Found problems: 85335
1976 Bundeswettbewerb Mathematik, 4
Each vertex of the 3-dimensional Euclidean space either is coloured red or blue. Prove that within those squares being possible in this space with edge length 1 there is at least one square either with three red vertices or four blue vertices !
2010 Korea - Final Round, 5
On a circular table are sitting $ 2n$ people, equally spaced in between. $ m$ cookies are given to these people, and they give cookies to their neighbors according to the following rule.
(i) One may give cookies only to people adjacent to himself.
(ii) In order to give a cookie to one's neighbor, one must eat a cookie.
Select arbitrarily a person $ A$ sitting on the table. Find the minimum value $ m$ such that there is a strategy in which $ A$ can eventually receive a cookie, independent of the distribution of cookies at the beginning.
Dumbest FE I ever created, 4.
Find all $f: \mathbb{R} \to \mathbb{Z^+}$ such that $$f(x+f(y))=f(x)+f(y)+1\quad\text{ or }\quad f(x)+f(y)-1$$
for all real number $x$ and $y$
1988 China Team Selection Test, 1
Let $f(x) = 3x + 2.$ Prove that there exists $m \in \mathbb{N}$ such that $f^{100}(m)$ is divisible by $1988$.
2016 PUMaC Team, 15
Compute the sum of all positive integers $n$ with the property that $x^n \equiv 1$ (mod $2016$) has $n$ solutions in $\{0, 1, 2, ... , 2015\}$.
2018 Polish Junior MO Finals, 5
Point $M$ is middle of side $AB$ of equilateral triangle $ABC$. Points $D$ and $E$ lie on segments $AC$ and $BC$, respectively and $\angle DME = 60 ^{\circ}$. Prove that, $AD + BE = DE + \frac{1}{2}AB$.
Durer Math Competition CD 1st Round - geometry, 2018.C+2
In an isosceles right-angled triangle $ABC$, the right angle is at $A$. $D$ lies so on the side $BC$ that $2CD = DB$. Let $E$ be the projection of $B$ onto $AD$. What is the measure fof angle $\angle CED $?
2011 Albania Team Selection Test, 3
In the acute angle triangle $ABC$ the point $O$ is the center of the circumscribed circle and the lines $OA,OB,OC$ intersect sides $BC,CA,AB$ respectively in points $M,N,P$ such that $\angle NMP=90^o$.
[b](a)[/b] Find the ratios $\frac{\angle AMN}{\angle NMC}$,$\frac{\angle AMP}{\angle PMB}$.
[b](b)[/b] If any of the angles of the triangle $ABC$ is $60^o$, find the two other angles.
2020 Thailand TSTST, 2
Let $x, y, z$ be positive real numbers such that $x^2+y^2+z^2=3$. Prove that $$\frac{x+1}{z+x+1}+\frac{y+1}{x+y+1}+\frac{z+1}{y+z+1}\geq\frac{(xy+yz+zx+\sqrt{xyz})^2}{(x+y)(y+z)(z+x)}.$$
2012 BMT Spring, 4
There are 1$2$ people labeled $1, ..., 12$ working together on $12$ missions, with people $1, ... , i $working on the $i$th mission. There is exactly one spy among them. If the spy is not working on a mission, it will be a huge success, but if the spy is working on the mission, it will fail with probability $1/2$. Given that the first $11$ missions succeed, and the $12$th mission fails, what is the probability that person $12$ is the spy?
2017 ASDAN Math Tournament, 2
Find the remainder of $7^{1010}+8^{2017}$ when divided by $57$.
2024 Kyiv City MO Round 2, Problem 2
You are given a positive integer $n$. What is the largest possible number of numbers that can be chosen from the set
$\{1, 2, \ldots, 2n\}$ so that there are no two chosen numbers $x > y$ for which $x - y = (x, y)$?
Here $(x, y)$ denotes the greatest common divisor of $x, y$.
[i]Proposed by Anton Trygub[/i]
2003 Federal Math Competition of S&M, Problem 1
Find the number of solutions to the equation$$x_1^4+x_2^4+\ldots+x_{10}^4=2011$$in the set of positive integers.
2024 Irish Math Olympiad, P3
Let $\mathbb{Z}_+=\{1,2,3,4...\}$ be the set of all positive integers. Determine all functions $f : \mathbb{Z}_+ \mapsto \mathbb{Z}_+$ that satisfy:
[list]
[*]$f(mn)+1=f(m)+f(n)$ for all positive integers $m$ and $n$;
[*]$f(2024)=1$;
[*]$f(n)=1$ for all positive $n\equiv22\pmod{23}$.
