Found problems: 85335
1994 All-Russian Olympiad, 2
Two circles $S_1$ and $S_2$ touch externally at $F$. their external common tangent touches $S_1$ at $A$ and $S_2$ at $B$. A line, parallel to $AB$ and tangent to $S_2$ at $C$, intersects $S_1$ at $D$ and $E$. Prove that points $A,F,C$ are collinear.
(A. Kalinin)
2016 Irish Math Olympiad, 6
Triangle $ABC$ has sides $a = |BC| > b = |AC|$. The points $K$ and $H$ on the segment $BC$ satisfy $|CH| = (a + b)/3$ and $|CK| = (a - b)/3$. If $G$ is the centroid of triangle $ABC$, prove that $\angle KGH = 90^o$.
2010 Contests, 3
Let $p$ be a prime number. Prove that from a $p^2\times p^2$ array of squares, we can select $p^3$ of the squares such that the centers of any four of the selected squares are not the vertices of a rectangle with sides parallel to the edges of the array.
1980 IMO Shortlist, 14
Let $\{x_n\}$ be a sequence of natural numbers such that \[(a) 1 = x_1 < x_2 < x_3 < \ldots; \quad (b) x_{2n+1} \leq 2n \quad \forall n.\] Prove that, for every natural number $k$, there exist terms $x_r$ and $x_s$ such that $x_r - x_s = k.$
2007 JBMO Shortlist, 3
Let $A$ be a set of positive integers containing the number $1$ and at least one more element. Given that for any two different elements $m, n$ of A the number $ \frac{m+1 }{(m+1,n+1) }$ is also an element of $A$, prove that $A$ coincides with the set of positive integers.
2014 Albania Round 2, 5
Prove that if the angles $\alpha$ and $\beta$ satisfy $\sin(\alpha + \beta) = 2 \sin \alpha$,
Then $$\alpha < \beta$$
2008 239 Open Mathematical Olympiad, 5
You are given a checkered square, the side of which is $n – 1$ long and contains $n \geq 10$ nodes. A non-return path is a path along edges, the intersection of which with any horizontal or vertical line is a segment, point or empty set, and which does not pass along any edge more than once. What is the smallest number of non-return paths that can cover all the edges?
(An edge is a unit segment between adjacent nodes.)
2004 Switzerland Team Selection Test, 12
Find all natural numbers which can be written in the form $\frac{(a+b+c)^2}{abc}$ , where $a,b,c \in N$.
2020 Durer Math Competition Finals, 13
In the game of Yahtzee , players have to achieve various combinations of values with $5$ dice. In a round, a player can roll the dice three times. At the second and third rolls, he can choose which dice to re-roll and which to keep. What is the probability that a player achieves at least four $6$’s in a round, given that he plays with the optimal strategy to maximise this probability?
Writing the answer as $p/q$ where $p$ and $q$ are coprime, you should submit the sum of all prime factors of $p$, counted with multiplicity. So for example if you obtained $\frac{p}{q} = \frac{3^4 \cdot 11}{ 2^5 \cdot 5}$ then the submitted answer should be $4 \cdot 3 + 11 = 23$.
2017 Sharygin Geometry Olympiad, 4
Given triangle $ABC$ and its incircle $\omega$ prove you can use just a ruler and drawing at most 8 lines to construct points$A',B',C'$ on $\omega$ such that $A,B',C'$ and $B,C',A'$ and $C,A',B'$ are collinear.
2004 National High School Mathematics League, 5
For a 3-digit-number $n=\overline{abc}$, if $a,b,c$ can be three sides of an isosceles triangle (regular triangle included), then the number of such numbers is
$\text{(A)}45\qquad\text{(B)}81\qquad\text{(C)}165\qquad\text{(D)}216$
2017 Brazil Undergrad MO, 3
Let $X = \{(x,y) \in \mathbb{R}^2 | y \geq 0, x^2+y^2 = 1\} \cup \{(x,0),-1\leq x\leq 1\} $ be the edge of the closed semicircle with radius 1.
a) Let $n>1$ be an integer and $P_1,P_2,\dots,P_n \in X$. Show that there exists a permutation $\sigma \colon \{1,2,\dots,n\}\to \{1,2,\dots,n\}$ such that
\[\sum_{j=1}^{n}|P_{\sigma(j+1)}-P_{\sigma(j)}|^2\leq 8\].
Where $\sigma(n+1) = \sigma(1)$.
b) Find all sets $\{P_1,P_2,\dots,P_n \} \subset X$ such that for any permutation $\sigma \colon \{1,2,\dots,n\}\to \{1,2,\dots,n\}$,
\[\sum_{j=1}^{n}|P_{\sigma(j+1)}-P_{\sigma(j)}|^2 \geq 8\].
Where $\sigma(n+1) = \sigma(1)$.
