Found problems: 85335
PEN J Problems, 2
Show that for all $n \in \mathbb{N}$, \[n = \sum^{}_{d \vert n}\phi(d).\]
2018 China Team Selection Test, 4
Suppose $A_1,A_2,\cdots ,A_n \subseteq \left \{ 1,2,\cdots ,2018 \right \}$ and $\left | A_i \right |=2, i=1,2,\cdots ,n$, satisfying that $$A_i + A_j, \; 1 \le i \le j \le n ,$$ are distinct from each other. $A + B = \left \{ a+b|a\in A,\,b\in B \right \}$. Determine the maximal value of $n$.
2015 USA TSTST, 1
Let $a_1, a_2, \dots, a_n$ be a sequence of real numbers, and let $m$ be a fixed positive integer less than $n$. We say an index $k$ with $1\le k\le n$ is good if there exists some $\ell$ with $1\le \ell \le m$ such that $a_k+a_{k+1}+...+a_{k+\ell-1}\ge0$, where the indices are taken modulo $n$. Let $T$ be the set of all good indices. Prove that $\sum\limits_{k \in T}a_k \ge 0$.
[i]Proposed by Mark Sellke[/i]
2017 Ukraine Team Selection Test, 9
There're two positive inegers $a_1<a_2$. For every positive integer $n \geq 3$ let $a_n$ be the smallest integer that bigger than $a_{n-1}$ and such that there's unique pair $1\leq i< j\leq n-1$ such that this number equals to $a_i+a_j$. Given that there're finitely many even numbers in this sequence. Prove that sequence $\{a_{n+1}-a_n \}$ is periodic starting from some element.
2012 Romania Team Selection Test, 3
Let $A$ and $B$ be finite sets of real numbers and let $x$ be an element of $A+B$. Prove that \[|A\cap (x-B)|\leq \frac{|A-B|^2}{|A+B|}\] where $A+B=\{a+b: a\in A, b\in B\}$, $x-B=\{x-b: b\in B\}$ and $A-B=\{a-b: a\in A, b\in B\}$.
2012 Grigore Moisil Intercounty, 3
Solve in the real numbers the equation $ (n+1)^x+(n+3)^x+\left( n^2+2n\right)^x=n^x+(n+2)^x+\left( n^2+4n+3\right)^x, $ wher $ n\ge 2 $ is a fixed natural number.
2022 Israel TST, 3
In triangle $ABC$, the angle bisectors are $BE$ and $CF$ (where $E, F$ are on the sides of the triangle), and their intersection point is $I$. Point $N$ lies on the circumcircle of $AEF$, and the angle $\angle IAN$ is right. The circumcircle of $AEF$ meets the line $NI$ a second time at the point $L$. Show that the circumcenter of $AIL$ lies on line $BC$.
1999 Mongolian Mathematical Olympiad, Problem 2
Can a square be divided into $10$ pairwise non-congruent triangles with the same area?
2011 Junior Balkan Team Selection Tests - Romania, 5
Consider $n$ persons, each of them speaking at most $3$ languages. From any $3$ persons there are at least two which speak a common language.
i) For $n \le 8$, exhibit an example in which no language is spoken by more than two persons.
ii) For $n \ge 9$, prove that there exists a language which is spoken by at least three persons
1966 IMO, 1
In a mathematical contest, three problems, $A,B,C$ were posed. Among the participants ther were 25 students who solved at least one problem each. Of all the contestants who did not solve problem $A$, the number who solved $B$ was twice the number who solved $C$. The number of students who solved only problem $A$ was one more than the number of students who solved $A$ and at least one other problem. Of all students who solved just one problem, half did not solve problem $A$. How many students solved only problem $B$?
2002 IMO, 1
Let $n$ be a positive integer. Each point $(x,y)$ in the plane, where $x$ and $y$ are non-negative integers with $x+y<n$, is coloured red or blue, subject to the following condition: if a point $(x,y)$ is red, then so are all points $(x',y')$ with $x'\leq x$ and $y'\leq y$. Let $A$ be the number of ways to choose $n$ blue points with distinct $x$-coordinates, and let $B$ be the number of ways to choose $n$ blue points with distinct $y$-coordinates. Prove that $A=B$.
2023 MMATHS, 2
$20$ players enter a chess tournament in which each player will play every other player exactly once. Some competitors are cheaters and will cheat in every game they play, but the rest of the competitors are not cheaters. A game is cheating if both players cheat, and a game is half-cheating if one player cheats and one player does not. If there were $68$ more half-cheating games than cheating games, how many of the players are cheaters?
2003 BAMO, 5
Let $ABCD$ be a square, and let $E$ be an internal point on side $AD$. Let $F$ be the foot of the perpendicular from $B$ to $CE$. Suppose $G$ is a point such that $BG = FG$, and the line through $G$ parallel to $BC$ passes through the midpoint of $EF$. Prove that $AC < 2 \cdot FG$.
