Found problems: 85335
2007 Iran MO (3rd Round), 1
Consider two polygons $ P$ and $ Q$. We want to cut $ P$ into some smaller polygons and put them together in such a way to obtain $ Q$. We can translate the pieces but we can not rotate them or reflect them. We call $ P,Q$ equivalent if and only if we can obtain $ Q$ from $ P$(which is obviously an equivalence relation).
[img]http://i3.tinypic.com/4lrb43k.png[/img]
a) Let $ P,Q$ be two rectangles with the same area(their sides are not necessarily parallel). Prove that $ P$ and $ Q$ are equivalent.
b) Prove that if two triangles are not translation of each other, they are not equivalent.
c) Find a necessary and sufficient condition for polygons $ P,Q$ to be equivalent.
1972 Spain Mathematical Olympiad, 6
Given three circumferences of radii $r$ , $r'$ and $r''$ , each tangent externally to the other two, calculate the radius of the circle inscribed in the triangle whose vertices are their three centers.
ICMC 8, 6
A set of points in the plane is called rigid if each point is equidistant from the three (or more) points nearest to it.
(a) Does there exist a rigid set of $9$ points?
(b) Does there exist a rigid set of $11$ points?
2024 Chile TST IMO, 2
Find all natural numbers that have a multiple consisting only of the digit 9.
Indonesia MO Shortlist - geometry, g8
Suppose the points $D, E, F$ lie on sides $BC, CA, AB$, respectively, so that $AD, BE, CF$ are angle bisectors. Define $P_1$, $P_2$, $P_3$ respectively as the intersection point of $AD$ with $EF$, $BE$ with $DF$, $CF$ with $DE$ respectively. Prove that
$$\frac{AD}{AP_1}+\frac{BE}{BP_2}+\frac{CF}{CP_3} \ge 6$$
2022 Math Prize for Girls Problems, 8
Let $S$ be the set of numbers of the form $n^5 - 5n^3 + 4n$, where $n$ is an integer that is not a multiple of $3$. What is the largest integer that is a divisor of every number in $S$?
2015 ITAMO, 4
Determine all pairs of integers $(a, b)$ that solve the equation $a^3 + b^3 + 3ab = 1$.
2020 Middle European Mathematical Olympiad, 3#
Let $ABC$ be an acute scalene triangle with circumcircle $\omega$ and incenter $I$. Suppose the orthocenter $H$ of $BIC$ lies inside $\omega$. Let $M$ be the midpoint of the longer arc $BC$ of $\omega$. Let $N$ be the midpoint of the shorter arc $AM$ of $\omega$.
Prove that there exists a circle tangent to $\omega$ at $N$ and tangent to the circumcircles of $BHI$ and $CHI$.
2003 AMC 10, 3
A solid box is $ 15$ cm by $ 10$ cm by $ 8$ cm. A new solid is formed by removing a cube $ 3$ cm on a side from each corner of this box. What percent of the original volume is removed?
$ \textbf{(A)}\ 4.5 \qquad
\textbf{(B)}\ 9 \qquad
\textbf{(C)}\ 12 \qquad
\textbf{(D)}\ 18 \qquad
\textbf{(E)}\ 24$
2023 Thailand Online MO, 3
Let $a$ and $n$ be positive integers such that the greatest common divisor of $a$ and $n!$ is $1$. Prove that $n!$ divides $a^{n!}-1$.
2007 Peru IMO TST, 3
Let $N$ be a natural number which can be expressed in the form $a^{2}+b^{2}+c^{2}$, where $a,b,c$ are integers divisible by 3.
Prove that $N$ can be expressed in the form $x^{2}+y^{2}+z^{2}$, where $x,y,z$ are integers and any of them are divisible by 3.
1988 AMC 12/AHSME, 25
$X$, $Y$ and $Z$ are pairwise disjoint sets of people. The average ages of people in the sets $X$, $Y$, $Z$, $X \cup Y$, $X \cup Z$ and $Y \cup Z$ are given in the table below.
\begin{tabular}{|c|c|c|c|c|c|c|} \hline
\rule{0pt}{1.1em} Set & $X$ & $Y$ & $Z$ & $X\cup Y$ & $X\cup Z$ & $Y\cup Z$\\[0.5ex] \hline \rule{0pt}{2.2em} \shortstack{Average age of \\ people in the set} & 37 & 23 & 41 & 29 & 39.5 & 33\\[1ex]\hline\end{tabular}
Find the average age of the people in set $X \cup Y \cup Z$.
