Found problems: 85335
It is known that the equation $x^3 + px^2 + q = 0$ where $q$ is non-zero, has three different integer roots, the absolute values of two of which are prime numbers. Find the roots of this equation.
We are given $n$ reals $a_1,a_2,\cdots , a_n$ such that the sum of any two of them is non-negative. Prove that the following statement and its converse are both true: if $n$ non-negative reals $x_1,x_2,\cdots ,x_n$ satisfy $x_1+x_2+\cdots +x_n=1$, then the inequality $a_1x_1+a_2x_2+\cdots +a_nx_n\ge a_1x^2_1+ a_2x^2_2+\cdots + a_nx^2_n$ holds.
Call a point in the Cartesian plane with integer coordinates a $lattice$ $point$. Given a finite set $\mathcal{S}$ of lattice points we repeatedly perform the following operation: given two distinct lattice points $A, B$ in $\mathcal{S}$ and two distinct lattice points $C, D$ not in $\mathcal{S}$ such that $ACBD$ is a parallelogram with $AB > CD$, we replace $A, B$ by $C, D$. Show that only finitely many such operations can be performed.
[I]Proposed by Joe Benton, United Kingdom.[/i]
If $\alpha $ is an acute angle and $a,b\geq 0$ then show that:
$\left( a+\frac{b}{\sin \alpha}\right)\left(b+\frac{a}{\cos \alpha}\right)\geq a^2+b^2+3ab$
Let $R > 0$, be an integer, and let $n(R)$ be the number um triples $(x, y, z) \in \mathbb{Z}^3$ such that $2x^2+3y^2+5z^2 = R$. What is the value of
$\lim_{ R \to \infty}\frac{n(1) + n(2) + \cdots + n(R)}{R^{3/2}}$?
The numbers $1, 2\ldots11$ are arranged in a line from left to right in a random order. It is observed that the middle number is larger than exactly one number to its left. Find the probability that it is larger than exactly one number to its right.
Determine the gcd of all numbers of the form $p^8-1$, with p a prime above 5.
Let a pair of positive integers $(n, m)$ that are relatively prime be called [i]intertwined[/i] if among any two divisors of $n$ greater than $1$, there exists a divisor of $m$ and among any two divisors of $m$ greater than $1$, there exists a divisor of $n$. For example, pair $(63, 64)$ is intertwined.
[b]a)[/b] Find the largest integer $k$ for which there exists an intertwined pair $(n, m)$ such that the product $nm$ is equal to the product of the first $k$ prime numbers.
[b]b)[/b] Prove that there does [b]not[/b] exist an intertwined pair $(n, m)$ such that the product $nm$ is the product of $2025$ distinct prime numbers.
[b]c)[/b] Prove that there exists an intertwined pair $(n, m)$ such that the number of divisors of $n$ is greater than $2025$.
[i]Proposed by Stijn Cambie, Belgium[/i]
A large candle is $119$ centimeters tall. It is designed to burn down more quickly when it is first lit and more slowly as it approaches its bottom. Specifically, the candle takes $10$ seconds to burn down the first centimeter from the top, $20$ seconds to burn down the second centimeter, and $10k$ seconds to burn down the $k$-th centimeter. Suppose it takes $T$ seconds for the candle to burn down completely. Then $\tfrac{T}{2}$ seconds after it is lit, the candle's height in centimeters will be $h$. Find $10h$.
Let $n$ be a positive integer. Each of the numbers $1,2,3,\ldots,100$ is painted with one of $n$ colors in such a way that two distinct numbers with a sum divisible by $4$ are painted with different colors. Determine the smallest value of $n$ for which such a situation is possible.
Let $f:[0,1)\to (0,1)$ a continous onto (surjective) function.
a) Prove that, for all $a\in(0,1)$, the function $f_a:(a,1)\to (0,1)$, given by $f_a(x) = f(x)$, for all $x\in(a,1)$ is onto;
b) Give an example of such a function.
a) Is it possible that the sum of all the positive divisors of two different natural numbers are equal?
b) Is it possible that the product of all the positive divisors of two different natural numbers are equal?
Let $a_1,a_2,a_3,a_4$ be strictly positive numbers. Then is the following inequality true:
$$4\left(a_1a_2^n+a_2a_3^n+a_3a_4^n+a_4a_1^n\right)^n\le\left(a_1^n+a_2^n+a_3^n+a_4^n\right)^{n+1}$$
for each $n\in\mathbb N$?
[i]Proposed by Mihály Bencze and Marius Drăgan[/i]
In the plane we consider rectangles whose sides are parallel to the coordinate axes and have positive length. Such a rectangle will be called a [i]box[/i]. Two boxes [i]intersect[/i] if they have a common point in their interior or on their boundary. Find the largest $ n$ for which there exist $ n$ boxes $ B_1$, $ \ldots$, $ B_n$ such that $ B_i$ and $ B_j$ intersect if and only if $ i\not\equiv j\pm 1\pmod n$.
