Found problems: 85335
2009 China Northern MO, 2
In an acute triangle $ABC$ , $AB>AC$ , $ \cos B+ \cos C=1$ , $E,F$ are on the extend line of $AB,AC$ such that $\angle ABF = \angle ACE = 90$ .
(1) Prove :$BE+CF=EF$ ;
(2) Assume the bisector of $\angle EBC$ meet $EF$ at $P$ , prove that $CP$ is the bisector of $\angle BCF$.
[img]https://cdn.artofproblemsolving.com/attachments/a/2/c554c2bc0b4e044c45f88138568f5234d544a8.png[/img]
Ukrainian TYM Qualifying - geometry, V.8
Let $X$ be a point inside an equilateral triangle $ABC$ such that $BX+CX <3 AX$. Prove that
$$3\sqrt3 \left( \cot \frac{\angle AXC}{2}+ \cot \frac{\angle AXB}{2}\right) +\cot \frac{\angle AXC}{2} \cot \frac{\angle AXB}{2} >5$$
2011 IMC, 4
Let $A_1,A_2,\dots, A_n$ be finite, nonempty sets. Define the function
\[f(t)=\sum_{k=1}^n \sum_{1\leq i_1<i_2<\dots<i_k\leq n} (-1)^{k-1}t^{|A_{i_1}\cup A_{i_2}\cup \dots\cup A_{i_k}|}.\]
Prove that $f$ is nondecreasing on $[0,1].$
($|A|$ denotes the number of elements in $A.$)
2000 IMO Shortlist, 6
Let $ p$ and $ q$ be relatively prime positive integers. A subset $ S$ of $ \{0, 1, 2, \ldots \}$ is called [b]ideal[/b] if $ 0 \in S$ and for each element $ n \in S,$ the integers $ n \plus{} p$ and $ n \plus{} q$ belong to $ S.$ Determine the number of ideal subsets of $ \{0, 1, 2, \ldots \}.$
2024 CMIMC Algebra and Number Theory, 6
Integers $a, b$ satisfy the following property: the line $y = 2x + ab$ passes through all intersection points of the two parabolas given by \[y = x^2 + 2x + a, \quad y = 2x^2 +bx,\] which intersect at least once. How many such $(a, b)$ satisfy $|ab| \leq 100$?
[i]Proposed by Justin Hsieh[/i]
2020 Serbian Mathematical Olympiad, Problem 4
In a trapezoid $ABCD$ such that the internal angles are not equal to $90^{\circ}$, the diagonals $AC$ and $BD$ intersect at the point $E$. Let $P$ and $Q$ be the feet of the altitudes from $A$ and $B$ to the sides $BC$ and $AD$ respectively. Circumscribed circles of the triangles $CEQ$ and $DEP$ intersect at the point $F\neq E$. Prove that the lines $AP$, $BQ$ and $EF$ are either parallel to each other, or they meet at exactly one point.
2008 Hanoi Open Mathematics Competitions, 1
How many integers are there in $(b,2008b]$, where $b$ ($b > 0$) is given.
2018 Canada National Olympiad, 1
Consider an arrangement of tokens in the plane, not necessarily at distinct points. We are allowed to apply a sequence of moves of the following kind: select a pair of tokens at points $A$ and $B$ and move both of them to the midpoint of $A$ and $B$.
We say that an arrangement of $n$ tokens is [i]collapsible[/i] if it is possible to end up with all $n$ tokens at the same point after a finite number of moves. Prove that every arrangement of $n$ tokens is collapsible if and only if $n$ is a power of $2$.
2020 LIMIT Category 2, 10
In a triangle $\triangle XYZ$, $\tan(x)\tan(z)=2$, $\tan(y)\tan(z)=18$. Then what is $\tan^2(z)$?
2008 JBMO Shortlist, 3
Let the real parameter $p$ be such that the system $\begin{cases} p(x^2 - y^2) = (p^2- 1)xy \\ |x - 1|+ |y| = 1 \end{cases}$ has at least three different real solutions. Find $p$ and solve the system for that $p$.
2002 All-Russian Olympiad, 3
Prove that if $0<x<\frac{\pi}{2}$ and $n>m$, where $n$,$m$ are natural numbers, \[ 2 \left| \sin^n x - \cos^n x \right| \le 3 \left| \sin^m x - \cos^m x \right|.\]
2006 Tournament of Towns, 1
Two regular polygons, a $7$-gon and a $17$-gon are given. For each of them two circles are drawn, an inscribed circle and a circumscribed circle. It happened that rings containing the polygons have equal areas. Prove that sides of the polygons are equal. (3)
2016 Romania National Olympiad, 1
Find all non-negative integers $n$ so that $\sqrt{n + 3}+ \sqrt{n +\sqrt{n + 3}} $ is an integer.
2015 Mathematical Talent Reward Programme, MCQ: P 7
How many $x$ are there such that $x,[x],\{x\}$ are in harmonic progression (i.e, the reciprocals are in arithmetic progression)? (Here $[x]$ is the largest integer less than equal to $x$ and $\{x\}=x-[ x]$ )
[list=1]
[*] 0
[*] 1
[*] 2
[*] 3
[/list]
2021 AMC 12/AHSME Fall, 19
Regular polygons with $5, 6, 7, $ and $8$ sides are inscribed in the same circle. No two of the polygons share a vertex, and no three of their sides intersect at a common point. At how many points inside the circle do two of their sides intersect?
