Found problems: 85335
2017 China Team Selection Test, 4
Given a circle with radius 1 and 2 points C, D given on it. Given a constant l with $0<l\le 2$. Moving chord of the circle AB=l and ABCD is a non-degenerated convex quadrilateral. AC and BD intersects at P. Find the loci of the circumcenters of triangles ABP and BCP.
PEN O Problems, 39
Find the smallest positive integer $n$ for which there exist $n$ different positive integers $a_{1}, a_{2}, \cdots, a_{n}$ satisfying [list] [*] $\text{lcm}(a_1,a_2,\cdots,a_n)=1985$,[*] for each $i, j \in \{1, 2, \cdots, n \}$, $gcd(a_i,a_j)\not=1$, [*] the product $a_{1}a_{2} \cdots a_{n}$ is a perfect square and is divisible by $243$, [/list] and find all such $n$-tuples $(a_{1}, \cdots, a_{n})$.
the 13th XMO, P3
Let O be the circumcenter of triangle ABC.
Let H be the orthocenter of triangle ABC.
the perpendicular bisector of AB meet AC,BC at D,E.
the circumcircle of triangle DEH meet AC,BC,OH again at F,G,L.
CH meet FG at T,and ABCT is concyclic.
Prove that LHBC is concyclic.
graph: https://cdn.luogu.com.cn/upload/image_hosting/w6z6mvm4.png
2011 ITAMO, 4
$ABCD$ is a convex quadrilateral. $P$ is the intersection of external bisectors of $\angle DAC$ and $\angle DBC$. Prove that $\angle APD = \angle BPC$ if and only if $AD+AC=BC+BD$
2012 India IMO Training Camp, 1
Determine all sequences $(x_1,x_2,\ldots,x_{2011})$ of positive integers, such that for every positive integer $n$ there exists an integer $a$ with \[\sum^{2011}_{j=1} j x^n_j = a^{n+1} + 1\]
[i]Proposed by Warut Suksompong, Thailand[/i]
2010 VJIMC, Problem 3
Prove that there exist positive constants $c_1$ and $c_2$ with the following properties:
a) For all real $k>1$,
$$\left|\int^1_0\sqrt{1-x^2}\cos(kx)\text dx\right|<\frac{c_1}{k^{3/2}}.$$b) For all real $k>1$,
$$\left|\int^1_0\sqrt{1-x^2}\sin(kx)\text dx\right|<\frac{c_2}k.$$
2001 AIME Problems, 12
Given a triangle, its midpoint triangle is obtained by joining the midpoints of its sides. A sequence of polyhedra $P_{i}$ is defined recursively as follows: $P_{0}$ is a regular tetrahedron whose volume is 1. To obtain $P_{i+1}$, replace the midpoint triangle of every face of $P_{i}$ by an outward-pointing regular tetrahedron that has the midpoint triangle as a face. The volume of $P_{3}$ is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.
2023 Brazil Cono Sur TST, 2
Define $d(n)$ as the number of positive divisors of $n\in\mathbb{Z_+^*}$. Let $a$ and $b$ be positive integers satisfying the equality $$a + d(a) = b^2 + 2$$ Prove that $a+b$ is even.
2001 Singapore MO Open, 2
Let $n$ be a positive integer, and let $a_1,a_2,...,a_n$ be $n$ positive real numbers such that $a_1+a_2+...+a_n = 1$. Is it true that $\frac{a_1^4}{a_1^2+a_2^2}+\frac{a_2^4}{a_2^2+a_3^2}+\frac{a_3^4}{a_3^2+a_4^2}+...+\frac{a_{n-1}^4}{a_{n-1}^2+a_n^2}+\frac{a_n^4}{a_n^2+a_1^2}\ge \frac{1}{2n}$ ?
Justify your answer.
2011 ELMO Shortlist, 2
A directed graph has each vertex with outdegree 2. Prove that it is possible to split the vertices into 3 sets so that for each vertex $v$, $v$ is not simultaneously in the same set with both of the vertices that it points to.
[i]David Yang.[/i]
[hide="Stronger Version"]See [url=http://www.artofproblemsolving.com/Forum/viewtopic.php?f=42&t=492100]here[/url].[/hide]
2021 Regional Olympiad of Mexico Southeast, 3
Let $a, b, c$ positive reals such that $a+b+c=1$. Prove that
$$\min\{a(1-b),b(1-c),c(1-a)\}\leq \frac{1}{4}$$
$$\max\{a(1-b),b(1-c),c(1-a)\}\geq \frac{2}{9}$$
1997 IMC, 5
Let $X$ be an arbitrary set and $f$ a bijection from $X$ to $X$. Show that there exist bijections $g,\ g':X\to X$ s.t. $f=g\circ g',\ g\circ g=g'\circ g'=1_X$.
