Found problems: 85335
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
$
2007 Estonia Team Selection Test, 3
Let $n$ be a natural number, $n > 2$. Prove that if $\frac{b^n-1}{b-1}$ is a prime power for some positive integer $b$ then $n$ is prime.
1995 Irish Math Olympiad, 1
There are $ n^2$ students in a class. Each week all the students participate in a table quiz. Their teacher arranges them into $ n$ teams of $ n$ players each. For as many weeks as possible, this arrangement is done in such a way that any pair of students who were members of the same team one week are not in the same team in subsequent weeks. Prove that after at most $ n\plus{}2$ weeks, it is necessary for some pair of students to have been members of the same team in at least two different weeks.
2019 Estonia Team Selection Test, 6
It is allowed to perform the following transformations in the plane with any integers $a$:
(1) Transform every point $(x, y)$ to the corresponding point $(x + ay, y)$,
(2) Transform every point $(x, y)$ to the corresponding point $(x, y + ax)$.
Does there exist a non-square rhombus whose all vertices have integer coordinates and which can be transformed to:
a) Vertices of a square,
b) Vertices of a rectangle with unequal side lengths?
1999 China Team Selection Test, 2
Find all prime numbers $p$ which satisfy the following condition: For any prime $q < p$, if $p = kq + r, 0 \leq r < q$, there does not exist an integer $q > 1$ such that $a^{2} \mid r$.
2001 Belarusian National Olympiad, 1
On the Cartesian coordinate plane, the graph of the parabola $y = x^2$ is drawn. Three distinct points $A$, $B$, and $C$ are marked on the graph with $A$ lying between $B$ and $C$. Point $N$ is marked on $BC$ so that $AN$ is parallel to the y-axis. Let $K_1$ and $K_2$ are the areas of triangles $ABN$ and $ACN$, respectively. Express $AN$ in terms of $K_1$ and $K_2$.