Found problems: 85335
1991 Tournament Of Towns, (284) 4
The number $123$ is shown on the screen of a computer. Each minute the computer adds $102$ to the number on the screen. The computer expert Misha may change the order of digits in the number on the screen whenever he wishes. Can he ensure that no four-digit number ever appears on the screen?
(F.L. Nazarov, Leningrad)
2001 Mediterranean Mathematics Olympiad, 3
Show that there exists a positive integer $N$ such that the decimal representation of $2000^N$ starts with the digits $200120012001.$
2018 Caucasus Mathematical Olympiad, 5
Baron Munсhausen discovered the following theorem: "For any positive integers $a$ and $b$ there exists a positive integer $n$ such that $an$ is a perfect cube, while $bn$ is a perfect fifth power". Determine if the statement of Baron’s theorem is correct.
2011 Romania National Olympiad, 4
[color=darkred]Let $A\, ,\, B\in\mathcal{M}_2(\mathbb{C})$ so that : $A^2+B^2=2AB$ .
[b]a)[/b] Prove that : $AB=BA$ .
[b]b)[/b] Prove that : $\text{tr}\, (A)=\text{tr}\, (B)$ .[/color]
2011 District Round (Round II), 3
Find all pairs $(m, n)$ of positive integers for which $4 (mn +1)$ is divisible by $(m + n)^2$.
1966 IMO Shortlist, 61
Prove that for every natural number $n$, and for every real number $x \neq \frac{k\pi}{2^t}$ ($t=0,1, \dots, n$; $k$ any integer) \[ \frac{1}{\sin{2x}}+\frac{1}{\sin{4x}}+\dots+\frac{1}{\sin{2^nx}}=\cot{x}-\cot{2^nx} \]
2018 India IMO Training Camp, 2
For an integer $n\ge 2$ find all $a_1,a_2,\cdots ,a_n, b_1,b_2,\cdots , b_n$ so that
(a) $0\le a_1\le a_2\le \cdots \le a_n\le 1\le b_1\le b_2\le \cdots \le b_n;$
(b) $\sum_{k=1}^n (a_k+b_k)=2n;$
(c) $\sum_{k=1}^n (a_k^2+b_k^2)=n^2+3n.$
2005 Purple Comet Problems, 12
Four mathletes and two coaches sit at a circular table. How many distinct arrangements are there of these six people if the two coaches sit opposite each other?
2025 6th Memorial "Aleksandar Blazhevski-Cane", P4
Prove that for all real numbers $a, b, c > 1$ the inequality
\[a(b^2 + c) + b(c^2 + a) + c(a^2 + b) \ge a^2 + b^2 + c^2 + 3abc\]
holds. When does equality hold?
Proposed by [i]Ilija Jovcevski[/i]
Ukrainian TYM Qualifying - geometry, VIII.3
Find the largest value of the expression $\frac{p}{R}\left( 1- \frac{r}{3R}\right)$ , where $p,R, r$ is, respectively, the perimeter, the radius of the circumscribed circle and the radius of the inscribed circle of a triangle.
PEN A Problems, 64
The last digit of the number $x^2 +xy+y^2$ is zero (where $x$ and $y$ are positive integers). Prove that two last digits of this numbers are zeros.
2015 ASDAN Math Tournament, 1
In his spare time, Thomas likes making rectangular windows. He builds windows by taking four $30\text{ cm}\times20\text{ cm}$ rectangles of glass and arranging them in a larger rectangle in wood. The window has an $x\text{ cm}$ wide strip of wood between adjacent glass pieces and an $x\text{ cm}$ wide strip of wood between each glass piece and the adjacent edge of the window. Given that the total area of the glass is equivalent to the total area of the wood, what is $x$?
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1981 Miklós Schweitzer, 9
Let $ n \geq 2$ be an integer, and let $ X$ be a connected Hausdorff space such that every point of $ X$ has a neighborhood homeomorphic to the Euclidean space $ \mathbb{R}^n$. Suppose that any discrete (not necessarily closed ) subspace $ D$ of $ X$ can be covered by a family of pairwise disjoint, open sets of $ X$ so that each of these open sets contains precisely one element of $ D$. Prove that $ X$ is a union of at most $ \aleph_1$ compact subspaces.
[i]Z. Balogh[/i]
2004 Germany Team Selection Test, 3
Let $ABC$ be an isosceles triangle with $AC=BC$, whose incentre is $I$. Let $P$ be a point on the circumcircle of the triangle $AIB$ lying inside the triangle $ABC$. The lines through $P$ parallel to $CA$ and $CB$ meet $AB$ at $D$ and $E$, respectively. The line through $P$ parallel to $AB$ meets $CA$ and $CB$ at $F$ and $G$, respectively. Prove that the lines $DF$ and $EG$ intersect on the circumcircle of the triangle $ABC$.
