Found problems: 85335
1949-56 Chisinau City MO, 19
The schoolchildren sat down on chairs located in transverse and longitudinal rows. The tallest student was chosen from each transverse row, and the lowest was chosen among them. Then the lowest student was selected from each longitudinal row, and the tallest was chosen among them. Which of these two students is higher?
2005 India IMO Training Camp, 2
Determine all positive integers $n > 2$ , such that \[ \frac{1}{2} \varphi(n) \equiv 1 ( \bmod 6) \]
2020 Bundeswettbewerb Mathematik, 1
Leo and Smilla find $2020$ gold nuggets with masses $1,2,\dots,2020$ gram, which they distribute to a red and a blue treasure chest according to the following rule:
First, Leo chooses one of the chests and tells its colour to Smilla. Then Smilla chooses one of the not yet distributed nuggets and puts it into this chest.
This is repeated until all the nuggets are distributed. Finally, Smilla chooses one of the chests and wins all the nuggets from this chest.
How many gram of gold can Smilla make sure to win?
2018 PUMaC Combinatorics B, 3
In an election between $\text{A}$ and $\text{B}$, during the counting of the votes, neither candidate was more than $2$ votes ahead, and the vote ended in a tie, $6$ votes to $6$ votes. Two votes for the same candidate are indistinguishable. In how many orders could the votes have been counted? One possibility is $\text{AABBABBABABA}$.
1961 AMC 12/AHSME, 25
Triangle $ABC$ is isosceles with base $AC$. Points $P$ and $Q$ are respectively in $CB$ and $AB$ and such that $AC=AP=PQ=QB$. The number of degrees in angle $B$ is:
${{ \textbf{(A)}\ 25 \frac{5}{7} \qquad\textbf{(B)}\ 26 \frac{1}{3} \qquad\textbf{(C)}\ 30\qquad\textbf{(D)}\ 40}\qquad\textbf{(E)}\ \text{Not determined by the information given} } $
1999 Gauss, 1
$1999-999+99$ equals
$\textbf{(A)}\ 901 \qquad \textbf{(B)}\ 1099 \qquad \textbf{(C)}\ 1000 \qquad \textbf{(D)}\ 199 \qquad \textbf{(E)}\ 99$
2007 Romania National Olympiad, 3
The plane is divided into strips of width $1$ by parallel lines (a strip - the region between two parallel lines). The points from the interior of each strip are coloured with red or white, such that in each strip only one color is used (the points of a strip are coloured with the same color). The points on the lines are not coloured. Show that there is an equilateral triangle of side-length $100$, with all vertices of the same colour.
2000 Denmark MO - Mohr Contest, 1
The quadrilateral $ABCD$ is a square of sidelength $1$, and the points $E, F, G, H$ are the midpoints of the sides. Determine the area of quadrilateral $PQRS$.
[img]https://1.bp.blogspot.com/--fMGH2lX6Go/XzcDqhgGKfI/AAAAAAAAMXo/x4NATcMDJ2MeUe-O0xBGKZ_B4l_QzROjACLcBGAsYHQ/s0/2000%2BMohr%2Bp1.png[/img]
2002 Abels Math Contest (Norwegian MO), 4
An integer is given $N> 1$. Arne and Britt play the following game:
(1) Arne says a positive integer $A$.
(2) Britt says an integer $B> 1$ that is either a divisor of $A$ or a multiple of $A$. ($A$ itself is a possibility.)
(3) Arne says a new number $A$ that is either $B - 1, B$ or $B + 1$.
The game continues by repeating steps 2 and 3. Britt wins if she is okay with being told the number $N$ before the $50$th has been said. Otherwise, Arne wins.
a) Show that Arne has a winning strategy if $N = 10$.
b) Show that Britt has a winning strategy if $N = 24$.
c) For which $N$ does Britt have a winning strategy?
2016 Harvard-MIT Mathematics Tournament, 4
A rectangular pool table has vertices at $(0, 0) (12, 0) (0, 10),$ and $(12, 10)$. There are pockets only in the four corners. A ball is hit from $(0, 0)$ along the line $y = x$ and bounces off several walls before eventually entering a pocket. Find the number of walls that the ball bounces off of before entering a pocket.
2022 Lusophon Mathematical Olympiad, 4
How many integer solutions exist that satisfy this equation?
$$x+4y-343\sqrt{x}-686\sqrt{y}+4\sqrt{xy}+2022=0$$.
2001 China Team Selection Test, 2
Let $\theta_i \in \left ( 0,\frac{\pi}{4} \right ]$ for $i=1,2,3,4$. Prove that:
$\tan \theta _1 \tan \theta _2 \tan \theta _3 \tan \theta _4 \le (\frac{\sin^8 \theta _1+\sin^8 \theta _2+\sin^8 \theta _3+\sin^8 \theta _4}{\cos^8 \theta _1+\cos^8 \theta _2+\cos^8 \theta _3+\cos^8 \theta _4})^\frac{1}{2}$
[hide=edit]@below, fixed now. There were some problems (weird characters) so aops couldn't send it.[/hide]
1980 IMO, 3
Two circles $C_1$ and $C_2$ are tangent at a point $P$. The straight line at $D$ is tangent at $A$ to one of the circles and cuts the other circle at the points $B$ and $C$. Prove that the straight line $PA$ is a bisector (interior or exterior) of the angle $BPC$.
