Found problems: 85335
2015 Spain Mathematical Olympiad, 1
All faces of a polyhedron are triangles. Each of the vertices of this polyhedron is assigned independently one of three colors : green, white or black. We say that a face is [i]Extremadura[/i] if its three vertices are of different colors, one green, one white and one black. Is it true that regardless of how the vertices's color, the number of [i]Extremadura[/i] faces of this polyhedron is always even?
2006 Irish Math Olympiad, 5
Find all functions $f : \mathbb{R} \mapsto \mathbb{R}$ such that $f(xy+f(x)) = xf(y) +f(x)$ for all $x,y \in \mathbb{R}$.
2015 BMT Spring, 8
Two players play a game with a pile of $N$ coins on a table. On a player's turn, if there are $n$ coins, the player can take at most $n/2+1$ coins, and must take at least one coin. The player who grabs the last coin wins. For how many values of $N$ between $1$ and $100$ (inclusive) does the first player have a winning strategy?
2002 Nordic, 2
In two bowls there are in total ${N}$ balls, numbered from ${1}$ to ${N}$. One ball is moved from one of the bowls into the other. The average of the numbers in the bowls is increased in both of the bowls by the same amount, ${x}$. Determine the largest possible value of ${x}$.
2004 Purple Comet Problems, 14
Two circles have radii $15$ and $95$. If the two external tangents to the circles intersect at $60$ degrees, how far apart are the centers of the circles?
1998 Irish Math Olympiad, 4
Show that a disk of radius $ 2$ can be covered by seven (possibly overlapping) disks of radius $ 1$.
2001 Dutch Mathematical Olympiad, 2
The function f has the following properties :
$f(x + y) = f(x) + f(y) + xy$ for all real $x$ and $y$
$f(4) = 10$
Calculate $f(2001)$.
2014 Federal Competition For Advanced Students, P2, 2
Let $S$ be the set of all real numbers greater than or equal to $1$.
Determine all functions$ f: S \to S$, so that for all real numbers $x ,y \in S$ with $x^2 -y^2 \in S$ the condition $f (x^2 -y^2) = f (xy)$ is fulfilled.
2012 India PRMO, 10
$ABCD$ is a square and $AB = 1$. Equilateral triangles $AYB$ and $CXD$ are drawn such that $X$ and $Y$ are inside the square. What is the length of $XY$?
1988 IMO Shortlist, 8
Let $ u_1, u_2, \ldots, u_m$ be $ m$ vectors in the plane, each of length $ \leq 1,$ with zero sum. Show that one can arrange $ u_1, u_2, \ldots, u_m$ as a sequence $ v_1, v_2, \ldots, v_m$ such that each partial sum $ v_1, v_1 \plus{} v_2, v_1 \plus{} v_2 \plus{} v_3, \ldots, v_1, v_2, \ldots, v_m$ has length less than or equal to $ \sqrt {5}.$
2013 Canadian Mathematical Olympiad Qualification Repechage, 1
Determine all real solutions to the following equation: \[2^{(2^x)}-3\cdot2^{(2^{x-1}+1)}+8=0.\]
2020 China Team Selection Test, 1
Let $\omega$ be a $n$ -th primitive root of unity. Given complex numbers $a_1,a_2,\cdots,a_n$, and $p$ of them are non-zero. Let
$$b_k=\sum_{i=1}^n a_i \omega^{ki}$$
for $k=1,2,\cdots, n$. Prove that if $p>0$, then at least $\tfrac{n}{p}$ numbers in $b_1,b_2,\cdots,b_n$ are non-zero.
2018 India Regional Mathematical Olympiad, 6
Define a sequence $\{a_n\}_{n\geq 1}$ of real numbers by \[a_1=2,\qquad a_{n+1} = \frac{a_n^2+1}{2}, \text{ for } n\geq 1.\] Prove that \[\sum_{j=1}^{N} \frac{1}{a_j + 1} < 1\] for every natural number $N$.
2013 Princeton University Math Competition, 3
Chris's pet tiger travels by jumping north and east. Chris wants to ride his tiger from Fine Hall to McCosh, which is $3$ jumps east and $10$ jumps north. However, Chris wants to avoid the horde of PUMaC competitors eating lunch at Frist, located $2$ jumps east and $4$ jumps north of Fine Hall. How many ways can he get to McCosh without going through Frist?
2017 AMC 10, 23
How many triangles with positive area have all their vertices at points $(i,j)$ in the coordinate plane, where $i$ and $j$ are integers between $1$ and $5$, inclusive?
