Found problems: 85335
2019 Iran RMM TST, 5
Edges of a planar graph $G$ are colored either with blue or red. Prove that there is a vertex like $v$ such that when we go around $v$ through a complete cycle, edges with the endpoint at $v$ change their color at most two times.
Clarifications for complete cycle:
If all the edges with one endpoint at $v$ are $(v,u_1),(v,u_2),\ldots,(v,u_k)$ such that $u_1,u_2,\ldots,u_k$ are clockwise with respect to $v$ then in the sequence of $(v,u_1),(v,u_2),\ldots,(v,u_k),(v,u_1)$ there are at most two $j$ such that colours of $(v,u_j),(v,u_{j+1})$ ($j \mod k$) differ.
2021 Latvia TST, 2.5
Suppose that $a,b,c,d$ are positive real numbers satisfying $(a+c)(b+d)=ac+bd$. Find the smallest possible value of
$$\frac{a}{b}+\frac{b}{c}+\frac{c}{d}+\frac{d}{a}.$$
[i]Israel[/i]
2010 Dutch IMO TST, 5
Find all triples $(x,y, z)$ of real (but not necessarily positive) numbers satisfying
$3(x^2 + y^2 + z^2) = 1$ , $x^2y^2 + y^2z^2 + z^2x^2 = xyz(x + y + z)^3$.
2024 Turkey MO (2nd Round), 6
Let $m,n\ge2$ be positive integers. On an $m\times n$ chessboard, some unit squares are occupied by rooks such that each rook attacked by odd number of other rooks. Determine the maximum number of rooks that can be placed on the chessboard.
2006 Sharygin Geometry Olympiad, 9.5
A straight line passing through the center of the circumscribed circle and the intersection point of the heights of the non-equilateral triangle $ABC$ divides its perimeter and area in the same ratio.Find this ratio.
2023 Bulgarian Autumn Math Competition, 11.4
Let $G$ be a complete bipartite graph with partition sets $A$ and $B$ of sizes $km$ and $kn$, respectively. The edges of $G$ are colored in $k$ colors. Prove that there exists a monochromatic connected component with at least $m+n$ vertices (which means that there exists a color and a set of vertices, such that between any two of them, there is a path consisting of edges only in that color).
V Soros Olympiad 1998 - 99 (Russia), 9.4
There are n points marked on the circle. It is known that among all possible distances between two marked points there are no more than $100$ different ones. What is the largest possible value for $n$?
1958 AMC 12/AHSME, 26
A set of $ n$ numbers has the sum $ s$. Each number of the set is increased by $ 20$, then multiplied by $ 5$, and then decreased by $ 20$. The sum of the numbers in the new set thus obtained is:
$ \textbf{(A)}\ s \plus{} 20n\qquad
\textbf{(B)}\ 5s \plus{} 80n\qquad
\textbf{(C)}\ s\qquad
\textbf{(D)}\ 5s\qquad
\textbf{(E)}\ 5s \plus{} 4n$
2021 Romanian Master of Mathematics, 4
Consider an integer \(n \ge 2\) and write the numbers \(1, 2, \ldots, n\) down on a board. A move consists in erasing any two numbers \(a\) and \(b\), then writing down the numbers \(a+b\) and \(\vert a-b \vert\) on the board, and then removing repetitions (e.g., if the board contained the numbers \(2, 5, 7, 8\), then one could choose the numbers \(a = 5\) and \(b = 7\), obtaining the board with numbers \(2, 8, 12\)). For all integers \(n \ge 2\), determine whether it is possible to be left with exactly two numbers on the board after a finite number of moves.
[i]Proposed by China[/i]
1996 India Regional Mathematical Olympiad, 5
Let $ABC$ be a triangle and $h_a$ be the altitude through $A$. Prove that \[ (b+c)^2 \geq a^2 + 4h_a ^2 . \]
2016 Greece JBMO TST, 4
Vaggelis has a box that contains $2015$ white and $2015$ black balls. In every step, he follows the procedure below:
He choses randomly two balls from the box. If they are both blacks, he paints one white and he keeps it in the box, and throw the other one out of the box. If they are both white, he keeps one in the box and throws the other out. If they are one white and one black, he throws the white out, and keeps the black in the box.
He continues this procedure, until three balls remain in the box. He then looks inside and he sees that there are balls of both colors. How many white balls does he see then, and how many black?
2000 South africa National Olympiad, 5
Find all functions $f: \mathbb{Z} \rightarrow \mathbb{Z}$ (where $\mathbb{Z}$ is the set of all integers) such that \[ 2000f(f(x)) - 3999f(x) + 1999x = 0\textrm{ for all }x \in \mathbb{Z}. \]
2009 239 Open Mathematical Olympiad, 8
Each of the $11$ girls wants to mail each of the other a gift for Christmas. The packages contain no more than two gifts. If they have enough time, what is the smallest possible number of packages that they have to send?
2016 Mathematical Talent Reward Programme, MCQ: P 11
In rectangle $ABCD$, $AD=1$, $P$ is on $AB$ and $DB$ and $DP$ trisect $\angle ADC$. What is the perimeter $\triangle BDP$
[list=1]
[*] $3+\frac{\sqrt{3}}{3}$
[*] $2+\frac{4\sqrt{3}}{3}$
[*] $2+2\sqrt{2}$
[*] $\frac{3+3\sqrt{5}}{2}$
[/list]
1941 Putnam, A7
Do either (1) or (2):
(1) Prove that the determinant of the matrix
$$\begin{pmatrix}
1+a^2 -b^2 -c^2 & 2(ab+c) & 2(ac-b)\\
2(ab-c) & 1-a^2 +b^2 -c^2 & 2(bc+a)\\
2(ac+b)& 2(bc-a) & 1-a^2 -b^2 +c^2
\end{pmatrix}$$
is given by $(1+a^2 +b^2 +c^2)^{3}$.
