Found problems: 85335
2011 AMC 10, 7
Which of the following equations does NOT have a solution?
$\textbf{ (A) }\:(x+7)^2=0$ $\textbf{(B) }\:|-3x|+5=0$ $\textbf{ (C) }\:\sqrt{-x}-2=0$ $\textbf{ (D) }\:\sqrt{x}-8=0$ $\textbf{ (E) }\:|-3x|-4=0 $
1954 Kurschak Competition, 3
A tournament is arranged amongst a finite number of people. Every person plays every other person just once and each game results in a win to one of the players (there are no draws). Show that there must a person $X$ such that, given any other person $Y$ in the tournament, either $X$ beat $Y$ , or $X$ beat $Z$ and $Z$ beat $Y$ for some $Z$.
1983 IMO Shortlist, 10
Let $p$ and $q$ be integers. Show that there exists an interval $I$ of length $1/q$ and a polynomial $P$ with integral coefficients such that
\[ \left|P(x)-\frac pq \right| < \frac{1}{q^2}\]for all $x \in I.$
2005 Sharygin Geometry Olympiad, 11.5
The angle and the point $K$ inside it are given on the plane. Prove that there is a point $M$ with the following property:
if an arbitrary line passing through intersects the sides of the angle at points $A$ and $B$, then $MK$ is the bisector of the angle $AMB$.
2013 Harvard-MIT Mathematics Tournament, 5
Let $a$ and $b$ be real numbers, and let $r$, $s$, and $t$ be the roots of $f(x)=x^3+ax^2+bx-1$. Also, $g(x)=x^3+mx^2+nx+p$ has roots $r^2$, $s^2$, and $t^2$. If $g(-1)=-5$, find the maximum possible value of $b$.
2024 Malaysian APMO Camp Selection Test, 3
Find all functions $f:\mathbb{Z}\rightarrow \mathbb{Z}$ such that for all integers $x$, $y$, $$f(x-f(y))=f(f(y))+f(x-2y)$$
[i]Proposed by Ivan Chan Kai Chin[/i]
2023 Bosnia and Herzegovina Junior BMO TST, 4.
Let $n$ be a positive integer. A board with a format $n*n$ is divided in $n*n$ equal squares.Determine all integers $n$≥3 such that the board can be covered in $2*1$ (or $1*2$) pieces so that there is exactly one empty square in each row and each column.
2017 Harvard-MIT Mathematics Tournament, 5
Let $ABCD$ be a quadrilateral with an inscribed circle $\omega$ and let $P$ be the intersection of its diagonals $AC$ and $BD$. Let $R_1$, $R_2$, $R_3$, $R_4$ be the circumradii of triangles $APB$, $BPC$, $CPD$, $DPA$ respectively. If $R_1=31$ and $R_2=24$ and $R_3=12$, find $R_4$.
1971 Spain Mathematical Olympiad, 1
Calculate $$\sum_{k=5}^{k=49}\frac{11_(k}{2\sqrt[3]{1331_(k}}$$ knowing that the numbers $11$ and $1331$ are written in base $k \ge 4$.
2015 Middle European Mathematical Olympiad, 2
Let $n\ge 3$ be an integer. An [i]inner diagonal[/i] of a [i]simple $n$-gon[/i] is a diagonal that is contained in the $n$-gon. Denote by $D(P)$ the number of all inner diagonals of a simple $n$-gon $P$ and by $D(n)$ the least possible value of $D(Q)$, where $Q$ is a simple $n$-gon. Prove that no two inner diagonals of $P$ intersect (except possibly at a common endpoint) if and only if $D(P)=D(n)$.
[i]Remark:[/i] A simple $n$-gon is a non-self-intersecting polygon with $n$ vertices. A polygon is not necessarily convex.
1984 IMO Longlists, 22
In a permutation $(x_1, x_2, \dots , x_n)$ of the set $1, 2, \dots , n$ we call a pair $(x_i, x_j )$ discordant if $i < j$ and $x_i > x_j$. Let $d(n, k)$ be the number of such permutations with exactly $k$ discordant pairs. Find $d(n, 2)$ and $d(n, 3).$
2018 District Olympiad, 2
Consider a right-angled triangle $ABC$, $\angle A = 90^{\circ}$ and points $D$ and $E$ on the leg $AB$ such that $\angle ACD \equiv \angle DCE \equiv \angle ECB$. Prove that if $3\overrightarrow{AD} = 2\overrightarrow{DE}$ and $\overrightarrow{CD} + \overrightarrow{CE} = 2\overrightarrow{CM}$ then $\overrightarrow{AB} = 4\overrightarrow{AM}$.