$\textbf{(A)}\ 2128 \qquad\textbf{(B)}\ 2148 \qquad\textbf{(C)}\ 2160 \qquad\textbf{(D)}\ 2200 \qquad\textbf{(E)}\ 2300$
2001 Saint Petersburg Mathematical Olympiad, 10.3
Let $I$ be the incenter of triangle $ABC$ and let $D$ be the midpoint of side $AB$. Prove that if the angle $\angle AOD$ is right, then $AB+BC=3AC$.
[I]Proposed by S. Ivanov[/i]
2005 MOP Homework, 6
Consider the three disjoint arcs of a circle determined by three points of the circle. We construct a circle around each of the midpoint of every arc which goes the end points of the arc. Prove that the three circles pass through a common point.
2002 Iran MO (3rd Round), 10
$H,I,O,N$ are orthogonal center, incenter, circumcenter, and Nagelian point of triangle $ABC$. $I_{a},I_{b},I_{c}$ are excenters of $ABC$ corresponding vertices $A,B,C$. $S$ is point that $O$ is midpoint of $HS$. Prove that centroid of triangles $I_{a}I_{b}I_{c}$ and $SIN$ concide.
2017 CIIM, Problem 6
Let $G$ be a simple, connected and finite grafo. A hunter and an invisible rabbit play in the graph $G$.
The rabbit is initially in a vertex $w_0$. In the $k$-th turn (for $k \geq 0$) the hunter picks freely a vertex $v_k$. If $v_k = w_k$, the rabbit is capture and the game ends. If not, the rabbit moves invisibly by an edge of $w_k$ to $w_{k+1}$ ($w_k$ and $w_{k+1}$ are adjacent and therefore distinct) and the game continues. The hunter knows these rules and the graph $G$. After the $k-$th turn he knows that $w_k \not = v_k$, but he gets no more information.
Characterize the graphs $G$ such that the hunter has an strategy that guaranties that he can capture the rabbit in at most $N$ turns for some positive integer $N$. Here $N$ must depend only on $G$ and the strategy should work independently of the initial position and trajectory of the rabbit.
2004 All-Russian Olympiad, 2
Let $ I(A)$ and $ I(B)$ be the centers of the excircles of a triangle $ ABC,$ which touches the sides $ BC$ and $ CA$ in its interior. Furthermore let $ P$ a point on the circumcircle $ \omega$ of the triangle $ ABC.$ Show that the center of the segment which connects the circumcenters of the triangles $ I(A)CP$ and $ I(B)CP$ coincides with the center of the circle $ \omega.$
MIPT student olimpiad spring 2024, 2
Let the matrix $S$ be orthogonal and the matrix $I-S$ be invertible, where I is the identity
matrix of the same size as $S$.
Find
$x^T(I-S)^{-1}x$
Where $x$ is a real unit vector.
1972 IMO Longlists, 8
We are given $3n$ points $A_1,A_2, \ldots , A_{3n}$ in the plane, no three of them collinear. Prove that one can construct $n$ disjoint triangles with vertices at the points $A_i.$
2020 AMC 10, 13
Andy the Ant lives on a coordinate plane and is currently at $(-20, 20)$ facing east (that is, in the positive $x$-direction). Andy moves $1$ unit and then turns $90^{\circ}$ degrees left. From there, Andy moves $2$ units (north) and then turns $90^{\circ}$ degrees left. He then moves $3$ units (west) and again turns $90^{\circ}$ degrees left. Andy continues his progress, increasing his distance each time by $1$ unit and always turning left. What is the location of the point at which Andy makes the $2020$th left turn?
$\textbf{(A)}\ (-1030, -994)\qquad\textbf{(B)}\ (-1030, -990)\qquad\textbf{(C)}\ (-1026, -994)\qquad\textbf{(D)}\ (-1026, -990)\qquad\textbf{(E)}\ (-1022, -994)$
1986 Tournament Of Towns, (112) 6
( "Sisyphian Labour" )
There are $1001$ steps going up a hill , with rocks on some of them {no more than 1 rock on each step ) . Sisyphus may pick up any rock and raise it one or more steps up to the nearest empty step . Then his opponent Aid rolls a rock (with an empty step directly below it) down one step . There are $500$ rocks, originally located on the first $500$ steps. Sisyphus and Aid move rocks in turn , Sisyphus making the first move . His goal is to place a rock on the top step.
Can Aid stop him?
( S . Yeliseyev)
2016 Argentina National Olympiad, 5
Let $a$ and $b$ be rational numbers such that $a+b=a^2+b^2$ . Suppose the common value $s=a+b=a^2+b^2$ is not an integer, and let's write it as an irreducible fraction: $s=\frac{m}{n}$. Let $p$ be the smallest prime divisor of $n$. Find the minimum value of $p$.