Found problems: 85335
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]
2019 Novosibirsk Oral Olympiad in Geometry, 6
Two turtles, the leader and the slave, are crawling along the plane from point $A$ to point $B$. They crawl in turn: first the leader crawls some distance, then the slave crawls some distance in a straight line towards the leading one. Then the leader crawls somewhere again, after which the slave crawls towards the leader, etc. Finally, they both crawl to $B$. Prove that the slave turtle crawled no more than the leading one.
1995 Niels Henrik Abels Math Contest (Norwegian Math Olympiad) Round 2, 2
On the figure, the quadrilateral $ ABCD$ is a rectangle, $ P$ lies on $ AD$ and $ Q$ on $ AB.$ The triangles $ PAQ, QBC,$ and $ PCD$ all have the same areas, and $ BQ \equal{} 2.$ How long is $ AQ$?
[img]http://i250.photobucket.com/albums/gg265/geometry101/NielsHenrikAbel1995Number2.jpg[/img]
A. 7/2
B. $ \sqrt{7}$
C. $ 2 \sqrt{3}$
D. $ 1 \plus{} \sqrt{5}$
E. Not uniquely determined
2014 BMT Spring, 3
Emma is seated on a train traveling at a speed of $120$ miles per hour. She notices distance markers are placed evenly alongside the track, with a constant distance $x$ between any two consecutive ones, and during a span of 6 minutes, she sees precisely $11$ markers pass by her. Determine the difference (in miles) between the largest and smallest possible values of $x$.
Kvant 2019, M2551
The vertices of a convex polygon with $n\geqslant 4$ sides are coloured with black and white. A diagonal is called [i]multicoloured[/i] if its vertices have different colours. A colouring of the vertices is [i]good[/i] if the polygon can be partitioned into triangles by using only multicoloured diagonals which do not intersect in the interior of the polygon. Find the number of good colourings.
[i]Proposed by S. Berlov[/i]