Found problems: 85335
2024 China Girls Math Olympiad, 7
Let $n$ be a positive integer. If $x_1, x_2, \ldots, x_n \geq 0$, $x_1+x_2+\ldots+x_n=1$ and, assuming $x_{n+1}=x_1$, find the maximal value of $$\sum_{k=1}^n \frac{1+x_k^2+x_k^4}{1+x_{k+1}+x_{k+1}^2+x_{k+1}^3+x_{k+1}^4}.$$
2021 IMC, 3
We say that a positive real number $d$ is $good$ if there exists an infinite squence $a_1,a_2,a_3,...\in (0,d)$ such that for each $n$, the points $a_1,a_2,...,a_n$ partition the interval $[0,d]$ into segments of length at most $\frac{1}{n}$ each . Find
$\text{sup}\{d| d \text{is good}\}$.
Estonia Open Junior - geometry, 1995.1.4
The midpoint of the hypotenuse $AB$ of the right triangle $ABC$ is $K$. The point $M$ on the side $BC$ is taken such that $BM = 2 \cdot MC$. Prove that $\angle BAM = \angle CKM$.
2023 Euler Olympiad, Round 2, 2
Let $n$ be a positive integer. The Georgian folk dance team consists of $2n$ dancers, with $n$ males and $n$ females. Each dancer, both male and female, is assigned a number from 1 to $n$. During one of their dances, all the dancers line up in a single line. Their wish is that, for every integer $k$ from 1 to $n$, there are exactly $k$ dancers positioned between the $k$th numbered male and the $k$th numbered female. Prove the following statements:
a) If $n \equiv 1 \text{ or } 2 \mod{4}$, then the dancers cannot fulfill their wish.
b) If $n \equiv 0 \text{ or } 3 \mod{4}$, then the dancers can fulfill their wish.
[i]Proposed by Giorgi Arabidze, Georgia[/i]
1986 IMO Longlists, 7
Let $f(n)$ be the least number of distinct points in the plane such that for each $k = 1, 2, \cdots, n$ there exists a straight line containing exactly $k$ of these points. Find an explicit expression for $f(n).$
[i]Simplified version.[/i]
Show that $f(n)=\left[\frac{n+1}{2}\right]\left[\frac{n+2}{2}\right].$ Where $[x]$ denoting the greatest integer not exceeding $x.$
2018 Math Prize for Girls Problems, 10
Let $T_1$ be an isosceles triangle with sides of length 8, 11, and 11. Let $T_2$ be an isosceles triangle with sides of length $b$, 1, and 1. Suppose that the radius of the incircle of $T_1$ divided by the radius of the circumcircle of $T_1$ is equal to the radius of the incircle of $T_2$ divided by the radius of the circumcircle of $T_2$. Determine the largest possible value of $b$.
2002 Tournament Of Towns, 4
Quadrilateral $ABCD$ is circumscribed about a circle $\Gamma$ and $K,L,M,N$ are points of tangency of sides $AB,BC,CD,DA$ with $\Gamma$ respectively. Let $S\equiv KM\cap LN$. If quadrilateral $SKBL$ is cyclic then show that $SNDM$ is also cyclic.
1998 Federal Competition For Advanced Students, Part 2, 1
Let $a \geq 0$ be a natural number. Determine all rational $x$, so that
\[\sqrt{1+(a-1)\sqrt[3]x}=\sqrt{1+(a-1)\sqrt x}\]
All occurring square roots, are not negative.
[b]Note.[/b] It seems the set of natural numbers = $\mathbb N = \{0,1,2,\ldots\}$ in this problem.
2013 India IMO Training Camp, 3
A marker is placed at the origin of an integer lattice. Calvin and Hobbes play the following game. Calvin starts the game and each of them takes turns alternatively. At each turn, one can choose two (not necessarily distinct) integers $a, b$, neither of which was chosen earlier by any player and move the marker by $a$ units in the horizontal direction and $b$ units in the vertical direction. Hobbes wins if the marker is back at the origin any time after the first turn. Prove or disprove that Calvin can prevent Hobbes from winning.
Note: A move in the horizontal direction by a positive quantity will be towards the right, and by a negative quantity will be towards the left (and similar directions in the vertical case as well).
2007 Kyiv Mathematical Festival, 2
Find all pairs of positive integers $(a,b)$ such that $\sqrt{a-1}+\sqrt{b-1}=\sqrt{ab-1}.$
1993 USAMO, 3
Consider functions $\, f: [0,1] \rightarrow \mathbb{R} \,$ which satisfy
(i) $f(x) \geq 0 \,$ for all $\, x \,$ in $\, [0,1],$
(ii) $f(1) = 1,$
(iii) $f(x) + f(y) \leq f(x+y)\,$ whenever $\, x, \, y, \,$ and $\, x + y \,$ are all in $\, [0,1]$.
Find, with proof, the smallest constant $\, c \,$ such that
\[ f(x) \leq cx \]
for every function $\, f \,$ satisfying (i)-(iii) and every $\, x \,$ in $\, [0,1]$.
