Found problems: 85335
For positive integers $m, n$ define the function $f_n(m) = 1^{2n} + 2^{2n} + 3^{2n} + \ldots +m^{2n}$. Prove that there are only finitely many pairs of positive integers $(a, b)$ such that $f_n(a) + f_n(b)$ is a prime number.
[i]Proposed by Nazar Serdyuk[/i]
Find all polynomials $P(x)$ with real coefficients such that
\[P(a)\in\mathbb{Z}\ \ \ \text{implies that}\ \ \ a\in\mathbb{Z}.\]
For any positive real numbers $a, b, c, d$, prove the following inequality:
$$(a^2+b^2)(b^2+c^2)(c^2+d^2)(d^2+a^2) \geq 64abcd|(a-b)(b-c)(c-d)(d-a)|$$
[i]Proposed by Anton Trygub[/i]
Let $n$ and $k$ be positive integers, with $k \le n.$ Define a (simple, undirected) graph $G_{n,k}$ as follows: its vertices are all of the binary strings of length $n,$ and there is an edge between two strings if and only if they differ in exactly $k$ positions. If $c_{n,k}$ denotes the number of connected components of $G_{n,k},$ compute $$\sum_{n=1}^{10} \sum_{k=1}^n c_{n,k}.$$ (For example, $G_{3,2}$ has two connected components.)
$\boxed{A5}$Let $n\in{N},n>2$,and suppose $a_1,a_2,...,a_{2n}$ is a permutation of the numbers $1,2,...,2n$ such that $a_1<a_3<...<a_{2n-1}$ and $a_2>a_4>...>a_{2n}.$Prove that
\[(a_1-a_2)^2+(a_3-a_4)^2+...+(a_{2n-1}-a_{2n})^2>n^3\]
Find all functions $f : \mathbb{Q} \to \mathbb{R}$ such that $f(x)f(y)f(x+y) = f(xy)(f(x) + f(y))$ for all $x,y\in\mathbb{Q}$.
[i]Sammy Luo and Alex Zhu.[/i]
Let $n$ be a natural number. To prove that the value of the expression
$$\prod^n_{i=0}\frac{x^{n-1}_i}{\prod_{j \ne i}(x_i - x_j)}$$
does not depend on the choice of the different real numbers $x_0, x_1, ... , x_n$.
It is given triangle $ABC$ and parallelogram $ASCR$ with diagonal $AC$. Let line constructed through point $B$ parallel with $CS$ intersects line $AS$ and $CR$ in $M$ and $P$, respectively. Let line constructed through point $B$ parallel with $AS$ intersects line $AR$ and $CS$ in $N$ and $Q$, respectively. Prove that lines $RS$, $MN$ and $PQ$ are concurrent
The set $N$ is partitioned into three subsets $A_1,A_2,A_3$.
Prove that at least one of them has the following property: There exists a positive number $m$ such that for any $k$ one can find numbers $a_1 < a_2 < ... < a_k$ in that subset satisfying $a_{j+1} -a_j \le m$ for $j = 1,...,k -1$.
(V.Protasov, 10--11) In the space, given two intersecting spheres of different radii and a point $ A$ belonging to both spheres. Prove that there is a point $ B$ in the space with the following property:
if an arbitrary circle passes through points $ A$ and $ B$ then the second points of its meet with the given spheres are equidistant from $ B$.
Two numbers are such that their difference, their sum, and their product are to one another as $1:7:24$. The product of the two numbers is:
$ \textbf{(A)}\ 6\qquad\textbf{(B)}\ 12\qquad\textbf{(C)}\ 24\qquad\textbf{(D)}\ 48\qquad\textbf{(E)}\ 96 $
Prove that if an arithmetic progression contains a perfect square, then it contains infinitely many perfect squares.
The sequence $\{a_n\}$ satisfies $a_0=1, a_1=2011,$ and $a_n=2a_{n-1}+a_{n-2}$ for all $n \geq 2$. Let
\[ S = \sum_{i=1}^{\infty} \frac{a_{i-1}}{a_i^2-a_{i-1}^2} \]
What is $\frac{1}{S}$?
[i]Author: Ray Li[/i]
Determine the maximum value of $m^2+n^2$, where $m$ and $n$ are integers in the range $1,2,\ldots,1981$ satisfying $(n^2-mn-m^2)^2=1$.
