Found problems: 85335
2003 IMO, 3
Each pair of opposite sides of a convex hexagon has the following property: the distance between their midpoints is equal to $\dfrac{\sqrt{3}}{2}$ times the sum of their lengths. Prove that all the angles of the hexagon are equal.
2010 IberoAmerican, 1
The arithmetic, geometric and harmonic mean of two distinct positive integers are different numbers. Find the smallest possible value for the arithmetic mean.
2013 AMC 8, 9
The Incredible Hulk can double the distance he jumps with each succeeding jump. If his first jump is 1 meter, the second jump is 2 meters, the third jump is 4 meters, and so on, then on which jump will he first be able to jump more than 1 kilometer?
$\textbf{(A)}\ 9^\text{th} \qquad \textbf{(B)}\ 10^\text{th} \qquad \textbf{(C)}\ 11^\text{th} \qquad \textbf{(D)}\ 12^\text{th} \qquad \textbf{(E)}\ 13^\text{th}$
2015 AMC 12/AHSME, 23
Let $S$ be a square of side length $1$. Two points are chosen independently at random on the sides of $S$. The probability that the straight-line distance between the points is at least $\tfrac12$ is $\tfrac{a-b\pi}c$, where $a$, $b$, and $c$ are positive integers and $\gcd(a,b,c)=1$. What is $a+b+c$?
$\textbf{(A) }59\qquad\textbf{(B) }60\qquad\textbf{(C) }61\qquad\textbf{(D) }62\qquad\textbf{(E) }63$
2007 Peru IMO TST, 4
Let be a board with $2007 \times 2007$ cells, we colour with black $P$ cells such that:
$\bullet$ there are no 3 colored cells that form a L-trinomes in any of its 4 orientations
Find the minimum value of $P$, such that when you colour one cell more, this configuration can't keep the condition above.
2019 IMO Shortlist, G4
Let $P$ be a point inside triangle $ABC$. Let $AP$ meet $BC$ at $A_1$, let $BP$ meet $CA$ at $B_1$, and let $CP$ meet $AB$ at $C_1$. Let $A_2$ be the point such that $A_1$ is the midpoint of $PA_2$, let $B_2$ be the point such that $B_1$ is the midpoint of $PB_2$, and let $C_2$ be the point such that $C_1$ is the midpoint of $PC_2$. Prove that points $A_2, B_2$, and $C_2$ cannot all lie strictly inside the circumcircle of triangle $ABC$.
(Australia)
2008 Postal Coaching, 3
Prove that for each natural number $m \ge 2$, there is a natural number $n$ such that $3^m$ divides $n^3 + 17$ but $3^{m+1}$ does not divide it.
2014 Costa Rica - Final Round, 1
Consider the following figure where $AC$ is tangent to the circle of center $O$, $\angle BCD = 35^o$, $\angle BAD = 40^o$ and the measure of the minor arc $DE$ is $70^o$. Prove that points $B, O, E$ are collinear.
[img]https://cdn.artofproblemsolving.com/attachments/4/0/fd5f8d3534d9d0676deebd696d174999c2ad75.png[/img]
2014 South East Mathematical Olympiad, 7
Show that there are infinitely many triples of positive integers $(a_i,b_i,c_i)$, $i=1,2,3,\ldots$, satisfying the equation $a^2+b^2=c^4$, such that $c_n$ and $c_{n+1}$ are coprime for any positive integer $n$.
2024 Stars of Mathematics, P3
Fix postive integer $n\geq 2$. Let $a_1,a_2,\dots ,a_n$ be real numbers in the interval $[1,2024]$. Prove that $$\sum_{i=1}^n\frac{1}{a_i}(a_1+a_2+\dots +a_i)>\frac{1}{44}n(n+33).$$
[i]Proposed by Radu-Andrei Lecoiu[/i]
2024 Chile Classification NMO Seniors, 2
Find all real numbers $x$ such that:
\[
2^x + 3^x + 6^x - 4^x - 9^x = 1,
\]
and prove that there are no others.
2020 Junior Macedonian National Olympiad, 3
Solve the following equation in the set of integers
$x^5 + 2 = 3 \cdot 101^y$.
2006 Tournament of Towns, 4
Is it possible to split a prism into disjoint set of pyramids so that each pyramid has its base on one base of the prism, while its vertex on another base of the prism ? (6)
2022/2023 Tournament of Towns, P4
In a checkered square, there is a closed door between any two cells adjacent by side. A beetle starts from some cell and travels through cells, passing through doors; she opens a closed door in the direction she is moving and leaves that door open. Through an open door, the beetle can only pass in the direction the door is opened. Prove that if at any moment the beetle wants to return to the starting cell, it is possible for her to do that.