2005 Sharygin Geometry Olympiad, 15
Given a circle centered at the origin.
Prove that there is a circle of smaller radius that has no less points with integer coordinates.
2019 Pan-African Shortlist, N2
Let $k$ be a positive integer. Consider $k$ not necessarily distinct prime numbers such that their product is ten times their sum. What are these primes and what is the value of $k$?
2008 District Round (Round II), 1
Let $n$ be an integer greater than $1$.Find all pairs of integers $(s,t)$ such that equations:
$x^n+sx=2007$
and
$x^n+tx=2008$
have at least one common real root.
2010 Brazil National Olympiad, 2
Determine all values of $n$ for which there is a set $S$ with $n$ points, with no 3 collinear, with the following property: it is possible to paint all points of $S$ in such a way that all angles determined by three points in $S$, all of the same color or of three different colors, aren't obtuse. The number of colors available is unlimited.
1989 Vietnam National Olympiad, 3
Let be given a parallelepiped $ ABCD.A'B'C'D'$. Show that if a line $ \Delta$ intersects three of the lines $ AB'$, $ BC'$, $ CD'$, $ DA'$, then it intersects also the fourth line.
2013 AMC 10, 18
Let points $ A = (0,0) , \ B = (1,2), \ C = (3,3), $ and $ D = (4,0) $. Quadrilateral $ ABCD $ is cut into equal area pieces by a line passing through $ A $. This line intersects $ \overline{CD} $ at point $ \left (\frac{p}{q}, \frac{r}{s} \right ) $, where these fractions are in lowest terms. What is $ p + q + r + s $?
$ \textbf{(A)} \ 54 \qquad \textbf{(B)} \ 58 \qquad \textbf{(C)} \ 62 \qquad \textbf{(D)} \ 70 \qquad \textbf{(E)} \ 75 $
1994 IMO, 2
Let $ ABC$ be an isosceles triangle with $ AB \equal{} AC$. $ M$ is the midpoint of $ BC$ and $ O$ is the point on the line $ AM$ such that $ OB$ is perpendicular to $ AB$. $ Q$ is an arbitrary point on $ BC$ different from $ B$ and $ C$. $ E$ lies on the line $ AB$ and $ F$ lies on the line $ AC$ such that $ E, Q, F$ are distinct and collinear. Prove that $ OQ$ is perpendicular to $ EF$ if and only if $ QE \equal{} QF$.
2012 Dutch BxMO/EGMO TST, 5
Let $A$ be a set of positive integers having the following property:
for each positive integer $n$ exactly one of the three numbers $n, 2n$ and $3n$ is an element of $A$.
Furthermore, it is given that $2 \in A$. Prove that $13824 \notin A$.
2021/2022 Tournament of Towns, P4
A rock travelled through an n x n board, stepping at each turn to the cell neighbouring the previous one by a side, so that each cell was visited once. Bob has put the integer numbers from 1 to n^2 into the cells, corresponding to the order in which the rook has passed them. Let M be the greatest difference of the numbers in neighbouring by side cells. What is the minimal possible value of M?
2001 Irish Math Olympiad, 3
Show that if an odd prime number $ p$ can be expressed in the form $ x^5\minus{}y^5$ for some integers $ x,y,$ then:
$ \sqrt{\frac{4p\plus{}1}{5}}\equal{}\frac{v^2\plus{}1}{2}$ for some odd integer $ v$.
1997 AMC 12/AHSME, 4
If $ a$ is $ 50\%$ larger than $ c$, and $ b$ is $ 25\%$ larger than $ c$,then $ a$ is what percent larger than $ b$?
$ \textbf{(A)}\ 20\%\qquad \textbf{(B)}\ 25\%\qquad \textbf{(C)}\ 50\%\qquad \textbf{(D)}\ 100\%\qquad \textbf{(E)}\ 200\%$
2020 AMC 12/AHSME, 7
Two nonhorizontal, non vertical lines in the $xy$-coordinate plane intersect to form a $45^{\circ}$ angle. One line has slope equal to $6$ times the slope of the other line. What is the greatest possible value of the product of the slopes of the two lines?
$\textbf{(A)}\ \frac16 \qquad\textbf{(B)}\ \frac23 \qquad\textbf{(C)}\ \frac32 \qquad\textbf{(D)}\ 3 \qquad\textbf{(E)}\ 6$
2023 Balkan MO Shortlist, N2
Find all positive integers, such that there exist positive integers $a, b, c$, satisfying $\gcd(a, b, c)=1$ and $n=\gcd(ab+c, ac-b)=a+b+c$.
2015 Middle European Mathematical Olympiad, 3
There are $n$ students standing in line positions $1$ to $n$. While the teacher looks away, some students change their positions. When the teacher looks back, they are standing in line again. If a student who was initially in position $i$ is now in position $j$, we say the student moved for $|i-j|$ steps. Determine the maximal sum of steps of all students that they can achieve.