2021 Kosovo National Mathematical Olympiad, 4
Let $ABCDE$ be a convex pentagon such that:
$\angle ABC=90,\angle BCD=135,\angle DEA=60$ and $AB=BC=CD=DE$. Find angle $\angle DAE$.
2017 Novosibirsk Oral Olympiad in Geometry, 6
In trapezoid $ABCD$, diagonal $AC$ is the bisector of angle $A$. Point $K$ is the midpoint of diagonal $AC$. It is known that $DC = DK$. Find the ratio of the bases $AD: BC$.
1940 Eotvos Mathematical Competition, 3
(a) Prove that for any triangle $H_1$, there exists a triangle $H_2$ whose side lengths are equal to the lengths of the medians of $H_1$.
(b) If $H_3$ is the triangle whose side lengths are equal to the lengths of the medians of $H_2$, prove that $H_1$ and $H_3$ are similar.
2019 Hong Kong TST, 2
Let $ABC$ be a triangle with $AB=AC$, and let $M$ be the midpoint of $BC$. Let $P$ be a point such that $PB<PC$ and $PA$ is parallel to $BC$. Let $X$ and $Y$ be points on the lines $PB$ and $PC$, respectively, so that $B$ lies on the segment $PX$, $C$ lies on the segment $PY$, and $\angle PXM=\angle PYM$. Prove that the quadrilateral $APXY$ is cyclic.
2017 Bosnia and Herzegovina Team Selection Test, Problem 1
Incircle of triangle $ ABC$ touches $ AB,AC$ at $ P,Q$. $ BI, CI$ intersect with $ PQ$ at $ K,L$. Prove that circumcircle of $ ILK$ is tangent to incircle of $ ABC$ if and only if $ AB\plus{}AC\equal{}3BC$.
2018 Costa Rica - Final Round, N1
Prove that there are only two sets of consecutive positive integers that satisfy that the sum of its elements is equal to $100$.
2020 Tournament Of Towns, 7
Gleb picked positive integers $N$ and $a$ ($a < N$). He wrote the number $a$ on a blackboard. Then each turn he did the following: he took the last number on the blackboard, divided the number $N$ by this last number with remainder and wrote the remainder onto the board. When he wrote the number $0$ onto the board, he stopped. Could he pick $N$ and $a$ such that the sum of the numbers on the blackboard would become greater than $100N$ ?
Ivan Mitrofanov
1982 Vietnam National Olympiad, 3
Let be given a triangle $ABC$. Equilateral triangles $BCA_1$ and $BCA_2$ are drawn so that $A$ and $A_1$ are on one side of $BC$, whereas $A_2$ is on the other side. Points $B_1,B_2,C_1,C_2$ are analogously defined. Prove that
\[S_{ABC} + S_{A_1B_1C_1} = S_{A_2B_2C_2}.\]
2002 Czech and Slovak Olympiad III A, 3
Show that a given natural number $A$ is the square of a natural number if and only if for any natural number $n$, at least one of the differences
\[(A + 1)^2 - A, (A + 2)^2 - A, (A + 3)^2 - A, \cdots , (A + n)^2 - A\]
is divisible by $n$.
2007 Estonia Team Selection Test, 4
In square $ABCD,$ points $E$ and $F$ are chosen in the interior of sides $BC$ and $CD$, respectively. The line drawn from $F$ perpendicular to $AE$ passes through the intersection point $G$ of $AE$ and diagonal $BD$. A point $K$ is chosen on $FG$ such that $|AK|= |EF|$. Find $\angle EKF.$
2018 Canadian Mathematical Olympiad Qualification, 6
Let $n \geq 2$ be a positive integer. Determine the number of $n$-tuples $(x_1, x_2, \ldots, x_n)$ such that $x_k \in \{0, 1, 2\}$ for $1 \leq k \leq n$ and $\sum_{k = 1}^n x_k - \prod_{k = 1}^n x_k$ is divisible by $3$.
2019 Durer Math Competition Finals, 5
Let $ABC$ be an acute triangle and let $X, Y , Z$ denote the midpoints of the shorter arcs $BC, CA, AB$ of its circumcircle, respectively. Let $M$ be an arbitrary point on side $BC$. The line through $M$, parallel to the inner angular bisector of $\angle CBA$ meets the outer angular bisector of $\angle BCA$ at point $N$. The line through $M$, parallel to the inner angular bisector of $\angle BCA$ meets the outer angular bisector of $\angle CBA$ at point $P$. Prove that lines $XM, Y N, ZP$ pass through a single point.
2001 Korea Junior Math Olympiad, 5
$A$ is a set satisfying the following the condition. Show that $2001+\sqrt{2001}$ is an element of $A$.
[b]Condition[/b]
(1) $1 \in A$
(2) If $x \in A$, then $x^2 \in A$.
(3) If $(x-3)^2 \in A$, then $x \in A$.