Found problems: 85335
2009 China Team Selection Test, 1
Let $ a > b > 1, b$ is an odd number, let $ n$ be a positive integer. If $ b^n|a^n\minus{}1,$ then $ a^b > \frac {3^n}{n}.$
2008 Estonia Team Selection Test, 3
Let $ n$ be a positive integer, and let $ x$ and $ y$ be a positive real number such that $ x^n \plus{} y^n \equal{} 1.$ Prove that
\[ \left(\sum^n_{k \equal{} 1} \frac {1 \plus{} x^{2k}}{1 \plus{} x^{4k}} \right) \cdot \left( \sum^n_{k \equal{} 1} \frac {1 \plus{} y^{2k}}{1 \plus{} y^{4k}} \right) < \frac {1}{(1 \minus{} x) \cdot (1 \minus{} y)}.
\]
[i]Author: Juhan Aru, Estonia[/i]
2015 JHMT, 5
Points $ABCDEF$ are evenly spaced on a unit circle and line segments $AD$, $DF$, $FB$, $BE$, $EC$, $CA$ are drawn. The line segments intersect each other at seven points inside the circle. Denote these intersections $p_1$, $p_2$, $...$,$p_7$, where $p_7$ is the center of the circle. What is the area of the $12$-sided shape $A_{p_1}B_{p_2}C_{p_3}D_{p_4}E_{p_5}F_{p_6}$?
[img]https://cdn.artofproblemsolving.com/attachments/9/2/645b3c31b0bb7f81d0c0b86e13b27ec5b32864.png[/img]
2017 Harvard-MIT Mathematics Tournament, 2
Determine the sum of all distinct real values of $x$ such that $||| \cdots ||x|+x| \cdots |+x|+x|=1$ where there are $2017$ $x$s in the equation.
2016 Brazil National Olympiad, 5
Consider the second-degree polynomial \(P(x) = 4x^2+12x-3015\). Define the sequence of polynomials
\(P_1(x)=\frac{P(x)}{2016}\) and \(P_{n+1}(x)=\frac{P(P_n(x))}{2016}\) for every integer \(n \geq 1\).
[list='a']
[*]Show that exists a real number \(r\) such that \(P_n(r) < 0\) for every positive integer \(n\).
[*]Find how many integers \(m\) are such that \(P_n(m)<0\) for infinite positive integers \(n\).
[/list]
Today's calculation of integrals, 894
Let $a$ be non zero real number. Find the area of the figure enclosed by the line $y=ax$, the curve $y=x\ln (x+1).$
2020 HMIC, 3
Let $P_1P_2P_3P_4$ be a tetrahedron in $\mathbb{R}^3$ and let $O$ be a point equidistant from each of its vertices. Suppose there exists a point $H$ such that for each $i$, the line $P_iH$ is perpendicular to the plane through the other three vertices. Line $P_1H$ intersects the plane through $P_2, P_3, P_4$ at $A$, and contains a point $B\neq P_1$ such that $OP_1=OB$. Show that $HB=3HA$.
[i]Michael Ren[/i]
1986 Austrian-Polish Competition, 7
Let $k$ and $n$ be integers with $0 < k < n^2/4$ such that k has no prime divisor greater than $n$. Prove that $k$ divides $n!$.
2005 IMO Shortlist, 2
Let $a_1,a_2,\ldots$ be a sequence of integers with infinitely many positive and negative terms. Suppose that for every positive integer $n$ the numbers $a_1,a_2,\ldots,a_n$ leave $n$ different remainders upon division by $n$.
Prove that every integer occurs exactly once in the sequence $a_1,a_2,\ldots$.
2006 VJIMC, Problem 2
Suppose that $(a_n)$ is a sequence of real numbers such that the series
$$\sum_{n=1}^\infty\frac{a_n}n$$is convergent. Show that the sequence
$$b_n=\frac1n\sum^n_{j=1}a_j$$is convergent and find its limit.
1993 APMO, 5
Let $P_1$, $P_2$, $\ldots$, $P_{1993} = P_0$ be distinct points in the $xy$-plane
with the following properties:
(i) both coordinates of $P_i$ are integers, for $i = 1, 2, \ldots, 1993$;
(ii) there is no point other than $P_i$ and $P_{i+1}$ on the line segment joining $P_i$ with $P_{i+1}$ whose coordinates are both integers, for $i = 0, 1, \ldots, 1992$.
Prove that for some $i$, $0 \leq i \leq 1992$, there exists a point $Q$ with coordinates $(q_x, q_y)$ on the line segment joining $P_i$ with $P_{i+1}$ such that both $2q_x$ and $2q_y$ are odd integers.
2016 Romania National Olympiad, 1
Let be a natural number $ n\ge 2 $ and $ n $ positive real numbers $ a_1,a_2,\ldots ,a_n $ whose product is $ 1. $
Prove that the function $ f:\mathbb{R}_{>0}\longrightarrow\mathbb{R} ,\quad f(x)=\prod_{i=1}^n \left( 1+a_i^x \right) , $ is nondecreasing.
1972 IMO Longlists, 7
$f$ and $g$ are real-valued functions defined on the real line. For all $x$ and $y, f(x+y)+f(x-y)=2f(x)g(y)$. $f$ is not identically zero and $|f(x)|\le1$ for all $x$. Prove that $|g(x)|\le1$ for all $x$.
