Found problems: 85335
2011 Kazakhstan National Olympiad, 2
Let $w$-circumcircle of triangle $ABC$ with an obtuse angle $C$ and $C '$symmetric point of point $C$ with respect to $AB$. $M$ midpoint of $AB$. $C'M$ intersects $w$ at $N$ ($C '$ between $M$ and $N$). Let $BC'$ second crossing point $w$ in $F$, and $AC'$ again crosses the $w$ at point $E$. $K$-midpoint $EF$. Prove that the lines $AB, CN$ and$ KC'$are concurrent.
PEN J Problems, 16
We say that an integer $m \ge 1$ is super-abundant if \[\frac{\sigma(m)}{m}>\frac{\sigma(k)}{k}\] for all $k \in \{1, 2,\cdots, m-1 \}$. Prove that there exists an infinite number of super-abundant numbers.
2016 Saudi Arabia IMO TST, 1
Define the sequence $a_1, a_2,...$ as follows: $a_1 = 1$, and for every $n \ge 2$, $a_n = n - 2$ if $a_{n-1} = 0$ and $a_n = a_{n-1} - 1$, otherwise. Find the number of $1 \le k \le 2016$ such that there are non-negative integers $r, s$ and a positive integer $n$ satisfying $k = r + s$ and $a_{n+r} = a_n + s$.
2020 MBMT, 29
The center of circle $\omega_1$ of radius $6$ lies on circle $\omega_2$ of radius $6$. The circles intersect at points $K$ and $W$. Let point $U$ lie on the major arc $\overarc{KW}$ of $\omega_2$, and point $I$ be the center of the largest circle that can be inscribed in $\triangle KWU$. If $KI+WI=11$, find $KI\cdot WI$.
[i]Proposed by Bradley Guo[/i]
2013 ELMO Shortlist, 2
Prove that for all positive reals $a,b,c$,
\[\frac{1}{a+\frac{1}{b}+1}+\frac{1}{b+\frac{1}{c}+1}+\frac{1}{c+\frac{1}{a}+1}\ge \frac{3}{\sqrt[3]{abc}+\frac{1}{\sqrt[3]{abc}}+1}. \][i]Proposed by David Stoner[/i]
XMO (China) 2-15 - geometry, 6.2
Assume that complex numbers $z_1,z_2,...,z_n$ satisfy $|z_i-z_j| \le 1$ for any $1 \le i <j \le n$. Let
$$S= \sum_{1 \le i <j \le n} |z_i-z_j|^2.$$
(1) If $n = 6063$, find the maximum value of $S$.
(2) If $n= 2021$, find the maximum value of $S$.
1976 IMO Longlists, 39
In $ ABC$, the inscribed circle is tangent to side $BC$ at$ X$. Segment $ AX$ is drawn. Prove that the line joining the midpoint of $ AX$ to the midpoint of side $ BC$ passes through center $ I$ of the inscribed circle.
2017-IMOC, G5
We have $\vartriangle ABC$ with $I$ as its incenter. Let $D$ be the intersection of $AI$ and $BC$ and define $E, F$ in a similar way. Furthermore, let $Y = CI \cap DE, Z = BI \cap DF$. Prove that if $\angle BAC = 120^o$, then $E, F, Y,Z$ are concyclic.
[img]https://1.bp.blogspot.com/-5IFojUbPE3o/XnSKTlTISqI/AAAAAAAALd0/0OwKMl02KJgqPs-SDOlujdcWXM0cWJiegCK4BGAYYCw/s1600/imoc2017%2Bg5.png[/img]
1998 Gauss, 25
Two natural numbers, $p$ and $q$, do not end in zero. The product of any pair, p and q, is a power of 10
(that is, $10, 100, 1000, 10 000$ , ...). If $p >q$, the last digit of $p – q$ cannot be
$\textbf{(A)}\ 1 \qquad \textbf{(B)}\ 3 \qquad \textbf{(C)}\ 5 \qquad \textbf{(D)}\ 7 \qquad \textbf{(E)}\ 9$
2022 APMO, 2
Let $ABC$ be a right triangle with $\angle B=90^{\circ}$. Point $D$ lies on the line $CB$ such that $B$ is between $D$ and $C$. Let $E$ be the midpoint of $AD$ and let $F$ be the seconf intersection point of the circumcircle of $\triangle ACD$ and the circumcircle of $\triangle BDE$. Prove that as $D$ varies, the line $EF$ passes through a fixed point.
1995 Moldova Team Selection Test, 4
Find all functions $f:\mathbb{Z}\rightarrow \mathbb{Z}$ satisfying the following:
$i)$ $f(1)=1$;
$ii)$ $f(m+n)(f(m)-f(n))=f(m-n)(f(m)+f(n))$ for all $m,n \in \mathbb{Z}$.
2014 Harvard-MIT Mathematics Tournament, 7
Find the largest real number $c$ such that \[\sum_{i=1}^{101}x_i^2\geq cM^2\] whenever $x_1,\ldots,x_{101}$ are real numbers such that $x_1+\cdots+x_{101}=0$ and $M$ is the median of $x_1,\ldots,x_{101}$.
