Found problems: 85335
2010 All-Russian Olympiad, 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.
2020 Argentina National Olympiad Level 2, 4
Juli has a deck of $54$ cards and proposes the following game to Bruno. Juli places the cards in a row, some face-up and others face-down. Bruno can repeatedly perform the following move: select a card and flip it along with its two neighbors (turning face-up cards face-down, and vice versa for face-down cards). Bruno wins if, through this process, he manages to turn all the cards face up. Otherwise, Juli wins. Determine which player has a winning strategy and explain it.
[b]Note:[/b] When Bruno selects the first or the last card in the row, he flips only two cards. In all other cases, he flips three cards.
2021 Regional Olympiad of Mexico Southeast, 2
Let $n\geq 2021$. Let $a_1<a_2<\cdots<a_n$ an arithmetic sequence such that $a_1>2021$ and $a_i$ is a prime number for all $1\leq i\leq n$. Prove that for all $p$ prime with $p<2021, p$ divides the diference of the arithmetic sequence.
2022 Bosnia and Herzegovina BMO TST, 3
Cyclic quadrilateral $ABCD$ is inscribed in circle $k$ with center $O$. The angle bisector of $ABD$ intersects $AD$ and $k$ in $K,M$ respectively, and the angle bisector of $CBD$ intersects $CD$ and $k$ in $L,N$ respectively. If $KL\parallel MN$ prove that the circumcircle of triangle $MON$ bisects segment $BD$.
2010 Purple Comet Problems, 20
How many of the rearrangements of the digits $123456$ have the property that for each digit, no more than two digits smaller than that digit appear to the right of that digit? For example, the rearrangement $315426$ has this property because digits $1$ and $2$ are the only digits smaller than $3$ which follow $3,$ digits $2$ and $4$ are the only digits smaller than $5$ which follow $5,$ and digit $2$ is the only digit smaller than $4$ which follows $4.$
2025 Harvard-MIT Mathematics Tournament, 11
Let $f(n)=n^2+100.$ Compute the remainder when $\underbrace{f(f(\cdots f(f(}_{2025 \ f\text{'s}}1))\cdots ))$ is divided by $10^4.$
2001 Belarusian National Olympiad, 4
The problem committee of a mathematical olympiad prepares some variants of the contest. Each variant contains $4$ problems, chosen from a shortlist of $n$ problems, and any two variants have at most one problem in common.
(a) If $n = 14$, determine the largest possible number of variants the problem committee can prepare.
(b) Find the smallest value of n such that it is possible to prepare ten variants of the contest.
2005 AMC 12/AHSME, 20
For each $ x$ in $ [0,1]$, define
\[ f(x)=\begin{cases}2x, &\text { if } 0 \leq x \leq \frac {1}{2}; \\
2 - 2x, &\text { if } \frac {1}{2} < x \leq 1. \end{cases}
\]Let $ f^{[2]}(x) = f(f(x))$, and $ f^{[n + 1]}(x) = f^{[n]}(f(x))$ for each integer $ n \geq 2$. For how many values of $ x$ in $ [0,1]$ is $ f^{[2005]}(x) = \frac {1}{2}$?
$ \textbf{(A)}\ 0 \qquad \textbf{(B)}\ 2005 \qquad \textbf{(C)}\ 4010 \qquad \textbf{(D)}\ 2005^2 \qquad \textbf{(E)}\ 2^{2005}$
2024 Azerbaijan IZhO TST, 2
Find all positive integers $n$ such that one can place checkers on a $n\times n$ checkerboard such that any square chosen from the checkerboard has exactly $2$ adjacent squares with checkers on it. Two squares are considered adjacent if they both share a common side
2024 Korea Winter Program Practice Test, Q7
Find all functions $f, g: \mathbb{R} \rightarrow \mathbb{R} $ satisfying the following conditions:
[list][*] $f$ is not a constant function and if $x \le y$ then $f(x)\le f(y)$
[*] For all real number $x$, $f(g(x))=g(f(x))=0$
[*] For all real numbers $x$ and $y$, $f(x)+f(y)+g(x)+g(y)=f(x+y)+g(x+y)$
[*] For all real numbers $x$ and $y$, $f(x)+f(y)+f(g(x)+g(y))=f(x+y)$
[/list]
1969 Leningrad Math Olympiad, 7.4*
There is a wolf in the centre of a square field, and four dogs in the corners. The wolf can easily kill one dog, but two dogs can kill the wolf. The wolf can run all over the field, and the dogs -- along the fence (border) only. Prove that if the dog's speed is $1.5$ times more than the wolf's, than the dogs can prevent the wolf escaping.
