Found problems: 85335
2015 Postal Coaching, Problem 6
Show that there are infinitely many natural numbers which are simultaneously a sum of two squares and a sum of two cubes but which are not a sum of two $6-$th powers.
1991 Brazil National Olympiad, 2
$P$ is a point inside the triangle $ABC$. The line through $P$ parallel to $AB$ meets $AC$ $A_0$ and $BC$ at $B_0$. Similarly, the line through $P$ parallel to $CA$ meets $AB$ at $A_1$ and $BC$ at $C_1$, and the line through P parallel to BC meets $AB$ at $B_2$ and $AC$ at $C_2$. Find the point $P$ such that $A_0B_0 = A_1B_1 = A_2C_2$.
2000 Tournament Of Towns, 3
The least common multiple of positive integers $a, b, c$ and $d$ is equal to $a + b + c + d$. Prove that $abcd$ is divisible by at least one of $3$ and $5$.
( V Senderov)
2007 Germany Team Selection Test, 3
Find all integer solutions of the equation \[\frac {x^{7} \minus{} 1}{x \minus{} 1} \equal{} y^{5} \minus{} 1.\]
2011 Belarus Team Selection Test, 2
The external angle bisector of the angle $A$ of an acute-angled triangle $ABC$ meets the circumcircle of $\vartriangle ABC$ at point $T$. The perpendicular from the orthocenter $H$ of $\vartriangle ABC$ to the line $TA$ meets the line $BC$ at point $P$. The line $TP$ meets the circumcircce of $\vartriangle ABC$ at point $D$. Prove that $AB^2+DC^2=AC^2+BD^2$
A. Voidelevich
2013 Oral Moscow Geometry Olympiad, 6
The trapezoid $ABCD$ is inscribed in the circle $w$ ($AD // BC$). The circles inscribed in the triangles $ABC$ and $ABD$ touch the base of the trapezoid $BC$ and $AD$ at points $P$ and $Q$ respectively. Points $X$ and $Y$ are the midpoints of the arcs $BC$ and $AD$ of circle $w$ that do not contain points $A$ and $B$ respectively. Prove that lines $XP$ and $YQ$ intersect on the circle $w$.
2020 USMCA, 16
Triangle $ABC$ has $BC = 7, CA = 8, AB = 9$. Let $D, E, F$ be the midpoints of $BC, CA, AB$ respectively, and let $G$ be the intersection of $AD$ and $BE$. $G'$ is the reflection of $G$ across $D$. Let $G'E$ meet $CG$ at $P$, and let $G'F$ meet $BG$ at $Q$. Determine the area of $APG'Q$.
2019 Baltic Way, 16
For a positive integer $N$, let $f(N)$ be the number of ordered pairs of positive integers $(a,b)$ such that the number
$$\frac{ab}{a+b}$$
is a divisor of $N$. Prove that $f(N)$ is always a perfect square.
2016 BMT Spring, 2
Find an integer pair of solutions $(x, y)$ to the following system of equations.
$$\log_2 (y^x) = 16$$
$$\log_2 (x^y) = 8$$
2011 Argentina National Olympiad, 2
Three players $A,B$ and $C$ take turns removing stones from a pile of $N$ stones. They move in the order $A,B,C,A,B,C,…A$. The game begins, and the one who takes out the last stone loses the game. The players $A$ and $C$ team up against $B$ , they agree on a joint strategy. $B$ can take in each play $1,2,3,4$ or $5$ stones, while $A$ and $C$, they can each get $1,2$ or $3$ stones each turn. Determine for what values of $N$ have winning strategy $A$ and $C$, and for what values the winning strategy is from $B$.
.
2024 Baltic Way, 4
Find the largest real number $\alpha$ such that, for all non-negative real numbers $x$, $y$ and $z$, the following inequality holds:
\[
(x+y+z)^3 + \alpha (x^2z + y^2x + z^2y) \geq \alpha (x^2y + y^2z + z^2x).
\]
1998 All-Russian Olympiad Regional Round, 9.8
The endpoints of a compass are at two lattice points of an infinite unit square
grid. It is allowed to rotate the compass around one of its endpoints, not varying
its radius, and thus move the other endpoint to another lattice point. Can the
endpoints of the compass change places after several such steps?
2011 Tournament of Towns, 4
A checkered table consists of $2012$ rows and $k > 2$ columns. A marker is placed in a cell of the left-most column. Two players move the marker in turns. During each move, the player moves the marker by $1$ cell to the right, up or down to a cell that had never been occupied by the marker before. The game is over when any of the players moves the marker to the right-most column. However, whether this player is to win or to lose, the players are advised only when the marker reaches the second column from the right. Can any player secure his win?
