Found problems: 85335
2014 Hanoi Open Mathematics Competitions, 4
If $p$ is a prime number such that there exist positive integers $a$ and $b$ such that $\frac{1}{p}=\frac{1}{a^2}+\frac{1}{b^2}$ then $p$ is
(A): $3$, (B): $5$, (C): $11$, (D): $7$, (E) None of the above.
2007 Singapore Senior Math Olympiad, 5
Find the maximum and minimum of $x + y$ such that $x + y = \sqrt{2x-1}+\sqrt{4y+3}$
2018 Harvard-MIT Mathematics Tournament, 6
Farmer James invents a new currency, such that for every positive integer $n\le 6$, there exists an $n$-coin worth $n!$ cents. Furthermore, he has exactly $n$ copies of each $n$-coin. An integer $k$ is said to be [i]nice[/i] if Farmer James can make $k$ cents using at least one copy of each type of coin. How many positive integers less than 2018 are nice?
2020 Taiwan TST Round 3, 6
Alice has a map of Wonderland, a country consisting of $n \geq 2$ towns. For every pair of towns, there is a narrow road going from one town to the other. One day, all the roads are declared to be “one way” only. Alice has no information on the direction of the roads, but the King of Hearts has offered to help her. She is allowed to ask him a number of questions. For each question in turn, Alice chooses a pair of towns and the King of Hearts tells her the direction of the road connecting those two towns.
Alice wants to know whether there is at least one town in Wonderland with at most one outgoing road. Prove that she can always find out by asking at most $4n$ questions.
1969 Spain Mathematical Olympiad, 1
Find the locus of the centers of the inversions that transform two points $A, B$ of a given circle $\gamma$ , at diametrically opposite points of the inverse circles of $\gamma$ .
2007 International Zhautykov Olympiad, 2
Let $ABCD$ be a convex quadrilateral, with $\angle BAC=\angle DAC$ and $M$ a point inside such that $\angle MBA=\angle MCD$ and $\angle MBC=\angle MDC$. Show that the angle $\angle ADC$ is equal to $\angle BMC$ or $\angle AMB$.
2005 France Team Selection Test, 2
Two right angled triangles are given, such that the incircle of the first one is equal to the circumcircle of the second one. Let $S$ (respectively $S'$) be the area of the first triangle (respectively of the second triangle).
Prove that $\frac{S}{S'}\geq 3+2\sqrt{2}$.
2016 LMT, 2
Mater is confused and starts going around the track in the wrong direction. He can go around 7 times in an hour. Lightning and Chick start in the same place at Mater and at the same time, both going the correct direction. Lightning can go around 91 times per hour, while Chick can go around 84 times per hour. When Lightning passes Chick for the third time, how many times will he have passed Mater (if Lightning is passing Mater just as he passes Chick for the third time, count this as passing Mater)?
[i]Proposed by Matthew Weiss
1980 Kurschak Competition, 1
The points of space are coloured with five colours, with all colours being used. Prove that some plane contains four points of different colours.
2010 IMO Shortlist, 4
Given a triangle $ABC$, with $I$ as its incenter and $\Gamma$ as its circumcircle, $AI$ intersects $\Gamma$ again at $D$. Let $E$ be a point on the arc $BDC$, and $F$ a point on the segment $BC$, such that $\angle BAF=\angle CAE < \dfrac12\angle BAC$. If $G$ is the midpoint of $IF$, prove that the meeting point of the lines $EI$ and $DG$ lies on $\Gamma$.
[i]Proposed by Tai Wai Ming and Wang Chongli, Hong Kong[/i]
1976 Vietnam National Olympiad, 3
$P$ is a point inside the triangle $ABC$. The perpendicular distances from $P$ to the three sides have product $p$. Show that $p \le \frac{ 8 S^3}{27abc}$, where $S =$ area $ABC$ and $a, b, c$ are the sides. Prove a similar result for a tetrahedron.
2025 Harvard-MIT Mathematics Tournament, 2
In a two-dimensional cave with a parallel floor and ceiling, two stalactites of lengths $16$ and $36$ hang perpendicularly from the ceiling, while two stalagmites of heights $25$ and $49$ grow perpendicularly from the ground. If the tips of these four structures form the vertices of a square in some order, compute the height of the cave.
PEN H Problems, 72
Find all pairs $(x, y)$ of positive rational numbers such that $x^{y}=y^{x}$.
1992 Nordic, 2
Let $n > 1$ be an integer and let $a_1, a_2,... , a_n$ be $n$ different integers. Show that the polynomial
$f(x) = (x -a_1)(x - a_2)\cdot ... \cdot (x -a_n) - 1$ is not divisible by any polynomial with integer coefficients
and of degree greater than zero but less than $n$ and such that the highest power of $x$ has coefficient $1$.
2013 Turkmenistan National Math Olympiad, 2
Sequence $x_1 , x_2 , ..., $ with $x_1=20$ ; $x_2=12$ for all $n\geq 1$ such that $x_{n+2}=x_n+x_{n+1}+2\sqrt{x_{n}*x_{n+1}+121} $then prove that $x_{2013}$ is an integer number.
