Found problems: 85335
1989 USAMO, 4
Let $ABC$ be an acute-angled triangle whose side lengths satisfy the inequalities $AB < AC < BC$. If point $I$ is the center of the inscribed circle of triangle $ABC$ and point $O$ is the center of the circumscribed circle, prove that line $IO$ intersects segments $AB$ and $BC$.
1989 Kurschak Competition, 2
For any positive integer $n$ denote $S(n)$ the digital sum of $n$ when represented in the decimal system. Find every positive integer $M$ for which $S(Mk)=S(M)$ holds for all integers $1\le k\le M$.
2009 Junior Balkan Team Selection Test, 1
Find all two digit numbers $ \overline{AB}$ such that $ \overline{AB}$ divides $ \overline{A0B}$.
2017 VJIMC, 3
Let $n \ge 2$ be an integer. Consider the system of equations
\begin{align} x_1+\frac{2}{x_2}=x_2+\frac{2}{x_3}=\dots=x_n+\frac{2}{x_1} \end{align}
1. Prove that $(1)$ has infinitely many real solutions $(x_1,\dotsc,x_n)$ such that the numbers $x_1,\dotsc,x_n$ are distinct.
2. Prove that every solution of $(1)$, such that the numbers $x_1,\dotsc,x_n$ are not all equal, satisfies $\vert x_1x_2\cdots x_n\vert=2^{n/2}$.
1973 Yugoslav Team Selection Test, Problem 2
A circle $k$ is drawn using a given disc (e.g. a coin). A point $A$ is chosen on $k$. Using just the given disc, determine the point $B$ on $k$ so that $AB$ is a diameter of $k$. (You are allowed to choose an arbitrary point in one of the drawn circles, and using the given disc it is possible to construct either of the two circles that passes through the points at a distance that is smaller than the radius of the circle.)
1978 AMC 12/AHSME, 3
For all non-zero numbers $x$ and $y$ such that $x = 1/y$, \[\left(x-\frac{1}{x}\right)\left(y+\frac{1}{y}\right)\] equals
$\textbf{(A) }2x^2\qquad\textbf{(B) }2y^2\qquad\textbf{(C) }x^2+y^2\qquad\textbf{(D) }x^2-y^2\qquad \textbf{(E) }y^2-x^2$
2022 MIG, 24
Cows Alpha and Beta are tied by eight-meter ropes, on the midpoints of adjacent sides of a rectangular fence. Both cows are outside the fence; Alpha can wander in a region with an area of $34\pi$ square meters and Beta can wander in a region with an area of $40\pi$ square meters. What is the area enclosed by the rectangular fence?
$\textbf{(A) }45\qquad\textbf{(B) }48\qquad\textbf{(C) }96\qquad\textbf{(D) }120\qquad\textbf{(E) }144$
2024 Korea Junior Math Olympiad (First Round), 1.
Find this:
$ (1+\frac{1}{5})(1+\frac{1}{6})...(1+\frac{1}{2023})(1+\frac{1}{2024}) $
1989 AMC 8, 7
If the value of $20$ quarters and $10$ dimes equals the value of $10$ quarters and $n$ dimes, then $n=$
$\text{(A)}\ 10 \qquad \text{(B)}\ 20 \qquad \text{(C)}\ 30 \qquad \text{(D)}\ 35 \qquad \text{(E)}\ 45$
1994 All-Russian Olympiad, 8
There are $30$ students in a class. In an examination, their results were all different from each other. It is given that everyone has the same number of friends. Find the maximum number of students such that each one of them has a better result than the majority of his friends.
PS. Here majority means larger than half.
2015 Middle European Mathematical Olympiad, 6
Let $I$ be the incentre of triangle $ABC$ with $AB>AC$ and let the line $AI$ intersect the side $BC$ at $D$. Suppose that point $P$ lies on the segment $BC$ and satisfies $PI=PD$. Further, let $J$ be the point obtained by reflecting $I$ over the perpendicular bisector of $BC$, and let $Q$ be the other intersection of the circumcircles of the triangles $ABC$ and $APD$. Prove that $\angle BAQ=\angle CAJ$.
2009 Princeton University Math Competition, 2
A triangle has sides of lengths 5, 6, 7. What is 60 times the square of the radius of the inscribed circle?
2017 ASDAN Math Tournament, 9
Eddy owns $5$ different cats, and has $9$ fish to distribute among the cats. Each cat gets at least $1$ fish and at most $3$ fish. If the fish are indistinguishable, how many ways can Eddy distribute the $9$ fish among the $5$ cats?
2012 Online Math Open Problems, 28
A fly is being chased by three spiders on the edges of a regular octahedron. The fly has a speed of $50$ meters per second, while each of the spiders has a speed of $r$ meters per second. The spiders choose their starting positions, and choose the fly's starting position, with the requirement that the fly must begin at a vertex. Each bug knows the position of each other bug at all times, and the goal of the spiders is for at least one of them to catch the fly. What is the maximum $c$ so that for any $r<c,$ the fly can always avoid being caught?