[i]Proposed by Hojoo Lee, Korea[/i]
2006 Croatia Team Selection Test, 1
Find all natural numbers that can be expressed in a unique way as a sum of five or less perfect squares.
MOAA Individual Speed General Rounds, 2023.4
A number is called \textit{super odd} if it is an odd number divisible by the square of an odd prime. For example, $2023$ is a \textit{super odd} number because it is odd and divisible by $17^2$. Find the sum of all \textit{super odd} numbers from $1$ to $100$ inclusive.
[i]Proposed by Andy Xu[/i]
2015 FYROM JBMO Team Selection Test, 5
$A$ and $B$ are two identical convex polygons, each with an area of $2015$. The polygon $A$ is divided into polygons $A_1, A_2,...,A_{2015}$, while $B$ is divided into polygons $B_1, B_2,...,B_{2015}$. Each of these smaller polygons has a positive area. Furthermore, $A_1, A_2,...,A_{2015}$ and $B_1, B_2,...,B_{2015}$ are colored in $2015$ distinct colors, such that $A_i$ and $A_j$ are differently colored for every distinct $i$ and $j$ and $B_i$ and $B_j$ are also differently colored for every distinct $i$ and $j$. After $A$ and $B$ overlap, we calculate the sum of the areas with the same colors. Prove that we can color the polygons such that this sum is at least $1$.
1993 Putnam, B2
A deck of $2n$ cards numbered from $1$ to $2n$ is shuffled and n cards are dealt to $A$ and $B$. $A$ and $B$ alternately discard a card face up, starting with $A$. The game when the sum of the discards is first divisible by $2n + 1$, and the last person to discard wins. What is the probability that $A$ wins if neither player makes a mistake?
2021 DIME, 5
Let $\mathcal{S}$ be the set of all positive integers which are both a multiple of $3$ and have at least one digit that is a $1$. For example, $123$ is in $\mathcal{S}$ and $450$ is not. The probability that a randomly chosen $3$-digit positive integer is in $\mathcal{S}$ can be written as $\tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.
[i]Proposed by GammaZero[/i]
2002 Finnish National High School Mathematics Competition, 5
There is a regular $17$-gon $\mathcal{P}$ and its circumcircle $\mathcal{Y}$ on the plane.
The vertices of $\mathcal{P}$ are coloured in such a way that $A,B \in \mathcal{P}$ are of diff erent colour, if the shorter arc connecting $A$ and $B$ on $\mathcal{Y}$ has $2^k+1$ vertices, for some $k \in \mathbb{N},$ including $A$ and $B.$
What is the least number of colours which suffices?
2013 IMC, 4
Does there exist an infinite set $\displaystyle{M}$ consisting of positive integers such that for any $\displaystyle{a,b \in M}$ with $\displaystyle{a < b}$ the sum $\displaystyle{a + b}$ is square-free?
[b]Note.[/b] A positive integer is called square-free if no perfect square greater than $\displaystyle{1}$ divides it.
[i]Proposed by Fedor Petrov, St. Petersburg State University.[/i]
2004 Manhattan Mathematical Olympiad, 4
We say that a circle is [i]half-inscribed[/i] in a triangle, if its center lies on one side of the triangle, and it is tangent to the other two sides. Show that a triangle that has two half-inscribed circles of equal radii, is isosceles. (Recall that a triangle is said to be [i]isosceles[/i], if it has two sides of equal length.)
2006 National Olympiad First Round, 26
For how many primes $p$, there exists an integr $m$ such that $m^3+3m-2 \equiv 0 \pmod p$ and $m^2+4m+5\equiv 0 \pmod p$?
$
\textbf{(A)}\ 1
\qquad\textbf{(B)}\ 2
\qquad\textbf{(C)}\ 3
\qquad\textbf{(D)}\ 4
\qquad\textbf{(E)}\ \text{Infinitely many}
$
2008 Regional Olympiad of Mexico Northeast, 3
Consider the sequence $1,9,8,3,4,3,…$ in which $a_{n+4}$ is the units digit of $a_n+a_{n+3}$, for $n$ positive integer. Prove that $a^2_{1985}+a^2_{1986}+…+a^2_{2000}$ is a multiple of $2$.
2016 Indonesia TST, 6
Let $ABC$ be an acute triangle and let $M$ be the midpoint of $AC$. A circle $\omega$ passing through $B$ and $M$ meets the sides $AB$ and $BC$ at points $P$ and $Q$ respectively. Let $T$ be the point such that $BPTQ$ is a parallelogram. Suppose that $T$ lies on the circumcircle of $ABC$. Determine all possible values of $\frac{BT}{BM}$.