1997 Moldova Team Selection Test, 4
Let $A=\{1,2,\ldots,1997\}$ be a set. Find the samllest integer $k>1$ such that in each subset $M{}$ of $A{}$, which cointain $k{}$ elements, there is a multiple of the smallest element from $M{}$, different from itself.
2021 Azerbaijan Junior NMO, 5
In $\triangle ABC\ T$ is a point lies on the internal angle bisector of $B$. Let $\omega$ be circle with diameter $BT$.
$\omega$ intersects with $BA$ and $BC$ at $P$ and $Q$,respectively. A circle passes through $A$ and tangent to $\omega$ at $P$ intersects with $AC$ again at $X$ . A circle passes through $B$ and tangent to $\omega$ at $Q$ intersects with $AC$ again at $Y$ . Prove that $TX=TY$
2009 China Team Selection Test, 2
Let $ n,k$ be given positive integers satisfying $ k\le 2n \minus{} 1$. On a table tennis tournament $ 2n$ players take part, they play a total of $ k$ rounds match, each round is divided into $ n$ groups, each group two players match. The two players in different rounds can match on many occasions. Find the greatest positive integer $ m \equal{} f(n,k)$ such that no matter how the tournament processes, we always find $ m$ players each of pair of which didn't match each other.
2009 Argentina Iberoamerican TST, 1
In the vertexes of a regular $ 31$-gon there are written the numbers from $ 1$ to $ 31$, ordered increasingly, clockwise oriented.
We are allowed to perform an operation which consists in taking any three vertexes, namely the ones who have written $ a$,$ b$, and $ c$ and change them into $ c$, $ a\minus{}\frac{1}{10}$ and $ b\plus{}\frac{1}{10}$ respectively ( $ a$ becomes $ c$, $ b$ becomes $ a\minus{}\frac{1}{10}$ and $ c$ turns into $ b\plus{}\frac{1}{10}$
Prove that after applying several operations we can reach the state in which the numbers in the vertexes are the numbers from $ 1$ to $ 31$, ordered increasingly,anti-clockwise oriented.
2022 Sharygin Geometry Olympiad, 10.7
Several circles are drawn on the plane and all points of their meeting or touching are marked. May be that each circle contains exactly four marked points and exactly four marked points lie on each circle?
2024 AMC 8 -, 7
A $3 \times 7$ is covered without overlap by $3$ shapes of tiles: $2 \times 2$, $1 \times 4$, and $1 \times 1$, shown below. What is the minimum possible number of $1 \times 1$ tiles used?
[center][img width=70]https://wiki-images.artofproblemsolving.com//e/ee/2024-AMC8-q7.png[/img][/center]
$\textbf{(A) }1\qquad\textbf{(B) }2\qquad\textbf{(C) }3\qquad\textbf{(D) }4\qquad\textbf{(E) }5$
1986 AMC 12/AHSME, 13
A parabola $y = ax^{2} + bx + c$ has vertex $(4,2)$. If $(2,0)$ is on the parabola, then $abc$ equals
$ \textbf{(A)}\ -12\qquad\textbf{(B)}\ -6\qquad\textbf{(C)}\ 0\qquad\textbf{(D)}\ 6\qquad\textbf{(E)}\ 12$
2005 Portugal MO, 5
Considers a quadrilateral $[ABCD]$ that has an inscribed circle and a circumscribed circle. The sides $[AD]$ and $[BC]$ are tangent to the circle inscribed at points $E$ and $F$, respectively. Prove that $AE \cdot F C = BF \cdot ED$.
[img]https://1.bp.blogspot.com/-6o1fFTdZ69E/X4XMo98ndAI/AAAAAAAAMno/7FXiJnWzJgcfSn-qSRoEAFyE8VgxmeBjwCLcBGAsYHQ/s0/2005%2BPortugal%2Bp5.png[/img]
2021-2022 OMMC, 22
A positive integer $N$ is [i]apt[/i] if for each integer $0 < k < 1009$, there exists exactly one divisor of $N$ with a remainder of $k$ when divided by $1009$. For a prime $p$, suppose there exists an [i]apt[/i] positive integer $N$ where $\tfrac Np$ is an integer but $\tfrac N{p^2}$ is not. Find the number of possible remainders when $p$ is divided by $1009$.
[i]Proposed by Evan Chang[/i]
LMT Theme Rounds, 2023F 4B
In triangle $ABC$, $AB = 13$, $BC = 14$, and $CA = 15$. Let $M$ be the midpoint of side $AB$, $G$ be the centroid of $\triangle ABC$, and $E$ be the foot of the altitude from $A$ to $BC$. Compute the area of quadrilateral $GAME$.
[i]Proposed by Evin Liang[/i]
[hide=Solution][i]Solution[/i]. $\boxed{23}$
Use coordinates with $A = (0,12)$, $B = (5,0)$, and $C = (-9,0)$. Then $M = \left(\dfrac{5}{2},6\right)$ and $E = (0,0)$. By shoelace, the area of $GAME$ is $\boxed{23}$.[/hide]