$ \textbf{(A)}\ 33\qquad\textbf{(B)}\ 33.5\qquad\textbf{(C)}\ 33.6\overline{6}\qquad\textbf{(D)}\ 33.83\overline{3}\qquad\textbf{(E)}\ 34 $
1958 Poland - Second Round, 2
Six equal disks are placed on a plane so that their centers lie at the vertices of a regular hexagon with sides equal to the diameter of the disks. How many revolutions will a seventh disk of the same size make when rolling in the same plane externally over the disks before returning to its initial position?
2009 Indonesia TST, 1
a. Does there exist 4 distinct positive integers such that the sum of any 3 of them is prime?
b. Does there exist 5 distinct positive integers such that the sum of any 3 of them is prime?
2022 Cono Sur, 1
A positive integer is [i]happy[/i] if:
1. All its digits are different and not $0$,
2. One of its digits is equal to the sum of the other digits.
For example, 253 is a [i]happy[/i] number. How many [i]happy[/i] numbers are there?
2009 Purple Comet Problems, 11
The four points $A(-1,2), B(3,-4), C(5,-6),$ and $D(-2,8)$ lie in the coordinate plane. Compute the minimum possible value of $PA + PB + PC + PD$ over all points P .
2021 Kyiv Mathematical Festival, 2
In 11 cells of a square grid there live hedgehogs. Every hedgehog divides the number of hedgehogs in its row by the number of hedgehogs in its column. Is it possible that all the hedgehogs get distinct numbers? (V.Brayman)
2023 Bulgarian Autumn Math Competition, 8.2
A quadrilateral is called $\textit{innovative}$ if its diagonals divide it into $4$ triangles, having the same sets of angle measures. Find the angle measures of an $\textit{innovative}$ quadrilateral, given that one of its angles has measure $13^{\circ}$.
1976 Spain Mathematical Olympiad, 2
Consider the set $C$ of all $r$ -tuple whose components are $1$ or $-1$. Calculate the sum of all the components of all the elements of $C$ excluding the $ r$ -tuple $(1, 1, 1, . . . , 1)$.
2022 Taiwan Mathematics Olympiad, 3
Find all functions $f,g:\mathbb{R}^2\to\mathbb{R}$ satisfying that
\[|f(a,b)-f(c,d)|+|g(a,b)-g(c,d)|=|a-c|+|b-d|\]
for all real numbers $a,b,c,d$.
[i]Proposed by usjl[/i]
2022 Brazil Team Selection Test, 3
Let $p$ be an odd prime number and suppose that $2^h \not \equiv 1 \text{ (mod } p\text{)}$ for all integer $1 \leq h \leq p-2$. Let $a$ be an even number such that $\frac{p}{2} < a < p$. Define the sequence $a_0, a_1, a_2, \ldots$ as $$a_0 = a, \qquad a_{n+1} = p -b_n, \qquad n = 0,1,2, \ldots,$$ where $b_n$ is the greatest odd divisor of $a_n$. Show that the sequence is periodic and determine its period.
2014 Contests, 2
Let $x_1$, $x_2$, …, $x_{10}$ be 10 numbers. Suppose that $x_i + 2 x_{i + 1} = 1$ for each $i$ from 1 through 9. What is the value of $x_1 + 512 x_{10}$?
2002 Turkey Team Selection Test, 3
Consider $2n+1$ points in space, no four of which are coplanar where $n>1$. Each line segment connecting any two of these points is either colored red, white or blue. A subset $M$ of these points is called a [i]connected monochromatic[/i] subset, if for each $a,b \in M$, there are points $a=x_0,x_1, \dots, x_l = b$ that belong to $M$ such that the line segments $x_0x_1, x_1x_2, \dots, x_{l-1}x_1$ are all have the same color. No matter how the points are colored, if there always exists a connected monochromatic $k-$subset, find the largest value of $k$. ($l > 1$)
2024 USAMTS Problems, 4
Let $x_1 \le x_2 \le \dots < x_n$ (with $n \ge 2$) and let $S$ be the set of all the $x_i$. Let $T$ be a randomly chosen subset of $S$. What is the expected value of the indexed alternating sum of $T$ ? Express your answer in terms of the $x_i$.
Note: We define the indexed alternating sum of $T$ as
\[
\sum_{i=1}^{|T|} (-1)^{i+1}(i) T[i],
\]
where $T[i]$ is the ith element of $T$ when listed in increasing order. For example, if $T = \{1, 3, 5\}$
then the indexed alternating sum of $T$ is
\[
1 \cdot 1 - 2 \cdot 3 + 3 \cdot 5 = 10.
\]
Alternating sums of empty sets are defined to be $0$.
PEN G Problems, 10
Show that $\frac{1}{\pi} \arccos \left( \frac{1}{\sqrt{2003}} \right)$ is irrational.