2003 Estonia Team Selection Test, 5
Let $a, b, c$ be positive real numbers satisfying the condition $\frac{1}{ab}+\frac{1}{ac}+\frac{1}{bc}=1$ . Prove the inequality $$\frac{a}{\sqrt{1+a^2}}+\frac{b}{\sqrt{1+b^2}}+\frac{c}{\sqrt{1+c^2}} \le \frac{3\sqrt3}{2}$$
When does the equality hold?
(L. Parts)
2015 China Team Selection Test, 6
There are some players in a Ping Pong tournament, where every $2$ players play with each other at most once. Given:
\\(1) Each player wins at least $a$ players, and loses to at least $b$ players. ($a,b\geq 1$)
\\(2) For any two players $A,B$, there exist some players $P_1,...,P_k$ ($k\geq 2$) (where $P_1=A$,$P_k=B$), such that $P_i$ wins $P_{i+1}$ ($i=1,2...,k-1$).
\\Prove that there exist $a+b+1$ distinct players $Q_1,...Q_{a+b+1}$, such that $Q_i$ wins $Q_{i+1}$ ($i=1,...,a+b$)
2019 Jozsef Wildt International Math Competition, W. 50
Let $x$, $y$, $z > 0$, $\lambda \in (-\infty, 0) \cup (1,+\infty)$ such that $x + y + z = 1$. Then$$\sum \limits_{cyc} x^{\lambda}y^{\lambda}\sum \limits_{cyc}\frac{1}{(x+y)^{2\lambda}}\geq 9\left(\frac{1}{4}-\frac{1}{9}\sum \limits_{cyc}\frac{1}{(x+1)^2} \right)^{\lambda}$$
1951 Putnam, A7
Show that if the series $a_1 + a_2 + a_3 + \cdots + a_n + \cdots$ converges, then the series $a_1 + a_2 / 2 + a_3 / 3 + \cdots + a_n / n + \cdots$ converges also.
2017 Purple Comet Problems, 28
Let $T_k = \frac{k(k+1)}{2}$ be the $k$-th triangular number. The in finite series
$$\sum_{k=4}^{\infty}\frac{1}{(T_{k-1} - 1)(Tk - 1)(T_{k+1} - 1)}$$
has the value $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Find $m + n$.
2014 Balkan MO Shortlist, N6
Let $ f: \mathbb{N} \rightarrow \mathbb{N} $ be a function from the positive integers to the positive integers for which $ f(1)=1,f(2n)=f(n) $ and $ f(2n+1)=f(n)+f(n+1) $ for all $ n\in \mathbb{N} $. Prove that for any natural number $ n $, the number of odd natural numbers $ m $ such that $ f(m)=n $ is equal to the number of positive integers not greater than $ n $ having no common prime factors with $ n $.
2018 Serbia National Math Olympiad, 4
Prove that there exists a uniqe $P(x)$ polynomial with real coefficients such that\\
$xy-x-y|(x+y)^{1000}-P(x)-P(y)$ for all real $x,y$.
2002 Regional Competition For Advanced Students, 3
In the convex $ABCDEF$ (has all interior angles less than $180^o$) with the perimeter $s$ the triangles $ACE$ and $BDF$ have perimeters $u$ and $v$ respectively.
a) Show the inequalities $\frac{1}{2} \le \frac{s}{u+v}\le 1$
b) Check whether $1$ is replaced by a smaller number or $1/2$ by a larger number can the inequality remains valid for all convex hexagons.
2023 Azerbaijan IMO TST, 3
For each $1\leq i\leq 9$ and $T\in\mathbb N$, define $d_i(T)$ to be the total number of times the digit $i$ appears when all the multiples of $1829$ between $1$ and $T$ inclusive are written out in base $10$.
Show that there are infinitely many $T\in\mathbb N$ such that there are precisely two distinct values among $d_1(T)$, $d_2(T)$, $\dots$, $d_9(T)$.
1970 IMO Longlists, 8
Consider a regular $2n$-gon and the $n$ diagonals of it that pass through its center. Let $P$ be a point of the inscribed circle and let $a_1, a_2, \ldots , a_n$ be the angles in which the diagonals mentioned are visible from the point $P$. Prove that
\[\sum_{i=1}^n \tan^2 a_i = 2n \frac{\cos^2 \frac{\pi}{2n}}{\sin^4 \frac{\pi}{2n}}.\]
2011 JHMT, 5
Let $ABCD$ be a unit square. Point $E$ is on $BC$, point $F$ is on $DC$, $\vartriangle AEF$ is equilateral, and $GHIJ$ is a square in $\vartriangle AEF$ such that $GH$ is on $EF$. Compute the area of square $GHIJ$.
2021 LMT Fall, 10
Convex cyclic quadrilateral $ABCD$ satisfies $AC \perp BD$ and $AC$ intersects $BD$ at $H$. Let the line through $H$ perpendicular to $AD$ and the line through $H$ perpendicular to $AB$ intersect $CB$ and $CD$ at $P$ and $Q$, respectively. The circumcircle of $\triangle CPQ$ intersects line $AC$ again at $X \ne C$. Given that $AB=13$, $BD=14$, and $AD=15$, the length of $AX$ can be written as $\frac{a}{b}$ where $a$ and $b$ are relatively prime positive integers. Find $a+b$.