Found problems: 85335
2010 Paenza, 6
In space are given two tetrahedra with the same barycenter such that one of them contains the other.
For each tetrahedron, we consider the octahedron whose vertices are the midpoints of the tetrahedron's edges.
Prove that one of this octahedra contains the other.
2011 Moldova Team Selection Test, 2
Find all pairs of real number $x$ and $y$ which simultaneously satisfy the following 2 relations:
$x+y+4=\frac{12x+11y}{x^2+y^2}$
$y-x+3=\frac{11x-12y}{x^2+y^2}$
2012 NIMO Problems, 6
The positive numbers $a, b, c$ satisfy $4abc(a+b+c) = (a+b)^2(a+c)^2$. Prove that $a(a+b+c)=bc$.
[i]Proposed by Aaron Lin[/i]
2010 India Regional Mathematical Olympiad, 1
Let $ABCDEF$ be a convex hexagon in which diagonals $AD, BE, CF$ are concurrent at $O$. Suppose $[OAF]$ is geometric mean of $[OAB]$ and $[OEF]$ and $[OBC]$ is geometric mean of $[OAB]$ and $[OCD]$. Prove that $[OED]$ is the geometric mean of $[OCD]$ and $[OEF]$.
(Here $[XYZ]$ denotes are of $\triangle XYZ$)
2019 Portugal MO, 6
A metro network with $n \ge 2$ stations, where each station is connected to each of the others by a one-way line, is said to be [i]dispersed [/i]i f there are two stations $A$ and $B$ such that it is not possible to go from $A$ to $B$ through is from the network. If a network is [i]dispersed[/i], but it is possible to choose a station $A$ and reverse the direction of all lines to and from $A$ so that the new network is no longer dispersed, the network is said to be [i]correctable[/i]. Indicates all integers $n$ for which there is a network with $n$ stations, [i]dispersed [/i]and not [i]correctable[/i].
1965 All Russian Mathematical Olympiad, 069
A spy airplane flies on the circle with the centre $A$ and radius $10$ km. Its speed is $1000$ km/h. At a certain moment, a rocket , that has same speed with the airplane, is launched from point $A$ and moves along on the straight line connecting the airplane and point $A$.How long after launch will the rocket hit the plane?
2005 Germany Team Selection Test, 3
For an ${n\times n}$ matrix $A$, let $X_{i}$ be the set of entries in row $i$, and $Y_{j}$ the set of entries in column $j$, ${1\leq i,j\leq n}$. We say that $A$ is [i]golden[/i] if ${X_{1},\dots ,X_{n},Y_{1},\dots ,Y_{n}}$ are distinct sets. Find the least integer $n$ such that there exists a ${2004\times 2004}$ golden matrix with entries in the set ${\{1,2,\dots ,n\}}$.
2019 Dutch IMO TST, 2
Write $S_n$ for the set $\{1, 2,..., n\}$. Determine all positive integers $n$ for which there exist functions $f : S_n \to S_n$ and $g : S_n \to S_n$ such that for every $x$ exactly one of the equalities $f(g(x)) = x$ and $g(f(x)) = x$ holds.
1985 IMO Longlists, 50
From each of the vertices of a regular $n$-gon a car starts to move with constant speed along the perimeter of the $n$-gon in the same direction. Prove that if all the cars end up at a vertex $A$ at the same time, then they never again meet at any other vertex of the $n$-gon. Can they meet again at $A \ ?$
1986 Canada National Olympiad, 1
In the diagram line segments $AB$ and $CD$ are of length 1 while angles $ABC$ and $CBD$ are $90^\circ$ and $30^\circ$ respectively. Find $AC$.
[asy]
import geometry;
import graph;
unitsize(1.5 cm);
pair A, B, C, D;
B = (0,0);
D = (3,0);
A = 2*dir(120);
C = extension(B,dir(30),A,D);
draw(A--B--D--cycle);
draw(B--C);
draw(arc(B,0.5,0,30));
label("$A$", A, NW);
label("$B$", B, SW);
label("$C$", C, NE);
label("$D$", D, SE);
label("$30^\circ$", (0.8,0.2));
label("$90^\circ$", (0.1,0.5));
perpendicular(B,NE,C-B);
[/asy]
2020 DMO Stage 1, 4.
[b]Q.[/b] We paint the numbers $1,2,3,4,5$ with red or blue. Prove that the equation $x+y=z$ have a monocolor solution (that is, all the 3 unknown there are the same color . It not needed that $x, y, z$ must be different!)
