Found problems: 85335
2020 USMCA, 12
Kelvin the Frog is playing the game of Survival. He starts with two fair coins. Every minute, he flips all his coins one by one, and throws a coin away if it shows tails. The game ends when he has no coins left, and Kelvin's score is the [i]square[/i] of the number of minutes elapsed. What is the expected value of Kelvin's score? For example, if Kelvin flips two tails in the first minute, the game ends and his score is 1.
2001 Tournament Of Towns, 2
One of the midlines of a triangle is longer than one of its medians. Prove that the triangle has an obtuse angle.
2024 AMC 12/AHSME, 25
A graph is $\textit{symmetric}$ about a line if the graph remains unchanged after reflection in that line. For how many quadruples of integers $(a,b,c,d)$, where $|a|,|b|,|c|,|d|\le5$ and $c$ and $d$ are not both $0$, is the graph of \[y=\frac{ax+b}{cx+d}\] symmetric about the line $y=x$?
$\textbf{(A) }1282\qquad\textbf{(B) }1292\qquad\textbf{(C) }1310\qquad\textbf{(D) }1320\qquad\textbf{(E) }1330$
2023 Junior Balkan Team Selection Tests - Moldova, 9
Let $ AD $, $ BE $ and $ CF $ be the altitudes of $ \Delta ABC $. The points $ P, \, \, Q, \, \, R $ and $ S $ are the feet of the perpendiculars drawn from the point $ D $ on the segments $ BA $, $ BE $, $ CF $ and $ CA $, respectively. Prove that the points $ P, \, \, Q, \, \, R $ and $ S $ are collinear.
2020 Hong Kong TST, 3
Given a list of integers $2^1+1, 2^2+1, \ldots, 2^{2019}+1$, Adam chooses two different integers from the list and computes their greatest common divisor. Find the sum of all possible values of this greatest common divisor.
Kvant 2021, M2680
Let $n>1$ be a natural number and $A_0A_1\ldots A_{2^n-2}$ be a regular polygon. Prove that \[\frac{1}{A_0A_1}=\frac{1}{A_0A_2}+\frac{1}{A_0A_4}+\frac{1}{A_0A_8}+\cdots+\frac{1}{A_0A_{2^{n-1}}}.\][i]Proposed by Le Hoang and Ngoc Thai (Vietnam)[/i]
1968 IMO Shortlist, 8
Given an oriented line $\Delta$ and a fixed point $A$ on it, consider all trapezoids $ABCD$ one of whose bases $AB$ lies on $\Delta$, in the positive direction. Let $E,F$ be the midpoints of $AB$ and $CD$ respectively. Find the loci of vertices $B,C,D$ of trapezoids that satisfy the following:
[i](i) [/i] $|AB| \leq a$ ($a$ fixed);
[i](ii) [/i] $|EF| = l$ ($l$ fixed);
[i](iii)[/i] the sum of squares of the nonparallel sides of the trapezoid is constant.
[hide="Remark"]
[b]Remark.[/b] The constants are chosen so that such trapezoids exist.[/hide]
2016 Brazil Team Selection Test, 2
For a finite set $A$ of positive integers, a partition of $A$ into two disjoint nonempty subsets $A_1$ and $A_2$ is $\textit{good}$ if the least common multiple of the elements in $A_1$ is equal to the greatest common divisor of the elements in $A_2$. Determine the minimum value of $n$ such that there exists a set of $n$ positive integers with exactly $2016$ good partitions.
PS. [url=https://artofproblemsolving.com/community/c6h1268855p6622233]2015 ISL C3 [/url] has 2015 instead of 2016
2012 All-Russian Olympiad, 2
Does there exist natural numbers $a,b,c$ all greater than $10^{10}$ such that their product is divisible by each of these numbers increased by $2012$?
2006 May Olympiad, 5
With $28$ points, a “triangular grid” of equal sides is formed, as shown in the figure.
One operation consists of choosing three points that are the vertices of an equilateral triangle and removing these three points from the grid. If after performing several of these operations there is only one point left, in what positions can that point remain?
Give all the possibilities and indicate in each case the operations carried out.
Justify why the remaining point cannot be in another position.
[img]https://cdn.artofproblemsolving.com/attachments/f/c/1cedfe0e1c5086b77151538265f8e253e93d2e.gif[/img]
2020 Peru Cono Sur TST., P5
Find the smallest positive integer $n$ such that for any $n$ distinct real numbers $b_1, b_2,\ldots ,b_n$ in the interval $[ 1, 1000 ]$ there always exist $b_i$ and $b_j$ such that:
$$0<b_i-b_j<1+3\sqrt[3]{b_ib_j}$$
2025 All-Russian Olympiad, 10.7
A competition consists of $25$ sports, each awarding one gold medal to a winner. $25$ athletes participate, each in all $25$ sports. There are also $25$ experts, each of whom must predict the number of gold medals each athlete will win. In each prediction, the medal counts must be non-negative integers summing to $25$. An expert is called competent if they correctly guess the number of gold medals for at least one athlete. What is the maximum number \( k \) such that the experts can make their predictions so that at least \( k \) of them are guaranteed to be competent regardless of the outcome? \\
2022 Czech-Austrian-Polish-Slovak Match, 6
Consider 26 letters $A,..., Z$. A string is a finite sequence consisting of those letters. We say that a string $s$ is nice if it contains each of the 26 letters at least once, and each permutation of letters $A,..., Z$ occurs in $s$ as a subsequences the same number of times. Prove that:
(a) There exists a nice string.
