Found problems: 85335
2005 Tournament of Towns, 1
The graphs of four functions of the form $y = x^2 + ax + b$, where a and b are real coefficients, are plotted on the coordinate plane. These graphs have exactly four points of intersection, and at each one of them, exactly two graphs intersect. Prove that the sum of the largest and the smallest $x$-coordinates of the points of intersection is equal to the sum of the other two.
[i](3 points)[/i]
2015 Geolympiad Spring, 2
Let $ABC$ be a triangle and $w$ its incircle. $w$ touches $BC,CA$ at $A_1,B_1$ respectively. The second intersection of $AA_1$ and $w$ is $A_2$, similarly define $B_2$. Then $AB,A_1B_1,A_2B_2$ concur at a point $C_3$.
2002 Moldova National Olympiad, 3
In a triangle $ ABC$, the bisectors of the angles at $ B$ and $ C$ meet the opposite sides $ B_1$ and $ C_1$, respectively. Let $ T$ be the midpoint $ AB_1$. Lines $ BT$ and $ B_1C_1$ meet at $ E$ and lines $ AB$ and $ CE$ meet at $ L$. Prove that the lines $ TL$ and $ B_1C_1$ have a point in common.
2002 China Team Selection Test, 1
In acute triangle $ ABC$, show that:
$ \sin^3{A}\cos^2{(B \minus{} C)} \plus{} \sin^3{B}\cos^2{(C \minus{} A)} \plus{} \sin^3{C}\cos^2{(A \minus{} B)} \leq 3\sin{A} \sin{B} \sin{C}$
and find out when the equality holds.
2004 IberoAmerican, 3
Let $ n$ and $ k$ be positive integers such as either $ n$ is odd or both $ n$ and $ k$ are even. Prove that exists integers $ a$ and $ b$ such as $ GCD(a,n) \equal{} GCD(b,n) \equal{} 1$ and $ k \equal{} a \plus{} b$
1990 Spain Mathematical Olympiad, 4
Prove that the sum $\sqrt[3]{\frac{a+1}{2}+\frac{a+3}{6}\sqrt{ \frac{4a+3}{3}}} +\sqrt[3]{\frac{a+1}{2}-\frac{a+3}{6}\sqrt{ \frac{4a+3}{3}}}$
is independent of $a$ for $ a \ge - \frac{3}{4}$ and evaluate it.
2023 AMC 8, 12
The figure below shows a large white circle with a number of smaller white and shaded circles in its interior. What fraction of the interior of the large white circle is shaded?
[asy]
size(5cm);
defaultpen(linewidth(1pt));
draw(circle((3,3),3));
filldraw(circle((5.5,3),0.5),mediumgray*0.5 + lightgray*0.5);
filldraw(circle((2,3),2),mediumgray*0.5 + lightgray*0.5);
filldraw(circle((1,3),1),white);
filldraw(circle((3,3),1),white);
add(grid(6,6,mediumgray*0.5+gray*0.5+linetype("4 4")));
filldraw(circle((4.5,4.5),0.5),mediumgray*0.5 + lightgray*0.5);
filldraw(circle((4.5,1.5),0.5),mediumgray*0.5 + lightgray*0.5);
[/asy]$\textbf{(A) } \dfrac14\qquad\textbf{(B) } \dfrac{11}{36}\qquad\textbf{(C) } \dfrac13\qquad\textbf{(D) } \dfrac{19}{36}\qquad\textbf{(E) } \dfrac59$
2020 Purple Comet Problems, 14
Let $x$ be a real number such that $3 \sin^4 x -2 \cos^6 x = -\frac{17}{25}$ . Then $3 \cos^4 x - 2 \sin^6 x = \frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Find $10m + n$.
2015 German National Olympiad, 1
Determine all pairs of real numbers $(x,y)$ satisfying
\begin{align*} x^3+9x^2y&=10,\\
y^3+xy^2 &=2.
\end{align*}
2018 ASDAN Math Tournament, 7
In $\vartriangle ABC$, $AB = 13$, $BC = 14$, and $AC = 15$. Draw the circumcircle of $\vartriangle ABC$, and suppose that the circumcircle has center $O$. Extend $AO$ past $O$ to a point $D$, $BO$ past $O$ to a point $E$, and $CO$ past $O$ to a point $F$ such that $D, E, F$ also lie on the circumcircle. Compute the area of the hexagon $AF BDCE$.
2018 IMO Shortlist, G3
A circle $\omega$ with radius $1$ is given. A collection $T$ of triangles is called [i]good[/i], if the following conditions hold:
[list=1]
[*] each triangle from $T$ is inscribed in $\omega$;
[*] no two triangles from $T$ have a common interior point.
[/list]
Determine all positive real numbers $t$ such that, for each positive integer $n$, there exists a good collection of $n$ triangles, each of perimeter greater than $t$.
1994 IMO Shortlist, 6
Define the sequence $ a_1, a_2, a_3, ...$ as follows. $ a_1$ and $ a_2$ are coprime positive integers and $ a_{n \plus{} 2} \equal{} a_{n \plus{} 1}a_n \plus{} 1$. Show that for every $ m > 1$ there is an $ n > m$ such that $ a_m^m$ divides $ a_n^n$. Is it true that $ a_1$ must divide $ a_n^n$ for some $ n > 1$?
