Found problems: 85335
2012 Belarus Team Selection Test, 3
Let $ABC$ be an acute triangle. Let $\omega$ be a circle whose centre $L$ lies on the side $BC$. Suppose that $\omega$ is tangent to $AB$ at $B'$ and $AC$ at $C'$. Suppose also that the circumcentre $O$ of triangle $ABC$ lies on the shorter arc $B'C'$ of $\omega$. Prove that the circumcircle of $ABC$ and $\omega$ meet at two points.
[i]Proposed by Härmel Nestra, Estonia[/i]
2008 Romania Team Selection Test, 1
Let $ ABCD$ be a convex quadrilateral and let $ O \in AC \cap BD$, $ P \in AB \cap CD$, $ Q \in BC \cap DA$. If $ R$ is the orthogonal projection of $ O$ on the line $ PQ$ prove that the orthogonal projections of $ R$ on the sidelines of $ ABCD$ are concyclic.
2025 Belarusian National Olympiad, 10.6
For a sequence of zeros and ones Vasya makes move of the following form until the process doesn't halt
1. If the first digit of the sequence is zero, this digit is erased.
2. If the first digit is one and there are at least two digits, Vasya swaps first two digits, reverses the sequence and replaces ones with zeros and zeros with ones.
3. If the sequence is empty or consists of a single one, the process stops.
Find the number of sequences of length 2025 starting from which Vasya can get to the empty sequence.
[i]M. Zorka[/i]
MBMT Team Rounds, 2020.12
Find the number of ways to partition $S = \{1, 2, 3, \dots, 2020\}$ into two disjoint sets $A$ and $B$ with $A \cup B = S$ so that if you choose an element $a$ from $A$ and an element $b$ from $B$, $a+b$ is never a multiple of $20$. $A$ or $B$ can be the empty set, and the order of $A$ and $B$ doesn't matter. In other words, the pair of sets $(A,B)$ is indistinguishable from the pair of sets $(B,A)$.
[i]Proposed by Timothy Qian[/i]
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.