Found problems: 85335
2006 MOP Homework, 1
There are $2005$ players in a chess tournament played a game. Every pair of players played a game against each other. At the end of the tournament, it turned out that if two players $A$ and $B$ drew the game between them, then every other player either lost to $A$ or to $B$. Suppose that there are at least two draws in the tournament. Prove that all players can be lined up in a single file from left to right in the such a way that every play won the game against the person immediately to his right.
2019 Tournament Of Towns, 2
Let $\omega$ be a circle with the center $O$ and $A$ and $C$ be two different points on $\omega$. For any third point $P$ of the circle let $X$ and $Y$ be the midpoints of the segments $AP$ and $CP$. Finally, let $H$ be the orthocenter (the point of intersection of the altitudes) of the triangle $OXY$ . Prove that the position of the point H does not depend on the choice of $P$.
(Artemiy Sokolov)
2003 China Team Selection Test, 2
Can we find positive reals $a_1, a_2, \dots, a_{2002}$ such that for any positive integer $k$, with $1 \leq k \leq 2002$, every complex root $z$ of the following polynomial $f(x)$ satisfies the condition $|\text{Im } z| \leq |\text{Re } z|$,
\[f(x)=a_{k+2001}x^{2001}+a_{k+2000}x^{2000}+ \cdots + a_{k+1}x+a_k,\] where $a_{2002+i}=a_i$, for $i=1,2, \dots, 2001$.
2015 AMC 12/AHSME, 25
A collection of circles in the upper half-plane, all tangent to the $x$-axis, is constructed in layers as follows. Layer $L_0$ consists of two circles of radii $70^2$ and $73^2$ that are externally tangent. For $k\geq 1$, the circles in $\textstyle\bigcup_{j=0}^{k-1} L_j$ are ordered according to their points of tangency with the $x$-axis. For every pair of consecutive circles in this order, a new circle is constructed externally tangent to each of the two circles in the pair. Layer $L_k$ consists of the $2^{k-1}$ circles constructed in this way. Let $S=\textstyle\bigcup_{j=0}^6 L_j$, and for every circle $C$ denote by $r(C)$ its radius. What is \[\sum_{C\in S}\dfrac1{\sqrt{r(C)}}?\]
[asy]
import olympiad;
size(350);
defaultpen(linewidth(0.7));
// define a bunch of arrays and starting points
pair[] coord = new pair[65];
int[] trav = {32,16,8,4,2,1};
coord[0] = (0,73^2); coord[64] = (2*73*70,70^2);
// draw the big circles and the bottom line
path arc1 = arc(coord[0],coord[0].y,260,360);
path arc2 = arc(coord[64],coord[64].y,175,280);
fill((coord[0].x-910,coord[0].y)--arc1--cycle,gray(0.78));
fill((coord[64].x+870,coord[64].y+425)--arc2--cycle,gray(0.78));
draw(arc1^^arc2);
draw((-930,0)--(70^2+73^2+850,0));
// We now apply the findCenter function 63 times to get
// the location of the centers of all 63 constructed circles.
// The complicated array setup ensures that all the circles
// will be taken in the right order
for(int i = 0;i<=5;i=i+1)
{
int skip = trav[i];
for(int k=skip;k<=64 - skip; k = k + 2*skip)
{
pair cent1 = coord[k-skip], cent2 = coord[k+skip];
real r1 = cent1.y, r2 = cent2.y, rn=r1*r2/((sqrt(r1)+sqrt(r2))^2);
real shiftx = cent1.x + sqrt(4*r1*rn);
coord[k] = (shiftx,rn);
}
// Draw the remaining 63 circles
}
for(int i=1;i<=63;i=i+1)
{
filldraw(circle(coord[i],coord[i].y),gray(0.78));
}[/asy]
$\textbf{(A) }\dfrac{286}{35}\qquad\textbf{(B) }\dfrac{583}{70}\qquad\textbf{(C) }\dfrac{715}{73}\qquad\textbf{(D) }\dfrac{143}{14}\qquad\textbf{(E) }\dfrac{1573}{146}$
2007 iTest Tournament of Champions, 1
Let $A$ be the area of the locus of points $z$ in the complex plane that satisfy $|z+12+9i| \leq 15$. Compute $\lfloor A\rfloor$.
2021 USAJMO, 4
Carina has three pins, labeled $A, B$, and $C$, respectively, located at the origin of the coordinate plane. In a [i]move[/i], Carina may move a pin to an adjacent lattice point at distance $1$ away. What is the least number of moves that Carina can make in order for triangle $ABC$ to have area 2021?
(A lattice point is a point $(x, y)$ in the coordinate plane where $x$ and $y$ are both integers, not necessarily positive.)
2005 India IMO Training Camp, 2
Find all functions $ f: \mathbb{N^{*}}\to \mathbb{N^{*}}$ satisfying
\[ \left(f^{2}\left(m\right)+f\left(n\right)\right) \mid \left(m^{2}+n\right)^{2}\]
for any two positive integers $ m$ and $ n$.
[i]Remark.[/i] The abbreviation $ \mathbb{N^{*}}$ stands for the set of all positive integers:
$ \mathbb{N^{*}}=\left\{1,2,3,...\right\}$.
By $ f^{2}\left(m\right)$, we mean $ \left(f\left(m\right)\right)^{2}$ (and not $ f\left(f\left(m\right)\right)$).
