Found problems: 85335
2024 Belarus - Iran Friendly Competition, 1.2
Given $n \geq 2$ positive real numbers $x_1 \leq x_2 \leq \ldots \leq x_n$ satisfying the equalities
$$x_1+x_2+\ldots+x_n=4n$$
$$\frac{1}{x_1}+\frac{1}{x_2}+\ldots+\frac{1}{x_n}=n$$
Prove that $\frac{x_n}{x_1} \geq 7+4\sqrt{3}$
2011 LMT, 7
A triangle $ABC$ has side lengths $AB=8$ and $BC=10.$ Given that the altitude to side $BC$ has length $4,$ what is the length of the altitude to side $AB?$
2022 Polish Junior Math Olympiad Second Round, 5.
Let $n\geq 3$ be an odd integer. On a line, $n$ points are marked in such a way that the distance between any two of them is an integer. It turns out that each marked point has an even sum of distances to the remaining $n-1$ marked points. Prove that the distance between any two marked points is even.
2024 Thailand TST, 1
Determine all polynomials $P$ with integer coefficients for which there exists an integer $a_n$ such that $P(a_n)=n^n$ for all positive integers $n$.
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?