2023 IFYM, Sozopol, 6
Let $S$ be a set of real numbers. We say that $S$ is [i]strong[/i] if for any two distinct $a$ and $b$ from $S$, the number $a^2 + b\sqrt{2023}$ is rational. We say that $S$ is [i]very strong[/i] if for every $a$ from $S$, the number $a\sqrt{2023}$ is rational.
a) Prove that if $S$ is a very strong set, then it is also strong.
b) Find the smallest natural number $k$ such that every strong set of $k$ distinct real numbers is very strong.
2023 CUBRMC, 9
Find the sum of all integers $n$ such that $1 < n < 30$ and $n$ divides
$$1+\sum^{n-1}_{k=1}k^{2k}.$$
2020 Tournament Of Towns, 5
Given are two circles which intersect at points $P$ and $Q$. Consider an arbitrary line $\ell$ through $Q$, let the second points of intersection of this line with the circles be $A$ and $B$ respectively. Let $C$ be the point of intersection of the tangents to the circles in those points. Let $D$ be the intersection of the line $AB$ and the bisector of the angle $CPQ$. Prove that all possible $D$ for any choice of $\ell$ lie on a single circle.
Alexey Zaslavsky
MIPT student olimpiad autumn 2022, 2
Let $n \geq 3$ be an integer. Find the minimum degree of one algebraic (polynomial) equation that defines the set of vertices of the correct $n$-gon on plane $R^2$.
1965 AMC 12/AHSME, 35
The length of a rectangle is $ 5$ inches and its width is less than $ 4$ inches. The rectangle is folded so that two diagonally opposite vertices coincide. If the length of the crease is $ \sqrt {6}$, then the width is:
$ \textbf{(A)}\ \sqrt {2} \qquad \textbf{(B)}\ \sqrt {3} \qquad \textbf{(C)}\ 2 \qquad \textbf{(D)}\ \sqrt {5} \qquad \textbf{(E)}\ \sqrt {\frac {11}{2}}$
2014 IMC, 4
We say that a subset of $\mathbb{R}^n$ is $k$-[i]almost contained[/i] by a hyperplane if there are less than $k$ points in that set which do not belong to the hyperplane. We call a finite set of points $k$-[i]generic[/i] if there is no hyperplane that $k$-almost contains the set. For each pair of positive integers $(k, n)$, find the minimal number of $d(k, n)$ such that every finite $k$-generic set in $\mathbb{R}^n$ contains a $k$-generic subset with at most $d(k, n)$ elements.
(Proposed by Shachar Carmeli, Weizmann Inst. and Lev Radzivilovsky, Tel Aviv Univ.)
2011 Regional Olympiad of Mexico Center Zone, 2
Let $ABC$ be a triangle and let $L$, $M$, $N$ be the midpoints of the sides $BC$, $CA$ and $AB$ , respectively. The points $P$ and $Q$ lie on $AB$ and $BC$, respectively; the points $R$ and $S$ are such that $N$ is the midpoint of $PR$ and $L$ is the midpoint of $QS$. Show that if $PS$ and $QR$ are perpendicular, then their intersection lies on in the circumcircle of triangle $LMN$.
2014 Brazil Team Selection Test, 1
Let $n$ be a positive integer. A [i]partition [/i] of $n$ is a multiset (set with repeated elements) whose sum of elements is $n$. For example, the partitions of $3$ are $\{1, 1, 1\}, \{1, 2\}$ and $\{3\}$. Each partition of $n$ is written as a non-descending sequence. For example, for $n = 3$, the list is $(1, 1, 1)$, $(1, 2)$ and $(3)$. For each sequence $x = (x_1, x_2, ..., x_k)$, define $f(x)=\prod_{i=1}^{k-1} {x_{i+1} \choose x_ i}$ . Furthermore, the $f$ of partition $\{n\}$ is $f((n)) = 1$. Prove that the sum of all $f$'s in the list is $2^{n-1}.$
2022 VJIMC, 3
Let $f:[0,1]\to\mathbb R$ be a given continuous function. Find the limit
$$\lim_{n\to\infty}(n+1)\sum_{k=0}^n\int^1_0x^k(1-x)^{n-k}f(x)dx.$$
2018 AMC 12/AHSME, 21
Which of the following polynomials has the greatest real root?
$\textbf{(A) } x^{19}+2018x^{11}+1 \qquad \textbf{(B) } x^{17}+2018x^{11}+1 \qquad \textbf{(C) } x^{19}+2018x^{13}+1 \qquad \textbf{(D) } x^{17}+2018x^{13}+1 \qquad \textbf{(E) } 2019x+2018 $
2018 IFYM, Sozopol, 6
Let $S$ be a real number. It is known that however we choose several numbers from the interval $(0, 1]$ with sum equal to $S$, these numbers can be separated into two subsets with the following property: The sum of the numbers in one of the subsets doesn’t exceed 1 and the sum of the numbers in the other subset doesn’t exceed 5.
Find the greatest possible value of $S$.
2022 Cono Sur, 6
On a blackboard the numbers $1,2,3,\dots,170$ are written. You want to color each of these numbers with $k$ colors $C_1,C_2, \dots, C_k$, such that the following condition is satisfied: for each $i$ with $1 \leq i < k$, the sum of all numbers with color $C_i$ divide the sum of all numbers with color $C_{i+1}$.
Determine the largest possible value of $k$ for which it is possible to do that coloring.