Found problems: 85335
1966 All Russian Mathematical Olympiad, 083
$20$ numbers are written on the board $1, 2, ... ,20$. Two players are putting signs before the numbers in turn ($+$ or $-$). The first wants to obtain the minimal possible absolute value of the sum. What is the maximal value of the absolute value of the sum that can be achieved by the second player?
2009 China Team Selection Test, 4
Let positive real numbers $ a,b$ satisfy $ b \minus{} a > 2.$ Prove that for any two distinct integers $ m,n$ belonging to $ [a,b),$ there always exists non-empty set $ S$ consisting of certain integers belonging to $ [ab,(a \plus{} 1)(b \plus{} 1))$ such that $ \frac {\displaystyle\prod_{x\in S}}{mn}$ is square of a rational number.
PEN M Problems, 19
A sequence with first two terms equal $1$ and $24$ respectively is defined by the following rule: each subsequent term is equal to the smallest positive integer which has not yet occurred in the sequence and is not coprime with the previous term. Prove that all positive integers occur in this sequence.
2023 MMATHS, 5
We call $\triangle{ABC}$ with centroid $G$ [i]balanced[/i] on side $AB$ if the foot of the altitude from $G$ onto line $\overline{AB}$ lies between $A$ and $B.$ $\triangle{XYZ},$ with $XY=2023$ and $\angle{ZXY}=120^\circ,$ is balanced on $XY.$ What is the maximum value of $XZ$?
2000 Stanford Mathematics Tournament, 13
How many permutations of $123456$ have exactly one number in the correct place?
2020 Romanian Master of Mathematics Shortlist, A2
Let $n>1$ be a positive integer and $\mathcal S$ be the set of $n^{\text{th}}$ roots of unity. Suppose $P$ is an $n$-variable polynomial with complex coefficients such that for all $a_1,\ldots,a_n\in\mathcal S$, $P(a_1,\ldots,a_n)=0$ if and only if $a_1,\ldots,a_n$ are all different. What is the smallest possible degree of $P$?
[i]Adam Ardeishar and Michael Ren[/i]
2016 European Mathematical Cup, 2
Two circles $C_{1}$ and $C_{2}$ intersect at points $A$ and $B$. Let $P$, $Q$ be points on circles $C_{1}$, $C_{2}$ respectively, such that $|AP| = |AQ|$. The segment $PQ$ intersects circles $C_{1}$ and $C_{2}$ in points $M$, $N$ respectively. Let $C$ be the center of the arc $BP$ of $C_{1}$ which does not contain point $A$ and let $D$ be the center of arc $BQ$ of $C_{2}$ which does not contain point $A$ Let $E$ be the intersection of $CM$ and $DN$. Prove that $AE$ is perpendicular to $CD$.
Proposed by Steve Dinh
2018 Sharygin Geometry Olympiad, 17
Let each of circles $\alpha, \beta, \gamma$ touches two remaining circles externally, and all of them touche a circle $\Omega$ internally at points $A_1, B_1, C_1$ respectively. The common internal tangent to $\alpha$ and $\beta$ meets the arc $A_1B_1$ not containing $C_1$ at point $C_2$. Points $A_2$, $B_2$ are defined similarly. Prove that the lines $A_1A_2, B_1B_2, C_1C_2$ concur.
2003 Korea - Final Round, 3
There are $n$ distinct points on a circumference. Choose one of the points. Connect this point and the $m$th point from the chosen point counterclockwise with a segment. Connect this $m$th point and the $m$th point from this $m$th point counterclockwise with a segment. Repeat such steps until no new segment is constructed. From the intersections of the segments, let the number of the intersections - which are in the circle - be $I$. Answer the following questions ($m$ and $n$ are positive integers that are relatively prime and they satisfy $6 \leq 2m < n$).
1) When the $n$ points take different positions, express the maximum value of $I$ in terms of $m$ and $n$.
2) Prove that $I \geq n$. Prove that there is a case, which is $I=n$, when $m=3$ and $n$ is arbitrary even number that satisfies the condition.
Kvant 2020, M2603
For an infinite sequence $a_1, a_2,. . .$ denote as it's [i]first derivative[/i] is the sequence $a'_n= a_{n + 1} - a_n$ (where $n = 1, 2,..$.), and her $k$- th derivative as the first derivative of its $(k-1)$-th derivative ($k = 2, 3,...$). We call a sequence [i]good[/i] if it and all its derivatives consist of positive numbers.
Prove that if $a_1, a_2,. . .$ and $b_1, b_2,. . .$ are good sequences, then sequence $a_1\cdot b_1, a_2 \cdot b_2,..$ is also a good one.
R. Salimov
2017 AMC 12/AHSME, 17
There are 24 different complex numbers $z$ such that $z^{24} = 1$. For how many of these is $z^6$ a real number?
$\textbf{(A) }1\qquad\textbf{(B) }3\qquad\textbf{(C) }6\qquad\textbf{(D) }12\qquad\textbf{(E) }24$
1978 Chisinau City MO, 163
On the plane $n$ points are selected that do not belong to one straight line. Prove that the shortest closed path passing through all these points is a non-self-intersecting polygon.
2004 VJIMC, Problem 2
Find all functions $f:\mathbb R_{\ge0}\times\mathbb R_{\ge0}\to\mathbb R_{\ge0}$ such that
$1$. $f(x,0)=f(0,x)=x$ for all $x\in\mathbb R_{\ge0}$,
$2$. $f(f(x,y),z)=f(x,f(y,z))$ for all $x,y,z\in\mathbb R_{\ge0}$ and
$3$. there exists a real $k$ such that $f(x+y,x+z)=kx+f(y,z)$ for all $x,y,z\in\mathbb R_{\ge0}$.
