Found problems: 85335
2004 China National Olympiad, 1
Let $EFGH,ABCD$ and $E_1F_1G_1H_1$ be three convex quadrilaterals satisfying:
i) The points $E,F,G$ and $H$ lie on the sides $AB,BC,CD$ and $DA$ respectively, and $\frac{AE}{EB}\cdot\frac{BF}{FC}\cdot \frac{CG}{GD}\cdot \frac{DH}{HA}=1$;
ii) The points $A,B,C$ and $D$ lie on sides $H_1E_1,E_1F_1,F_1,G_1$ and $G_1H_1$ respectively, and $E_1F_1||EF,F_1G_1||FG,G_1H_1||GH,H_1E_1||HE$.
Suppose that $\frac{E_1A}{AH_1}=\lambda$. Find an expression for $\frac{F_1C}{CG_1}$ in terms of $\lambda$.
[i]Xiong Bin[/i]
May Olympiad L2 - geometry, 2017.3
Let $ABCD$ be a quadrilateral such that $\angle ABC = \angle ADC = 90º$ and $\angle BCD$ > $90º$. Let $P$ be a point inside of the $ABCD$ such that $BCDP$ is parallelogram, the line $AP$ intersects $BC$ in $M$. If $BM = 2, MC = 5, CD = 3$. Find the length of $AM$.
2018 PUMaC Geometry A, 4
Triangle $ABC$ has $\angle{A}=90^\circ$, $\angle{C}=30^\circ$, and $AC=12$. Let the circumcircle of this triangle
be $W$. Define $D$ to be the point on arc $BC$ not containing $A$ so that $\angle{CAD}=60^\circ$. Define
points $E$ and $F$ to be the foots of the perpendiculars from $D$ to lines $AB$ and $AC$, respectively.
Let $J$ be the intersection of line $EF$ with $W$, where $J$ is on the minor arc $AC$. The line $DF$
intersects $W$ at $H$ other than $D$. The area of the triangle $FHJ$ is in the form $\frac{a}{b}(\sqrt{c}-\sqrt{d})$
for positive integers $a,b,c,d,$ where $a,b$ are relatively prime, and the sum of $a,b,c,d$ is minimal.
Find $a+b+c+d$.
2006 IberoAmerican Olympiad For University Students, 5
A regular $n$-gon is inscribed in a circle of radius $1$. Let $a_1,\cdots,a_{n-1}$ be the distances of one of the vertices of the polygon to all the other vertices. Prove that
\[(5-a_1^2)\cdots(5-a_{n-1}^2)=F_n^2\]
where $F_n$ is the $n^{th}$ term of the Fibonacci sequence $1,1,2,\cdots$
2023 Switzerland - Final Round, 1
Let $ABC$ be an acute triangle with incenter $I$. On its circumcircle, let $M_A$, $M_B$ and $M_C$ be the midpoints of minor arcs $BC, CA$ and $AB$, respectively. Prove that the reflection $M_A$ over the line $IM_B$ lies on the circumcircle of the triangle $IM_BM_C$.
2016 German National Olympiad, 5
Let $A,B,C,D$ be points on a circle with radius $r$ in this order such that $|AB|=|BC|=|CD|=s$ and $|AD|=s+r$. Find all possible values of the interior angles of the quadrilateral $ABCD$.
2019 Turkey EGMO TST, 1
$A_1, A_2, ..., A_n$ are the subsets of $|S|=2019$ such that union of any three of them gives $S$ but if we combine two of subsets it doesn't give us $S$. Find the maximum value of $n$.
2021 Girls in Math at Yale, 8
Let $A$ and $B$ be digits between $0$ and $9$, and suppose that the product of the two-digit numbers $\overline{AB}$ and $\overline{BA}$ is equal to $k$. Given that $k+1$ is a multiple of $101$, find $k$.
[i]Proposed by Andrew Wu[/i]
2012 Waseda University Entrance Examination, 2
Consider a sequence $\{a_n\}_{n\geq 0}$ such that $a_{n+1}=a_n-\lfloor{\sqrt{a_n}}\rfloor\ (n\geq 0),\ a_0\geq 0$.
(1) If $a_0=24$, then find the smallest $n$ such that $a_n=0$.
(2) If $a_0=m^2\ (m=2,\ 3,\ \cdots)$, then for $j$ with $1\leq j\leq m$, express $a_{2j-1},\ a_{2j}$ in terms of $j,\ m$.
(3) Let $m\geq 2$ be integer and for integer $p$ with $1\leq p\leq m-1$, let $a\0=m^2-p$. Find $k$ such that $a_k=(m-p)^2$, then
find the smallest $n$ such that $a_n=0$.
1990 Tournament Of Towns, (247) 1
Find the maximum number of parts into which the $Oxy$-plane can be divided by $100$ graphs of different quadratic functions of the form $y = ax^2 + bx + c$.
(N.B. Vasiliev, Moscow)
1976 Kurschak Competition, 2
A lottery ticket is a choice of $5$ distinct numbers from $1, 2,3,...,90$. Suppose that $5^5$ distinct lottery tickets are such that any two of them have a common number. Prove that one can find four numbers such that every ticket contains at least one of the four.
