Found problems: 85335
2011 IFYM, Sozopol, 7
We define the sequence
$x_1=n,y_1=1,x_{i+1}=[\frac{x_i+y_i}{2}],y_{i+1}=[\frac{n}{x_{i+1}} ]$.
Prove that $min\{ x_1, x_2, ..., x_n\}=[\sqrt{n}]$ .
2014 All-Russian Olympiad, 3
Positive rational numbers $a$ and $b$ are written as decimal fractions and each consists of a minimum period of 30 digits. In the decimal representation of $a-b$, the period is at least $15$. Find the minimum value of $k\in\mathbb{N}$ such that, in the decimal representation of $a+kb$, the length of period is at least $15$.
[i]A. Golovanov[/i]
2024 Putnam, A1
Determine all positive integers $n$ for which there exists positive integers $a$, $b$, and $c$ satisfying
\[
2a^n+3b^n=4c^n.
\]
2024 China Girls Math Olympiad, 8
It is known that there are $2024$ pairs of friends among $100$ people. Show that is possible to split them into $50$ pairs so that:
(a) There are at most $20$ pairs that are friends with each other;
(b) There are at least $23$ pairs that are friends with each other;
(c) There are exactly $22$ pairs that are friends with each other.
1991 IMO Shortlist, 30
Two students $ A$ and $ B$ are playing the following game: Each of them writes down on a sheet of paper a positive integer and gives the sheet to the referee. The referee writes down on a blackboard two integers, one of which is the sum of the integers written by the players. After that, the referee asks student $ A:$ “Can you tell the integer written by the other student?” If A answers “no,” the referee puts the same question to student $ B.$ If $ B$ answers “no,” the referee puts the question back to $ A,$ and so on. Assume that both students are intelligent and truthful. Prove that after a finite number of questions, one of the students will answer “yes.”
2015 China Northern MO, 8
Given a positive integer $n \ge 3$. Find the smallest real number $k$ such that for any positive real number except $a_1, a_2,..,a_n$, $$\sum_{i=1}^{n-1}\frac{a_i}{ s-a_i}+\frac{ka_n}{s-a_n} \ge \frac{n-1}{n-2}$$ where, $s=a_1+a_2+..+a_n$
2022 Belarusian National Olympiad, 11.2
Two perpendicular lines pass through the point $F(1;1)$ of coordinate plane. One of them intersects hyperbola $y=\frac{1}{2x}$ at $A$ and $C$ ($C_x>A_x$), and the other one intersects the left part of hyperbola at $B$ and the right at $D$. Let $m=(C_x-A_x)(D_x-B_x)$
Find the area of non-convex quadraliteral $ABCD$ (in terms of $m$)
1996 IMO Shortlist, 3
Let $ k,m,n$ be integers such that $ 1 < n \leq m \minus{} 1 \leq k.$ Determine the maximum size of a subset $ S$ of the set $ \{1,2,3, \ldots, k\minus{}1,k\}$ such that no $ n$ distinct elements of $ S$ add up to $ m.$
1995 Turkey MO (2nd round), 1
Let $m_{1},m_{2},\ldots,m_{k}$ be integers with $2\leq m_{1}$ and $2m_{1}\leq m_{i+1}$ for all $i$. Show that for any integers $a_{1},a_{2},\ldots,a_{k}$ there are infinitely many integers $x$ which do not satisfy any of the congruences \[x\equiv a_{i}\ (\bmod \ m_{i}),\ i=1,2,\ldots k.\]
2020 Macedonia Additional BMO TST, 4
There's a group of $21$ people such that each person has no more than $7$ friends among the others and any two friends have a different number of total friends. Prove that there are $6$ people, none of which knows the others.
2020 BAMO, C/1
Find all real numbers $x$ that satisfy the equation $$\frac{x-2020}{1}+\frac{x-2019}{2}+\cdots+\frac{x-2000}{21}=\frac{x-1}{2020}+\frac{x-2}{2019}+\cdots+\frac{x-21}{2000},$$ and simplify your answer(s) as much as possible. Justify your solution.
1988 IMO Shortlist, 2
Let $ n$ be a positive integer. Find the number of odd coefficients of the polynomial
\[ u_n(x) \equal{} (x^2 \plus{} x \plus{} 1)^n.
\]
1960 AMC 12/AHSME, 10
Given the following six statements:
$\text{(1) All women are good drivers}$
$\text{(2) Some women are good drivers}$
$\text{(3) No men are good drivers}$
$\text{(4) All men are bad drivers}$
$\text{(5) At least one man is a bad driver}$
$\text{(6) All men are good drivers.}$
The statement that negates statement $\text{(6)}$ is:
$ \textbf{(A) }(1)\qquad\textbf{(B) }(2)\qquad\textbf{(C) }(3)\qquad\textbf{(D) }(4)\qquad\textbf{(E) }(5) $
2010 Brazil Team Selection Test, 2
Let $k > 1$ be a fixed integer. Prove that there are infinite positive integers $n$ such that
$$ lcm \, (n, n + 1, n + 2, ... , n + k) > lcm \, (n + 1, n + 2, n + 3,... , n + k + 1).$$
2013 Stanford Mathematics Tournament, 10
Given a complex number $z$ such that $z^{13}=1$, find all possible value of $z+z^3+z^4+z^9+z^{10}+z^{12}$.