(2) A solid is formed by rotating the first quadrant of the ellipse $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$ around the $x$-axis. Prove that this solid can rest in stable equilibrium on its vertex if and only if $\frac{a}{b}\leq \sqrt{\frac{8}{5}}$.
2020 HK IMO Preliminary Selection Contest, 14
In $\Delta ABC$, $\angle ABC=120^\circ$. The internal bisector of $\angle B$ meets $AC$ at $D$. If $BD=1$, find the smallest possible value of $4BC+AB$.
2016 CMIMC, 10
Given $x_0\in\mathbb R$, $f,g:\mathbb R\to\mathbb R$, we define the $\emph{non-redundant binary tree}$ $T(x_0,f,g)$ in the following way:
[list=1]
[*]The tree $T$ initially consists of just $x_0$ at height $0$.
[*]Let $v_0,\dots,v_k$ be the vertices at height $h$. Then the vertices of height $h+1$ are added to $T$ by: for $i=0,1,\dots,k$, $f(v_i)$ is added as a child of $v_i$ if $f(v_i)\not\in T$, and $g(v_i)$ is added as a child of $v_i$ if $g(v_i)\not\in T$.
[/list]
For example, if $f(x)=x+1$ and $g(x)=x-1$, then the first three layers of $T(0,f,g)$ look like:
[asy]
size(100);
draw((-0.1,-0.2)--(-0.4,-0.8),EndArrow(size=3));
draw((0.1,-0.2)--(0.4,-0.8),EndArrow(size=3));
draw((-0.6,-1.2)--(-0.9,-1.8),EndArrow(size=3));
draw((0.6,-1.2)--(0.9,-1.8),EndArrow(size=3));
label("$0$",(0,0));
label("$1$",(-.5,-1));
label("$-1$",(.5,-1));
label("$2$",(-1,-2));
label("$-2$",(1,-2));[/asy]
If $f(x)=1024x-2047\lfloor x/2\rfloor$ and $g(x)=2x-3\lfloor x/2\rfloor+2\lfloor x/4\rfloor$, then how many vertices are in $T(2016,f,g)$?
2020 CHMMC Winter (2020-21), 7
Given $10$ points on a plane such that no three are collinear, we connect each pair of points with a segment and color each segment either red or blue. Assume that there exists some point $A$ among the $10$ points such that:
1. There is an odd number of red segments connected to $A$}
2. The number of red segments connected to each of the other points are all different
Find the number of red triangles (i.e, a triangle whose three sides are all red segments) on the plane.
2021 BMT, 23
Alireza is currently standing at the point $(0, 0)$ in the $x-y$ plane. At any given time, Alireza can move from the point $(x, y)$ to the point $(x + 1, y)$ or the point $(x, y + 1)$. However, he cannot move to any point of the form $(x, y)$ where $y \equiv 2x\,\, (\mod \,\,5)$. Let $p_k$ be the number of paths Alireza can take starting from the point $(0, 0)$ to the point $(k + 1, 2k + 1)$. Evaluate the sum $$\sum^{\infty}_{k=1} \frac{p_k}{5^k}.$$.
2024 Tuymaada Olympiad, 6
The triangle $ABC$ is given. On the arc $BC$ of its circumscribed circle, which does not contain point $A$, the variable point $X$ is selected, and on the rays $XB$ and $XC$, the variable points $Y$ and $Z$, respectively, so that $XA = XY = XZ$. Prove that the line $YZ$ passes through a fixed point.
[i]Proposed by A. Kuznetsov[/i]
2022 Belarusian National Olympiad, 10.6
Circles $\omega_1$ and $\omega_2$ intersect at $X$ and $Y$. Through point $Y$ two lines pass, one of which intersects $\omega_1$ and $\omega_2$ for the second time at $A$ and $B$, and the other at $C$ and $D$. Line $AD$ intersects for the second time circles $\omega_1$ and $\omega_2$ at $P$ and $Q$. It turned out that $YP=YQ$
Prove that the circumcircles of triangles $BCY$ and $PQY$ are tangent to each other.
2009 Harvard-MIT Mathematics Tournament, 3
Let $T$ be a right triangle with sides having lengths $3$, $4$, and $5$. A point $P$ is called [i]awesome[/i] if P is the center of a parallelogram whose vertices all lie on the boundary of $T$. What is the area of the set of awesome points?
2012 Princeton University Math Competition, A3
Six ants are placed on the vertices of a regular hexagon with an area of $12$. At each point in time, each ant looks at the next ant in the hexagon (in counterclockwise order), and measures the distance, $s$, to the next ant. Each ant then proceeds towards the next ant at a speed of $\frac{s}{100}$ units per year. After T years, the ants’ new positions are the vertices of a new hexagon with an area of $4$. T is of the form $a \ln b$, where $b$ is square-free. Find $a + b$.
2022 IMAR Test, 2
Let $n, k$ be natural numbers, $1 \leq k < n$. In each vertex of a regular polygon with $n$ sides is written $1$ or $-1$. At each step we choose $k$ consecutive vertices and change their signs. Is it possible that, starting from a certain configuration and by doing the operation a few times to obtain any other configuration?
2011 China Team Selection Test, 2
Let $a_1,a_2,\ldots,a_n,\ldots$ be any permutation of all positive integers. Prove that there exist infinitely many positive integers $i$ such that $\gcd(a_i,a_{i+1})\leq \frac{3}{4} i$.