Found problems: 85335
2014 Math Prize For Girls Problems, 14
A triangle has area 114 and sides of integer length. What is the perimeter of the triangle?
2015 British Mathematical Olympiad Round 1, 6
A positive integer is called [i]charming[/i] if it is equal to $2$ or is of the form $3^{i}5^{j}$ where $i$ and $j$ are non-negative integers. Prove that every positive integer can be written as a sum of different charming numbers.
2019 Korea National Olympiad, 2
Triangle $ABC$ is an scalene triangle. Let $I$ the incenter, $\Omega$ the circumcircle, $E$ the $A$-excenter of triangle $ABC$. Let $\Gamma$ the circle centered at $E$ and passes $A$. $\Gamma$ and $\Omega$ intersect at point $D(\neq A)$, and the perpendicular line of $BC$ which passes $A$ meets $\Gamma$ at point $K(\neq A)$. $L$ is the perpendicular foot from $I$ to $AC$. Now if $AE$ and $DK$ intersects at $F$, prove that $BE\cdot CI=2\cdot CF\cdot CL$.
2023 JBMO Shortlist, A7
Let $a_1,a_2,a_3,\ldots,a_{250}$ be real numbers such that $a_1=2$ and
$$a_{n+1}=a_n+\frac{1}{a_n^2}$$
for every $n=1,2, \ldots, 249$. Let $x$ be the greatest integer which is less than
$$\frac{1}{a_1}+\frac{1}{a_2}+\ldots+\frac{1}{a_{250}}$$
How many digits does $x$ have?
[i]Proposed by Miroslav Marinov, Bulgaria[/i]
2019 Brazil National Olympiad, 5
(a) Prove that given constants $a,b$ with $1<a<2<b$, there is no partition of the set of positive integers into two subsets $A_0$ and $A_1$ such that: if $j \in \{0,1\}$ and $m,n$ are in $A_j$, then either $n/m <a$ or $n/m>b$.
(b) Find all pairs of real numbers $(a,b)$ with $1<a<2<b$ for which the following property holds: there exists a partition of the set of positive integers into three subsets $A_0, A_1, A_2$ such that if $j \in \{0,1,2\}$ and $m,n$ are in $A_j$, then either $n/m <a$ or $n/m>b$.
2021 USAJMO, 5
A finite set $S$ of positive integers has the property that, for each $s \in S,$ and each positive integer divisor $d$ of $s$, there exists a unique element $t \in S$ satisfying $\text{gcd}(s, t) = d$. (The elements $s$ and $t$ could be equal.)
Given this information, find all possible values for the number of elements of $S$.
2013 Stanford Mathematics Tournament, 3
Nine people are practicing the triangle dance, which is a dance that requires a group of three people. During each round of practice, the nine people split off into three groups of three people each, and each group practices independently. Two rounds of practice are different if there exists some person who does not dance with the same pair in both rounds. How many different rounds of practice can take place?
1995 Czech And Slovak Olympiad IIIA, 6
Find all real parameters $p$ for which the equation $x^3 -2p(p+1)x^2+(p^4 +4p^3 -1)x-3p^3 = 0$
has three distinct real roots which are sides of a right triangle.
Ukraine Correspondence MO - geometry, 2011.11
In a quadrilateral $ABCD$, the diagonals are perpendicular and intersect at the point $S$. Let $K, L, M$, and $N$ be points symmetric to $S$ with respect to the lines $AB, BC, CD$, and $DA$, respectively, $BN$ intersects the circumcircle of the triangle $SKN$ at point $E$, and $BM$ intersects circumscribed the circle of the triangle $SLM$ at the point $F$. Prove that the quadrilateral $EFLK$ is cyclic .
2009 China Team Selection Test, 1
Given that points $ D,E$ lie on the sidelines $ AB,BC$ of triangle $ ABC$, respectively, point $ P$ is in interior of triangle $ ABC$ such that $ PE \equal{} PC$ and $ \bigtriangleup DEP\sim \bigtriangleup PCA.$ Prove that $ BP$ is tangent of the circumcircle of triangle $ PAD.$
2013 AMC 10, 11
Real numbers $x$ and $y$ satisfy the equation $x^2+y^2=10x-6y-34$. What is $x+y$?
$ \textbf{(A) }1\qquad\textbf{(B) }2\qquad\textbf{(C) }3\qquad\textbf{(D) }6\qquad\textbf{(E) }8 $
2009 AIME Problems, 2
There is a complex number $ z$ with imaginary part $ 164$ and a positive integer $ n$ such that
\[ \frac {z}{z \plus{} n} \equal{} 4i.
\]Find $ n$.
2011 Morocco TST, 2
For positive integers $m$ and $n$, find the smalles possible value of $|2011^m-45^n|$.
