Found problems: 85335
2010 Contests, 1
Compute
\[\sqrt{1+\sqrt{1+\sqrt{1+\sqrt{1+\sqrt{1+\sqrt{1+\cdots}}}}}}\]
May Olympiad L2 - geometry, 2008.4
In the plane we have $16$ lines(not parallel and not concurrents), we have $120$ point(s) of intersections of this lines.
Sebastian has to paint this $120$ points such that in each line all the painted points are with colour differents, find the minimum(quantity) of colour(s) that Sebastian needs to paint this points.
If we have have $15$ lines(in this situation we have $105$ points), what's the minimum(quantity) of colour(s)?
2014 BMT Spring, 5
Alice, Bob, and Chris each roll $4$ dice. Each only knows the result of their own roll. Alice claims that there are at least $5$ multiples of $3$ among the dice rolled. Bob has $1$ six and no threes, and knows that Alice wouldn’t claim such a thing unless he had at least $2$ multiples of $3$. Bob can call Alice a liar, or claim that there are at least $6$ multiples of $3$, but Chris says that he will immediately call Bob a liar if he makes this claim. Bob wins if he calls Alice a liar and there aren't at least $5$ multiples of $3$, or if he claims there are at least $6$ multiples of $3$, and there are. What is the probability that Bob loses no matter what he does?
1978 USAMO, 5
Nine mathematicians meet at an international conference and discover that among any three of them, at least two speak a common language. If each of the mathematicians speak at most three languages, prove that there are at least three of the mathematicians who can speak the same language.
2023 Czech and Slovak Olympiad III A., 1
Alice and Bob are playing a game on a plane consisting of $72$ cells arranged in circle. At the beginning of the game, Bob places a stone on some of the cells. Then, in every round first Alice picks one empty cell and then Bob must move a stone from one of the two neighboring cells on this cell. If he is unable to do that, game ends. Determine the smallest number of stones he has to place in the beginning so he has a strategy to make the game last for at least $2023$ rounds.
2010 APMO, 2
For a positive integer $k,$ call an integer a $pure$ $k-th$ $power$ if it can be represented as $m^k$ for some integer $m.$ Show that for every positive integer $n,$ there exists $n$ distinct positive integers such that their sum is a pure $2009-$th power and their product is a pure $2010-$th power.
2017 ITAMO, 2
Let $n\geq 2$ be an integer. Consider the solutions of the system
$$\begin{cases}
n=a+b-c \\
n=a^2+b^2-c^2
\end{cases}$$
where $a,b,c$ are integers. Show that there is at least one solution and that the solutions are finitely many.
2007 Sharygin Geometry Olympiad, 3
Given a hexagon $ABCDEF$ such that $AB=BC$, $CD=DE$ , $EF=FA$ and $\angle A = \angle C = \angle E $ Prove that $AD, BE, CF$ are concurrent.
2025 Euler Olympiad, Round 2, 2
Points $A$, $B$, $C$, and $D$ lie on a line in that order, and points $E$ and $F$ are located outside the line such that $EA=EB$, $FC=FD$ and $EF \parallel AD$. Let the circumcircles of triangles $ABF$ and $CDE$ intersect at points $P$ and $Q$, and the circumcircles of triangles $ACF$ and $BDE$ intersect at points $M$ and $N$. Prove that the lines $PQ$ and $MN$ pass through the midpoint of segment $EF$.
[i]
Proposed by Giorgi Arabidze, Georgia[/i]
2025 Kyiv City MO Round 1, Problem 3
Point \( H \) is the orthocenter of the acute triangle \( ABC \), and \( AD \) is its altitude. Tangents are drawn from points \( B \) and \( C \) to the circle with center \( A \) and radius \( AD \), which do not coincide with the line \( BC \). These tangents intersect at point \( P \). Prove that the radius of the incircle of \( \triangle BCP \) is equal to \( HD \).
[i]Proposed by Danylo Khilko[/i]
LMT Accuracy Rounds, 2021 F2
A random rectangle (not necessarily a square) with positive integer dimensions is selected from the $2\times4$ grid below. The probability that the selected rectangle contains only white squares can be written as $\frac{a}{b}$ where $a$ and $b$ are relatively prime positive integers. Find $a+b$.
[asy]
fill((2,0)--(3,0)--(3,1)--(2,1)--cycle,blue);
fill((2,1)--(3,1)--(3,2)--(2,2)--cycle,blue);
draw((0,0)--(4,0),black);
draw((0,0)--(0,2),black);
draw((4,0)--(4,2),black);
draw((4,2)--(0,2),black);
draw((0,1)--(4,1),black);
draw((1,0)--(1,2),black);
draw((2,0)--(2,2),black);
draw((3,0)--(3,2),black);
[/asy]
2007 Bulgaria Team Selection Test, 3
Let $I$ be the center of the incircle of non-isosceles triangle $ABC,A_{1}=AI\cap BC$ and $B_{1}=BI\cap AC.$ Let $l_{a}$ be the line through $A_{1}$ which is parallel to $AC$ and $l_{b}$ be the line through $B_{1}$ parallel to $BC.$ Let $l_{a}\cap CI=A_{2}$ and $l_{b}\cap CI=B_{2}.$ Also $N=AA_{2}\cap BB_{2}$ and $M$ is the midpoint of $AB.$ If $CN\parallel IM$ find $\frac{CN}{IM}$.
