Found problems: 85335
2024 Portugal MO, 3
A sequence composed by $0$s and $1$s has at most two consecutive $0$s. How many sequences of length $10$ exist?
2013 Germany Team Selection Test, 1
Two concentric circles $\omega, \Omega$ with radii $8,13$ are given. $AB$ is a diameter of $\Omega$ and the tangent from $B$ to $\omega$ touches $\omega$ at $D$. What is the length of $AD$.
2012 Iran MO (3rd Round), 2
Suppose $s,k,t\in \mathbb N$. We've colored each natural number with one of the $k$ colors, such that each color is used infinitely many times. We want to choose a subset $\mathcal A$ of $\mathbb N$ such that it has $t$ disjoint monochromatic $s$-element subsets. What is the minimum number of elements of $A$?
[i]Proposed by Navid Adham[/i]
2005 Estonia National Olympiad, 5
Does there exist an integer $n > 1$ such that $2^{2^n-1} -7$ is not a perfect square?
1988 IMO Shortlist, 24
Let $ \{a_k\}^{\infty}_1$ be a sequence of non-negative real numbers such that:
\[ a_k \minus{} 2 a_{k \plus{} 1} \plus{} a_{k \plus{} 2} \geq 0
\]
and $ \sum^k_{j \equal{} 1} a_j \leq 1$ for all $ k \equal{} 1,2, \ldots$. Prove that:
\[ 0 \leq a_{k} \minus{} a_{k \plus{} 1} < \frac {2}{k^2}
\]
for all $ k \equal{} 1,2, \ldots$.
2010 Bosnia Herzegovina Team Selection Test, 4
Convex quadrilateral is divided by diagonals into four triangles with congruent inscribed circles. Prove that this quadrilateral is rhombus.
2006 National Olympiad First Round, 26
For how many primes $p$, there exists an integr $m$ such that $m^3+3m-2 \equiv 0 \pmod p$ and $m^2+4m+5\equiv 0 \pmod p$?
$
\textbf{(A)}\ 1
\qquad\textbf{(B)}\ 2
\qquad\textbf{(C)}\ 3
\qquad\textbf{(D)}\ 4
\qquad\textbf{(E)}\ \text{Infinitely many}
$
2022 Indonesia TST, G
Let $AB$ be the diameter of circle $\Gamma$ centred at $O$. Point $C$ lies on ray $\overrightarrow{AB}$. The line through $C$ cuts circle $\Gamma$ at $D$ and $E$, with point $D$ being closer to $C$ than $E$ is. $OF$ is the diameter of the circumcircle of triangle $BOD$. Next, construct $CF$, cutting the circumcircle of triangle $BOD$ at $G$. Prove that $O,A,E,G$ are concyclic.
(Possibly proposed by Pak Wono)
1959 Polish MO Finals, 2
In an equilateral triangle $ ABC $, point $ O $ is chosen and perpendiculars $ OM $, $ ON $, $ OP $ are dropped to the sides $ BC $, $ CA $, $ AB $, respectively. Prove that the sum of the segments $ AP $, $ BM $, $ CN $ does not depend on the position of point $ O $.
2005 Greece Team Selection Test, 3
Let the polynomial $P(x)=x^3+19x^2+94x+a$ where $a\in\mathbb{N}$. If $p$ a prime number, prove that no more than three numbers of the numbers $P(0), P(1),\ldots, P(p-1)$ are divisible by $p$.
2014 Contests, 1b
Find all functions $f : R-\{0\} \to R$ which satisfy $(1 + y)f(x) - (1 + x)f(y) = yf(x/y) - xf(y/x)$ for all real $x, y \ne 0$, and which take the values $f(1) = 32$ and $f(-1) = -4$.
2017 Iran Team Selection Test, 1
$ABCD$ is a trapezoid with $AB \parallel CD$. The diagonals intersect at $P$. Let $\omega _1$ be a circle passing through $B$ and tangent to $AC$ at $A$. Let $\omega _2$ be a circle passing through $C$ and tangent to $BD$ at $D$. $\omega _3$ is the circumcircle of triangle $BPC$.
Prove that the common chord of circles $\omega _1,\omega _3$ and the common chord of circles $\omega _2, \omega _3$ intersect each other on $AD$.
[i]Proposed by Kasra Ahmadi[/i]
1977 Kurschak Competition, 2
$ABC$ is a triangle with orthocenter $H$. The median from $A$ meets the circumcircle again at $A_1$, and $A_2$ is the reflection of $A_1$ in the midpoint of $BC$. The points$ B_2$ and $C_2$ are defined similarly. Show that $H$, $A_2$, $B_2$ and $C_2$ lie on a circle.
[img]https://cdn.artofproblemsolving.com/attachments/f/1/192d14a0a7a9aa9ac7b38763e6ea6a4a95be55.png[/img]
2006 Cono Sur Olympiad, 3
Let $n$ be a natural number. The finite sequence $\alpha$ of positive integer terms, there are $n$ different numbers ($\alpha$ can have repeated terms). Moreover, if from one from its terms any we subtract 1, we obtain a sequence which has, between its terms, at least $n$ different positive numbers. What's the minimum value of the sum of all the terms of $\alpha$?
