Found problems: 85335
2012 IMO Shortlist, N2
Find all triples $(x,y,z)$ of positive integers such that $x \leq y \leq z$ and
\[x^3(y^3+z^3)=2012(xyz+2).\]
2014 Saudi Arabia Pre-TST, 1.2
Let $D$ be the midpoint of side $BC$ of triangle $ABC$ and $E$ the midpoint of median $AD$. Line $BE$ intersects side $CA$ at $F$. Prove that the area of quadrilateral $CDEF$ is $\frac{5}{12}$ the area of triangle $ABC$.
2014 Purple Comet Problems, 5
The diagram below shows a large triangle with area $72$. Each side of the triangle has been trisected, and line segments have been drawn between these trisection points parallel to the sides of the triangle. Find the area of the shaded region.
[asy]
size(4cm);
pair A,B1,B2,B3,C1,C2,C3,M,I,J;
A=origin;
dotfactor=4;
B1=dir(49);
B2=2*B1;
B3=3*B1;
C1=1.35*dir(127);
C2=2*C1;
C3=3*C1;
M=(B2+C2)/2;
I=B1+C2;
J=C1+B2;
pair d[] = {A,B1,B2,B3,C1,C2,C3,M,I,J};
filldraw(C1--B1--B2--J--I--C2--cycle,rgb(.76,.76,.76));
draw(A--C3--B3--cycle);
draw(C1--J^^C2--B2^^B1--I);
for(int i=0;i<10;++i){
dot(d[i]);
}
[/asy]
2015 India Regional MathematicaI Olympiad, 5
Two circles X and Y in the plane intersect at two distinct points A and B such that the centre of Y lies on X. Let points C and D be on X and Y respectively, so that C, B and D are collinear. Let point E on Y be such that DE is parallel to AC. Show that AE = AB.
2021 Brazil Team Selection Test, 6
Let $\mathcal{S}$ be a set consisting of $n \ge 3$ positive integers, none of which is a sum of two other distinct members of $\mathcal{S}$. Prove that the elements of $\mathcal{S}$ may be ordered as $a_1, a_2, \dots, a_n$ so that $a_i$ does not divide $a_{i - 1} + a_{i + 1}$ for all $i = 2, 3, \dots, n - 1$.
2023 Estonia Team Selection Test, 3
In the acute-angled triangle $ABC$, the point $F$ is the foot of the altitude from $A$, and $P$ is a point on the segment $AF$. The lines through $P$ parallel to $AC$ and $AB$ meet $BC$ at $D$ and $E$, respectively. Points $X \ne A$ and $Y \ne A$ lie on the circles $ABD$ and $ACE$, respectively, such that $DA = DX$ and $EA = EY$.
Prove that $B, C, X,$ and $Y$ are concyclic.
2024 ISI Entrance UGB, P1
Find, with proof, all possible values of $t$ such that
\[\lim_{n \to \infty} \left( \frac{1 + 2^{1/3} + 3^{1/3} + \dots + n^{1/3}}{n^t} \right ) = c\]
for some real $c>0$. Also find the corresponding values of $c$.
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 $\angle ABC=2\angle DFE$.