Found problems: 85335
2019 JBMO Shortlist, C1
Let $S$ be a set of $100$ positive integer numbers having the following property:
“Among every four numbers of $S$, there is a number which divides each of the other three
or there is a number which is equal to the sum of the other three.”
Prove that the set $S$ contains a number which divides all other $99$ numbers of $S$.
[i]Proposed by Tajikistan[/i]
1983 Brazil National Olympiad, 5
Show that $1 \le n^{1/n} \le 2$ for all positive integers $n$.
Find the smallest $k$ such that $1 \le n ^{1/n} \le k$ for all positive integers $n$.
KoMaL A Problems 2022/2023, A. 853
Let points $A, B, C, A', B', C'$ be chosen in the plane such that no three of them are collinear, and let lines $AA'$, $BB'$ and $CC'$ be tangent to a given equilateral hyperbola at points $A$, $B$ and $C$, respectively. Assume that the circumcircle of $A'B'C'$ is the same as the nine-point circle of triangle $ABC$. Let $s(A')$ be the Simson line of point $A'$ with respect to the orthic triangle of $ABC$. Let $A^*$ be the intersection of line $B'C'$ and the perpendicular on $s(A')$ from the point $A$. Points $B^*$ and $C^*$ are defined in a similar manner. Prove that points $A^*$, $B^*$ and $C^*$ are collinear.
[i]Submitted by Áron Bán-Szabó, Budapest[/i]
2022 Saint Petersburg Mathematical Olympiad, 5
Let $n$ be a positive integer and let $a_1, a_2, \cdots a_k$ be all numbers less than $n$ and coprime to $n$ in increasing order. Find the set of values the function $f(n)=gcd(a_1^3-1, a_2^3-1, \cdots, a_k^3-1)$.
2017 IMO Shortlist, A1
Let $a_1,a_2,\ldots a_n,k$, and $M$ be positive integers such that
$$\frac{1}{a_1}+\frac{1}{a_2}+\cdots+\frac{1}{a_n}=k\quad\text{and}\quad a_1a_2\cdots a_n=M.$$
If $M>1$, prove that the polynomial
$$P(x)=M(x+1)^k-(x+a_1)(x+a_2)\cdots (x+a_n)$$
has no positive roots.
2007 ITest, 49
How many 7-element subsets of $\{1, 2, 3,\ldots , 14\}$ are there, the sum of whose elements is divisible by $14$?
2003 Junior Macedonian Mathematical Olympiad, Problem 2
There are $2003$ coins distributed in several bags. The bags are then distributed in several pockets. It is known that the total number of bags is greater than the number of coins in each of the pockets. Is it true that the total number of pockets is greater than the number of coins in some of the bags?
2014 USAMTS Problems, 2:
Let $a, b, c, x$ and $y$ be positive real numbers such that $ax + by \leq bx + cy \leq cx + ay$.
Prove that $b \leq c$.
2012 USAMO, 1
Find all integers $n \geq 3$ such that among any $n$ positive real numbers $a_1, a_2, \hdots, a_n$ with $\text{max}(a_1,a_2,\hdots,a_n) \leq n \cdot \text{min}(a_1,a_2,\hdots,a_n)$, there exist three that are the side lengths of an acute triangle.
2024 Malaysian Squad Selection Test, 8
Given a triangle $ABC$, let $I$ be the incenter, and $J$ be the $A$-excenter. A line $\ell$ through $A$ perpendicular to $BC$ intersect the lines $BI$, $CI$, $BJ$, $CJ$ at $P$, $Q$, $R$, $S$ respectively. Suppose the angle bisector of $\angle BAC$ meet $BC$ at $K$, and $L$ is a point such that $AL$ is a diameter in $(ABC)$.
Prove that the line $KL$, $\ell$, and the line through the centers of circles $(IPQ)$ and $(JRS)$, are concurrent.
[i]Proposed by Chuah Jia Herng & Ivan Chan Kai Chin[/i]
VI Soros Olympiad 1999 - 2000 (Russia), 11.3
The numbers $a, b$ and $c$ are such that $a^2 + b^2 + c^2 = 1$. Prove that $$a^4 + b^4 + c^4 + 2(ab^2 + bc^2 + ca^2)^2\le 1. $$ At what $a, b$ and $c$ does inequality turn into equality?
2023 ELMO Shortlist, N2
Determine the greatest positive integer \(n\) for which there exists a sequence of distinct positive integers \(s_1\), \(s_2\), \(\ldots\), \(s_n\) satisfying \[s_1^{s_2}=s_2^{s_3}=\cdots=s_{n-1}^{s_n}.\]
[i]Proposed by Holden Mui[/i]
2008 Philippine MO, 4
Let $f:\mathbb{R}\rightarrow \mathbb{R}$ be a function defined by $f(x)=\frac{2008^{2x}}{2008+2008^{2x}}$. Prove that
\[\begin{aligned}
f\left(\frac{1}{2007}\right)+f\left(\frac{2}{2007}\right)+\cdots+f\left(\frac{2005}{2007}\right)+f\left(\frac{2006}{2007}\right)=1003.
