Found problems: 85335
1981 All Soviet Union Mathematical Olympiad, 318
The points $C_1, A_1, B_1$ belong to $[AB], [BC], [CA]$ sides, respectively, of the triangle $ABC$ .
$$\frac{|AC_1|}{|C_1B| }=\frac{ |BA_1|}{|A_1C| }= \frac{|CB_1|}{|B_1A| }= \frac{1}{3}$$
Prove that the perimeter $P$ of the triangle $ABC$ and the perimeter $p$ of the triangle $A_1B_1C_1$ , satisfy inequality $$\frac{P}{2} < p < \frac{3P}{4}$$
2008 Sharygin Geometry Olympiad, 6
(A. Myakishev, 8--9) In the plane, given two concentric circles with the center $ A$. Let $ B$ be an arbitrary point on some of these circles, and $ C$ on the other one. For every triangle $ ABC$, consider two equal circles mutually tangent at the point $ K$, such that one of these circles is tangent to the line $ AB$ at point $ B$ and the other one is tangent to the line $ AC$ at point $ C$. Determine the locus of points $ K$.
2012 Sharygin Geometry Olympiad, 16
Given right-angled triangle $ABC$ with hypothenuse $AB$. Let $M$ be the midpoint of $AB$ and $O$ be the center of circumcircle $\omega$ of triangle $CMB$. Line $AC$ meets $\omega$ for the second time in point $K$. Segment $KO$ meets the circumcircle of triangle $ABC$ in point $L$. Prove that segments $AL$ and $KM$ meet on the circumcircle of triangle $ACM$.
1961 AMC 12/AHSME, 8
Let the two base angles of a triangle be $A$ and $B$, with $B$ larger than $A$. The altitude to the base divides the vertex angle $C$ into two parts, $C_1$ and $C_2$, with $C_2$ adjacent to side $a$. Then:
${{{ \textbf{(A)}\ C_1+C_2=A+B \qquad\textbf{(B)}\ C_1-C_2=B-A \qquad\textbf{(C)}\ C_1-C_2=A-B} \qquad\textbf{(D)}\ C_1+C_2=B-A}\qquad\textbf{(E)}\ C_1-C_2=A+B} $
2004 Harvard-MIT Mathematics Tournament, 1
Find the largest number $n$ such that $(2004!)!$ is divisible by $((n!)!)!$.
2022 Durer Math Competition Finals, 12
Csongi taught Benedek how to fold a duck in 8 steps from a $24$ cm $\times 24$ cm piece of paper. The paper is meant to be folded along the dashed line in the direction of the arrow. Once Benedek folded the duck, he undid all the steps, finding crease lines on the square sheet of paper. On one side of the paper, he drew in blue the folds which opened towards Benedek, and in red the folds which opened toward the table. How many cm is the difference between the total length of the blue lines and the red lines?
[img]https://cdn.artofproblemsolving.com/attachments/0/1/358a3b2c3b959a85406b94e34c182fd1c2e28d.png[/img]
2017 Junior Balkan Team Selection Tests - Romania, 2
Let $A$ be a point outside the circle $\omega$ . The tangents from $A$ touch the circle at $B$ and $C$. Let $P$ be an arbitrary point on extension of $AC$ towards $C$, $Q$ the projection of $C$ onto $PB$ and $E$ the second intersection point of the circumcircle of $ABP$ with the circle $\omega$ . Prove that $\angle PEQ = 2\angle APB$
2004 Alexandru Myller, 1
Show that the equation $ (x+y)^{-1}=x^{-1}+y^{-1} $ has a solution in the field of integers modulo $ p $ if and only if $ p $ is a prime congruent to $ 1 $ modulo $ 3. $
[i]Mihai Piticari[/i]
2004 China National Olympiad, 1
Let $EFGH,ABCD$ and $E_1F_1G_1H_1$ be three convex quadrilaterals satisfying:
i) The points $E,F,G$ and $H$ lie on the sides $AB,BC,CD$ and $DA$ respectively, and $\frac{AE}{EB}\cdot\frac{BF}{FC}\cdot \frac{CG}{GD}\cdot \frac{DH}{HA}=1$;
ii) The points $A,B,C$ and $D$ lie on sides $H_1E_1,E_1F_1,F_1,G_1$ and $G_1H_1$ respectively, and $E_1F_1||EF,F_1G_1||FG,G_1H_1||GH,H_1E_1||HE$.
Suppose that $\frac{E_1A}{AH_1}=\lambda$. Find an expression for $\frac{F_1C}{CG_1}$ in terms of $\lambda$.
[i]Xiong Bin[/i]
1999 Tournament Of Towns, 5
A square is cut into $100$ rectangles by $9$ straight lines parallel to one of the sides and $9$ lines parallel to another. If exactly $9$ of the rectangles are actually squares, prove that at least two of these $9$ squares are of the same size .
(V Proizvolov)
2014 Contests, 1
Let $D$ be the midpoint of the side $BC$ of a triangle $ABC$ and $AD$ intersect the circumcircle of $ABC$ for the second time at $E$. Let $P$ be the point symmetric to the point $E$ with respect to the point $D$ and $Q$ be the point of intersection of the lines $CP$ and $AB$. Prove that if $A,C,D,Q$ are concyclic, then the lines $BP$ and $AC$ are perpendicular.