1989 National High School Mathematics League, 6
Set $M=\{u|u=12m+8n+4l,m,n,l\in\mathbb{Z}\},N=\{u|u=20p+16q+12x,p,q,x\in\mathbb{Z}\}$. Then
$\text{(A)}M=N\qquad\text{(B)}M\not\subset N,N\not\subset M\qquad\text{(C)}M\subset N\qquad\text{(D)}N\subset M$
2020 Jozsef Wildt International Math Competition, W3
Let $n \geq 2$ be an integer. Calculate$$\int \limits_{0}^{\frac{\pi}{2}}\frac{\sin x}{\sin^{2n-1}x+\cos^{2n-1}x}dx$$
2014 Singapore Senior Math Olympiad, 18
Given that in the expansion of $(2+3x)^n$, the coefficients of $x^3$ and $x^4$ are in the ratio $8:15$. Find the value of $n$.
2024 Taiwan TST Round 2, 2
Let $n$ be a positive integer. Prove that the inequality
\[n \sum_{i=1}^n \sum_{j = 1}^n \sum_{k=1}^n \frac{3}{a_ja_k + a_ka_i + a_i a_j} \ge \left(\sum_{j=1}^n \sum_{k=1}^n \frac{2}{a_j + a_k}\right)^2 \]
holds for any positive real numbers $a_1$, $a_2$, $\dots$, $a_n$.
[i]Proposed by Li4 and Ming Hsiao.[/i]
2011 Sharygin Geometry Olympiad, 2
Quadrilateral $ABCD$ is circumscribed. Its incircle touches sides $AB, BC, CD, DA$ in points $K, L, M, N$ respectively. Points $A', B', C', D'$ are the midpoints of segments $LM, MN, NK, KL$. Prove that the quadrilateral formed by lines $AA', BB', CC', DD'$ is cyclic.
2020 Adygea Teachers' Geometry Olympiad, 1
In planimetry, criterions of congruence of triangles with two sides and a larger angle, with two sides and the median drawn to the third side are known. Is it true that two triangles are congruent if they have two sides equal and the height drawn to the third side?
2016 Switzerland - Final Round, 10
Find all functions $f : R \to R$ such that for all $x, y \in R$:
$$f(x + yf(x + y)) = y^2 + f(xf(y + 1)).$$
2019 IMC, 8
Let $x_1,\ldots,x_n$ be real numbers. For any set $I\subset\{1,2,…,n\}$ let $s(I)=\sum_{i\in I}x_i$. Assume that the function $I\to s(I)$ takes on at least $1.8^n$ values where $I$ runs over all $2^n$ subsets of $\{1,2,…,n\}$. Prove that the number of sets $I\subset \{1,2,…,n\}$ for which $s(I)=2019$ does not exceed $1.7^n$.
[i]Proposed by Fedor Part and Fedor Petrov, St. Petersburg State University[/i]
2001 Denmark MO - Mohr Contest, 1
For the Georg Mohr game, a playing piece is used, a Georg Mohr cube (i.e. a die whose six sides show the letters G, E, O, R, M and H) as well as a game board:
[img]https://cdn.artofproblemsolving.com/attachments/0/9/30ca5cd2579bfcc1d702b40f3ef58916ac768f.png[/img]
With each stroke, you advance to the next field with that letter the cube shows; if it is not possible to advance, one remains standing. Peter playing the georg mohr game. Determine the probability that he completes played in two strokes.
2017 Romanian Masters In Mathematics, 2
Determine all positive integers $n$ satisfying the following condition: for every monic polynomial $P$ of degree at most $n$ with integer coefficients, there exists a positive integer $k\le n$ and $k+1$ distinct integers $x_1,x_2,\cdots ,x_{k+1}$ such that \[P(x_1)+P(x_2)+\cdots +P(x_k)=P(x_{k+1})\].
[i]Note.[/i] A polynomial is [i]monic[/i] if the coefficient of the highest power is one.
2003 AIME Problems, 14
Let $A=(0,0)$ and $B=(b,2)$ be points on the coordinate plane. Let $ABCDEF$ be a convex equilateral hexagon such that $\angle FAB=120^\circ,$ $\overline{AB}\parallel \overline{DE},$ $\overline{BC}\parallel \overline{EF,}$ $\overline{CD}\parallel \overline{FA},$ and the y-coordinates of its vertices are distinct elements of the set $\{0,2,4,6,8,10\}.$ The area of the hexagon can be written in the form $m\sqrt{n},$ where $m$ and $n$ are positive integers and n is not divisible by the square of any prime. Find $m+n.$
2005 MOP Homework, 2
Let $x$, $y$, $z$ be positive real numbers and $x+y+z=1$. Prove that
$\sqrt{xy+z}+\sqrt{yz+x}+\sqrt{zx+y} \ge 1+\sqrt{xy}+\sqrt{yz}+\sqrt{zx}$.
2009 German National Olympiad, 3
Let $ ABCD$ be a (non-degenerate) quadrangle and $ N$ the intersection of $ AC$ and $ BD$. Denote by $ a,b,c,d$ the length of the altitudes from $ N$ to $ AB,BC,CD,DA$, respectively.
Prove that $ \frac{1}{a}\plus{}\frac{1}{c} \equal{} \frac{1}{b}\plus{}\frac{1}{d}$ if $ ABCD$ has an incircle.
Extension: Prove that the converse is true, too.
[If this has already been posted, I humbly apologize. A quick search turned up nothing.]