2020 Caucasus Mathematical Olympiad, 2
Let $\omega_1$ and $\omega_2$ be two non-intersecting circles. Let one of its internal tangents touches $\omega_1$ and $\omega_2$ at $A_1$ and $A_2$, respectively, and let one of its external tangents touches $\omega_1$ and $\omega_2$ at $B_1$ and $B_2$, respectively. Prove that if $A_1B_2 = A_2B_1$, then $A_1B_2 \perp A_2B_1$.
MOAA Gunga Bowls, 2023.21
In obtuse triangle $ABC$ where $\angle B > 90^\circ$ let $H$ and $O$ be its orthocenter and circumcenter respectively. Let $D$ be the foot of the altitude from $A$ to $HC$ and $E$ be the foot of the altitude from $B$ to $AC$ such that $O,E,D$ lie on a line. If $OC=8$ and $OE=4$, find the area of triangle $HAB$.
[i]Proposed by Harry Kim[/i]
1997 China National Olympiad, 3
Prove that there are infinitely many natural numbers $n$ such that we can divide $1,2,\ldots ,3n$ into three sequences $(a_n),(b_n)$ and $(c_n)$, with $n$ terms in each, satisfying the following conditions:
i) $a_1+b_1+c_1= a_2+b_2+c_2=\ldots =a_n+b_n+c_n$ and $a_1+b_1+c_1$ is divisible by $6$;
ii) $a_1+a_2+\ldots +a_n= b_1+b_2+\ldots +b_n=c_1+c_2+\ldots +c_n,$ and $a_1+a_2+\ldots +a_n$ is divisible by $6$.
2010 Lithuania National Olympiad, 1
$a,b$ are real numbers such that:
\[ a^3+b^3=8-6ab. \]
Find the maximal and minimal value of $a+b$.
2010 Indonesia TST, 4
Prove that for all integers $ m$ and $ n$, the inequality
\[ \dfrac{\phi(\gcd(2^m \plus{} 1,2^n \plus{} 1))}{\gcd(\phi(2^m \plus{} 1),\phi(2^n \plus{} 1))} \ge \dfrac{2\gcd(m,n)}{2^{\gcd(m,n)}}\]
holds.
[i]Nanang Susyanto, Jogjakarta [/i]
2009 F = Ma, 15
A $\text{22.0 kg}$ suitcase is dragged in a straight line at a constant speed of $\text{1.10 m/s}$ across a level airport floor by a student on the way to the 40th IPhO in Merida, Mexico. The individual pulls with a $\text{1.00} \times \text{10}^2 \text{N}$ force along a handle with makes an upward angle of $\text{30.0}$ degrees with respect to the horizontal. What is the coefficient of kinetic friction between the suitcase and the floor?
(A) $\mu_\text{k} = \text{0.013}$
(B) $\mu_\text{k} = \text{0.394}$
(C) $\mu_\text{k} = \text{0.509}$
(D) $\mu_\text{k} = \text{0.866}$
(E) $\mu_\text{k} = \text{1.055}$
2015 Paraguay Mathematical Olympiad, 1
Alexa wrote the first $16$ numbers of a sequence:
\[1, 2, 2, 3, 4, 4, 5, 6, 6, 7, 8, 8, 9, 10, 10, 11, …\]
Then she continued following the same pattern, until she had $2015$ numbers in total.
What was the last number she wrote?
2009 Finnish National High School Mathematics Competition, 1
In a plane, the point $(x,y)$ has temperature $x^2+y^2-6x+4y$. Determine the coldest point of the plane and its temperature.
2015 IMC, 4
Determine whether or not there exist 15 integers $m_1,\ldots,m_{15}$
such that~
$$\displaystyle \sum_{k=1}^{15}\,m_k\cdot\arctan(k) = \arctan(16). \eqno(1)$$
(Proposed by Gerhard Woeginger, Eindhoven University of Technology)
1989 IMO Longlists, 12
Let $ P(x)$ be a polynomial such that the following inequalities are satisfied:
\[ P(0) > 0;\]\[ P(1) > P(0);\]\[ P(2) > 2P(1) \minus{} P(0);\]\[ P(3) > 3P(2) \minus{} 3P(1) \plus{} P(0);\]
and also for every natural number $ n,$ \[ P(n\plus{}4) > 4P(n\plus{}3) \minus{} 6P(n\plus{}2)\plus{}4P(n \plus{} 1) \minus{} P(n).\]
Prove that for every positive natural number $ n,$ $ P(n)$ is positive.
2021 Sharygin Geometry Olympiad, 7
The incircle of triangle $ABC$ centered at $I$ touches $CA,AB$ at points $E,F$ respectively. Let points $M,N$ of line $EF$ be such that $CM=CE$ and $BN=BF$. Lines $BM$ and $CN$ meet at point $P$. Prove that $PI$ bisects segment $MN$.
2010 Balkan MO Shortlist, C2
A grasshopper jumps on the plane from an integer point (point with integer coordinates) to another integer point according to the following rules: His first jump is of length $\sqrt{98}$, his second jump is of length $\sqrt{149}$, his next jump is of length $\sqrt{98}$, and so on, alternatively. What is the least possible odd number of moves in which the grasshopper could return to his starting point?