[i]Proposed by Gerhard Woeginger, Netherlands[/i]
Find the largest integer $ n$ satisfying the following conditions:
(i) $ n^2$ can be expressed as the difference of two consecutive cubes;
(ii) $ 2n\plus{}79$ is a perfect square.
For how many primes $ p$, there exits unique integers $ r$ and $ s$ such that for every integer $ x$ $ x^{3} \minus{} x \plus{} 2\equiv \left(x \minus{} r\right)^{2} \left(x \minus{} s\right)\pmod p$?
$\textbf{(A)}\ 0 \qquad\textbf{(B)}\ 1 \qquad\textbf{(C)}\ 2 \qquad\textbf{(D)}\ 3 \qquad\textbf{(E)}\ \text{None}$
Let $p\ge 17$ be a prime. Prove that $t=3$ is the largest positive integer which satisfies the following condition:
For any integers $a,b,c,d$ such that $abc$ is not divisible by $p$ and $(a+b+c)$ is divisible by $p$, there exists integers $x,y,z$ belonging to the set $\{0,1,2,\ldots , \left\lfloor \frac{p}{t} \right\rfloor - 1\}$ such that $ax+by+cz+d$ is divisible by $p$.
Prove that there are $2012$ points in the plane, none of which are three on one straight line and in pairs have integer distances .
Let $ (a_n)^{\infty}_{n\equal{}1}$ be a sequence of integers with $ a_{n} < a_{n\plus{}1}, \quad \forall n \geq 1.$ For all quadruple $ (i,j,k,l)$ of indices such that $ 1 \leq i < j \leq k < l$ and $ i \plus{} l \equal{} j \plus{} k$ we have the inequality $ a_{i} \plus{} a_{l} > a_{j} \plus{} a_{k}.$ Determine the least possible value of $ a_{2008}.$
Let $a$ and $b$ be natural numbers. Prove that the number of polynomials $P(x)$ with integer coefficients such that $|P(n)| \leq a^n$ for every natural number $n \geq b$ is finite.
Let $ n\ge 3 $ be a natural number and $ A_0A_1...A_{n-1} $ a regular polygon. Consider $ B_0 $ on the segment $ A_0A_1 $ such that $ A_0B_0<\frac{1}{2}A_0A_1; B_1 $ on $ A_1A_2 $ so that $ A_1B_1<\frac{1}{2} A_1A_2; $ etc.; $ B_{n-2} $ on $ A_{n-2}A_{n-1} $ so that $ A_{n-2}B_{n-2} <\frac{1}{2} A_{n-2}A_{n-1} , $ and $ B_{n-1} $ on $ A_{n-1}A_0 $ with $ A_{n-1}B_{n-1} <\frac{1}{2} A_{n-1}A_{0} . $
Show that the perimeter of any ploygon that has its vertices on the segments $ A_1B_1,A_2B_2,...,A_{n-1}B_{n-1}, $ is equal or greater than the perimeter of $ B_0B_1...B_{n-1} . $
$ P$ is the intersection point of diagonals of cyclic $ ABCD$. The circumcenters of $ \triangle APB$ and $ \triangle CPD$ lie on circumcircle of $ ABCD$. If $ AC \plus{} BD \equal{} 18$, then area of $ ABCD$ is ?
$\textbf{(A)}\ 36 \qquad\textbf{(B)}\ \frac {81}{2} \qquad\textbf{(C)}\ \frac {36\sqrt 3}{2} \qquad\textbf{(D)}\ \frac {81\sqrt 3}{4} \qquad\textbf{(E)}\ \text{None}$
Let $ABCD$ be a convex quadrilateral with $\angle ABC=\angle ADC=120^{\circ}$. The point $E$ lies on the segment $AD$ and is such that $AE \cdot BC=AB \cdot DE$ and similarly the point $F$ lies on the segment $BC$ and satisfies $BF \cdot CD=AD \cdot CF$. Show that $BE$ and $DF$ are parallel.
If $x$ and $y$ are nonnegative real numbers with $x+y= 2$, show that $x^2y^2(x^2+y^2)\le 2$.
(i) Find the angles of $\triangle ABC$ if the length of the altitude through $B$ is equal to the length of the median through $C$ and the length of the altitude through $C$ is equal to the length of the median through $B$.
(ii) Find all possible values of $\angle ABC$ of $\triangle ABC$ if the length of the altitude through $A$ is equal to the length of the median through $C$ and the length of the altitude through $C$ is equal to the length of the median through $B$.