Found problems: 85335
Let $ a, b, c$ be positive real numbers such that
\[ a+b+c=\frac{1}{a^2} + \frac{1}{b^2} + \frac{1}{c^2} . \]
Prove that
\[ 2(a+b+c) \ge \sqrt[3]{7 a^2 b +1 } + \sqrt[3]{7 b^2 c +1 } + \sqrt[3]{7 c^2 a +1 } . \]
Find all triples $ (a,b,c) $ for which equality holds.
Set $\mathbb Q_1=\{x\in\mathbb Q\mid x\ge1\}$. Suppose that a function $f:\mathbb Q_1\to\mathbb R$ satisfies the inequality $\left|f(x+y)-f(x)-f(y)\right|<\epsilon$ for all $x,y\in\mathbb Q_1$, where $\epsilon>0$ is given. Prove that there exists a real number $q$ such that
$$\left|\frac{f(x)}x-q\right|<2\epsilon\qquad\text{for all }x\in\mathbb Q_1.$$
If $n$ is a positive integer, let $A = \{n,n+1,...,n+17 \}$.
Does there exist some values of $n$ for which we can divide $A$ into two disjoints subsets $B$ and $C$ such that the product of the elements of $B$ is equal to the product of the elements of $C$?
A circle has center $ (-10,-4) $ and radius $13$. Another circle has center $(3,9) $ and radius $\sqrt{65}$. The line passing through the two points of intersection of the two circles has equation $x+y=c$. What is $c$?
$\textbf{(A)} \text{ 3} \qquad \textbf{(B)} \text{ } 3 \sqrt{3} \qquad \textbf{(C)} \text{ } 4\sqrt{2} \qquad \textbf{(D)} \text{ 6} \qquad \textbf{(E)} \text{ }\frac{13}{2}$
Determine the number of pairs $(a, b)$, where $1 \le a \le b \le 100$ are positive integers, so that $\frac{a^3+b^3}{a^2+b^2}$ is an integer.
There are representants from $n$ different countries sit around a circular table ($n\geq2$), in such way that if two representants are from the same country, then, their neighbors to the right are not from the same country. Find, for every $n$, the maximal number of people that can be sit around the table.
Let $ABCD$ be a square. Let $N,P$ be two points on sides $AB, AD$, respectively such that $NP=NC$, and let $Q$ be a point on $AN$ such that $\angle QPN = \angle NCB$. Prove that \[ \angle BCQ = \dfrac{1}{2} \angle AQP .\]
Let $\sigma$ be a permutation of the set $S := \{1, 2, \ldots , 100\},$ such that $\sigma(a+b) \equiv \sigma(a)+\sigma(b) \pmod{100}$ whenever $a, b, a + b \in S.$ Denote by $f(s)$ the sum of the distinct values $\sigma(s)$ can take over all possible $\sigma$s satisfying the given condition. What is the nonnegative difference between the maximum and minimum value $f$ takes on when ranging over all $s \in S$?
Given $4$ lines in Euclidean $3-$space:
$L_1: x = 1, y = 0;$
$L_2: y = 1, z = 0;$
$L_3: x = 0, z = 1;$
$L_4: x = y, y = -6z.$
Find the equations of the two lines which both meet all of the $L_i.$
Find all natural $n$ such that for some different natural $a, b, c$ and $d$ among numbers
$$\frac{(a-c)(b-d)}{(b-c)(a-d)} ,
\frac{(b-c)(a-d)}{(a-c)(b-d)} ,
\frac{(a-b)(d-c)}{(a-d)(b-c)} ,
\frac{(a-c)(b-d)}{(a-b)(c-d)} ,$$
there are at least two numbers equal to $n$.
The system of inequalities
$$a-b^2\ge\frac14$$$$b-c^2\ge\frac14$$$$c-d^2\ge\frac14$$$$d-a^2\ge\frac14$$where $a,b,c,d$ are real numbers has
$\textbf{(A)}~\text{no solutions}$
$\textbf{(B)}~\text{exactly one solution}$
$\textbf{(C)}~\text{exactly two solutions}$
$\textbf{(D)}~\text{infinitely many solutions}$
Let $ABC$ be an acute triangle and $(O)$ be its circumcircle. Denote by $H$ its orthocenter and $I$ the midpoint of $BC$. The lines $BH, CH$ intersect $AC,AB$ at $E, F$ respectively. The circles $(IBF$) and $(ICE)$ meet again at $D$.
a) Prove that $D, I,A$ are collinear and $HD, EF, BC$ are concurrent.
b) Let $L$ be the foot of the angle bisector of $\angle BAC$ on the side $BC$. The circle $(ADL)$ intersects $(O)$ again at $K$ and intersects the line $BC$ at $S$ out of the side $BC$. Suppose that $AK,AS$ intersects the circles $(AEF)$ again at $G, T$ respectively. Prove that $TG = TD$.
