Found problems: 85335
2013 Korea Junior Math Olympiad, 3
$\{a_n\}$ is a positive integer sequence such that $a_{i+2} = a_{i+1} +a_i$ (for all $i \ge 1$).
For positive integer $n$, de fine as $$b_n=\frac{1}{a_{2n+1}}\Sigma_{i=1}^{4n-2}a_i$$
Prove that $b_n$ is positive integer.
1984 Brazil National Olympiad, 3
Given a regular dodecahedron of side $a$. Take two pairs of opposite faces: $E, E' $ and $F, F'$. For the pair $E, E'$ take the line joining the centers of the faces and take points $A$ and $C$ on the line each a distance $m$ outside one of the faces. Similarly, take $B$ and $D$ on the line joining the centers of $F, F'$ each a distance $m$ outside one of the faces. Show that $ABCD$ is a rectangle and find the ratio of its side lengths.
2002 China Western Mathematical Olympiad, 2
Given a positive integer $ n$, find all integers $ (a_{1},a_{2},\cdots,a_{n})$ satisfying the following conditions:
$ (1): a_{1}\plus{}a_{2}\plus{}\cdots\plus{}a_{n}\ge n^2;$
$ (2): a_{1}^2\plus{}a_{2}^2\plus{}\cdots\plus{}a_{n}^2\le n^3\plus{}1.$
2023 Polish Junior Math Olympiad First Round, 3.
Let $ABCD$ be a rectangle. Point $E$ lies on side $AB$, and point $F$ lies on segment $CE$. Prove that if triangles $ADE$ and $CDF$ have equal areas, then triangles $BCE$ and $DEF$ also have equal areas.
2015 Saudi Arabia IMO TST, 1
Let $ABC$ be an acute-angled triangle inscribed in the circle $(O)$, $H$ the foot of the altitude of $ABC$ at $A$ and $P$ a point inside $ABC$ lying on the bisector of $\angle BAC$. The circle of diameter $AP$ cuts $(O)$ again at $G$. Let $L$ be the projection of $P$ on $AH$. Prove that if $GL$ bisects $HP$ then $P$ is the incenter of the triangle $ABC$.
Lê Phúc Lữ
2023 UMD Math Competition Part I, #9
The Amazing Prime company ships its products in boxes whose length, width, and height (in inches) are prime numbers. If the volume of one of their boxes is $105$ cubic inches, what is its surface area (that is, the sum of the areas of the 6 sides of the box) in square inches?
$$
\mathrm a. ~ 21\qquad \mathrm b.~71\qquad \mathrm c. ~77 \qquad \mathrm d. ~05 \qquad \mathrm e. ~142
$$
2016 India Regional Mathematical Olympiad, 2
Let $a,b,c$ be positive real numbers such that $$\frac{a}{1+a}+\frac{b}{1+b}+\frac{c}{1+c}=1.$$ Prove that $abc \le \frac{1}{8}$.
1999 Singapore Senior Math Olympiad, 2
In $\vartriangle ABC$ with edges $a, b$ and $c$, suppose $b + c = 6$ and the area $S$ is $a^2 - (b -c)^2$. Find the value of $\cos A$ and the largest possible value of $S$.
2025 Kosovo National Mathematical Olympiad`, P4
Let $D$ be a point on the side $AC$ of triangle $\triangle ABC$ such that $AB=AD=DC$ and let $E$ be a point on the side $BC$ such that $BE=2CE$. Prove that $\angle BDE = 90 ^{\circ}$.
XMO (China) 2-15 - geometry, 11.4
We define a beehive of order $n$ as follows:
a beehive of order 1 is one hexagon
To construct a beehive of order $n$, take a beehive of order $n-1$ and draw a layer of hexagons in the exterior of these hexagons. See diagram for examples of $n=2,3$
Initially some hexagons are infected by a virus. If a hexagon has been infected, it will always be infected. Otherwise, it will be infected if at least 5 out of the 6 neighbours are infected.
Let $f(n)$ be the minimum number of infected hexagons in the beginning so that after a finite time, all hexagons become infected. Find $f(n)$.
2011 Denmark MO - Mohr Contest, 3
Determine all the ways in which the fraction $\frac{1}{11}$ can be written as $\frac{1}{n}+\frac{1}{m}$ , where $n$ and $m$ are two different positive integers.
2009 Postal Coaching, 3
Let $ABC$ be a triangle with circumcentre $O$ and incentre $I$ such that $O$ is different from $I$. Let $AK, BL, CM$ be the altitudes of $ABC$, let $U, V , W$ be the mid-points of $AK, BL, CM$ respectively. Let $D, E, F$ be the points at which the in-circle of $ABC$ respectively touches the sides $BC, CA, AB$. Prove that the lines $UD, VE, WF$ and $OI$ are concurrent.
1962 Poland - Second Round, 6
Find a three-digit number with the property that the number represented by these digits and in the same order, but with a numbering base different than $ 10 $, is twice as large as the given number.
1991 Bundeswettbewerb Mathematik, 2
Let $g$ be an even positive integer and $f(n) = g^n + 1$ , $(n \in N^* )$.
