Found problems: 85335
2019 CCA Math Bonanza, TB4
The number $28!$ ($28$ in decimal) has base $30$ representation \[28!=Q6T32S??OCLQJ6000000_{30}\] where the seventh and eighth digits are missing. What are the missing digits? In base $30$, we have that the digits $A=10$, $B=11$, $C=12$, $D=13$, $E=14$, $F=15$, $G=16$, $H=17$, $I=18$, $J=19$, $K=20$, $L=21$, $M=22$, $N=23$, $O=24$, $P=25$, $Q=26$, $R=27$, $S=28$, $T=29$.
[i]2019 CCA Math Bonanza Tiebreaker Round #4[/i]
2022 Girls in Math at Yale, 6
Carissa is crossing a very, very, very wide street, and did not properly check both ways before doing so. (Don't be like Carissa!) She initially begins walking at $2$ feet per second. Suddenly, she hears a car approaching, and begins running, eventually making it safely to the other side, half a minute after she began crossing. Given that Carissa always runs $n$ times as fast as she walks and that she spent $n$ times as much time running as she did walking, and given that the street is $260$ feet wide, find Carissa's running speed, in feet per second.
[i]Proposed by Andrew Wu[/i]
2014 Contests, 3
Let $B$ and $C$ be two fixed points on a circle centered at $O$ that are not diametrically opposed. Let $A$ be a variable point on the circle distinct from $B$ and $C$ and not belonging to the perpendicular bisector of $BC$. Let $H$ be the orthocenter of $\triangle ABC$, and $M$ and $N$ be the midpoints of the segments $BC$ and $AH$, respectively. The line $AM$ intersects the circle again at $D$, and finally, $NM$ and $OD$ intersect at $P$. Determine the locus of points $P$ as $A$ moves around the circle.
Durer Math Competition CD Finals - geometry, 2018.C+2
Given an $ABC$ triangle. Let $D$ be an extension of section $AB$ beyond $A$ such that that $AD = BC$ and $E$ is the extension of the section $BC$ beyond $B$ such that $BE = AC$. Prove that the circumcircle of triangle $DEB$ passes through the center of the inscribed circle of triangle $ABC$.
1963 Miklós Schweitzer, 8
Let the Fourier series \[ \frac{a_0}{2}+ \sum _{k\geq 1}(a_k\cos kx+b_k \sin kx)\] of a function $ f(x)$ be
absolutely convergent, and let \[ a^2_k+b^2_k \geq a_{k+1}^2+b_{k+1}^2 \;(k=1,2,...)\ .\] Show that \[ \frac1h \int_0^{2\pi} (f(x+h)-f(x-h))^2dx \;(h>0)\] is uniformly bounded in $ h$. [K. Tandori]
2008 Indonesia TST, 2
Let $S = \{1, 2, 3, ..., 100\}$ and $P$ is the collection of all subset $T$ of $S$ that have $49$ elements, or in other words: $$P = \{T \subset S : |T| = 49\}.$$ Every element of $P$ is labelled by the element of $S$ randomly (the labels may be the same). Show that there exist subset $M$ of $S$ that has $50$ members such that for every $x \in M$, the label of $M -\{x\}$ is not equal to $x$
2023 Polish MO Finals, 2
Given an acute triangle $ABC$ with their incenter $I$. Point $X$ lies on $BC$ on the same side as $B$ wrt $AI$. Point $Y$ lies on the shorter arc $AB$ of the circumcircle $ABC$. It is given that $$\angle AIX = \angle XYA = 120^\circ.$$
Prove that $YI$ is the angle bisector of $XYA$.
1981 Austrian-Polish Competition, 1
Find the smallest $n$ for which we can find $15$ distinct elements $a_{1},a_{2},...,a_{15}$ of $\{16,17,...,n\}$ such that $a_{k}$ is a multiple of $k$.
1982 IMO Longlists, 40
We consider a game on an infinite chessboard similar to that of solitaire: If two adjacent fields are occupied by pawns and the next field is empty (the three fields lie on a vertical or horizontal line), then we may remove these two pawns and put one of them on the third field. Prove that if in the initial position pawns fill a $3k \times n$ rectangle, then it is impossible to reach a position with only one pawn on the board.
2018 PUMaC Team Round, 5
There exist real numbers $a$, $b$, $c$, $d$, and $e$ such that for all positive integers $n$, we have
$$\sqrt{n}=\sum_{i=0}^{n-1}\sqrt[5]{\sqrt{ai^5+bi^4+ci^3+di^2+ei+1}-\sqrt{ai^5+bi^4+ci^3+di^2+ei}}.$$
Find $a+b+c+d$.
1991 China National Olympiad, 5
Find all natural numbers $n$, such that $\min_{k\in \mathbb{N}}(k^2+[n/k^2])=1991$. ($[n/k^2]$ denotes the integer part of $n/k^2$.)