2021 CMIMC, 2
You are initially given the number $n=1$. Each turn, you may choose any positive divisor $d\mid n$, and multiply $n$ by $d+1$. For instance, on the first turn, you must select $d=1$, giving $n=1\cdot(1+1)=2$ as your new value of $n$. On the next turn, you can select either $d=1$ or $2$, giving $n=2\cdot(1+1)=4$ or $n=2\cdot(2+1)=6$, respectively, and so on.
Find an algorithm that, in at most $k$ steps, results in $n$ being divisible by the number $2021^{2021^{2021}} - 1$.
An algorithm that completes in at most $k$ steps will be awarded:
1 pt for $k>2021^{2021^{2021}}$
20 pts for $k=2021^{2021^{2021}}$
50 pts for $k=10^{10^4}$
75 pts for $k=10^{10}$
90 pts for $k=10^5$
95 pts for $k=6\cdot10^4$
100 pts for $k=5\cdot10^4$
2014 National Olympiad First Round, 27
Let $f$ be a function defined on positive integers such that $f(1)=4$, $f(2n)=f(n)$ and $f(2n+1)=f(n)+2$ for every positive integer $n$. For how many positive integers $k$ less than $2014$, it is $f(k)=8$?
$
\textbf{(A)}\ 45
\qquad\textbf{(B)}\ 120
\qquad\textbf{(C)}\ 165
\qquad\textbf{(D)}\ 180
\qquad\textbf{(E)}\ 215
$
1999 Romania National Olympiad, 3
Let $a,b,c \in \mathbb{C}$ and $a \neq 0$. The roots $z_1$ and $z_2$ of the equation $az^2+bz+c=0$ satisfy $|z_1|<1$ and $|z_2|<1$. Prove that the roots $z_3$ and $z_4$ of the equation $$(a+\overline{c})z^2+(b+\overline{b})z+\overline{a}+c=0$$
satisfy $|z_3|=|z_4|=1$
JOM 2014, 3.
There is a complete graph $G$ with $4027$ vertices drawn on the whiteboard. Ivan paints all the edges by red or blue colour. Find all ordered pairs $(r, b)$ such that Ivan can paint the edges so that every vertex is connected to exactly $r$ red edges and $b$ blue edges.
2023 Philippine MO, 4
In chess, a knight placed on a chess board can move by jumping to an adjacent square in one direction (up, down, left, or right) then jumping to the next two squares in a perpendicular direction. We then say that a square in a chess board [i]can be attacked[/i] by a knight if the knight can end up on that square after a move. Thus, depending on where a knight is placed, it can attack as many as eight squares, or maybe even less.
In a $10 \times 10$ chess board, what is the maximum number of knights that can be placed such that each square on the board can be attacked by at most one knight?
2007 Princeton University Math Competition, 2
Find the largest integer $n$ which equals the product of its leading digit and the sum of its digits.
2008 Princeton University Math Competition, B7
In this problem, we consider only polynomials with integer coeffients. Call two polynomials $p$ and $q$ [i]really close[/i] if $p(2k + 1) \equiv q(2k + 1)$ (mod $210$) for all $k \in Z^+$. Call a polynomial $p$ [i]partial credit[/i] if no polynomial of lesser degree is [i]really close[/i] to it. What is the maximum possible degree of [i]partial credit[/i]?
1982 IMO Longlists, 21
Al[u][b]l[/b][/u] edges and all diagonals of regular hexagon $A_1A_2A_3A_4A_5A_6$ are colored blue or red such that each triangle $A_jA_kA_m, 1 \leq j < k < m\leq 6$ has at least one red edge. Let $R_k$ be the number of red segments $A_kA_j, (j \neq k)$. Prove the inequality
\[\sum_{k=1}^6 (2R_k-7)^2 \leq 54.\]
1939 Moscow Mathematical Olympiad, 048
Factor $a^{10} + a^5 + 1$ into nonconstant polynomials with integer coefficients
2006 Abels Math Contest (Norwegian MO), 2
a) Let $a$ and $b$ be two non-negative real numbers. Show that $a+b \ge \sqrt{\frac{a^2+b^2}{2}}+ \sqrt{ab}$
b) Let $a$ and $b$ be two real numbers in $[0, 3]$. Show that $\sqrt{\frac{a^2+b^2}{2}}+ \sqrt{ab} \ge \frac{(a+b)^2}{2}$
2018 Harvard-MIT Mathematics Tournament, 8
Let $ABC$ be an equilateral triangle with side length $8.$ Let $X$ be on side $AB$ so that $AX=5$ and $Y$ be on side $AC$ so that $AY=3.$ Let $Z$ be on side $BC$ so that $AZ,BY,CX$ are concurrent. Let $ZX,ZY$ intersect the circumcircle of $AXY$ again at $P,Q$ respectively. Let $XQ$ and $YP$ intersect at $K.$ Compute $KX\cdot KQ.$
2022 Tuymaada Olympiad, 2
Two circles $w_{1}$ and $w_{2}$ of different radii touch externally at $L$. A line touches $w_{1}$ at $A$ and $w_{2}$ at $B$ (the points $A$ and $B$ are different from $L$). A point $X$ is chosen in the plane. $Y$ and $Z$ are the second points of intersection of the lines $XA$ and $XB$ with $w_{1}$ and $w_{2}$ respectively. Prove that all $X$ such that $AB||Y Z$ belong to one circle.