[i](Swiss Mathematical Olympiad, Final round, problem 3)[/i]
2022 Purple Comet Problems, 26
Antonio plays a game where he continually flips a fair coin to see the sequence of heads ($H$) and tails ($T$) that he flips. Antonio wins the game if he sees on four consecutive flips the sequence $TTHT$ before he sees the sequence $HTTH$. The probability that Antonio wins the game is $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Find $m + n$.
2017 Saudi Arabia JBMO TST, 4
Find the number of ways one can put numbers $1$ or $2$ in each cell of an $8\times 8$ chessboard in such a way that the sum of the numbers in each column and in each row is an odd number. (Two ways are considered different if the number in some cell in the first way is different from the number in the cell situated in the corresponding position in the second way)
2002 Greece JBMO TST, 3
Let $ABC$ be a triangle with $\angle A=60^o, AB\ne AC$ and let $AD$ be the angle bisector of $\angle A$. Line $(e)$ that is perpendicular on the angle bisector $AD$ at point $A$, intersects the extension of side $BC$ at point $E$ and also $BE=AB+AC$. Find the angles $\angle B$ and $\angle C$ of the triangle $ABC$.
1948 Moscow Mathematical Olympiad, 151
The distance between the midpoints of the opposite sides of a convex quadrilateral is equal to a half sum of lengths of the other two sides. Prove that the first pair of sides is parallel.
2010 Stanford Mathematics Tournament, 9
For an acute triangle $ABC$ and a point $X$ satisfying $\angle{ABX}+\angle{ACX}=\angle{CBX}+\angle{BCX}$.
Fi nd the minimum length of $AX$ if $AB=13$, $BC=14$, and $CA=15$.
2017 QEDMO 15th, 5
Let $F$ be a finite subset of the integer numbers. We define a new subset $s(F)$ in that $a\in Z$ lies in $s (F)$ if and only if exactly one of the numbers $a$ and $a -1$ in $F$. In the same way one gets from $s (F)$ the set $s^2(F) = s (s (F))$ and by $n$-fold application of $s$ then iteratively further subsets $s^n (F)$.
Prove there are infinitely many natural numbers $n$ for which $s^n (F) = F\cup \{a + n|a \in F\}$.
2011 AMC 10, 24
Two distinct regular tetrahedra have all their vertices among the vertices of the same unit cube. What is the volume of the region formed by the intersection of the tetrahedra?
$ \textbf{(A)}\ \frac{1}{12}\qquad\textbf{(B)}\ \frac{\sqrt{2}}{12}\qquad\textbf{(C)}\ \frac{\sqrt{3}}{12}\qquad\textbf{(D)}\ \frac{1}{6}\qquad\textbf{(E)}\ \frac{\sqrt{2}}{6} $
LMT Guts Rounds, 13
A circle with center $O$ has radius $5,$ and has two points $A,B$ on the circle such that $\angle AOB = 90^{\circ}.$ Rays $OA$ and $OB$ are extended to points $C$ and $D,$ respectively, such that $AB$ is parallel to $CD,$ and the length of $CD$ is $200\%$ more than the radius of circle $O.$ Determine the length of $AC.$
2006 Kazakhstan National Olympiad, 3
The racing tournament has $12$ stages and $ n $ participants. After each stage, all participants, depending on the occupied place $ k $, receive points $ a_k $ (the numbers $ a_k $ are natural and $ a_1> a_2> \dots> a_n $). For what is the smallest $ n $ the tournament organizer can choose the numbers $ a_1 $, $ \dots $, $ a_n $ so that after the penultimate stage for any possible distribution of places at least two participants had a chance to take first place.
2020 Serbia National Math Olympiad, 3
We are given a triangle $ABC$. Points $D$ and $E$ on the line $AB$ are such that $AD=AC$ and $BE=BC$, with the arrangment of points $D - A - B - E$. The circumscribed circles of the triangles $DBC$ and $EAC$ meet again at the point $X\neq C$, and the circumscribed circles of the triangles $DEC$ and $ABC$ meet again at the point $Y\neq C$. Find the measure of $\angle ACB$ given the condition $DY+EY=2XY$.
2005 MOP Homework, 6
A computer network is formed by connecting $2004$ computers by cables. A set $S$ of these computers is said to be independent if no pair of computers of $S$ is connected by a cable. Suppose that the number of cables used is the minimum number possible such that the size of any independent set is at most $50$. Let $c(L)$ be the number of cables connected to computer $L$. Show that for any distinct computers $A$ and $B$, $c(A)=c(B)$ if they are connected by a cable and $|c(A)-c(B)| \le 1$ otherwise. Also, find the number of cables used in the network.
2013 Polish MO Finals, 4
Given is a tetrahedron $ABCD$ in which $AB=CD$ and the sum of measures of the angles $BAD$ and $BCD$ equals $180$ degrees. Prove that the measure of the angle $BAD$ is larger than the measure of the angle $ADC$.