2011 Junior Balkan MO, 1
Let $a,b,c$ be positive real numbers such that $abc = 1$. Prove that:
$\displaystyle\prod(a^5+a^4+a^3+a^2+a+1)\geq 8(a^2+a+1)(b^2+b+1)(c^2+c+1)$
2015 Portugal MO, 6
For what values of $n$ is it possible to mark $n$ points on the plane so that each point has at least three other points at distance $1$?
2007 Mongolian Mathematical Olympiad, Problem 4
Let $ a,b,c>0$. Prove that
$ \frac{a}{b}\plus{}\frac{b}{c}\plus{}\frac{c}{a}\geq 3\sqrt{\frac{a^2\plus{}b^2\plus{}c^2}{ab\plus{}bc\plus{}ca}}$
2013 Saint Petersburg Mathematical Olympiad, 1
Find the minimum positive noninteger root of $ \sin x=\sin \lfloor x \rfloor $.
F. Petrov
2014 BMT Spring, 5
Call two regular polygons supplementary if the sum of an internal angle from each polygon adds up to $180^o$. For instance, two squares are supplementary because the sum of the internal angles is $90^o + 90^o = 180^o$. Find the other pair of supplementary polygons. Write your answer in the form $(m, n)$ where m and n are the number of sides of the polygons and $m < n$.
2000 Harvard-MIT Mathematics Tournament, 8
A sphere is inscribed inside a pyramid with a square as a base whose height is $\frac{\sqrt{15}}{2}$ times the length of one edge of the base. A cube is inscribed inside the sphere. What is the ratio of the volume of the pyramid to the volume of the cube?
2014 All-Russian Olympiad, 1
Does there exist positive $a\in\mathbb{R}$, such that
\[|\cos x|+|\cos ax| >\sin x +\sin ax \]
for all $x\in\mathbb{R}$?
[i]N. Agakhanov[/i]
1998 Niels Henrik Abels Math Contest (Norwegian Math Olympiad) Round 2, 10
The number of pairs of integers $ (m,n)$ satisfying the equation
\[ m^3 \plus{} 6m^2 \plus{} 5m \equal{} 27n^3 \plus{} 9n^2 \plus{} 9n \plus{} 1\]
is
$ \text{(A)}\ 0 \qquad \text{(B)}\ 1 \qquad \text{(C)}\ 2 \qquad \text{(D)}\ 3 \qquad \text{(E)}\ \text{Infinitely many}$
Estonia Open Junior - geometry, 1995.1.2
Two circles of equal radius intersect at two distinct points $A$ and $B$. Let their radii $r$ and their midpoints respectively be $O_1$ and $O_2$. Find the greatest possible value of the area of the rectangle $O_1AO_2B$.
2017 Junior Balkan Team Selection Tests - Moldova, Problem 8
The bottom line of a $2\times 13$ rectangle is filled with $13$ tokens marked with the numbers $1, 2, ..., 13$ and located in that order. An operation is a move of a token from its cell into a free adjacent cell (two cells are called adjacent if they have a common side). What is the minimum number of operations needed to rearrange the chips in reverse order in the bottom line of the rectangle?
2022 Rioplatense Mathematical Olympiad, 2
Eight teams play a rugby tournament in which each team plays exactly one match against each of the remaining seven teams. In each match, if it's a tie each team gets $1$ point and if it isn't a tie then the winner gets $2$ points and the loser gets $0$ points. After the tournament it was observed that each of the eight teams had a different number of points and that the number of points of the winner of the tournament was equal to the sum of the number of points of the last four teams.
Give an example of a tournament that satisfies this conditions, indicating the number of points obtained by each team and the result of each match.
2008 Saint Petersburg Mathematical Olympiad, 6
A diagonal of a 100-gon is called good if it divides the 100-gon into two polygons each with an odd number of sides. A 100-gon was split into triangles with non-intersecting diagonals, exactly 49 of which are good. The triangles are colored into two colors such that no two triangles that border each other are colored with the same color. Prove that there is the same number of triangles colored with one color as with the other.
Fresh translation; slightly reworded.
2021 Portugal MO, 4
Pedro and Tiago are playing a game with a deck of n cards, numbered from $1$ to $n$. Starting with Pedro, they choose cards alternately, and receive the number of points indicated by the cards. However, whenever the player chooses the card with the highest number among those remaining in the deck, he is forced to pass his next turn, not choosing any card. When the deck runs out, the player with the most points wins. Knowing that Tiago can at least draw, regardless of Pedro's moves, how many cards are in the deck? Indicates all possibilities,
2021 Kyiv City MO Round 1, 10.1
Prove the following inequality:
$$\sin{1} + \sin{3} + \ldots + \sin{2021} > \frac{2\sin{1011}^2}{\sqrt{3}}$$
[i]Proposed by Oleksii Masalitin[/i]