Found problems: 260
a. Does there exist 4 distinct positive integers such that the sum of any 3 of them is prime?
b. Does there exist 5 distinct positive integers such that the sum of any 3 of them is prime?
Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a continuous function with the following property: for all $\alpha \in \mathbb{R}_{>0}$, the sequence $(a_n)_{n \in \mathbb{N}}$ defined as $a_n = f(n\alpha)$ satisfies $\lim_{n \to \infty} a_n = 0$. Is it necessarily true that $\lim_{x \to +\infty} f(x) = 0$?
Prove that \[ 1 < \frac{1}{1001} + \frac{1}{1002} + \frac{1}{1003} + \cdots + \frac{1}{3001} < 1 \frac{1}{3}. \]
Hamilton Avenue has eight houses. On one side of the street are the houses
numbered 1,3,5,7 and directly opposite are houses 2,4,6,8 respectively. An
eccentric postman starts deliveries at house 1 and delivers letters to each of
the houses, finally returning to house 1 for a cup of tea. Throughout the
entire journey he must observe the following rules. The numbers of the houses
delivered to must follow an odd-even-odd-even pattern throughout, each house
except house 1 is visited exactly once (house 1 is visited twice) and the postman
at no time is allowed to cross the road to the house directly opposite. How
many different delivery sequences are possible?
A circle which is tangent to the sides $ [AB]$ and $ [BC]$ of $ \triangle ABC$ is also tangent to its circumcircle at the point $ T$. If $ I$ is the incenter of $ \triangle ABC$ , show that $ \widehat{ATI}\equal{}\widehat{CTI}$
Within a group of $ 2009$ people, every two people has exactly one common friend. Find the least value of the difference between the person with maximum number of friends and the person with minimum number of friends.
Let $ f$ be a function from the real numbers to the real numbers such that $ f(1) \equal{} 1, f(a\plus{}b) \equal{} f(a)\plus{}f(b)$ for all $ a, b,$ and $ f(x)f \left( \frac{1}{x} \right) \equal{} 1$ for all $ x \neq 0.$ Prove that $ f(x) \equal{} x$ for all real numbers $ x.$
Find the value of $x+y$ for which the expression
\[\frac{6x^2}{y^6} + \frac{6y^2}{x^6}+9x^2y^2+\frac{4}{x^6y^6}\]
is minimized.
Circle $ A$ has radius $ 100$. Circle $ B$ has an integer radius $ r<100$ and remains internally tangent to circle $ A$ as it rolls once around the circumference of circle $ A$. The two circles have the same points of tangency at the beginning and end of circle $ B$'s trip. How many possible values can $ r$ have?
$ \textbf{(A)}\ 4 \qquad
\textbf{(B)}\ 8 \qquad
\textbf{(C)}\ 9 \qquad
\textbf{(D)}\ 50 \qquad
\textbf{(E)}\ 90$
Find all polynomials $f\in \mathbb{Z}[X]$ such that if $p$ is prime then $f(p)$ is also prime.
Find all pairs $ (a, b)$ of real numbers such that whenever $ \alpha$ is a root of $ x^{2} \plus{} ax \plus{} b \equal{} 0$, $ \alpha^{2} \minus{} 2$ is also a root of the equation.
[b][Weightage 17/100][/b]
Let $a,b,c,d$ be positive reals such taht $a+b+c+d=1$.
Prove that: \[6(a^{3}+b^{3}+c^{3}+d^{3})\geq a^{2}+b^{2}+c^{2}+d^{2}+\frac{1}{8}.\]
Determine whether there exist two reals $x,y$ and a sequence $\{a_n\}_{n=0}^{\infty}$ of nonzero reals such that $a_{n+2}=xa_{n+1}+ya_n$ for all $n\ge0$ and for every positive real number $r$, there exist positive integers $i,j$ such that $|a_i|<r<|a_j|$.
[i]Alex Zhu.[/i]
Prove that there exist positive constants $ c$ and $ n_0$ with the following property. If $ A$ is a finite set of integers, $ |A| \equal{} n > n_0$, then
\[ |A \minus{} A| \minus{} |A \plus{} A| \leq n^2 \minus{} c n^{8/5}.\]
Initially numbers from 1 to 1000000 are all colored black. A move consists of picking one number, then change the color (black to white or white to black) of itself and all other numbers NOT coprime with the chosen number. Can all numbers become white after finite numbers of moves?
Edited by pbornsztein
Three circles touch each other externally and all these cirlces also touch a fixed straight line. Let $A,B,C$ be the mutual points of contact of these circles. If $\omega$ denotes the Brocard angle of the triangle $ABC$, prove that $\cot{\omega}$ = 2.
For each positive integer $n$, let $S(n)$ denote the sum of the digits of $n.$ For how many values of $n$ is $n + S(n) + S(S(n)) = 2007?$
$\mathrm{(A)}\ 1 \qquad \mathrm{(B)}\ 2 \qquad \mathrm{(C)}\ 3 \qquad \mathrm{(D)}\ 4 \qquad \mathrm{(E)}\ 5$
Line $l$ tangents unit circle $S$ in point $P$. Point $A$ and circle $S$ are on the same side of $l$, and the distance from $A$ to $l$ is $h$ ($h > 2$). Two tangents of circle $S$ are drawn from $A$, and intersect line $l$ at points $B$ and $C$ respectively.
Find the value of $PB \cdot PC$.
Find all functions $~$ $f: \mathbb{R} \to \mathbb{R}$ $~$ which satisfy the equation
\[ f(x+yf(x))\, =\, f(f(x)) + xf(y)\, . \]
A positive integer $ n$ is given. In an association consisting of $ n$ members work $ 6$ commissions. Each commission contains at least $ \large \frac{n}{4}$ persons. Prove that there exist two commissions containing at least $ \large \frac{n}{30}$ persons in common.
Find the sum of all positive rational numbers that are less than $10$ and that have denominator $30$ when written in lowest terms.
For any positive numbers $a, b, c$ prove the inequalities
\[\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\ge \frac{2}{a+b}+\frac{2}{b+c}+\frac{2}{c+a}\ge \frac{9}{a+b+c}.\]
Suppose that every positve integer has been given one of the colors red, blue,arbitrarily. Prove that there exists an infinite sequence of positive integers $ a_{1} < a_{2} < a_{3} < \cdots < a_{n} < \cdots,$ such that inifinite sequence of positive integers $ a_{1},\frac {a_{1} \plus{} a_{2}}{2},a_{2},\frac {a_{2} \plus{} a_{3}}{2},a_{3},\frac {a_{3} \plus{} a_{4}}{2},\cdots$ has the same color.
Find the sum of all positive integers $n$ such that \[ \frac{2n+1}{n(n-1)} \] has a terminating decimal representation.
[i]Proposed by Evan Chen[/i]
There are $ 2009$ pebbles in some points $ (x,y)$ with both coordinates integer. A operation consists in choosing a point $ (a,b)$ with four or more pebbles, removing four pebbles from $ (a,b)$ and putting one pebble in each of the points
\[ (a,b\minus{}1),\ (a,b\plus{}1),\ (a\minus{}1,b),\ (a\plus{}1,b)\]
Show that after a finite number of operations each point will necessarily have at most three pebbles. Prove that the final configuration doesn't depend on the order of the operations.