2016 PUMaC Team, 7
In triangle $ABC$, let $S$ be on $BC$ and $T$ be on $AC$ so that $AS \perp BC$ and $BT \perp AC$, and let $AS$ and $BT$ intersect at $H$. Let $O$ be the center of the circumcircle of $\vartriangle AHT, P$ be the center of the circumcircle of $\vartriangle BHS$, and $G$ be the other point of intersection (besides $H$) of the two circles. Let $GH$ and $OP$ intersect at $X$. If $AB = 14, BH = 6$, and HA = 11, then $XO - XP$ can be written in simplest form as $\frac{m}{n}$ . Find $m + n$.
2023 JBMO Shortlist, C1
Given is a square board with dimensions $2023 \times 2023$, in which each unit cell is colored blue or red. There are exactly $1012$ rows in which the majority of cells are blue, and exactly $1012$ columns in which the majority of cells are red.
What is the maximal possible side length of the largest monochromatic square?
2018 Cono Sur Olympiad, 2
Prove that every positive integer can be formed by the sums of powers of 3, 4 and 7, where do not appear two powers of the same number and with the same exponent.
Example: $2= 7^0 + 7^0$ and $22=3^2 + 3^2+4^1$ are not valid representations, but $2=3^0+7^0$ and $22=3^2+3^0+4^1+4^0+7^1$ are valid representations.
2022 Cono Sur, 3
Prove that for every positive integer $n$ there exists a positive integer $k$, such that each of the numbers $k, k^2, \dots, k^n$ have at least one block of $2022$ in their decimal representation.
For example, the numbers 4[b]2022[/b]13 and 544[b]2022[/b]1[b]2022[/b] have at least one block of $2022$ in their decimal representation.
2012 Romania National Olympiad, 2
[color=darkred] Let $(R,+,\cdot)$ be a ring and let $f$ be a surjective endomorphism of $R$ such that $[x,f(x)]=0$ for any $x\in R$ , where $[a,b]=ab-ba$ , $a,b\in R$ . Prove that:
[list]
[b]a)[/b] $[x,f(y)]=[f(x),y]$ and $x[x,y]=f(x)[x,y]$ , for any $x,y\in R\ ;$
[b]b)[/b] If $R$ is a division ring and $f$ is different from the identity function, then $R$ is commutative.
[/list]
[/color]
2018 Hong Kong TST, 2
For which natural number $n$ is it possible to place natural number from 1 to $3n$ on the edges of a right $n$-angled prism (on each edge there is exactly one number placed and each one is used exactly 1 time) in such a way, that the sum of all the numbers, that surround each face is the same?
2023 BMT, 22
Let $d_n(x)$ be the $n$-th decimal digit (after the decimal point) of $x$. For example, $d_3(\pi) = 1$ because $\pi = 3.14\underline{1}5...$ For a positive integer $k$, let $f(k) = p^4_k$, where $p_k$ is the $k$-th prime number. Compute the value of $$\sum^{2023}_{i=1} d_{f(i)} \left( \frac{1}{1275}\right).$$
1965 Czech and Slovak Olympiad III A, 1
Show that the number $5^{2n+1}2^{n+2}+3^{n+2}2^{2n+1}$ is divisible by $19$ for every non-negative integer $n$.
2010 VJIMC, Problem 4
For every positive integer $n$ let $\sigma(n)$ denote the sum of all its positive divisors. A number $n$ is called weird if $\sigma(n)\ge2n$ and there exists no representation
$$n=d_1+d_2+\ldots+d_r,$$where $r>1$ and $d_1,\ldots,d_r$ are pairwise distinct positive divisors of $n$.
Prove that there are infinitely many weird numbers.
2018 Sharygin Geometry Olympiad, 5
The side $AB$ of a square $ABCD$ is the base of an isosceles triangle $ABE$ such that $AE=BE$ lying outside the square. Let $M$ be the midpoint of $AE$, $O$ be the intersection of $AC$ and $BD$. $K$ is the intersection of $OM$ and $ED$. Prove that $EK=KO$.
2002 All-Russian Olympiad, 2
A quadrilateral $ABCD$ is inscribed in a circle $\omega$. The tangent to $\omega$ at $A$ intersects the ray $CB$ at $K$, and the tangent to $\omega$ at $B$ intersects the ray $DA$ at $M$. Prove that if $AM=AD$ and $BK=BC$, then $ABCD$ is a trapezoid.
2019-2020 Fall SDPC, 7
Find all pairs of positive integers $a,b$ with $$a^a+b^b \mid (ab)^{|a-b|}-1.$$
2003 National Olympiad First Round, 8
Let $P$ be a polynomial such that $(x-4)P(2x) = 4(x-1)P(x)$, for every real $x$. If $P(0) \neq 0$, what is the degree of $P$?
$
\textbf{(A)}\ 0
\qquad\textbf{(B)}\ 1
\qquad\textbf{(C)}\ 2
\qquad\textbf{(D)}\ 3
\qquad\textbf{(E)}\ \text{None of the preceding}
$