2015 Polish MO Finals, 1
In triangle $ABC$ the angle $\angle A$ is the smallest. Points $D, E$ lie on sides $AB, AC$ so that $\angle CBE=\angle DCB=\angle BAC$. Prove that the midpoints of $AB, AC, BE, CD$ lie on one circle.
2004 Greece National Olympiad, 2
If $m\geq 2$ show that there does not exist positive integers $x_1, x_2, ..., x_m,$ such that \[x_1< x_2<...< x_m \ \ \text{and} \ \ \frac{1}{x_1^3}+\frac{1}{x_2^3}+...+\frac{1}{x_m^3}=1.\]
2021 Yasinsky Geometry Olympiad, 2
In the quadrilateral $ABCD$ it is known that $\angle A = 90^o$, $\angle C = 45^o$ . Diagonals $AC$ and $BD$ intersect at point $F$, and $BC = CF$, and the diagonal $AC$ is the bisector of angle $A$. Determine the other two angles of the quadrilateral $ABCD$.
(Maria Rozhkova)
2006 Estonia Math Open Senior Contests, 10
Let $ n \ge 2$ be a fixed integer and let $ a_{i,j} (1 \le i < j \le n)$ be some positive integers. For a sequence $ x_1, ... , x_n$ of reals, let $ K(x_1, .... , x_n)$ be the product of all expressions $ (x_i \minus{} x_j)^{a_{i,j}}$ where $ 1 \le i < j \le n$. Prove that if the inequality $ K(x_1, .... , x_n) \ge 0$ holds independently of the choice of the sequence $ x_1, ... , x_n$ then all integers $ a_{i,j}$ are even.
2019 USAMO, 2
Let $ABCD$ be a cyclic quadrilateral satisfying $AD^2 + BC^2 = AB^2$. The diagonals of $ABCD$ intersect at $E$. Let $P$ be a point on side $\overline{AB}$ satisfying $\angle APD = \angle BPC$. Show that line $PE$ bisects $\overline{CD}$.
[i]Proposed by Ankan Bhattacharya[/i]
2015 AIME Problems, 13
With all angles measured in degrees, the product $\prod_{k=1}^{45} \csc^2(2k-1)^\circ=m^n$, where $m$ and $n$ are integers greater than 1. Find $m+n$.
2011 Purple Comet Problems, 15
A pyramid has a base which is an equilateral triangle with side length $300$ centimeters. The vertex of the pyramid is $100$ centimeters above the center of the triangular base. A mouse starts at a corner of the base of the pyramid and walks up the edge of the pyramid toward the vertex at the top. When the mouse has walked a distance of $134$ centimeters, how many centimeters above the base of the pyramid is the mouse?
2015 BMT Spring, 4
Triangle $ABC$ has side lengths $AB = 3$, $BC = 4$, and $CD = 5$. Draw line $\ell_A$ such that $\ell_A$ is parallel to $BC$ and splits the triangle into two polygons of equal area. Define lines $\ell_B$ and $\ell_C$ analogously. The intersection points of $\ell_A$, $\ell_B$, and $\ell_C$ form a triangle. Determine its area.
2022 Moldova EGMO TST, 7
Find all triplets of nonnegative integers $(x, y, z)$ that satisfy: $x^2-3y^2=y^2-3z^2=22$.
2014 Harvard-MIT Mathematics Tournament, 2
There are $10$ people who want to choose a committee of 5 people among them. They do this by first electing a set of $1, 2, 3,$ or $4$ committee leaders, who then choose among the remaining people to complete the 5-person committee. In how many ways can the committee be formed, assuming that people are distinguishable? (Two committees that have the same members but different sets of leaders are considered to be distinct.)
2018 Sharygin Geometry Olympiad, 7
A convex quadrilateral $ABCD$ is circumscribed about a circle of radius $r$. What is the maximum value of $\frac{1}{AC^2}+\frac{1}{BD^2}$?
2024 Regional Olympiad of Mexico West, 5
Consider a sequence of positive integers $a_1,a_2,a_3,...$ such that $a_1>1$ and
$$a_{n+1}=\frac{a_n}{p}+p,$$
where $p$ is the greatest prime factor of $a_n$. Prove that for any choice of $a_1$, the sequence $a_1,a_2,a_3,...$ has an infinite terms that are equal between them.
2021 Indonesia MO, 2
Let $ABC$ be an acute triangle. Let $D$ and $E$ be the midpoint of segment $AB$ and $AC$ respectively. Suppose $L_1$ and $L_2$ are circumcircle of triangle $ABC$ and $ADE$ respectively. $CD$ intersects $L_1$ and $L_2$ at $M (M \not= C)$ and $N (N \not= D)$. If $DM = DN$, prove that $\triangle ABC$ is isosceles.
2000 China Second Round Olympiad, 1
In acute-angled triangle $ABC,$ $E,F$ are on the side $BC,$ such that $\angle BAE=\angle CAF,$ and let $M,N$ be the projections of $F$ onto $AB,AC,$ respectively. The line $AE$ intersects $ \odot (ABC) $ at $D$(different from point $A$).
Prove that $S_{AMDN}=S_{\triangle ABC}.$