D$ meets $AC$ again at $M$. Line $BM$ meets $\Gamma$ again at $P$. Prove that line $AP$ is a median of $\triangle ABD.$
2009 Ukraine National Mathematical Olympiad, 4
Let $ABCD$ be a parallelogram with $\angle BAC = 45^\circ,$ and $AC > BD .$ Let $w_1$ and $w_2$ be two circles with diameters $AC$ and $DC,$ respectively. The circle $w_1$ intersects $AB$ at $E$ and the circle $w_2$ intersects $AC$ at $O$ and $C$, and $AD$ at $F.$ Find the ratio of areas of triangles $AOE$ and $COF$ if $AO = a,$ and $FO = b .$
2019 Durer Math Competition Finals, 12
$P$ and $Q$ are two different non-constant polynomials such that $P(Q(x)) = P(x)Q(x)$ and $P(1) = P(-1) = 2019$. What are the last four digits of $Q(P(-1))$?
2009 IberoAmerican, 1
Given a positive integer $ n\geq 2$, consider a set of $ n$ islands arranged in a circle. Between every two neigboring islands two bridges are built as shown in the figure.
Starting at the island $ X_1$, in how many ways one can one can cross the $ 2n$ bridges so that no bridge is used more than once?
2011 Switzerland - Final Round, 2
Let $\triangle{ABC}$ be an acute-angled triangle and let $D$, $E$, $F$ be points on $BC$, $CA$, $AB$, respectively, such that \[\angle{AFE}=\angle{BFD}\mbox{,}\quad\angle{BDF}=\angle{CDE}\quad\mbox{and}\quad\angle{CED}=\angle{AEF}\mbox{.}\] Prove that $D$, $E$ and $F$ are the feet of the perpendiculars through $A$, $B$ and $C$ on $BC$, $CA$ and $AB$, respectively.
[i](Swiss Mathematical Olympiad 2011, Final round, problem 2)[/i]
2017 IFYM, Sozopol, 3
Let $n$ be a composite number and $a_1,a_2… a_k\in \mathbb{N}$ are the numbers smaller than $n$ and not coprime with it (in this case $k=n-\phi (n)$). Let $b_1,b_2…b_k$ be a permutation of $a_1,a_2… a_k$ Prove that there exist indexes $i$ and $j$, $i\neq j$ for which $a_i b_i\equiv a_j b_j (mod $ $n)$.
2022 AMC 10, 9
A rectangle is partitioned into 5 regions as shown. Each region is to be painted a solid color - red, orange, yellow, blue, or green - so that regions that touch are painted different colors, and colors can be used more than once. How many different colorings are possible?
[asy]
size(5.5cm);
draw((0,0)--(0,2)--(2,2)--(2,0)--cycle);
draw((2,0)--(8,0)--(8,2)--(2,2)--cycle);
draw((8,0)--(12,0)--(12,2)--(8,2)--cycle);
draw((0,2)--(6,2)--(6,4)--(0,4)--cycle);
draw((6,2)--(12,2)--(12,4)--(6,4)--cycle);
[/asy]
$\textbf{(A) }120\qquad\textbf{(B) }270\qquad\textbf{(C) }360\qquad\textbf{(D) }540\qquad\textbf{(E) }720$
2013 Portugal MO, 6
In each side of a regular polygon with $n$ sides, we choose a point different from the vertices and we obtain a new polygon of $n$ sides. For which values of $n$ can we obtain a polygon such that the internal angles are all equal but the polygon isn't regular?
1981 AMC 12/AHSME, 11
The three sides of a right triangle have integral lengths which form an arithmetic progression. One of the sides could have length
$\text{(A)}\ 22 \qquad \text{(B)}\ 58 \qquad \text{(C)}\ 81 \qquad \text{(D)}\ 91 \qquad \text{(E)}\ 361$
2022 BMT, Tie 4
How many positive integers less than $2022$ contain at least one digit less than $5$ and also at least one digit greater than $4$?
1935 Moscow Mathematical Olympiad, 020
How many ways are there of representing a positive integer $n$ as the sum of three positive integers? Representations which differ only in the order of the summands are considered to be distinct.
1979 IMO Longlists, 48
In the plane a circle $C$ of unit radius is given. For any line $l$, a number $s(l)$ is defined in the following way: If $l$ and $C$ intersect in two points, $s(l)$ is their distance; otherwise, $s(l) = 0$. Let $P$ be a point at distance $r$ from the center of $C$. One defines $M(r)$ to be the maximum value of the sum $s(m) + s(n)$, where $m$ and $n$ are variable mutually orthogonal lines through $P$. Determine the values of $r$ for which $M(r) > 2$.
II Soros Olympiad 1995 - 96 (Russia), 9.5
Angle $A$ of triangle $ABC$ is $33^o$. A straight line passing through $A$ perpendicular to $AC$ intersects straight line $BC$ at point $D$ so that $CD = 2AB$. What is angle $C$ of triangle $ABC$? (Please list all options.)