(b) Any nice string contains at least $2022$ letters.
1927 Eotvos Mathematical Competition, 1
Let the integers $a, b, c, d$ be relatively prime to $$m = ad - bc.$$
Prove that the pairs of integers $(x,y)$ for which $ax+by$ is a multiple of $m$ are identical with those for which $cx + dy$ is a multiple of $m$.
2007 Vietnam Team Selection Test, 1
Given two sets $A, B$ of positive real numbers such that: $|A| = |B| =n$; $A \neq B$ and $S(A)=S(B)$, where $|X|$ is the number of elements and $S(X)$ is the sum of all elements in set $X$. Prove that we can fill in each unit square of a $n\times n$ square with positive numbers and some zeros such that:
a) the set of the sum of all numbers in each row equals $A$;
b) the set of the sum of all numbers in each column equals $A$.
c) there are at least $(n-1)^{2}+k$ zero numbers in the $n\times n$ array with $k=|A \cap B|$.
2015 Korea National Olympiad, 3
A positive integer $n$ is given. If there exists sets $F_1, F_2, \cdots F_m$ satisfying the following conditions, prove that $m \le n$. (For sets $A, B$, $|A|$ is the number of elements of $A$. $A-B$ is the set of elements that are in $A$ but not $B$. $\text{min}(x,y)$ is the number that is not larger than the other.)
(i): For all $1 \le i \le m$, $F_i \subseteq \{1,2,\cdots,n\}$
(ii): For all $1 \le i < j \le m$, $\text{min}(|F_i-F_j|,|F_j-F_i|) = 1$
2004 Irish Math Olympiad, 1
1. (a) For which positive integers n, does 2n divide the sum of the first n positive
integers?
(b) Determine, with proof, those positive integers n (if any) which have the
property that 2n + 1 divides the sum of the first n positive integers.
Estonia Open Junior - geometry, 2020.1.5
A circle $c$ with center $A$ passes through the vertices $B$ and $E$ of a regular pentagon $ABCDE$. The line $BC$ intersects the circle $c$ for second time at point $F$. Prove that the lines $DE$ and $EF$ are perpendicular.
2014 IMAC Arhimede, 5
Let $p$ be a prime number. The natural numbers $m$ and $n$ are written in the system with the base $p$ as $n = a_0 + a_1p +...+ a_kp^k$ and $m = b_0 + b_1p +..+ b_kp^k$. Prove that
$${n \choose m} \equiv \prod_{i=0}^{k}{a_i \choose b_i} (mod p)$$
1986 Bundeswettbewerb Mathematik, 4
Given the finite set $M$ with $m$ elements and $1986$ further sets $M_1,M_2,M_3,...,M_{1986}$, each of which contains more than $\frac{m}{2}$ elements from $M$ . Show that no more than ten elements need to be marked in order for any set $M_i$ ($i =1, 2, 3,..., 1986$) contains at least one marked element.
2009 Abels Math Contest (Norwegian MO) Final, 1b
Show that the sum of three consecutive perfect cubes can always be written as the difference between two perfect squares.
2008 Greece Team Selection Test, 2
In a village $X_0$ there are $80$ tourists who are about to visit $5$ nearby villages $X_1,X_2,X_3,X_4,X_5$.Each of them has chosen to visit only one of them.However,there are cases when the visit in a village forces the visitor to visit other villages among $X_1,X_2,X_3,X_4,X_5$.Each tourist visits only the village he has chosen and the villages he is forced to.If $X_1,X_2,X_3,X_4,X_5$ are totally visited by $40,60,65,70,75$ tourists respectively,then find how many tourists had chosen each one of them and determine all the ordered pairs $(X_i,X_j):i,j\in \{1,2,3,4,5\}$ which are such that,the visit in $X_i$ forces the visitor to visit $X_j$ as well.
1997 IMC, 3
Let $A,B \in \mathbb{R}^{n\times n}$ with $A^2+B^2=AB$. Prove that if $BA-AB$ is invertible then $3|n$.
2006 Kyiv Mathematical Festival, 2
See all the problems from 5-th Kyiv math festival [url=http://www.mathlinks.ro/Forum/viewtopic.php?p=506789#p506789]here[/url]
2006 equilateral triangles are located in the square with side 1. The sum of their perimeters is equal to 300. Prove that at least three of them have a common point.
1982 Poland - Second Round, 3
Prove that for every natural number $ n \geq 2 $ the inequality holds
$$
\log_n 2 \cdot \log_n 4 \cdot \log_n 6 \ldots \log_n (2n - 2) \leq 1.$$