2021 BMT, 4
Derek and Julia are two of 64 players at a casual basketball tournament. The players split up into 8 teams of 8 players at random. Each team then randomly selects 2 captains among their players. What is the probability that both Derek and Julia are captains?
2005 MOP Homework, 4
A convex $2004$-sided polygon $P$ is given such that no four vertices are cyclic. We call a triangle whose vertices are vertices of $P$ thick if all other $2001$ vertices of $P$ lie inside the circumcircle of the triangle, and thin if they all lie outside its circumcircle. Prove that the number of thick triangles is equal to the number of thin triangles.
2020 LMT Spring, 23
Let $\triangle ABC$ be a triangle such that $AB=AC=40$ and $BC=79.$ Let $X$ and $Y$ be the points on segments $AB$ and $AC$ such that $AX=5, AY=25.$ Given that $P$ is the intersection of lines $XY$ and $BC,$ compute $PX\cdot PY-PB\cdot PC.$
2010 Canadian Mathematical Olympiad Qualification Repechage, 3
Prove that there is no real number $x$ satisfying both equations \begin{align*}2^x+1=2\sin x \\ 2^x-1=2\cos x.\end{align*}
XMO (China) 2-15 - geometry, 7.1
As shown in the figure, it is known that $BC = AC$ in $ABC$, $M$ is the midpoint of $AB$, points $D$ and $E$ lie on $AB$ satisfying $\angle DCE = \angle MCB$, the circumscribed circle of $\vartriangle BDC$ and the circumscribed circle of $\vartriangle AEC$ intersect at point $F$ (different from point $C$), point $H$ lies on $AB$ such that the straight line $CM$ bisects the line segment $HF$. Let the circumcenters of $\vartriangle HFE$ and $\vartriangle BFM$ be $O_1$ and $O_2$ respectively. Prove that $O_1O_2\perp CF$.
[img]https://cdn.artofproblemsolving.com/attachments/e/4/e8fc62735b8cfbd382e490617f26d335c46823.png[/img]
2022 Olympic Revenge, Problem 3
positive real $C$ is $n-vengeful$ if it is possible to color the cells of an $n \times n$ chessboard such that:
i) There is an equal number of cells of each color.
ii) In each row or column, at least $Cn$ cells have the same color.
a) Show that $\frac{3}{4}$ is $n-vengeful$ for infinitely many values of $n$.
b) Show that it does not exist $n$ such that $\frac{4}{5}$ is $n-vengeful$.
1974 AMC 12/AHSME, 15
If $ x<\minus{}2$ then $ |1\minus{}|1\plus{}x|$ $ |$ equals
$ \textbf{(A)}\ 2\plus{}x \qquad
\textbf{(B)}\ \minus{}2\minus{}x \qquad
\textbf{(C)}\ x \qquad
\textbf{(D)}\ \minus{}x \qquad
\textbf{(E)}\ \minus{}2$
1994 Vietnam Team Selection Test, 2
Determine all functions $f: \mathbb{R} \mapsto \mathbb{R}$ satisfying
\[f\left(\sqrt{2} \cdot x\right) + f\left(4 + 3 \cdot \sqrt{2} \cdot x \right) = 2 \cdot f\left(\left(2 + \sqrt{2}\right) \cdot x\right)\]
for all $x$.
2013 BMT Spring, 10
In a far away kingdom, there exist $k^2$ cities subdivided into k distinct districts, such that in the $i^ {th}$ district, there exist $2i - 1$ cities. Each city is connected to every city in its district but no cities outside of its district. In order to improve transportation, the king wants to add $k - 1$ roads such that all cities will become connected, but his advisors tell him there are many ways to do this. Two plans are different if one road is in one plan that is not in the other. Find the total number of possible plans in terms of $k$.
2023 ELMO Shortlist, G8
Convex quadrilaterals \(ABCD\), \(A_1B_1C_1D_1\), and \(A_2B_2C_2D_2\) are similar with vertices in order. Points \(A\), \(A_1\), \(B_2\), \(B\) are collinear in order, points \(B\), \(B_1\), \(C_2\), \(C\) are collinear in order, points \(C\), \(C_1\), \(D_2\), \(D\) are collinear in order, and points \(D\), \(D_1\), \(A_2\), \(A\) are collinear in order. Diagonals \(AC\) and \(BD\) intersect at \(P\), diagonals \(A_1C_1\) and \(B_1D_1\) intersect at \(P_1\), and diagonals \(A_2C_2\) and \(B_2D_2\) intersect at \(P_2\). Prove that points \(P\), \(P_1\), and \(P_2\) are collinear.
[i]Proposed by Holden Mui[/i]