Found problems: 85335
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 $i$, $j$.
1999 Israel Grosman Mathematical Olympiad, 1
For any $16$ positive integers $n,a_1,a_2,...,a_{15}$ we define $T(n,a_1,a_2,...,a_{15}) = (a_1^n+a_2^n+ ...+a_{15}^n)a_1a_2...a_{15}$.
Find the smallest $n$ such that $T(n,a_1,a_2,...,a_{15})$ is divisible by $15$ for any choice of $a_1,a_2,...,a_{15}$.
1986 AMC 12/AHSME, 9
The product \[\left(1 - \frac{1}{2^{2}}\right)\left(1 - \frac{1}{3^{2}}\right)\ldots\left(1 - \frac{1}{9^{2}}\right)\left(1 - \frac{1}{10^{2}}\right)\] equals
$ \textbf{(A)}\ \frac{5}{12}\qquad\textbf{(B)}\ \frac{1}{2}\qquad\textbf{(C)}\ \frac{11}{20}\qquad\textbf{(D)}\ \frac{2}{3}\qquad\textbf{(E)}\ \frac{7}{10} $
2007 Italy TST, 1
We have a complete graph with $n$ vertices. We have to color the vertices and the edges in a way such that: no two edges pointing to the same vertice are of the same color; a vertice and an edge pointing him are coloured in a different way. What is the minimum number of colors we need?
2012 AMC 10, 23
A solid tetrahedron is sliced off a solid wooden unit cube by a plane passing through two nonadjacent vertices on one face and one vertex on the opposite face not adjacent to either of the first two vertices. The tetrahedron is discarded and the remaining portion of the cube is placed on a table with the cut surface face down. What is the height of this object?
$ \textbf{(A)}\ \dfrac{\sqrt{3}}{3}\qquad\textbf{(B)}\ \dfrac{2\sqrt{2}}{3}\qquad\textbf{(C)}\ 1\qquad\textbf{(D)}\ \dfrac{2\sqrt{3}}{3}\qquad\textbf{(E)}\ \sqrt{2} $
1997 Korea National Olympiad, 8
For any positive integers $x,y,z$ and $w,$ prove that $x^2,y^2,z^2$ and $w^2$ cannot be four consecutive terms of arithmetic sequence.
2013 Kyiv Mathematical Festival, 1
There are $24$ apples in $4$ boxes. An optimistic worm is convinced that he can eat no more than half of the apples such that there will be $3$ boxes with equal number of apples. Is it possible that he is wrong?
2006 VJIMC, Problem 2
Let $(G,\cdot)$ be a finite group of order $n$. Show that each element of $G$ is a square if and only if $n$ is odd.
2010 IMO, 1
Find all function $f:\mathbb{R}\rightarrow\mathbb{R}$ such that for all $x,y\in\mathbb{R}$ the following equality holds \[
f(\left\lfloor x\right\rfloor y)=f(x)\left\lfloor f(y)\right\rfloor \] where $\left\lfloor a\right\rfloor $ is greatest integer not greater than $a.$
[i]Proposed by Pierre Bornsztein, France[/i]
2015 AMC 12/AHSME, 7
A regular $15$-gon has $L$ lines of symmetry, and the smallest positive angle for which it has rotational symmetry is $R$ degrees. What is $L+R$?
$\textbf{(A) }24\qquad\textbf{(B) }27\qquad\textbf{(C) }32\qquad\textbf{(D) }39\qquad\textbf{(E) }54$