Found problems: 85335
2005 AIME Problems, 8
Circles $C_1$ and $C_2$ are externally tangent, and they are both internally tangent to circle $C_3$. The radii of $C_1$ and $C_2$ are $4$ and $10$, respectively, and the centers of the three circles are all collinear. A chord of $C_3$ is also a common external tangent of $C_1$ and $C_2$. Given that the length of the chord is $\frac{m\sqrt{n}}{p}$ where $m,n,$ and $p$ are positive integers, $m$ and $p$ are relatively prime, and $n$ is not divisible by the square of any prime, find $m+n+p$.
2018 Polish MO Finals, 3
Find all real numbers $c$ for which there exists a function $f\colon\mathbb R\rightarrow \mathbb R$ such that for each $x, y\in\mathbb R$ it's true that
$$f(f(x)+f(y))+cxy=f(x+y).$$
Russian TST 2018, P1
The natural numbers $a > b$ are such that $a-b=5b^2-4a^2$. Prove that the number $8b + 1$ is composite.
2011 Rioplatense Mathematical Olympiad, Level 3, 4
We consider $\Gamma_1$ and $\Gamma_2$ two circles that intersect at points $P$ and $Q$ . Let $A , B$ and $C$ be points on the circle $\Gamma_1$ and $D , E$ and $F$ points on the circle $\Gamma_2$ so that the lines $A E$ and $B D$ intersect at $P$ and the lines $A F$ and $C D$ intersect at $Q$. Denote $M$ and $N$ the intersections of lines $A B$ and $D E$ and of lines $A C$ and $D F$ , respectively. Show that $A M D N$ is a parallelogram.
2021 DIME, 13
Let $\triangle ABC$ have side lengths $AB=7$, $BC=8$, and $CA=9$. Let $D$ be the projection from $A$ to $\overline{BC}$ and $D'$ be the reflection of $D$ over the perpendicular bisector of $\overline{BC}$. Let $P$ and $Q$ be distinct points on the line through $D'$ parallel to $\overline{AC}$ such that $\angle APB = \angle AQB = 90^{\circ}$. The value of $AP+AQ$ can be written as $\tfrac{a+b\sqrt{c}}{d}$, where $a$, $b$, $c$, and $d$ are positive integers such that $b$ and $d$ are relatively prime, and $c$ is not divisible by the square of any prime. Find $a+b+c+d$.
[i]Proposed by i3435[/i]
2020 DMO Stage 1, 5.
[b]Q.[/b] Let $ABC$ be a triangle, where $L_A, L_B, L_C$ denote the internal angle bisectors of $\angle BAC, \angle ABC, \angle ACB$ respectively and $\ell_A, \ell_B, \ell_C$, the altitudes from the corresponding vertices. Suppose $ L_A\cap \overline{BC} = \{A_1\}$, $\ell_A \cap \overline{BC} = \{A_2\}$ and the circumcircle of $\triangle AA_1A_2$ meets $AB$ and $AC$ at $S$ and $T$ respectively. If $\overline{ST} \cap \overline{BC} = \{A'\}$, prove that $A',B',C'$ are collinear, where $B'$ and $C'$ are defined in a similar manner.
[i]Proposed by Functional_equation[/i]
2014 Purple Comet Problems, 23
Suppose $x$ is a real number satisfying $x^2-990x+1=(x+1)\sqrt x$. Find $\sqrt x+\tfrac1{\sqrt x}$.
2005 Purple Comet Problems, 24
$\triangle ABC$ has area $240$. Points $X, Y, Z$ lie on sides $AB$, $BC$, and $CA$, respectively. Given that $\frac{AX}{BX} = 3$, $\frac{BY}{CY} = 4$, and $\frac{CZ}{AZ} = 5$, find the area of $\triangle XYZ$.
[asy]
size(175);
defaultpen(linewidth(0.8));
pair A=(0,15),B=(0,-5),C=(25,0.5),X=origin,Y=(4C+B)/5,Z=(5A+C)/6;
draw(A--B--C--cycle^^X--Y--Z--cycle);
label("$A$",A,N);
label("$B$",B,S);
label("$C$",C,E);
label("$X$",X,W);
label("$Y$",Y,S);
label("$Z$",Z,NE);[/asy]
2001 IMC, 4
$p(x)$ is a polynomial of degree $n$ with every coefficient $0 $ or $\pm1$, and $p(x)$ is divisible by $ (x - 1)^k$ for some integer $ k > 0$. $q$ is a prime such that $\frac{q}{\ln q}< \frac{k}{\ln n+1}$. Show that the complex $q$-th roots of unity must be roots of $ p(x). $
2014 JHMMC 7 Contest, 24
When submitting problems, Steven the troll likes to submit silly names rather than his own. On day $1$, he gives no
name at all. Every day after that, he alternately adds $2$ words and $4$ words to his name. For example, on day $4$ he
submits an $8\text{-word}$ name. On day $n$ he submits the $44\text{-word name}$ “Steven the AJ Dennis the DJ Menace the Prince of Tennis the Merchant of Venice the Hygienist the Evil Dentist the Major Premise the AJ Lettuce the Novel’s Preface the Core Essence the Young and the Reckless the Many Tenants the Deep, Dark Crevice”. Compute $n$.
