Found problems: 85335
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}
$
2003 Germany Team Selection Test, 1
At a chess tournament the winner gets 1 point and the defeated one 0 points. A tie makes both obtaining $\frac{1}{2}$ points. 14 players, none of them equally aged, participated in a competition where everybody played against all the other players. After the competition a ranking was carried out. Of the two players with the same number of points the younger received the better ranking. After the competition Jan realizes that the best three players together got as many points as the last 9 players obtained points together. And Joerg noted that the number of ties was maximal. Determine the number of ties.
2018 Macedonia JBMO TST, 3
Let $x$, $y$, and $z$ be positive real numbers such that $x + y + z = 1$. Prove that
$\frac{(x+y)^3}{z} + \frac{(y+z)^3}{x} + \frac{(z+x)^3}{y} + 9xyz \ge 9(xy + yz + zx)$.
When does equality hold?
2001 India IMO Training Camp, 3
In a triangle $ABC$ with incircle $\omega$ and incenter $I$ , the segments $AI$ , $BI$ , $CI$ cut $\omega$ at $D$ , $E$ , $F$ , respectively. Rays $AI$ , $BI$ , $CI$ meet the sides $BC$ , $CA$ , $AB$ at $L$ , $M$ , $N$ respectively. Prove that:
\[AL+BM+CN \leq 3(AD+BE+CF)\]
When does equality occur?
2009 Swedish Mathematical Competition, 3
An urn contain a number of yellow and green balls. You extract two balls from the urn (without adding them back) and calculate the probability of both balls being green. Can you choose the number of yellow and green balls such that this probability to be $\frac{1}{4}$?
1964 AMC 12/AHSME, 27
If $x$ is a real number and $|x-4|+|x-3|<a$ where $a>0$, then:
$ \textbf{(A)}\ 0<a<.01\qquad\textbf{(B)}\ .01<a<1 \qquad\textbf{(C)}\ 0<a<1\qquad$
$\textbf{(D)}\ 0<a \le 1\qquad\textbf{(E)}\ a>1 $
2022 Math Prize for Girls Problems, 3
Let $ABCD$ be a square face of a cube with edge length $2$. A plane $P$ that contains $A$ and the midpoint of $\overline{BC}$ splits the cube into two pieces of the same volume. What is the square of the area of the intersection of $P$ and the cube?