2019 Czech-Polish-Slovak Junior Match, 4
Let $k$ be a circle with diameter $AB$. A point $C$ is chosen inside the segment $AB$ and a point $D$ is chosen on $k$ such that $BCD$ is an acute-angled triangle, with circumcentre denoted by $O$. Let $E$ be the intersection of the circle $k$ and the line $BO$ (different from $B$). Show that the triangles $BCD$ and $ECA$ are similar.
2006 MOP Homework, 5
Let $n$ be a nonnegative integer, and let $p$ be a prime number that is congruent to $7$ modulo $8$. Prove that
$$\sum_{k=1}^{p} \left\{ \frac{k^{2n}}{p} - \frac{1}{2} \right\} = \frac{p-1}{2}$$
2005 Slovenia National Olympiad, Problem 4
Several teams from Littletown and Bigtown took part on a tournament. There were nine more teams from Bigtown than those from Littletown. Any two teams played exactly one match, and the winner and loser got 1 and 0 points respectively (no ties). The teams from Bigtown in total gained nine times more points than those from Littletown. What is the maximum possible number of wins of the best team from Littletown?
1995 ITAMO, 6
Find all pairs of positive integers $x,y$ such that $x^2 +615 = 2^y$
2013 Germany Team Selection Test, 2
Given a $m\times n$ grid rectangle with $m,n \ge 4$ and a closed path $P$ that is not self intersecting from inner points of the grid, let $A$ be the number of points on $P$ such that $P$ does not turn in them and let $B$ be the number of squares that $P$ goes through two non-adjacent sides of them furthermore let $C$ be the number of squares with no side in $P$. Prove that $$A=B-C+m+n-1.$$
2004 Purple Comet Problems, 14
A polygon has five times as many diagonals as it has sides. How many vertices does the polygon have?
2024 All-Russian Olympiad Regional Round, 9.5
Let $ABC$ be an isosceles triangle with $BA=BC$. The points $D, E$ lie on the extensions of $AB, BC$ beyond $B$ such that $DE=AC$. The point $F$ lies on $AC$ is such that $\angle CFE=\angle DEF$. Show that $\angle ABC=2\angle DFE$.
2023 All-Russian Olympiad, 8
In a country, there are ${}N{}$ cities and $N(N-1)$ one-way roads: one road from $X{}$ to $Y{}$ for each ordered pair of cities $X \neq Y$. Every road has a maintenance cost. For each $k = 1,\ldots, N$ let's consider all the ways to select $k{}$ cities and $N - k{}$ roads so that from each city it is possible to get to some selected city, using only selected roads.
We call such a system of cities and roads with the lowest total maintenance cost $k{}$[i]-optimal[/i]. Prove that cities can be numbered from $1{}$ to $N{}$ so that for each $k = 1,\ldots, N$ there is a $k{}$-optimal system of roads with the selected cities numbered $1,\ldots, k$.
[i]Proposed by V. Buslov[/i]
1959 Miklós Schweitzer, 3
[b]3.[/b]Let $G$ be an arbitrary group, $H_1,\dots ,H_n$ some (not necessarily distinet) subgroup of $G$ and $g_1, \dots , g_n$ elements of $G$ such that each element of $G$ belongs at least to one of the right cosets $H_1 g_1, \dots , H_n g_n$. Show that if, for any $k$, the set-union of the cosets $H_i g_i (i=1, \dots , k-1, k+1, \dots , n)$ differs from $G$, then every $H_k (k=1, \dots , n)$ is of finite index in $G$. [b](A. 15)[/b]
1956 Moscow Mathematical Olympiad, 331
Given a closed broken line $A_1A_2A_3...A_n$ in space and a plane intersecting all its segments, $A_1A_2$ at $B_1, A_2A_3$ at $B_2$ ,$... $, $A_nA_1$ at $B_n$, prove that
$$\frac{A_1B_1}{B_1A_2}\cdot \frac{A_2B_2}{B_2A_3}\cdot \frac{A_3B_3}{B_3A_4}\cdot ...\cdot \frac{A_nB_n}{B_nA_1}= 1$$.
2023 Bundeswettbewerb Mathematik, 1
Tick, Trick and Track have 20, 23 and 25 tickets respectively for the carousel at the fair in Duckburg. They agree that they will only ride all three together, for which they must each give up one of their tickets. Also, before a ride, if they want, they can redistribute tickets among themselves as many times as they want according to the following rule: If one has an even number of tickets, he can give half of his tickets to any of the other two.
Can it happen that after any trip: (a) exactly one has no ticket left,
(b) exactly two have no ticket left,
(c) all tickets are given away?
2015 Geolympiad Summer, 4.
Let $ABC$ be a triangle and $I$ be its incenter. Let $D$ be the intersection of the exterior bisectors of $\angle BAC$ and $\angle BIC$, $E$ be the intersection of the exterior bisectors of $\angle ABC$ and $\angle AIC$, and $F$ be the intersection of the exterior bisectors of $\angle ACB$ and $\angle AIB$. Prove that $D$, $E$, $F$ are collinear
2017 Harvard-MIT Mathematics Tournament, 10
Let $ABCD$ be a quadrilateral with an inscribed circle $\omega$. Let $I$ be the center of $\omega$, and let $IA=12,$ $IB=16,$ $IC=14,$ and $ID=11$. Let $M$ be the midpoint of segment $AC$. Compute the ratio $\frac{IM}{IN}$, where $N$ is the midpoint of segment $BD$.
1997 Romania National Olympiad, 2
Find the range of the function $f: \mathbb{R} \to \mathbb{R},$ $$f(x)=\frac{3+2\sin x}{\sqrt{1+\cos x}+\sqrt{1-\cos x}}.$$