2009 Bosnia Herzegovina Team Selection Test, 2
Line $p$ intersects sides $AB$ and $BC$ of triangle $\triangle ABC$ at points $M$ and $K.$ If area of triangle $\triangle MBK$ is equal to area of quadrilateral $AMKC,$ prove that \[\frac{\left|MB\right|+\left|BK\right|}{\left|AM\right|+\left|CA\right|+\left|KC\right|}\geq\frac{1}{3}\]
2024 IFYM, Sozopol, 1
Find all functions \( f: \mathbb{R}^{+} \to \mathbb{R}^{+} \) such that:
\[
f(x^2 + y) = xf(x) + \frac{f(y^2)}{y}
\]
for any positive real numbers \( x \) and \( y \).
2024 ELMO Shortlist, N3
Given a positive integer $k$, find all polynomials $P$ of degree $k$ with integer coefficients such that for all positive integers $n$ where all of $P(n)$, $P(2024n)$, $P(2024^2n)$ are nonzero, we have
$$\frac{\gcd(P(2024n), P(2024^2n))}{\gcd(P(n), P(2024n))}=2024^k.$$
[i]Allen Wang[/i]
2020 Yasinsky Geometry Olympiad, 1
In the rectangle $ABCD$, $AB = 2BC$. An equilateral triangle $ABE$ is constructed on the side $AB$ of the rectangle so that its sides $AE$ and $BE$ intersect the segment $CD$. Point $M$ is the midpoint of $BE$. Find the $\angle MCD$.
2018 lberoAmerican, 5
Let $n$ be a positive integer. For a permutation $a_1, a_2, \dots, a_n$ of the numbers $1, 2, \dots, n$ we define
$$b_k = \min_{1 \leq i \leq k} a_i + \max_{1 \leq j \leq k} a_j$$
We say that the permutation $a_1, a_2, \dots, a_n$ is [i]guadiana[/i] if the sequence $b_1, b_2, \dots, b_n$ does not contain two consecutive equal terms. How many guadiana permutations exist?
2014 IFYM, Sozopol, 6
Is it true that for each natural number $n$ there exist a circle, which contains exactly $n$ points with integer coordinates?
Geometry Mathley 2011-12, 14.3
Let $ABC$ be a triangle inscribed in circle $(I)$ that is tangent to the sides $BC,CA,AB$ at points $D,E, F$ respectively. Assume that $L$ is the intersection of $BE$ and $CF,G$ is the centroid of triangle $DEF,K$ is the symmetric point of $L$ about $G$. If $DK$ meets $EF$ at $P, Q$ is on $EF$ such that $QF = PE$, prove that $\angle DGE + \angle FGQ = 180^o$.
Nguyễn Minh Hà
2001 India Regional Mathematical Olympiad, 3
Find the number of positive integers $x$ such that \[ \left[ \frac{x}{99} \right] = \left[ \frac{x}{101} \right] . \]
2006 Harvard-MIT Mathematics Tournament, 6
For how many ordered triplets $(a,b,c)$ of positive integers less than $10$ is the product $a\times b\times c$ divisible by $20$?
2006 AMC 12/AHSME, 7
Mr. and Mrs. Lopez have two children. When they get into their family car, two people sit in the front, and the other two sit in the back. Either Mr. Lopez or Mrs. Lopez must sit in the driver's seat. How many seating arrangements are possible?
$ \textbf{(A) } 4 \qquad \textbf{(B) } 12 \qquad \textbf{(C) } 16 \qquad \textbf{(D) } 24 \qquad \textbf{(E) } 48$
1997 Argentina National Olympiad, 3
Let $x_1,x_2,x_3,\ldots ,x_{100}$ be one hundred real numbers greater than or equal to $0$ and less than or equal to $1$. Find the maximum possible value of the sum$$S=x_1(1-x_2)+x_2(1-x_3)+x_3(1-x_4)+\cdots +x_{99}(1-x_{100})+x_ {100}(1-x_1).$$
2003 Moldova National Olympiad, 8.5
$\text{Prove that each integer}$ $n\ge3$ can be written as a sum of some consecutive natural numbers only and only if it isn't a power of 2
Geometry Mathley 2011-12, 2.2
Let $ABC$ be a scalene triangle. A circle $(O)$ passes through $B,C$, intersecting the line segments $BA,CA$ at $F,E$ respectively. The circumcircle of triangle $ABE$ meets the line $CF$ at two points $M,N$ such that $M$ is between $C$ and $F$. The circumcircle of triangle $ACF$ meets the line $BE$ at two points $P,Q$ such that $P$ is betweeen $B$ and $E$. The line through $N$ perpendicular to $AN$ meets $BE$ at $R$, the line through $Q$ perpendicular to $AQ$ meets $CF$ at $S$. Let $U$ be the intersection of $SP$ and $NR, V$ be the intersection of $RM$ and $QS$. Prove that three lines $NQ,UV$ and $RS$ are concurrent.
Trần Quang Hùng
2002 China Girls Math Olympiad, 6
Find all pairs of positive integers $ (x,y)$ such that
\[ x^y \equal{} y^{x \minus{} y}.
\]
[i]Albania[/i]