1993 Greece National Olympiad, 2
During a recent campaign for office, a candidate made a tour of a country which we assume lies in a plane. On the first day of the tour he went east, on the second day he went north, on the third day west, on the fourth day south, on the fifth day east, etc. If the candidate went $n^2/2$ miles on the $n^{\text{th}}$ day of this tour, how many miles was he from his starting point at the end of the $40^{\text{th}}$ day?
2023 Tuymaada Olympiad, 5
A small ship sails on an infinite coordinate sea. At the moment $t$ the ship is at the point with coordinates $(f(t), g(t))$, where $f$ and $g$ are two polynomials of third degree. Yesterday at $14:00$ the ship was at the same point as at $13:00$, and at $20:00$, it was at the same point as at $19:00$. Prove that the ship sails along a straight line.
2017 Miklós Schweitzer, 4
Let $K$ be a number field which is neither $\mathbb{Q}$ nor a quadratic imaginary extension of $\mathbb{Q}$. Denote by $\mathcal{L}(K)$ the set of integers $n\ge 3$ for which we can find units $\varepsilon_1,\ldots,\varepsilon_n\in K$ for which
$$\varepsilon_1+\dots+\varepsilon_n=0,$$but $\displaystyle\sum_{i\in I}\varepsilon_i\neq 0$ for any nonempty proper subset $I$ of $\{1,2,\dots,n\}$. Prove that $\mathcal{L}(K)$ is infinite, and that its smallest element can be bounded from above by a function of the degree and discriminant of $K$. Further, show that for infinitely many $K$, $\mathcal{L}(K)$ contains infinitely many even and infinitely many odd elements.
1979 IMO Longlists, 11
Prove that a pyramid $A_1A_2 \ldots A_{2k+1}S$ with equal lateral edges and equal space angles between adjacent lateral walls is regular.
2015 CCA Math Bonanza, I6
How many positive integers less than or equal to $1000$ are divisible by $2$ and $3$ but not by $5$?
[i]2015 CCA Math Bonanza Individual Round #6[/i]
2017 Saint Petersburg Mathematical Olympiad, 5
Given a scalene triangle $ABC$ with $\angle B=130^{\circ}$. Let $H$ be the foot of altitude from $B$. $D$ and $E$ are points on the sides $AB$ and $BC$, respectively, such that $DH=EH$ and $ADEC$ is a cyclic quadrilateral. Find $\angle{DHE}$.
1992 IMO Longlists, 17
In the plane let $\,C\,$ be a circle, $\,L\,$ a line tangent to the circle $\,C,\,$ and $\,M\,$ a point on $\,L$. Find the locus of all points $\,P\,$ with the following property: there exists two points $\,Q,R\,$ on $\,L\,$ such that $\,M\,$ is the midpoint of $\,QR\,$ and $\,C\,$ is the inscribed circle of triangle $\,PQR$.
2023 Brazil National Olympiad, 4
Let $x, y, z$ be three real distinct numbers such that
$$\begin{cases} x^2-x=yz \\ y^2-y=zx \\ z^2-z=xy \end{cases}$$ Show that $-\frac{1}{3} < x,y,z < 1$.
Mathley 2014-15, 2
Let $n$ be a positive integer and $p$ a prime number $p > n + 1$.
Prove that the following equation does not have integer solution $$1 + \frac{x}{n + 1} + \frac{x^2}{2n + 1} + ...+ \frac{x^p}{pn + 1} = 0$$
Luu Ba Thang, Department of Mathematics, College of Education
1989 USAMO, 3
Let $P(z)= z^n + c_1 z^{n-1} + c_2 z^{n-2} + \cdots + c_n$ be a polynomial in the complex variable $z$, with real coefficients $c_k$. Suppose that $|P(i)| < 1$. Prove that there exist real numbers $a$ and $b$ such that $P(a + bi) = 0$ and $(a^2 + b^2 + 1)^2 < 4 b^2 + 1$.
2020 BMT Fall, 1
How many permutations of the set $\{B, M, T, 2,0\}$ do not have $B$ as their fi rst element?
2016 Kyrgyzstan National Olympiad, 3
Given a $\triangle ABC$ with sides $a,b,c.$
Three tangents are drawn to the incircle of $\triangle ABC$ parallel to the sides of $\triangle ABC$.These tangents cut [b]three new little triangles[/b].Three little incircles are drawn into new little triangles.[b][u]Find the sum of the area of these 4 incircles.[/u][/b]
1985 Brazil National Olympiad, 2
Given $n$ points in the plane, show that we can always find three which give an angle $\le \pi / n$.