Found problems: 85335
1993 China Team Selection Test, 3
A graph $G=(V,E)$ is given. If at least $n$ colors are required to paints its vertices so that between any two same colored vertices no edge is connected, then call this graph ''$n-$colored''. Prove that for any $n \in \mathbb{N}$, there is a $n-$colored graph without triangles.
1972 IMO Longlists, 43
A fixed point $A$ inside a circle is given. Consider all chords $XY$ of the circle such that $\angle XAY$ is a right angle, and for all such chords construct the point $M$ symmetric to $A$ with respect to $XY$ . Find the locus of points $M$.
2008 Harvard-MIT Mathematics Tournament, 15
In a game show, Bob is faced with $ 7$ doors, $ 2$ of which hide prizes. After he chooses a door, the host opens three other doors, of which one is hiding a prize. Bob chooses to switch to another door. What is the probability that his new door is hiding a prize?
1905 Eotvos Mathematical Competition, 3
Let $C_1$ be any point on side $AB$ of a triangle $ABC$, and draw $C_1C$. Let $A_1$ be the intersection of $BC$ extended and the line through $A$ parallel to $CC_1$, similarly let $B_1$ be the intersection of $AC$ extended and the line through $B$ parallel to $CC_1$. Prove that $$\frac{1}{AA_1}+\frac{1}{BB_1}=\frac{1}{CC_1}.$$
1991 Kurschak Competition, 2
A convex polyhedron has two triangle and three quadrilateral faces. Connect every vertex of one of the triangle faces with the intersection point of the diagonals in the quadrilateral face opposite to it. Show that the resulting three lines are concurrent.
1991 AMC 8, 3
Two hundred thousand times two hundred thousand equals
$\text{(A)}\ \text{four hundred thousand} \qquad \text{(B)}\ \text{four million} \qquad \text{(C)}\ \text{forty thousand} \\ \text{(D)}\ \text{four hundred million} \qquad \text{(E)}\ \text{forty billion}$
2008 Grigore Moisil Intercounty, 2
Given a convex quadrilateral $ ABCD, $ find the locus of points $ X $ that verify the qualities:
$$ XA^2+XB^2+CD^2=XB^2+XC^2+DA^2=XC^2+XD^2+AB^2=XD^2+XA^2+BC^2 $$
[i]Maria Pop[/i]
LMT Team Rounds 2021+, 5
Let $H$ be a regular hexagon with side length $1$. The sum of the areas of all triangles whose vertices are all vertices of $H$ can be expressed as $A\sqrt{B}$ for positive integers $A$ and $B$ such that $B$ is square-free. What is $1000A +B$?
1995 Niels Henrik Abels Math Contest (Norwegian Math Olympiad) Round 2, 5
Let $ f(x) \equal{} \frac{x}{1 \minus{} x}$ and let $ a$ be a real number. If $ x_0 \equal{} a, x_1 \equal{} f(x_0), x_2 \equal{} f(x_1), ...., x_{1996} \equal{} f(x_{1995})$ and $ x_{1996} \equal{} 1,$ what is $ a$?
A. 0
B. 1/1997
C. 1995
D. 1995/1996
E. None of these
2024 Bulgarian Autumn Math Competition, 12.3
Let $n \geq 2$ be a positive integer. If $m$ is a positive integer, for which all of its positive divisors can be split into $n$ disjoint sets of equal sum, prove that $m \geq 2^{n+1}-2$
2004 India IMO Training Camp, 1
Let $ABC$ be a triangle and $I$ its incentre. Let $\varrho_1$ and $\varrho_2$ be the inradii of triangles $IAB$ and $IAC$ respectively.
(a) Show that there exists a function $f: ( 0, \pi ) \mapsto \mathbb{R}$ such that \[ \frac{ \varrho_1}{ \varrho_2} = \frac{f(C)}{f(B)} \] where $B = \angle ABC$ and $C = \angle BCA$
(b) Prove that \[ 2 ( \sqrt{2} -1 ) < \frac{ \varrho_1} { \varrho_2} < \frac{ 1 + \sqrt{2}}{2} \]
2010 Romania Team Selection Test, 3
Let $n$ be a positive integer number. If $S$ is a finite set of vectors in the plane, let $N(S)$ denote the number of two-element subsets $\{\mathbf{v}, \mathbf{v'}\}$ of $S$ such that
\[4\,(\mathbf{v} \cdot \mathbf{v'}) + (|\mathbf{v}|^2 - 1)(|\mathbf{v'}|^2 - 1) < 0. \]
Determine the maximum of $N(S)$ when $S$ runs through all $n$-element sets of vectors in the plane.
[i]***[/i]
1986 Spain Mathematical Olympiad, 5
Consider the curve $\Gamma$ defined by the equation $y^2 = x^3 +bx+b^2$, where $b$ is a nonzero rational constant. Inscribe in the curve $\Gamma$ a triangle whose vertices have rational coordinates.