[i]Proposed by Mohsen Jamali, Iran[/i]
2022 Durer Math Competition Finals, 6
In Kacs Aladár street, houses are only found on one side of the road, so that only odd house numbers are found along the street. There are an odd number of allotments, as well. The middle three allotments belong to Scrooge McDuck, so he only put up the smallest of the three house numbers. The numbering of the other houses is standard, and the numbering begins with $1$. What is the largest number in the street if the sum of house numbers put up is $3133$?
2025 Al-Khwarizmi IJMO, 3
On a circle are arranged $100$ baskets, each containing at least one candy. The total number of candies is $780$. Asad and Sevinch make moves alternatingly, with Asad going first. On one move, Asad takes all the candies from $9$ consecutive non-empty baskets, while Sevinch takes all the candies from a single non-empty basket that has at least one empty neighboring basket. Prove that Asad can take overall at least $700$ candies, regardless of the initial distribution of candies and Sevinch's actions.
[i] Shubin Yakov, Russia [/i]
2018 Abels Math Contest (Norwegian MO) Final, 4
Find all polynomials $P$ such that
$P(x) + \binom{2018}{2}P(x+2)+...+\binom{2018}{2106}P(x+2016)+P(x+2018)=$
$=\binom{2018}{1}P(x+1)+\binom{2018}{3}P(x+3)+...+\binom{2018}{2105}P(x+2015)+\binom{2018}{2107}P(x+2017)$
for all real numbers $x$.
PEN A Problems, 116
What is the smallest positive integer that consists base 10 of each of the ten digits, each used exactly once, and is divisible by each of the digits $2$ through $9$?
LMT Team Rounds 2010-20, B28
There are $2500$ people in Lexington High School, who all start out healthy. After $1$ day, $1$ person becomes infected with coronavirus. Each subsequent day, there are twice as many newly infected people as on the previous day. How many days will it be until over half the school is infected?
2001 Manhattan Mathematical Olympiad, 4
How many digits has the number $2^{100}$?
2017 Tuymaada Olympiad, 4
A right triangle has all its sides rational numbers and the area $S $. Prove that there exists a right triangle, different from the original one, such that all its sides are rational numbers and its area is $S $.
Tuymaada 2017 Q4 Juniors
2024 Korea Junior Math Olympiad (First Round), 1.
Find this:
$ (1+\frac{1}{5})(1+\frac{1}{6})...(1+\frac{1}{2023})(1+\frac{1}{2024}) $
IV Soros Olympiad 1997 - 98 (Russia), 9.8
The equation $P(x) = 0$, where $P(x) = x^2+bx+c$, has a single root, and the equation $P(P(P(x))) = 0$ has exactly three different roots. Solve the equation $P(P(P(x))) = 0.$
Novosibirsk Oral Geo Oly VII, 2020.4
The altitudes $AN$ and $BM$ are drawn in triangle $ABC$. Prove that the perpendicular bisector to the segment $NM$ divides the segment $AB$ in half.
1978 Yugoslav Team Selection Test, Problem 2
Let $k_0$ be a unit semi-circle with diameter $AB$. Assume that $k_1$ is a circle of radius
$r_1=\frac12$ that is tangent to both $k_0$ and $AB$. The circle $k_{n+1}$ of radius $r_{n+1}$ touches
$k_n,k_0$, and $AB$. Prove that:
(a) For each $n\in\{2,3,\ldots\}$ it holds that $\frac1{r_{n+1}}+\frac1{r_{n-1}}=\frac6{r_n}-4$.
(b) $\frac1{r_n}$ is either a square of an even integer, or twice a square of an odd integer.
2004 Regional Competition For Advanced Students, 2
Solve the following equation for real numbers:
$ \sqrt{4\minus{}x\sqrt{4\minus{}(x\minus{}2)\sqrt{1\plus{}(x\minus{}5)(x\minus{}7)}}}\equal{}\frac{5x\minus{}6\minus{}x^2}{2}$
(all square roots are non negative)
1949-56 Chisinau City MO, 34
Construct a triangle by its altitude , median and angle bisector originating from one vertex.
1967 Putnam, A4
Show that if $\lambda > \frac{1}{2}$ there does not exist a real-valued function $u(x)$ such that for all $x$ in the closed interval $[0,1]$ the following holds:
$$u(x)= 1+ \lambda \int_{x}^{1} u(y) u(y-x) \; dy.$$
2005 IMC, 3
What is the maximal dimension of a linear subspace $ V$ of the vector space of real $ n \times n$ matrices such that for all $ A$ in $ B$ in $ V$, we have $ \text{trace}\left(AB\right) \equal{} 0$ ?
2016 PUMaC Combinatorics B, 3
Chitoge is painting a cube; she can paint each face either black or white, but she wants no vertex of the cube to be touching three faces of the same color. In how many ways can Chitoge paint the cube? Two paintings of a cube are considered to be the same if you can rotate one cube so that it looks like the other cube.
2018 Harvard-MIT Mathematics Tournament, 5
Compute the smallest positive integer $n$ for which $$\sqrt{100+\sqrt{n}}+\sqrt{100-\sqrt{n}}$$ is an integer.
2004 Tournament Of Towns, 3
We have a number of towns, with bus routes between some of them (each bus route goes from a town to another town without any stops). It is known that you can get from any town to any other by bus (possibly changing buses several times). Mr. Ivanov bought one ticket for each of the bus routes (a ticket allows single travel in either direction, but not returning on the same route). Mr. Petrov bought n tickets for each of the bus routes. Both Ivanov and Petrov started at town A. Ivanov used up all his tickets without buying any new ones and finished his travel at town B. Petrov, after using some of his tickets, got stuck at town X: he can not leave it without buying a new ticket. Prove that X is either A or B.