2001 Slovenia National Olympiad, Problem 1
None of the positive integers $k,m,n$ are divisible by $5$. Prove that at least one of the numbers $k^2-m^2,m^2-n^2,n^2-k^2$ is divisible by $5$.
2005 AIME Problems, 8
The equation \[2^{333x-2}+2^{111x+2}=2^{222x+1}+1\] has three real roots. Given that their sum is $m/n$ where $m$ and $n$ are relatively prime positive integers, find $m+n$.
MOAA Individual Speed General Rounds, 2021.6
Suppose $(a,b)$ is an ordered pair of integers such that the three numbers $a$, $b$, and $ab$ form an arithmetic progression, in that order. Find the sum of all possible values of $a$.
[i]Proposed by Nathan Xiong[/i]
1990 IMO Longlists, 7
Let $S$ be the incenter of triangle $ABC$. $A_1, B_1, C_1$ are the intersections of $AS, BS, CS$ with the circumcircle of triangle $ABC$ respectively. Prove that $SA_1 + SB_1 + SC_1 \geq SA + SB + SC.$
2017 Switzerland - Final Round, 7
Let $n$ be a natural number such that there are exactly$ 2017$ distinct pairs of natural numbers $(a, b)$,
which the equation $$\frac{1}{a}+\frac{1}{b}=\frac{1}{n}$$ fulfilld. Show that $n$ is a perfect square .
Remark: $(7, 4) \ne (4, 7)$
2005 AIME Problems, 2
For each positive integer $k$, let $S_k$ denote the increasing arithmetic sequence of integers whose first term is $1$ and whose common difference is $k$. For example, $S_3$ is the sequence $1,4,7,10,...$. For how many values of $k$ does $S_k$ contain the term $2005$?
2019 China Girls Math Olympiad, 4
Given parallelogram $OABC$ in the coodinate with $O$ the origin and $A,B,C$ be lattice points. Prove that for all lattice point $P$ in the internal or boundary of $\triangle ABC$, there exists lattice points $Q,R$(can be the same) in the internal or boundary of $\triangle OAC$ with $\overrightarrow{OP}=\overrightarrow{OQ}+\overrightarrow{OR}$.
2001 National Olympiad First Round, 14
Let $x_1, x_2, \dots, x_n$ be a positive integer sequence such that each term is less than or equal to $2001$ and for every $i\geq 3$, $x_i = |x_{i-1}-x_{i-2}|$. What is the largest possible value of $n$?
$
\textbf{(A)}\ 1000
\qquad\textbf{(B)}\ 2001
\qquad\textbf{(C)}\ 3002
\qquad\textbf{(D)}\ 4003
\qquad\textbf{(E)}\ \text{None of the preceding}
$
2011 Indonesia TST, 2
A graph $G$ with $n$ vertex is called [i]good [/i] if every vertex could be labelled with distinct positive integers which are less than or equal $\lfloor \frac{n^2}{4} \rfloor$ such that there exists a set of nonnegative integers $D$ with the following property: there exists an edge between $2$ vertices if and only if the difference of their labels is in $D$.
Show that there exists a positive integer $N$ such that for every $n \ge N$, there exist a not-good graph with $n$ vertices.
2010 Saudi Arabia Pre-TST, 3.1
Let $a \ge b \ge c > 0$. Prove that $$(a-b+c)\left(\frac{1}{a}-\frac{1}{b}+\frac{1}{c}\right) \ge 1$$
1966 IMO Shortlist, 7
For which arrangements of two infinite circular cylinders does their intersection lie in a plane?
2019 Online Math Open Problems, 14
The sum \[\displaystyle\sum_{i=0}^{1000} \dfrac{\dbinom{1000}{i}}{\dbinom{2019}{i}}\] can be expressed in the form $\dfrac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Compute $p+q$.
[i]Proposed by James Lin[/i]
2023 China Northern MO, 1
As shown in the figure, $AB$ is the diameter of circle $\odot O$, and chords $AC$ and $BD$ intersect at point $E$, $EF\perp AB$ intersects at point $F$, and $FC$ intersects $BD$ at point $G$. Point $M$ lies on $AB$ such that $MD=MG$ . Prove that points $F$, $M$, $D$, $G$ lies on a circle.
[img]https://cdn.artofproblemsolving.com/attachments/2/3/614ef5b9e8c8b16a29b8b960290ef9d7297529.jpg[/img]