Found problems: 85335
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 $a+b$.
2013 IFYM, Sozopol, 2
Do there exist natural numbers $a, b$ and $c$ such that $a^2+b^2+c^2$ is divisible by $2013(ab+bc+ca)$?
[i]Proposed by Mahan Malihi[/i]
1982 IMO Longlists, 45
Let $ABCD$ be a convex quadrilateral and draw regular triangles $ABM, CDP, BCN, ADQ$, the first two outward and the other two inward. Prove that $MN = AC$. What can be said about the quadrilateral $MNPQ$?
2010 Stanford Mathematics Tournament, 4
Compute $\sqrt{1+\sqrt{1+\sqrt{1+\sqrt{1+\sqrt{1}}}}...}$
Denmark (Mohr) - geometry, 2014.3
The points $C$ and $D$ lie on a halfline from the midpoint $M$ of a segment $AB$, so that $|AC| = |BD|$. Prove that the angles $u = \angle ACM$ and $v = \angle BDM$ are equal.
[img]https://1.bp.blogspot.com/-tQEJ1VBCa8U/XzT7IhwlZHI/AAAAAAAAMVI/xpRdlj5Rl64VUt_tCRsQ1UxIsv_SGrMlACLcBGAsYHQ/s0/2014%2BMohr%2Bp3.png[/img]
2006 Oral Moscow Geometry Olympiad, 6
Given triangle $ABC$ and points $P$. Let $A_1,B_1,C_1$ be the second points of intersection of straight lines $AP, BP, CP$ with the circumscribed circle of $ABC$. Let points $A_2, B_2, C_2$ be symmetric to $A_1,B_1,C_1$ wrt $BC,CA,AB$, respectively. Prove that the triangles $A_1B_1C_1$ and $A_2B_2C_2$ are similar.
(A. Zaslavsky)
2018 Tournament Of Towns, 6.
In the land of knights (who always tell the truth) and liars (who always lie), 10 people sit at a round table, each at a vertex of an inscribed regular 10-gon, at least one of them is a liar. A traveler can stand at any point outside the table and ask the people: ”What is the distance from me to the nearest liar at the table?” After that each person at the table gives him an answer. What is the minimal number of questions the traveler has to ask to determine which people at the table are liars? (Both the people at the table and the traveler should be considered as points, and everyone can compute the distance between any two points) (10 points)
Maxim Didin