$\textbf{(A)}\ 52 \qquad\textbf{(B)}\ 56 \qquad\textbf{(C)}\ 60 \qquad\textbf{(D)}\
64 \qquad\textbf{(E)}\ 68$
1971 IMO Longlists, 41
Let $L_i,\ i=1,2,3$, be line segments on the sides of an equilateral triangle, one segment on each side, with lengths $l_i,\ i=1,2,3$. By $L_i^{\ast}$ we denote the segment of length $l_i$ with its midpoint on the midpoint of the corresponding side of the triangle. Let $M(L)$ be the set of points in the plane whose orthogonal projections on the sides of the triangle are in $L_1,L_2$, and $L_3$, respectively; $M(L^{\ast})$ is defined correspondingly. Prove that if $l_1\ge l_2+l_3$, we have that the area of $M(L)$ is less than or equal to the area of $M(L^{\ast})$.
PEN M Problems, 28
Let $\{u_{n}\}_{n \ge 0}$ be a sequence of integers satisfying the recurrence relation $u_{n+2}=u_{n+1}^2 -u_{n}$ $(n \in \mathbb{N})$. Suppose that $u_{0}=39$ and $u_{1}=45$. Prove that $1986$ divides infinitely many terms of this sequence.
1941 Putnam, B4
Given two perpendicular diameters $AB$ and $CD$ of an ellipse, we say that the diameter $A'B'$ is conjugate to $AB$ if $A'B'$ is parallel to the tangent to the ellipse at $A$. Let $A'B'$ be conjugate to $AB$ and $C'D'$ be conjugate to $CD$.
Prove that the rectangular hyperbola through $A', B', C'$ and $D'$ passes through the foci of the ellipse.
2021 Iberoamerican, 3
Let $a_1,a_2,a_3, \ldots$ be a sequence of positive integers and let $b_1,b_2,b_3,\ldots$ be the sequence of real numbers given by
$$b_n = \dfrac{a_1a_2\cdots a_n}{a_1+a_2+\cdots + a_n},\ \mbox{for}\ n\geq 1$$
Show that, if there exists at least one term among every million consecutive terms of the sequence $b_1,b_2,b_3,\ldots$ that is an integer, then there exists some $k$ such that $b_k > 2021^{2021}$.
1994 Tournament Of Towns, (428) 5
The periods of two periodic sequences are $7$ and $13$. What is the maximal length of initial sections of the two sequences which can coincide? (The period $p$ of a sequence $a_1$,$a_2$, $...$ is the minimal $p$ such that $a_n = a_{n+p}$ for all $n$.)
(AY Belov)
2014 Contests, 3
Let $\mathbb{Z}$ be the set of integers. Find all functions $f : \mathbb{Z} \rightarrow \mathbb{Z}$ such that \[xf(2f(y)-x)+y^2f(2x-f(y))=\frac{f(x)^2}{x}+f(yf(y))\] for all $x, y \in \mathbb{Z}$ with $x \neq 0$.
PEN M Problems, 24
Let $k$ be a given positive integer. The sequence $x_n$ is defined as follows: $x_1 =1$ and $x_{n+1}$ is the least positive integer which is not in $\{x_{1}, x_{2},..., x_{n}, x_{1}+k, x_{2}+2k,..., x_{n}+nk \}$. Show that there exist real number $a$ such that $x_n = \lfloor an\rfloor$ for all positive integer $n$.
2012 Peru MO (ONEM), 1
For each positive integer $n$ whose canonical decomposition is $n = p_1^{a_1} \cdot p_2^{a_2} \cdot\cdot\cdot p_k^{a_k}$, we define $t(n) = (p_1 + 1) \cdot (p_2 + 1) \cdot\cdot\cdot (p_k + 1)$. For example, $t(20) = t(2^2\cdot 5^1) = (2 + 1) (5 + 1) = 18$, $t(30) = t(2^1\cdot 3^1\cdot 5^1) = (2 + 1) (3 + 1) (5 + 1) = 72$ and $t(125) = t(5^3) = (5 + 1) = 6$ .
We say that a positive integer $n$ is [i]special [/i]if $t(n)$ is a divisor of $n$. How many positive divisors of the number $54610$ are special?
2017 Purple Comet Problems, 2
The figure below was made by gluing together 5 non-overlapping congruent squares. The figure has area 45. Find the perimeter of the figure.
[center][img]https://snag.gy/ZeKf4q.jpg[/center][/img]
1994 Austrian-Polish Competition, 2
The sequences $(a_n)$ and (c_n) are given by $a_0 =\frac12$, $c_0=4$ , and for $n \ge 0$ , $a_{n+1}=\frac{2a_n}{1+a_n^2}$, $c_{n+1}=c_n^2-2c_n+2$
Prove that for all $n\ge 1$, $a_n=\frac{2c_0c_1...c_{n-1}}{c_n}$