2009 Iran Team Selection Test, 5
$ ABC$ is a triangle and $ AA'$ , $ BB'$ and $ CC'$ are three altitudes of this triangle . Let $ P$ be the feet of perpendicular from $ C'$ to $ A'B'$ , and $ Q$ is a point on $ A'B'$ such that $ QA \equal{} QB$ . Prove that :
$ \angle PBQ \equal{} \angle PAQ \equal{} \angle PC'C$
2010 F = Ma, 12
A ball with mass $m$ projected horizontally off the end of a table with an initial kinetic energy $K$. At a time $t$ after
it leaves the end of the table it has kinetic energy $3K$. What is $t$? Neglect air resistance.
(A) $(3/g)\sqrt{K/m}$
(B) $(2/g)\sqrt{K/m}$
(C) $(1/g)\sqrt{8K/m}$
(D) $(K/g)\sqrt{6/m}$
(E) $(2K/g)\sqrt{1/m}$
2016 Baltic Way, 10
Let $a_{0,1}, a_{0,2}, . . . , a_{0, 2016}$ be positive real numbers. For $n\geq 0$ and $1 \leq k < 2016$ set $$a_{n+1,k} = a_{n,k} +\frac{1}{2a_{n,k+1}} \ \ \text{and} \ \ a_{n+1,2016} = a_{n,2016} +\frac{1}{2a_{n,1}}.$$
Show that $\max_{1\leq k \leq 2016} a_{2016,k} > 44.$
2023 Tuymaada Olympiad, 8
Given is a positive integer $n$. Let $A$ be the set of points $x \in (0;1)$ such that $|x-\frac{p} {q}|>\frac{1}{n^3}$ for each rational fraction $\frac{p} {q}$ with denominator $q \leq n^2$. Prove that $A$ is a union of intervals with total length not exceeding $\frac{100}{n}$.
Proposed by Fedor Petrov
1967 All Soviet Union Mathematical Olympiad, 086
a) A lamp of a lighthouse enlights an angle of $90$ degrees. Prove that you can turn the lamps of four arbitrary posed lighthouses so, that all the plane will be enlightened.
b) There are eight lamps in eight points of the space. Each can enlighten an octant (three-faced space polygon with three mutually orthogonal edges). Prove that you can turn them so, that all the space will be enlightened.
2015 Thailand Mathematical Olympiad, 1
Let $p$ be a prime, and let $a_1, a_2, a_3, . . .$ be a sequence of positive integers so that $a_na_{n+2} = a^2_{n+1} + p$ for all positive integers $n$. Show that $a_{n+1}$ divides $a_n + a_{n+2}$ for all positive integers $n$.
2018 Balkan MO Shortlist, G6
In a triangle $ABC$ with $AB=AC$, $\omega$ is the circumcircle and $O$ its center. Let $D$ be a point on the extension of $BA$ beyond $A$. The circumcircle $\omega_{1}$ of triangle $OAD$ intersects the line $AC$ and the circle $\omega$ again at points $E$ and $G$, respectively. Point $H$ is such that $DAEH$ is a parallelogram. Line $EH$ meets circle $\omega_{1}$ again at point $J$. The line through $G$ perpendicular to $GB$ meets $\omega_{1}$ again at point $N$ and the line through $G$ perpendicular to $GJ$ meets $\omega$ again at point $L$. Prove that the points $L, N, H, G$ lie on a circle.
2019 LIMIT Category C, Problem 4
Let $X,Y$ be i.i.d $\operatorname{Geom}(p)$. What is the conditional distribution of $X|X+Y=k$?
$\textbf{(A)}~\operatorname{Uniform}\left\{1,2,\ldots,\left\lfloor\frac k2\right\rfloor\right\}$
$\textbf{(B)}~\operatorname{Uniform}\left\{1,2,\ldots,k\right\}$
$\textbf{(C)}~\operatorname{Uniform}\left\{1,2,\ldots,\left\lfloor\frac k2\right\rfloor+1\right\}$
$\textbf{(D)}~\text{None of the above}$
2007 iTest Tournament of Champions, 5
Convex quadrilateral $ABCD$ has the property that the circles with diameters $AB$ and $CD$ are tangent at point $X$ inside the quadrilateral, and likewise, the circles with diameters $BC$ and $DA$ are tangent at a point $Y$ inside the quadrilateral. Given that the perimeter of $ABCD$ is $96$, and the maximum possible length of $XY$ is $m$, find $\lfloor 2007m\rfloor$.
2002 India IMO Training Camp, 14
Let $p$ be an odd prime and let $a$ be an integer not divisible by $p$. Show that there are $p^2+1$ triples of integers $(x,y,z)$ with $0 \le x,y,z < p$ and such that $(x+y+z)^2 \equiv axyz \pmod p$
2020 Jozsef Wildt International Math Competition, W38
Let $(a_n)_{n\in\mathbb N}$ be a sequence, given by the recurrence:
$$ma_{n+1}+(m-2)a_n-a_{n-1}=0$$
where $m\in\mathbb R$ is a parameter and the first two terms of $a_n$ are fixed known real numbers. Find $m\in\mathbb R$, so that
$$\lim_{n\to\infty}a_n=0$$
[i]Proposed by Laurențiu Modan[/i]