2000 French Mathematical Olympiad, Exercise 2
Let $A,B,C$ be three distinct points in space, $(A)$ the sphere with center $A$ and radius $r$. Let $E$ be the set of numbers $R>0$ for which there is a sphere $(H)$ with center $H$ and radius $R$ such that $B$ and $C$ are outside the sphere, and the points of the sphere $(A)$ are strictly inside it.
(a) Suppose that $B$ and $C$ are on a line with $A$ and strictly outside $(A)$. Show that $E$ is nonempty and bounded, and determine its supremum in terms of the given data.
(b) Find a necessary and sufficient condition for $E$ to be nonempty and bounded
(c) Given $r$, compute the smallest possible supremum of $E$, if it exists.
2004 Cuba MO, 1
A square is divided into $25$ small squares, equal to each other, drawing lines parallel to the sides of the square. Some are drawn diagonals of small squares so that there are no two diagonals with a common point. What is the maximum number of diagonals that can be traced?
2022 Indonesia TST, A
Given a monic quadratic polynomial $Q(x)$, define \[ Q_n (x) = \underbrace{Q(Q(\cdots(Q(x))\cdots))}_{\text{compose $n$ times}} \]
for every natural number $n$. Let $a_n$ be the minimum value of the polynomial $Q_n(x)$ for every natural number $n$. It is known that $a_n > 0$ for every natural number $n$ and there exists some natural number $k$ such that $a_k \neq a_{k+1}$.
(a) Prove that $a_n < a_{n+1}$ for every natural number $n$.
(b) Is it possible to satisfy $a_n < 2021$ for every natural number $n$?
[i]Proposed by Fajar Yuliawan[/i]
2013 Argentina National Olympiad Level 2, 2
Let $ABC$ be a right triangle. It is known that there are points $D$ on the side $AC$ and $E$ on the side $BC$ such that $AB = AD = BE$ and $BD$ is perpendicular to $DE$. Calculate the ratios $\frac{AB}{BC}$ and $\frac{BC}{CA}$.
2024 CMIMC Algebra and Number Theory, 2
Suppose $P(x)=x^2+Ax+B$ for real $A$ and $B$. If the sum of the roots of $P(2x)$ is $\tfrac 12$ and the product of the roots of $P(3x)$ is $\tfrac 13$, find $A+B$.
[i]Proposed by Connor Gordon[/i]
Russian TST 2022, P3
Write the natural numbers from left to right in ascending order. Every minute, we perform an operation. After $m$ minutes, we divide the entire available series into consecutive blocks of $m$ numbers. We leave the first block unchanged and in each of the other blocks we move all the numbers except the first one one place to the left, and move the first one to the end of the block. Prove that throughout the process, each natural number will only move a finite number of times.
1993 AMC 8, 18
The rectangle shown has length $AC=32$, width $AE=20$, and $B$ and $F$ are midpoints of $\overline{AC}$ and $\overline{AE}$, respectively. The area of quadrilateral $ABDF$ is
[asy]
pair A,B,C,D,EE,F;
A = (0,20); B = (16,20); C = (32,20); D = (32,0); EE = (0,0); F = (0,10);
draw(A--C--D--EE--cycle);
draw(B--D--F);
dot(A); dot(B); dot(C); dot(D); dot(EE); dot(F);
label("$A$",A,NW);
label("$B$",B,N);
label("$C$",C,NE);
label("$D$",D,SE);
label("$E$",EE,SW);
label("$F$",F,W);
[/asy]
$\text{(A)}\ 320 \qquad \text{(B)}\ 325 \qquad \text{(C)}\ 330 \qquad \text{(D)}\ 335 \qquad \text{(E)}\ 340$
2020 Argentina National Olympiad Level 2, 2
Let $n$ be a positive integer. There are $n$ colors available. Each of the integers from $1$ to $1000$ must be painted with one of the $n$ colors such that any two different numbers, if one divides the other, are painted in different colors. Determine the smallest value of $n$ for which this is possible.
2023 Korea Junior Math Olympiad, 8
A red equilateral triangle $T$ with side length $1$ is drawn on a plane. For a positive real $c$, we place three blue equilateral triangle shaped paper with side length $c$ on a plane to cover $T$ completely. Find the minimum value of $c$. As shown in the picture, it doesn't matter if the blue papers overlap each other or stick out from $T$. Folding or tearing the paper is not allowed.
2019 Philippine TST, 2
Four positive integers $x,y,z$ and $t$ satisfy the relations
\[ xy - zt = x + y = z + t. \]
Is it possible that both $xy$ and $zt$ are perfect squares?
2003 Bulgaria Team Selection Test, 3
Some of the vertices of a convex $n$-gon are connected by segments, such that any two of them have no common interior point. Prove that, for any $n$ points in general position, there exists a one-to-one correspondence between the points and the vertices of the $n$ gon, such that any two segments between the points, corresponding to the respective segments from the $n$ gon, have no common interior point.
2013 India PRMO, 4
Three points $X, Y,Z$ are on a striaght line such that $XY = 10$ and $XZ = 3$. What is the product of all possible values of $YZ$?