[i]Proposed by TuZo[/i]
2019 PUMaC Team Round, 4
What is the sum of the leading (first) digits of the integers from $ 1$ to $2019$ when the integers are written in base $3$? Give your answer in base $10$.
2011 IFYM, Sozopol, 1
In the cells of a square table $n$ x $n$ the numbers $1,2,...,n^2$ are written in an arbitrary way. Prove that there exist two adjacent cells, for which the difference between the numbers written in them is no lesser than $n$.
II Soros Olympiad 1995 - 96 (Russia), 10.4
Find the equation of the line tangent to the parabola $y = 1/3(x^2-2x+4)$ and a circle of unit radius centered at the origin. (List all solutions.)
2020 Junior Balkаn MO, 1
Find all triples $(a,b,c)$ of real numbers such that the following system holds:
$$\begin{cases} a+b+c=\frac{1}{a}+\frac{1}{b}+\frac{1}{c} \\a^2+b^2+c^2=\frac{1}{a^2}+\frac{1}{b^2}+\frac{1}{c^2}\end{cases}$$
[i]Proposed by Dorlir Ahmeti, Albania[/i]
1991 Brazil National Olympiad, 1
At a party every woman dances with at least one man, and no man dances with every woman. Show that there are men M and M' and women W and W' such that M dances with W, M' dances with W', but M does not dance with W', and M' does not dance with W.
2019 AMC 12/AHSME, 14
Let $S$ be the set of all positive integer divisors of $100,000.$ How many numbers are the product of two distinct elements of $S?$
$\textbf{(A) }98\qquad\textbf{(B) }100\qquad\textbf{(C) }117\qquad\textbf{(D) }119\qquad\textbf{(E) }121$
2024 Vietnam National Olympiad, 3
Let $ABC$ be an acute triangle with circumcenter $O$. Let $A'$ be the center of the circle passing through $C$ and tangent to $AB$ at $A$, let $B'$ be the center of the circle passing through $A$ and tangent to $BC$ at $B$, let $C'$ be the center of the circle passing through $B$ and tangent to $CA$ at $C$.
a) Prove that the area of triangle $A'B'C'$ is not less than the area of triangle $ABC$.
b) Let $X, Y, Z$ be the projections of $O$ onto lines $A'B', B'C', C'A'$. Given that the circumcircle of triangle $XYZ$ intersects lines $A'B', B'C', C'A'$ again at $X', Y', Z'$ ($X' \neq X, Y' \neq Y, Z' \neq Z$), prove that lines $AX', BY', CZ'$ are concurrent.
2018 Estonia Team Selection Test, 8
Find all integers $k \ge 5$ for which there is a positive integer $n$ with exactly $k$ positive divisors
$1 = d_1 <d_2 < ... <d_k = n$ and $d_2d_3 + d_3d_5 + d_5d_2 = n$.
1991 ITAMO, 3
We consider the sums of the form $\pm 1 \pm 4 \pm 9\pm ... \pm n^2$. Show that every integer can be represented in this form for some $n$. (For example, $3 = -1 + 4$ and $8 = 1-4-9+16+25-36-49+64$.)
2016 Iran MO (2nd Round), 6
Find all functions $f: \mathbb N \to \mathbb N$ Such that:
1.for all $x,y\in N$:$x+y|f(x)+f(y)$
2.for all $x\geq 1395$:$x^3\geq 2f(x)$
2023 Ukraine National Mathematical Olympiad, 9.8
What is the largest possible number of edges in a graph on $2n$ nodes, if there exists exactly one way to split its nodes into $n$ pairs so that the nodes from each pair are connected by an edge?
[i]Proposed by Anton Trygub[/i]
2019 Yasinsky Geometry Olympiad, p3
Let $ABCDEF$ be the regular hexagon. It is known that the area of the triangle $ACD$ is equal to $8$. Find the hexagonal area of $ABCDEF$.
2017 Estonia Team Selection Test, 10
Let $ABC$ be a triangle with $AB = \frac{AC}{2 }+ BC$. Consider the two semicircles outside the triangle with diameters $AB$ and $BC$. Let $X$ be the orthogonal projection of $A$ onto the common tangent line of those semicircles. Find $\angle CAX$.
2019 Ramnicean Hope, 2
Calculate $ \int_1^4 \frac{\ln x}{(1+x)(4+x)} dx . $
[i]Ovidiu Țâțan[/i]