1994 Niels Henrik Abels Math Contest (Norwegian Math Olympiad) Round 2, 4
Two circles with radii 1 and 2 touch each other and a line as in the figure. In the region between the circles and the line, there is a circle with radius $ r$ which touches the two circles and the line. What is $ r$?
[img]http://i250.photobucket.com/albums/gg265/geometry101/GeometryImage2.jpg[/img]
A. 1/3
B. $ \frac {1}{\sqrt {5}}$
C. $ \sqrt {3} \minus{} \sqrt {2}$
D. $ 6 \minus{} 4 \sqrt {2}$
E. None of these
VMEO III 2006, 12.1
Given a circle $(O)$ and a point $P$ outside that circle. $M$ is a point running on the circle $(O)$. The circle with center $I$ and diameter $PM$ intersects circle $(O)$ again at $N$. The tangent of $(I)$ at $P$ intersects $MN$ at $Q$. The line through $Q$ perpendicular to $PO$ intersects $PM$ at $ A$. $AN$ intersects $(O)$ further at $ B$. $BM$ intersects $PO$ at $C$. Prove that $AC$ is perpendicular to $OQ$.
2014 SDMO (Middle School), 4
Let $a$, $b$, and $c$ be nonzero real numbers. Prove that $a+b+c$ and $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}$ cannot both be $0$.
1998 Iran MO (2nd round), 3
Let $n$ be a positive integer. We call $(a_1,a_2,\cdots,a_n)$ a [i]good[/i] $n-$tuple if $\sum_{i=1}^{n}{a_i}=2n$ and there doesn't exist a set of $a_i$s such that the sum of them is equal to $n$. Find all [i]good[/i] $n-$tuple.
(For instance, $(1,1,4)$ is a [i]good[/i] $3-$tuple, but $(1,2,1,2,4)$ is not a [i]good[/i] $5-$tuple.)
2005 Swedish Mathematical Competition, 1
Find all integer solutions $x$,$y$ of the equation $(x+y^2)(x^2+y)=(x+y)^3$.
2008 Korean National Olympiad, 3
Points $A,B,C,D,E$ lie in a counterclockwise order on a circle $O$, and $AC = CE$
$P=BD \cap AC$, $Q=BD \cap CE$
Let $O_1$ be the circle which is tangent to $\overline {AP}, \overline {BP}$ and arc $AB$ (which doesn't contain $C$)
Let $O_2$ be the circle which is tangent $\overline {DQ}, \overline {EQ}$ and arc $DE$ (which doesn't contain $C$)
Let $O_1 \cap O = R, O_2 \cap O = S, RP \cap QS = X$
Prove that $XC$ bisects $\angle ACE$
2021 Greece JBMO TST, 1
If positive reals $x,y$ are such that $2(x+y)=1+xy$, find the minimum value of expression $$A=x+\frac{1}{x}+y+\frac{1}{y}$$
2020 Princeton University Math Competition, B1
The function $f(x) = x^2 + (2a + 3)x + (a^2 + 1)$ only has real zeroes. Suppose the smallest possible value of $a$ can be written in the form $p/q$, where $p, q$ are relatively prime integers. Find $|p| + |q|$.
2018 Romania National Olympiad, 4
Let $n$ be an integer with $n \geq 2$ and let $A \in \mathcal{M}_n(\mathbb{C})$ such that $\operatorname{rank} A \neq \operatorname{rank} A^2.$ Prove that there exists a nonzero matrix $B \in \mathcal{M}_n(\mathbb{C})$ such that $$AB=BA=B^2=0$$
[i]Cornel Delasava[/i]
2024 UMD Math Competition Part II, #1
Find the largest positive integer $n$ satisfying the following:
[center]
"There are precisely $53$ integers in the list of integers $1, 2, \ldots, n$ that are either perfect squares, perfect cubes or both."[/center]
2000 ITAMO, 5
A man disposes of sufficiently many metal bars of length $2$ and wants to construct a grill of the shape of an $n \times n$ unit net. He is allowed to fold up two bars at an endpoint or to cut a bar into two equal pieces, but two bars may not overlap or intersect. What is the minimum number of pieces he must use?
2018 Purple Comet Problems, 13
Five lighthouses are located, in order, at points $A, B, C, D$, and $E$ along the shore of a circular lake with a diameter of $10$ miles. Segments $AD$ and $BE$ are diameters of the circle. At night, when sitting at $A$, the lights from $B, C, D$, and $E$ appear to be equally spaced along the horizon. The perimeter in miles of pentagon $ABCDE$ can be written $m +\sqrt{n}$, where $m$ and $n$ are positive integers. Find $m + n$.