[i]Author: Anderson Wang[/i]
2007 China Team Selection Test, 1
$ u,v,w > 0$,such that $ u \plus{} v \plus{} w \plus{} \sqrt {uvw} \equal{} 4$
prove that $ \sqrt {\frac {uv}{w}} \plus{} \sqrt {\frac {vw}{u}} \plus{} \sqrt {\frac {wu}{v}}\geq u \plus{} v \plus{} w$
2019 Iran RMM TST, 5
Edges of a planar graph $G$ are colored either with blue or red. Prove that there is a vertex like $v$ such that when we go around $v$ through a complete cycle, edges with the endpoint at $v$ change their color at most two times.
Clarifications for complete cycle:
If all the edges with one endpoint at $v$ are $(v,u_1),(v,u_2),\ldots,(v,u_k)$ such that $u_1,u_2,\ldots,u_k$ are clockwise with respect to $v$ then in the sequence of $(v,u_1),(v,u_2),\ldots,(v,u_k),(v,u_1)$ there are at most two $j$ such that colours of $(v,u_j),(v,u_{j+1})$ ($j \mod k$) differ.
VMEO III 2006 Shortlist, A10
Let ${a_n}$ be a sequence defined by $a_1=2$, $a_{n+1}=\left[ \frac {3a_n}{2}\right]$ $\forall n \in \mathbb N$
$0.a_1a_2...$ rational or irrational?
2007 Bulgaria National Olympiad, 1
The quadrilateral $ABCD$, where $\angle BAD+\angle ADC>\pi$, is inscribed a circle with centre $I$. A line through $I$ intersects $AB$ and $CD$ in points $X$ and $Y$ respectively such that $IX=IY$. Prove that $AX\cdot DY=BX\cdot CY$.
2006 China Second Round Olympiad, 6
Let $S$ be the set of all those 2007 place decimal integers $\overline{2s_1a_2a_3 \ldots a_{2006}}$ which contain odd number of digit $9$ in each sequence $a_1, a_2, a_3, \ldots, a_{2006}$. The cardinal number of $S$ is
${ \textbf{(A)}\ \frac{1}{2}(10^{2006}+8^{2006})\qquad\textbf{(B)}\ \frac{1}{2}(10^{2006}-8^{2006})\qquad\textbf{(C)}\ 10^{2006}+8^{2006}\qquad
\textbf{(D)}}\ 10^{2006}-8^{2006}\qquad $
2021 Thailand Online MO, P2
Determine all integers $n>1$ that satisfy the following condition: for any positive integer $x$, if gcd$(x,n)=1$, then gcd$(x+101,n)=1$.
2011 National Olympiad First Round, 12
Each of 100 students sends messages to 50 different students. What is the least number of pairs of students who send messages to each other?
$\textbf{(A)}\ 100 \qquad\textbf{(B)}\ 75 \qquad\textbf{(C)}\ 50 \qquad\textbf{(D)}\ 25 \qquad\textbf{(E)}\ \text{None}$
2005 Tuymaada Olympiad, 5
You have $2$ columns of $11$ squares in the middle, in the right and in the left you have columns of $9$ squares (centered on the ones of $11$ squares), then columns of $7,5,3,1$ squares. (This is the way it was explained in the original thread, http://www.artofproblemsolving.com/Forum/viewtopic.php?t=44430 ; anyway, i think you can understand how it looks)
Several rooks stand on the table and beat all the squares ( a rook beats the square it stands in, too). Prove that one can remove several rooks such that not more than $11$ rooks are left and still beat all the table.
[i]Proposed by D. Rostovsky, based on folklore[/i]
2007 Cuba MO, 2
A prism is called [i]binary [/i] if it can be assigned to each of its vertices a number from the set $\{-1, 1\}$, such that the product of the numbers assigned to the vertices of each face is equal to $-1$.
a) Prove that the number of vertices of the binary prisms is divisible for $8$.
b) Prove that a prism with $2000$ vertices is binary.
2008 Estonia Team Selection Test, 2
Let $ABCD$ be a cyclic quadrangle whose midpoints of diagonals $AC$ and $BD$ are $F$ and $G$, respectively.
a) Prove the following implication: if the bisectors of angles at $B$ and $D$ of the quadrangle intersect at diagonal $AC$ then $\frac14 \cdot |AC| \cdot |BD| = | AG| \cdot |BF| \cdot |CG| \cdot |DF|$.
b) Does the converse implication also always hold?
2023 Cono Sur Olympiad, 5
Let $ABC$ be an acute triangle and $D, E, F$ are the midpoints of $BC, CA, AB$, respectively. The circle with diameter $AD$ intersects the lines $AB$ and $AC$ at points $P$ and $Q$ , respectively. The lines through $P$ and $Q$ parallel to $BC$ intersect $DE$ at point $R$ and $DF$ at point $S$, respectively. The circumcircle of $DPR$ intersects $AB$ at $X$, the circumcircle of $DQS$ intersects $AC$ in $Y$, and these two circles intersect again point $Z$. Prove that $Z$ is the midpoint of $XY$.