\end{aligned}\]
2013 International Zhautykov Olympiad, 1
A quadratic trinomial $p(x)$ with real coefficients is given. Prove that there is a positive integer $n$ such that the equation $p(x) = \frac{1}{n}$ has no rational roots.
1953 Putnam, A3
$a, b, c$ are real, and the sum of any two is greater than the third.
Show that $\frac{2(a + b + c)(a^2 + b^2 + c^2)}{3} > a^3 + b^3 + c^3 + abc$.
2020 BMT Fall, 8
Let triangle $ \vartriangle ABC$ have $AB = 17$, $BC = 14$, $CA = 12$. Let $M_A$, $M_B$, $M_C$ be midpoints of $\overline{BC}$, $\overline{AC}$, and $\overline{AB}$ respectively. Let the angle bisectors of $ A$, $ B$, and $C$ intersect $\overline{BC}$, $\overline{AC}$, and $\overline{AB}$ at $P$, $Q$, and $R$, respectively. Reflect $M_A$ about $\overline{AP}$, $M_B$ about $\overline{BQ}$, and $M_C$ about $\overline{CR}$ to obtain $M'_A$, $M'_B$, $M'_C$, respectively. The lines $AM'_A$, $BM'_B$, and $CM'_C$ will then intersect $\overline{BC}$, $\overline{AC}$, and $\overline{AB}$ at $D$, $E$, and $F$, respectively. Given that $\overline{AD}$, $\overline{BE}$, and $\overline{CF}$ concur at a point $K$ inside the triangle, in simplest form, the ratio $[KAB] : [KBC] : [KCA]$ can be written in the form $p : q : r$, where $p$, $q$ and $ r$ are relatively prime positive integers and $[XYZ]$ denotes the area of $\vartriangle XYZ$. Compute $p + q + r$.
2009 Baltic Way, 17
Find the largest integer $n$ for which there exist $n$ different integers such that none of them are divisible by either of $7,11$ or $13$, but the sum of any two of them is divisible by at least one of $7,11$ and $13$.
2019 CMI B.Sc. Entrance Exam, 5
Three positive reals $x , y , z $ satisfy \\
$x^2 + y^2 = 3^2 \\
y^2 + yz + z^2 = 4^2 \\
x^2 + \sqrt{3}xz + z^2 = 5^2 .$ \\
Find the value of $2xy + xz + \sqrt{3}yz$
2008 Indonesia MO, 2
Prove that for $ x,y\in\mathbb{R^ \plus{} }$,
$ \frac {1}{(1 \plus{} \sqrt {x})^{2}} \plus{} \frac {1}{(1 \plus{} \sqrt {y})^{2}} \ge \frac {2}{x \plus{} y \plus{} 2}$
1999 Mongolian Mathematical Olympiad, Problem 5
Let $D$ be a point in the angle $ABC$. A circle $\gamma$ passing through $B$ and $D$ intersects the lines $AB$ and $BC$ at $M$ and $N$ respectively. Find the locus of the midpoint of $MN$ when circle $\gamma$ varies.
2010 Sharygin Geometry Olympiad, 22
A circle centered at a point $F$ and a parabola with focus $F$ have two common points. Prove that there exist four points $A, B, C, D$ on the circle such that the lines $AB, BC, CD$ and $DA$ touch the parabola.
2005 Junior Balkan MO, 4
Find all 3-digit positive integers $\overline{abc}$ such that \[ \overline{abc} = abc(a+b+c) , \] where $\overline{abc}$ is the decimal representation of the number.
2017 NIMO Problems, 3
Suppose there exist constants $A$, $B$, $C$, and $D$ such that \[n^4=A\binom n4+B\binom n3+C\binom n2 + D\binom n1\] holds true for all positive integers $n\geq 4$. What is $A+B+C+D$?
[i]Proposed by David Altizio[/i]
1965 Dutch Mathematical Olympiad, 2
Prove that $S_1 = (n + 1)^2 + (n + 2)^2 +...+ (n + 5)^2$ is divisible by $5$ for every $n$.
Prove that for no $n$: $\sum_{\ell=1}^5 (n+\ell)^2$ is a perfect square.
Let $S_2=(n + 6)^2 + (n + 7)^2 + ... + (n + 10)^2$. Prove that $S_1 \cdot S_2$ is divisible by $150$.
2022 Romania National Olympiad, P3
Let $Z\subset \mathbb{C}$ be a set of $n$ complex numbers, $n\geqslant 2.$ Prove that for any positive integer $m$ satisfying $m\leqslant n/2$ there exists a subset $U$ of $Z$ with $m$ elements such that\[\Bigg|\sum_{z\in U}z\Bigg|\leqslant\Bigg|\sum_{z\in Z\setminus U}z\Bigg|.\][i]Vasile Pop[/i]