[/list]
1987 AMC 8, 8
If $\text{A}$ and $\text{B}$ are nonzero digits, then the number of digits (not necessarily different) in the sum of the three whole numbers is
\[\begin{tabular}[t]{cccc}
9 & 8 & 7 & 6 \\
& A & 3 & 2 \\
& & B & 1 \\ \hline
\end{tabular}\]
$\text{(A)}\ 4 \qquad \text{(B)}\ 5 \qquad \text{(C)}\ 6 \qquad \text{(D)}\ 9 \qquad \text{(E)}\ \text{depends on the values of A and B}$
2016 Latvia National Olympiad, 2
Triangle $ABC$ has median $AF$, and $D$ is the midpoint of the median. Line $CD$ intersects $AB$ in $E$. Prove that $BD = BF$ implies $AE = DE$!
2005 AMC 10, 15
An envelope contains eight bills: $ 2$ ones, $ 2$ fives, $ 2$ tens, and $ 2$ twenties. Two bills are drawn at random without replacement. What is the probability that their sum is $ \$ 20$ or more?
$ \textbf{(A)}\ \frac {1}{4}\qquad
\textbf{(B)}\ \frac {2}{7}\qquad
\textbf{(C)}\ \frac {3}{7}\qquad
\textbf{(D)}\ \frac {1}{2}\qquad
\textbf{(E)}\ \frac {2}{3}$
2014 AMC 10, 12
The largest divisor of $2,014,000,000$ is itself. What is its fifth largest divisor?
$\textbf{(A) }125,875,000\qquad\textbf{(B) }201,400,000\qquad\textbf{(C) }251,750,000\qquad\textbf{(D) }402,800,000\qquad\textbf{(E) }503,500,000\qquad$
2014 JHMMC 7 Contest, 2
2. What’s the closest number to $169$ that’s divisible by $9$?
2019 South Africa National Olympiad, 3
Let $A$, $B$, $C$ be points on a circle whose centre is $O$ and whose radius is $1$, such that $\angle BAC = 45^\circ$. Lines $AC$ and $BO$ (possibly extended) intersect at $D$, and lines $AB$ and $CO$ (possibly extended) intersect at $E$. Prove that $BD \cdot CE = 2$.
2015 IFYM, Sozopol, 4
Let $k$ be a natural number. For each natural number $n$ we define $f_k (n)$ to be the least number, greater than $kn$, for which $nf_k (n)$ is a perfect square. Prove that $f_k (n)$ is injective.
2011 Regional Olympiad of Mexico Center Zone, 6
Given a circle $C$ and a diameter $AB$ in it, mark a point $P$ on $AB$ different from the ends. In one of the two arcs determined by $AB$ choose the points $M$ and $N$ such that $\angle APM = 60 ^ \circ = \angle BPN$. The segments $MP$ and $NP$ are drawn to obtain three curvilinear triangles; $APM $, $MPN$ and $NPB$ (the sides of the curvilinear triangle $APM$ are the segments $AP$ and $PM$ and the arc $AM$). In each curvilinear triangle a circle is inscribed, that is, a circle is built tangent to the three sides. Show that the sum of the radii of the three inscribed circles is less than or equal to the radius of $C$.
2010 Harvard-MIT Mathematics Tournament, 8
Let $O$ be the point $(0,0)$. Let $A$, $B$, $C$ be three points in the plane such that $AO=15$, $BO = 15$, and $CO = 7$, and such that the area of triangle $ABC$ is maximal. What is the length of the shortest side of $ABC$?
1997 AMC 12/AHSME, 29
Call a positive real number special if it has a decimal representation that consists entirely of digits $ 0$ and $ 7$. For example, $ \frac{700}{99} \equal{} 7.\overline{07} \equal{} 7.070707\cdots$ and $ 77.007$ are special numbers. What is the smallest $ n$ such that $ 1$ can be written as a sum of $ n$ special numbers?
$ \textbf{(A)}\ 7\qquad
\textbf{(B)}\ 8\qquad
\textbf{(C)}\ 9\qquad
\textbf{(D)}\ 10\qquad\\
\textbf{(E)}\ \text{The number 1 cannot be represented as a sum of finitely many special numbers.}$
India EGMO 2021 TST, 1
Mad scientist Kyouma writes $N$ positive integers on a board. Each second, he chooses two numbers $x, y$ written on the board with $x > y$, and writes the number $x^2-y^2$ on the board. After some time, he sends the list of all the numbers on the board to Christina. She notices that all the numbers from 1 to 1000 are present on the list. Aid Christina in finding the minimum possible value of N.