PEN N Problems, 3
Let $\,n>6\,$ be an integer and $\,a_{1},a_{2},\ldots,a_{k}\,$ be all the natural numbers less than $n$ and relatively prime to $n$. If \[a_{2}-a_{1}=a_{3}-a_{2}=\cdots =a_{k}-a_{k-1}>0,\] prove that $\,n\,$ must be either a prime number or a power of $\,2$.
1995 Bundeswettbewerb Mathematik, 2
A line $g$ and a point $A$ outside $g$ are given in a plane. A point $P$ moves along $g$.
Find the locus of the third vertices of equilateral triangles whose two vertices are $A$ and $P$.
1952 AMC 12/AHSME, 47
In the set of equations $ z^x \equal{} y^{2x}, 2^z \equal{} 2\cdot4^x, x \plus{} y \plus{} z \equal{} 16$, the integral roots in the order $ x,y,z$ are:
$ \textbf{(A)}\ 3,4,9 \qquad\textbf{(B)}\ 9, \minus{} 5 \minus{} ,12 \qquad\textbf{(C)}\ 12, \minus{} 5,9 \qquad\textbf{(D)}\ 4,3,9 \qquad\textbf{(E)}\ 4,9,3$
2023 ELMO Shortlist, C5
Define the [i]mexth[/i] of \(k\) sets as the \(k\)th smallest positive integer that none of them contain, if it exists. Does there exist a family \(\mathcal F\) of sets of positive integers such that [list] [*]for any nonempty finite subset \(\mathcal G\) of \(\mathcal F\), the mexth of \(\mathcal G\) exists, and [*]for any positive integer \(n\), there is exactly one nonempty finite subset \(\mathcal G\) of \(\mathcal F\) such that \(n\) is the mexth of \(\mathcal G\). [/list]
[i]Proposed by Espen Slettnes[/i]
Kvant 2020, M2599
Henry invited $2N$ guests to his birthday party. He has $N$ white hats and $N$ black hats. He wants to place hats on his guests and split his guests into one or several dancing circles so that in each circle there would be at least two people and the colors of hats of any two neighbours would be different. Prove that Henry can do this in exactly $(2N)!$ different ways. (All the hats with the same color are identical, all the guests are obviously distinct, $(2N)! = 1 \cdot 2 \cdot . . . \cdot (2N)$.)
Gleb Pogudin
2018 Hanoi Open Mathematics Competitions, 8
Let $k$ be a positive integer such that $1 +\frac12+\frac13+ ... +\frac{1}{13}=\frac{k}{13!}$. Find the remainder when $k$ is divided by $7$.
2010 Today's Calculation Of Integral, 662
In $xyz$ space, let $A$ be the solid generated by a rotation of the figure, enclosed by the curve $y=2-2x^2$ and the $x$-axis about the $y$-axis.
(1) When the solid is cut by the plane $x=a\ (|a|\leq 1)$, find the inequality which expresses the figure of the cross-section.
(2) Denote by $L$ the distance between the point $(a,\ 0,\ 0)$ and the point on the perimeter of the cross-section found in (1), find the maximum value of $L$.
(3) Find the volume of the solid by a rotation of the solid $A$ about the $x$-axis.
[i]1987 Sophia University entrance exam/Science and Technology[/i]
2000 Harvard-MIT Mathematics Tournament, 43
Box A contains $3$ black and $4$ blue marbles. Box B has $7$ black and $1$ blue, whereas Box C has $2$ black, $3$ blue and $1$ green marble. I close my eyes and pick two marbles from $2$ different boxes. If it turns out that I get $1$ black and $1$ blue marble, what is the probability that the black marble is from box A and the blue one is from C?
2020 MMATHS, I6
Prair has a box with some combination of red and green balls. If she randomly draws two balls out of the box (without replacement), the probability of drawing two balls of the same color is equal to the probability of drawing two balls of different colors! How many possible values between $200$ and $1000$ are there for the total number of balls in the box?
[i]Proposed by Andrew Wu[/i]
2012 India IMO Training Camp, 1
Let $ABCD$ be a trapezium with $AB\parallel CD$. Let $P$ be a point on $AC$ such that $C$ is between $A$ and $P$; and let $X, Y$ be the midpoints of $AB, CD$ respectively. Let $PX$ intersect $BC$ in $N$ and $PY$ intersect $AD$ in $M$. Prove that $MN\parallel AB$.
2021 Sharygin Geometry Olympiad, 10-11.1
.Let $CH$ be an altitude of right-angled triangle $ABC$ ($\angle C = 90^o$), $HA_1$, $HB_1$ be the bisectors of angles $CHB$, $AHC$ respectively, and $E, F$ be the midpoints of $HB_1$ and $HA_1$ respectively. Prove that the lines $AE$ and $BF$ meet on the bisector of angle $ACB$.