Found problems: 85335
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.]
1988 China Team Selection Test, 1
Let $f(x) = 3x + 2.$ Prove that there exists $m \in \mathbb{N}$ such that $f^{100}(m)$ is divisible by $1988$.
2021 HMNT, 9
Let $N$ be the smallest positive integer for which $x^2 + x + 1$ divides $166 -\sum_{d|N, d>0} x^d$. Find the remainder when $N$ is divided by $1000$.
2011 Baltic Way, 3
A sequence $a_1,a_2,a_3,\ldots $ of non-negative integers is such that $a_{n+1}$ is the last digit of $a_n^n+a_{n-1}$ for all $n>2$. Is it always true that for some $n_0$ the sequence $a_{n_0},a_{n_0+1},a_{n_0+2},\ldots$ is periodic?
2024 ELMO Shortlist, G6
In triangle $ABC$ with $AB<AC$ and $AB+AC=2BC$, let $M$ be the midpoint of $\overline{BC}$. Choose point $P$ on the extension of $\overline{BA}$ past $A$ and point $Q$ on segment $\overline{AC}$ such that $M$ lies on $\overline{PQ}$. Let $X$ be on the opposite side of $\overline{AB}$ from $C$ such that $\overline{AX} \parallel \overline{BC}$ and $AX=AP=AQ$. Let $\overline{BX}$ intersect the circumcircle of $BMQ$ again at $Y \neq B$, and let $\overline{CX}$ intersect the circumcircle of $CMP$ again at $Z \neq C$. Prove that $A$, $Y$, and $Z$ are collinear.
[i]Tiger Zhang[/i]
1988 Kurschak Competition, 1
Prove that if there exists a point $P$ inside the convex quadrilateral $ABCD$ such that the triangles $PAB$, $PBC$, $PCD$, $PDA$ have the same area, then one of the diagonals of $ABCD$ bisects the area of the quadrilateral.
2019 South East Mathematical Olympiad, 6
In $\triangle ABC$, $AB>AC$, the bisectors of $\angle ABC, \angle ACB$ meet sides $AC,AB$ at $D,E$ respectively. The tangent at $A$ to the circumcircle of $\triangle ABC$ intersects $ED$ extended at $P$. Suppose that $AP=BC$. Prove that $BD\parallel CP$.
1960 AMC 12/AHSME, 24
If $\log_{2x}216 = x$, where $x$ is real, then $x$ is:
$ \textbf{(A)}\ \text{A non-square, non-cube integer} \qquad$
$\textbf{(B)}\ \text{A non-square, non-cube, non-integral rational number} \qquad$
$\textbf{(C)}\ \text{An irrational number} \qquad$
$\textbf{(D)}\ \text{A perfect square}\qquad$
$\textbf{(E)}\ \text{A perfect cube} $
2022-2023 OMMC, 25
A clock has a second, minute, and hour hand. A fly initially rides on the second hand of the clock starting at noon. Every time the hand the fly is currently riding crosses with another, the fly will then switch to riding the other hand. Once the clock strikes midnight, how many revolutions has the fly taken?
$\emph{(Observe that no three hands of a clock coincide between noon and midnight.)}$
1995 Romania Team Selection Test, 4
A convex set $S$ on a plane, not lying on a line, is painted in $p$ colors.
Prove that for every $n \ge 3$ there exist infinitely many congruent $n$-gons whose vertices are of the same color.
2022 AIME Problems, 11
Let $ABCD$ be a convex quadrilateral with $AB=2, AD=7,$ and $CD=3$ such that the bisectors of acute angles $\angle{DAB}$ and $\angle{ADC}$ intersect at the midpoint of $\overline{BC}.$ Find the square of the area of $ABCD.$
1991 Arnold's Trivium, 54
$\ddot{x}=3x-x^3-1$. In which of the potential wells is the period of oscillation greater (in the shallower or the deeper) with equal values of the total energy?
2022 Grosman Mathematical Olympiad, P1
For each positive integer $n$ denote:
\[n!=1\cdot 2\cdot 3\dots n\]
Find all positive integers $n$ for which $1!+2!+3!+\cdots+n!$ is a perfect square.
2016 Online Math Open Problems, 21
Say a real number $r$ is \emph{repetitive} if there exist two distinct complex numbers $z_1,z_2$ with $|z_1|=|z_2|=1$ and $\{z_1,z_2\}\neq\{-i,i\}$ such that
\[
z_1(z_1^3+z_1^2+rz_1+1)=z_2(z_2^3+z_2^2+rz_2+1).
\]
There exist real numbers $a,b$ such that a real number $r$ is \emph{repetitive} if and only if $a < r\le b$. If the value of $|a|+|b|$ can be expressed in the form $\frac{p}{q}$ for relatively prime positive integers $p$ and $q$, find $100p+q$.
[i]Proposed by James Lin[/i]
KoMaL A Problems 2018/2019, A. 749
Given are two polyominos, the first one is an L-shape consisting of three squares, the other one contains at least two squares. Prove that if $n$ and $m$ are coprime then at most one of the $n\times n$ and $m\times m$ boards can be tiled by translated copies of the two polyominos.
[i]Proposed by: András Imolay, Dávid Matolcsi, Ádám Schweitzer and Kristóf Szabó, Budapest[/i]
2022 All-Russian Olympiad, 3
Initially, a pair of numbers $(1,1)$ is written on the board. If for some $x$ and $y$ one of the pairs $(x, y-1)$ and $(x+y, y+1)$ is written on the board, then you can add the other one. Similarly for $(x, xy)$ and $(\frac {1} {x}, y)$. Prove that for each pair that appears on the board, its first number will be positive.
2019 Iran MO (3rd Round), 1
$a,b$ and $c$ are positive real numbers so that $\sum_{\text{cyc}} (a+b)^2=2\sum_{\text{cyc}} a +6abc$. Prove that
$$\sum_{\text{cyc}} (a-b)^2\leq\left|2\sum_{\text{cyc}} a -6abc\right|.$$
2019 China Team Selection Test, 4
Find all functions $f: \mathbb{R}^2 \rightarrow \mathbb{R}$, such that
1) $f(0,x)$ is non-decreasing ;
2) for any $x,y \in \mathbb{R}$, $f(x,y)=f(y,x)$ ;
3) for any $x,y,z \in \mathbb{R}$, $(f(x,y)-f(y,z))(f(y,z)-f(z,x))(f(z,x)-f(x,y))=0$ ;
4) for any $x,y,a \in \mathbb{R}$, $f(x+a,y+a)=f(x,y)+a$ .
2010 Contests, 2
There are $100$ random, distinct real numbers corresponding to $100$ points on a circle. Prove that you can always choose $4$ consecutive points in such a way that the sum of the two numbers corresponding to the points on the outside is always greater than the sum of the two numbers corresponding to the two points on the inside.
2005 China Team Selection Test, 2
In acute angled triangle $ABC$, $BC=a$,$CA=b$,$AB=c$, and $a>b>c$. $I,O,H$ are the incentre, circumcentre and orthocentre of $\triangle{ABC}$ respectively. Point $D \in BC$, $E \in CA$ and $AE=BD$, $CD+CE=AB$. Let the intersectionf of $BE$ and $AD$ be $K$. Prove that $KH \parallel IO$ and $KH = 2IO$.