Prove that for every positive integer $n$ we have:
a) $f(n)$ divides each of the numbers $f(3n), f(5n), f(7n)$
b) $f(n)$ is relative prime to each of the numbers $f(2n), f(4n),f(6n),...$
2017 Saudi Arabia BMO TST, 3
Let $ABC$ be an acute triangle and $(O)$ be its circumcircle. Denote by $H$ its orthocenter and $I$ the midpoint of $BC$. The lines $BH, CH$ intersect $AC,AB$ at $E, F$ respectively. The circles $(IBF$) and $(ICE)$ meet again at $D$.
a) Prove that $D, I,A$ are collinear and $HD, EF, BC$ are concurrent.
b) Let $L$ be the foot of the angle bisector of $\angle BAC$ on the side $BC$. The circle $(ADL)$ intersects $(O)$ again at $K$ and intersects the line $BC$ at $S$ out of the side $BC$. Suppose that $AK,AS$ intersects the circles $(AEF)$ again at $G, T$ respectively. Prove that $TG = TD$.
2023 Yasinsky Geometry Olympiad, 5
Let $ABC$ be a scalene triangle. Given the center $I$ of the inscribe circle and the points $K_1$, $K_2$ and $K_3$ where the inscribed circle is tangent to the sides $BC$, $AC$ and $AB$. Using only a ruler, construct the center of the circumscribed circle of triangle $ABC$.
(Hryhorii Filippovskyi)
1953 AMC 12/AHSME, 38
If $ f(a)\equal{}a\minus{}2$ and $ F(a,b)\equal{}b^2\plus{}a$, then $ F(3,f(4))$ is:
$ \textbf{(A)}\ a^2\minus{}4a\plus{}7 \qquad\textbf{(B)}\ 28 \qquad\textbf{(C)}\ 7 \qquad\textbf{(D)}\ 8 \qquad\textbf{(E)}\ 11$
2018 China Team Selection Test, 2
A number $n$ is [i]interesting[/i] if 2018 divides $d(n)$ (the number of positive divisors of $n$). Determine all positive integers $k$ such that there exists an infinite arithmetic progression with common difference $k$ whose terms are all interesting.
2024 BAMO, C/1
Sugar Station sells $44$ different kinds of candies, packaged one to a box. Each box is priced at a positive integer number of cents, and it costs $\$1.51$ to buy one of every kind. (There is no discount based on the number of candies in a purchase.) Unfortunately, Anna only has $\$0.75$.
[list=a]
[*] Show that Anna can buy at least $22$ boxes, each containing a different candy.
[*] Show that Anna can do even better, buying at least $25$ boxes, each containing a different candy.
[/list]
2016 Online Math Open Problems, 15
Two bored millionaires, Bilion and Trilion, decide to play a game. They each have a sufficient supply of $\$ 1, \$ 2,\$ 5$, and $\$ 10$ bills. Starting with Bilion, they take turns putting one of the bills they have into a pile. The game ends when the bills in the pile total exactly $\$1{,}000{,}000$, and whoever makes the last move wins the $\$1{,}000{,}000$ in the pile (if the pile is worth more than $\$1{,}000{,}000$ after a move, then the person who made the last move loses instead, and the other person wins the amount of cash in the pile). Assuming optimal play, how many dollars will the winning player gain?
[i]Proposed by Yannick Yao[/i]
2006 Estonia Team Selection Test, 3
A grid measuring $10 \times 11$ is given. How many "crosses" covering five unit squares can be placed on the grid?
(pictured right) so that no two of them cover the same square?
[img]https://cdn.artofproblemsolving.com/attachments/a/7/8a5944233785d960f6670e34ca7c90080f0bd6.png[/img]
2016 Israel Team Selection Test, 1
Let $a,b,c$ be positive numbers satisfying $ab+bc+ca+2abc=1$. Prove that $4a+b+c \geq 2$.
2013 CentroAmerican, 3
Let $ABCD$ be a convex quadrilateral and let $M$ be the midpoint of side $AB$. The circle passing through $D$ and tangent to $AB$ at $A$ intersects the segment $DM$ at $E$. The circle passing through $C$ and tangent to $AB$ at $B$ intersects the segment $CM$ at $F$. Suppose that the lines $AF$ and $BE$ intersect at a point which belongs to the perpendicular bisector of side $AB$. Prove that $A$, $E$, and $C$ are collinear if and only if $B$, $F$, and $D$ are collinear.
2021 LMT Spring, A 24
A Haiku is a Japanese poem of seventeen syllables, in three lines of five, seven, and five.
Using the four words
“Hi”, “hey”, “hello”, and “haiku”,
How many haikus
Can somebody make?
(Repetition is allowed,
Order does matter.)
[i]Proposed by Jeff Lin[/i]
2007 Hanoi Open Mathematics Competitions, 7
Find all sequences of integer $x_1,x_2,..,x_n,...$ such that $ij$ divides $x_i+x_j$ for any distinct positive integer $i$, $j$.