2003 Romania National Olympiad, 1
Let $ m,n$ be positive integers. Prove that the number $ 5^n\plus{}5^m$ can be represented as sum of two perfect squares if and only if $ n\minus{}m$ is even.
[i]Vasile Zidaru[/i]
2011 All-Russian Olympiad Regional Round, 11.6
$\omega$ is the circumcirle of an acute triangle $ABC$. The tangent line passing through $A$ intersects the tangent lines passing through points $B$ and $C$ at points $K$ and $L$, respectively. The line parallel to $AB$ through $K$ and the line parallel to $AC$ through $L$ intersect at point $P$. Prove that $BP=CP$.
(Author: P. Kozhevnikov)
2007 Kyiv Mathematical Festival, 1
Is it possible to cut the table of size $2007\times2007$ into figures shown here, if one has to use
at least one figure of each sort? $\begin{picture}(45,25) \put(5,5){\put(0,0){\line(1,0){16}}\put(0,8){\line(1,0){24}}\put(0,16){\line(1,0){24}}\put(8,24){\line(1,0){16}}\put(0,0){\line(0,1){16}}\put(8,0){\line(0,1){24}}\put(16,0){\line(0,1){24}}\put(24,8){\line(0,1){16}}}\put(35,5){\put(0,0){\line(1,0){8}}\put(0,8){\line(1,0){8}}\put(0,16){\line(1,0){8}}\put(0,24){\line(1,0){8}}\put(0,0){\line(0,1){24}}\put(8,0){\line(0,1){24}}}\end{picture}$
2022 HMNT, 23
Let $ABC$ be a triangle with $AB = 2021, AC = 2022,$ and $BC = 2023.$ Compute the minimum value of $AP +2BP +3CP$ over all points $P$ in the plane.
2005 Vietnam National Olympiad, 2
Let $(O)$ be a fixed circle with the radius $R$. Let $A$ and $B$ be fixed points in $(O)$ such that $A,B,O$ are not collinear. Consider a variable point $C$ lying on $(O)$ ($C\neq A,B$). Construct two circles $(O_1),(O_2)$ passing through $A,B$ and tangent to $BC,AC$ at $C$, respectively. The circle $(O_1)$ intersects the circle $(O_2)$ in $D$ ($D\neq C$). Prove that:
a) \[ CD\leq R \]
b) The line $CD$ passes through a point independent of $C$ (i.e. there exists a fixed point on the line $CD$ when $C$ lies on $(O)$).
2018 Korea Winter Program Practice Test, 3
Let $n$ be a "Good Number" if sum of all divisors of $n$ is less than $2n$ for $n\in \mathbb{Z}.$
Does there exist an infinite set $M$ that satisfies the following?
For all $a,b\in M,$ $a+b$ is good number. ($a=b$ is allowed.)
2022 Iranian Geometry Olympiad, 1
Find the angles of the pentagon $ABCDE$ in the figure below.
1983 Bulgaria National Olympiad, Problem 2
Let $b_1\ge b_2\ge\ldots\ge b_n$ be nonnegative numbers, and $(a_1,a_2,\ldots,a_n)$ be an arbitrary permutation of these numbers. Prove that for every $t\ge0$,
$$(a_1a_2+t)(a_3a_4+t)\cdots(a_{2n-1}a_{2n}+t)\le(b_1b_2+t)(b_3b_4+t)\cdots(b_{2n-1}b_{2n}+t).$$
2014 BMT Spring, P2
Let $ABC$ be a fixed scalene triangle. Suppose that $X, Y$ are variable points on segments $AB$, $AC$, respectively such that $BX = CY$ . Prove that the circumcircle of $\vartriangle AXY$ passes through a fixed point other than $A$.
2011 Tuymaada Olympiad, 3
In a word of more than $10$ letters, any two consecutive letters are different. Prove that one can change places of two consecutive letters so that the resulting word is not [i]periodic[/i], that is, cannot be divided into equal subwords.
2000 USA Team Selection Test, 4
Let $n$ be a positive integer. Prove that
\[ \binom{n}{0}^{-1} + \binom{n}{1}^{-1} + \cdots + \binom{n}{n}^{-1} = \frac{n+1}{2^{n+1}} \left( \frac{2}{1} + \frac{2^2}{2} + \cdots + \frac{2^{n+1}}{n+1} \right). \]
1994 Tournament Of Towns, (419) 7
Consider an arbitrary “figure” $F$ (non convex polygon). A chord of $F$ is defined to be a segment which lies entirely within $ F$ and whose ends are on its boundary.
(a) Does there always exist a chord of $F$ that divides its area in half?
(b) Prove that for any $F$ there exists a chord such that the area of each of the two parts of $F$ is not less than $ 1/3$ of the area of $F$.
(c) Can the number $1/3$ in (b) be changed to a greater one?
(V Proizvolov)
STEMS 2021 CS Cat B, Q1
We are given $k$ colors and we have to assign a single color to every vertex. An edge is [u][b]satisfied[/b][/u] if the vertices on that edge, are of different colors.
[list]
[*]Prove that you can always find an algorithm which assigns colors to vertices so that at least $\frac{k - 1}{k}|E|$ edges are satisfied where \(|E|\) is the cardinality of the edges in the graph.[/*]
[*]Prove that there is a poly time deterministic algorithm for this [/*]
[/list]