If the circumradius of any three consecutive vertices of a convex polygon is at most $ 1, $ show that the discs of radius $ 1 $ centered at each vertex cover the polygon and its interior.
Find the smallest positive integer $n$ or show no such $n$ exists, with the following property: there are infinitely many distinct $n$-tuples of positive rational numbers $(a_1, a_2, \ldots, a_n)$ such that both
$$a_1+a_2+\dots +a_n \quad \text{and} \quad \frac{1}{a_1} + \frac{1}{a_2} + \dots + \frac{1}{a_n}$$
are integers.
A certain town is represented as an infinite plane, which is divided by straight lines into squares. The lines are streets, while the squares are blocks. Along a certain street there stands a policeman on each $100$th intersection . Somewhere in the town there is a bandit , whose position and speed are unknown, but he can move only along the streets. The aim of the police is to see the bandit . Does there exist an algorithm available to the police to enable them to achieve their aim?
(A. Andjans, Riga)
Let $p > 7$ be a prime which leaves residue 1 when divided by 6. Let $m=2^p-1,$ then prove $2^{m-1}-1$ can be divided by $127m$ without residue.
There are $n>1$ cities in Jansonland, with two-way roads joining certain pairs of cities. Janson will send a few robots one-by-one to build more roads. The robots operate as such:
1. Janson first selects an integer $k$ and a list of cities $a_0, a_1, \dots, a_k$ (cities can repeat).
2. The robot begins at $a_0$ and goes to $a_1$, then $a_2$, and so on until $a_k$.
3. When the robot goes from $a_i$ to $a_{i+1}$, if there is no road then the robot builds a road, but if there is a road then the robot destroys the road.
In terms of $n$, determine the smallest constant $k$ such that Janson can always achieve a configuration such that every pair of cities has a road connecting them using no more than $k$ robots.
[i](Proposed by Ho Janson)[/i]
Prove that for every prime number $p$ and positive integer $a$, there exists a natural number $n$ such that $p^n$ contains $a$ consecutive equal digits.
After multiplying out and simplifying polynomial $ (x \minus{} 1)(x^2 \minus{} 1)(x^3 \minus{} 1)\cdots(x^{2007} \minus{} 1),$ getting rid of all terms whose powers are greater than $ 2007,$ we acquire a new polynomial $ f(x).$ Find its degree and the coefficient of the term having the highest power. Find the degree of $ f(x) \equal{} (1 \minus{} x)(1 \minus{} x^{2})...(1 \minus{} x^{2007})$ $ (mod$ $ x^{2008}).$
Players $A$ and $B$ play a "paintful" game on the real line. Player $A$ has a pot of paint with four units of black ink. A quantity $p$ of this ink suffices to blacken a (closed) real interval of length $p$. In every round, player $A$ picks some positive integer $m$ and provides $1/2^m $ units of ink from the pot. Player $B$ then picks an integer $k$ and blackens the interval from $k/2^m$ to $(k+1)/2^m$ (some parts of this interval may have been blackened before). The goal of player $A$ is to reach a situation where the pot is empty and the interval $[0,1]$ is not completely blackened.
Decide whether there exists a strategy for player $A$ to win in a finite number of moves.
All numbers from $1$ to $32$ a written on the stars from the picture below, such that each number is written once. Can all sums of the numbers written in each square that is not divided in smaller squares be equal?
Let $S$ be a set of $n$ points in the plane such that no four points are collinear. Let $\{d_1,d_2,\cdots ,d_k\}$ be the set of distances between pairs of distinct points in $S$, and let $m_i$ be the multiplicity of $d_i$, i.e. the number of unordered pairs $\{P,Q\}\subseteq S$ with $|PQ|=d_i$. Prove that $\sum_{i=1}^k m_i^2\leq n^3-n^2$.
If $a, b, c$ are non-zero reals, prove that $|\frac{b} {a}-\frac{b} {c}|+|\frac{c} {a}-\frac{c}{b}|+|bc+1|>1$.
Square $ABCD$ has side length $1$; circle $\Gamma$ is centered at $A$ with radius $1$. Let $M$ be the midpoint of $BC$, and let $N$ be the point on segment $CD$ such that $MN$ is tangent to $\Gamma$. Compute $MN$.
[i]2018 CCA Math Bonanza Individual Round #11[/i]
Show that there do not exist non-zero real numbers $a,b,c$ such that the following statements hold simultaneously:
$\bullet$ the equation $ax^2 + bx + c = 0$ has two distinct roots $x_1,x_2$;
$\bullet$ the equation $bx^2 + cx + a = 0$ has two distinct roots $x_2,x_3$;
$\bullet$ the equation $cx^2 + ax + b = 0$ has two distinct roots $x_3,x_1$.
(Note that $x_1,x_2,x_3$ may be real or complex numbers.)