2012 Belarus Team Selection Test, 4
Given $0 < a < b < c$ prove that $$ a^{20}b^{12} + b^{20}c^{12 }+ c^{20}a^{12} <b^{20}a^{12}+ a^{20}c^{12} + c^{20}b^{12} $$
(I. Voronovich)
2003 Italy TST, 3
Let $p(x)$ be a polynomial with integer coefficients and let $n$ be an integer. Suppose that there is a positive integer $k$ for which $f^{(k)}(n) = n$, where $f^{(k)}(x)$ is the polynomial obtained as the composition of $k$ polynomials $f$. Prove that $p(p(n)) = n$.
2003 Junior Balkan Team Selection Tests - Romania, 1
Suppose $ABCD$ and $AEFG$ are rectangles such that the points $B,E,D,G$ are collinear (in this order). Let the lines $BC$ and $GF$ intersect at point $T$ and let the lines $DC$ and $EF$ intersect at point $H$. Prove that points $A, H$ and $T$ are collinear.
2017 Israel National Olympiad, 2
Denote by $P(n)$ the product of the digits of a positive integer $n$. For example, $P(1948)=1\cdot9\cdot4\cdot8=288$.
[list=a]
[*] Evaluate the sum $P(1)+P(2)+\dots+P(2017)$.
[*] Determine the maximum value of $\frac{P(n)}{n}$ where $2017\leq n\leq5777$.
[/list]
1989 IMO Longlists, 28
In a triangle $ ABC$ for which $ 6(a\plus{}b\plus{}c)r^2 \equal{} abc$ holds and where $ r$ denotes the inradius of $ ABC,$ we consider a point M on the inscribed circle and the projections $ D,E, F$ of $ M$ on the sides $ BC\equal{}a, AC\equal{}b,$ and $ AB\equal{}c$ respectively. Let $ S, S_1$ denote the areas of the triangles $ ABC$ and $ DEF$ respectively. Find the maximum and minimum values of the quotient $ \frac{S}{S_1}$
1966 German National Olympiad, 5
Prove that \[\tan 7 30^{\prime }=\sqrt{6}+\sqrt{2}-\sqrt{3}-2.\]
2003 JHMMC 8, 15
Evaluate $\frac{100-99+98-97\cdots +4-3+2-1}{1-2+3-4\cdots +97-98+99-100}$.
1998 South africa National Olympiad, 6
You are given $n$ squares, not necessarily all of the same size, which have total area 1. Is it always possible to place them without overlapping in a square of area 2?
Ukrainian TYM Qualifying - geometry, 2011.2
Eight circles of radius $r$ located in a right triangle $ABC$ (angle $C$ is right) as shown in figure (each of the circles touches the respactive sides of the triangle and the other circles). Find the radius of the inscribed circle of triangle $ABC$.
[img]https://cdn.artofproblemsolving.com/attachments/4/7/1b1cd7d6bc7f5004b8e94468d723ed16e9a039.png[/img]
2011 F = Ma, 5
A crude approximation is that the Earth travels in a circular orbit about the Sun at constant speed, at a distance of $\text{150,000,000 km}$ from the Sun. Which of the following is the closest for the acceleration of the Earth in this orbit?
(A) $\text{exactly 0 m/s}^2$
(B) $\text{0.006 m/s}^2$
(C) $\text{0.6 m/s}^2$
(D) $\text{6 m/s}^2$
(E) $\text{10 m/s}^2$
2014 Bosnia And Herzegovina - Regional Olympiad, 2
Let $a$, $b$ and $c$ be positive real numbers such that $ab+bc+ca=1$. Prove the inequality: $$\frac{1}{a}+\frac{1}{b}+\frac{1}{c} \geq 3(a+b+c)$$
2018 Dutch IMO TST, 3
Let $ABC$ be an acute triangle, and let $D$ be the foot of the altitude through $A$. On $AD$, there are distinct points $E$ and $F$ such that $|AE| = |BE|$ and $|AF| =|CF|$. A point$ T \ne D$ satis es $\angle BTE = \angle CTF = 90^o$. Show that $|TA|^2 =|TB| \cdot |TC|$.
2024 MMATHS, 1
On a planet, far, far away, the Yaliens have defined: $x$ "equals" $y$ if and only if $|x-y| \le 3.$ Let $S$ be a set of positive integers. What is the smallest possible number of elements in $S$ such that, for any positive integer $r,$ where $1 \le r \le 2024,$ $r$ "equals" some element in $S$?
Russian TST 2019, P1
Point $M{}$ is the middle of the side side $AB$ of the isosceles triangle $ABC$. On the extension of the base $AC$, point $D{}$ is marked such that $C{}$ is between $A{}$ and $D{}$, and point $E{}$ is marked on the segment $BM$. The circumcircle of the triangle $CDE$ intersects the segment $ME$ a second time at point $F$. Prove that it is possible to make a triangle from the segments $AD, DE$ and $AF$.
2010 National Olympiad First Round, 20
Starting from $0$, at each step we take $1$ more or $2$ times of the previous number. Which one below can be get in a less number of steps?
$ \textbf{(A)}\ 2011
\qquad\textbf{(B)}\ 2010
\qquad\textbf{(C)}\ 2009
\qquad\textbf{(D)}\ 2008
\qquad\textbf{(E)}\ 2007
$