Vasya has $n{}$ candies of several types, where $n>145$. It is known that for any group of at least 145 candies, there is a type of candy which appears exactly 10 times. Find the largest possible value of $n{}$.
[i]Proposed by A. Antropov[/i]
Find all $f:\mathbb{R}\to\mathbb{R}$ satisfying:
$$f(xf(y)-y)+f(xy-x)+f(x+y)=2xy,\quad\forall x,y\in\mathbb{R}.$$
How many solutions does
\[
x^2 - [x^2] = \left(x - [x]\right)^2
\]
have satisfying $1 \leq x \leq n$?
Jack had a bag of $128$ apples. He sold $25\% $ of them to Jill. Next he sold $25\% $ of those remaining to June. Of those apples still in his bag, he gave the shiniest one to his teacher. How many apples did Jack have then?
$\text{(A)}\ 7 \qquad \text{(B)}\ 63 \qquad \text{(C)}\ 65 \qquad \text{(D)}\ 71 \qquad \text{(E)}\ 111$
Let $n\ge 2$ be an integer. Thibaud the Tiger lays $n$ $2\times 2$ overlapping squares out on a table, such that the centers of the squares are equally spaced along the line $y=x$ from $(0,0)$ to $(1,1)$ (including the two endpoints). For example, for $n=4$ the resulting figure is shown below, and it covers a total area of $\frac{23}{3}$.
[asy]
fill((0,0)--(2,0)--(2,.333333333333)--(0.333333333333,0.333333333333)--(0.333333333333,2)--(0,2)--cycle, lightgrey);
fill((0.333333333333,0.333333333333)--(2.333333333333,0.333333333333)--(2.333333333333,.6666666666666)--(0.666666666666,0.666666666666666)--(0.66666666666,2.33333333333)--(.333333333333,2.3333333333333)--cycle, lightgrey);
fill((0.6666666666666,.6666666666666)--(2.6666666666666,.6666666666)--(2.6666666666666,.6666666666666)--(2.6666666666666,1)--(1,1)--(1,2.6666666666666)--(0.6666666666666,2.6666666666666)--cycle, lightgrey);
fill((1,1)--(3,1)--(3,3)--(1,3)--cycle, lightgrey);
draw((0.33333333333333,2)--(2,2)--(2,0.333333333333), dashed+grey+linewidth(0.4));
draw((0.66666666666666,2.3333333333333)--(2.3333333333333,2.3333333333333)--(2.3333333333333,0.66666666666), dashed+grey+linewidth(0.4));
draw((1,2.666666666666)--(2.666666666666,2.666666666666)--(2.666666666666,1), dashed+grey+linewidth(0.4));
draw((0,0)--(2,0)--(2,.333333333333)--(0.333333333333,0.333333333333)--(0.333333333333,2)--(0,2)--(0,0),linewidth(0.4));
draw((0.333333333333,0.333333333333)--(2.333333333333,0.333333333333)--(2.333333333333,.6666666666666)--(0.666666666666,0.666666666666666)--(0.66666666666,2.33333333333)--(.333333333333,2.3333333333333)--(0.333333333333,.333333333333),linewidth(0.4));
draw((0.6666666666666,.6666666666666)--(2.6666666666666,.6666666666)--(2.6666666666666,.6666666666666)--(2.6666666666666,1)--(1,1)--(1,2.6666666666666)--(0.6666666666666,2.6666666666666)--(0.6666666666666,0.6666666666666),linewidth(0.4));
draw((1,1)--(3,1)--(3,3)--(1,3)--cycle,linewidth(0.4));
[/asy]
Find, with proof, the minimum $n$ such that the figure covers an area of at least $\sqrt{63}$.
Inscribed Triangle $ABC$ on circle $\odot O$. Bisector of $\angle ABC$ intersects $\odot O$ at $D$. Two lines $PB$ and $PC$ that are tangent to $\odot O$ intersect at $P$. $PD$ intersects $AC$ at $E$, $\odot O$ at $F$. $M$ is the midpoint of $BC$. Prove that $M,F,C,E$ are concyclic.
Solve the function $f: \Re \to \Re$:
\[ f( x^{3} )+ f(y^{3}) = (x+y)(f(x^{2} )+f(y^{2} )-f(xy))\]
Let $ABC$ be a triangle with $AB=13$, $AC=25$, and $\tan A = \frac{3}{4}$. Denote the reflections of $B,C$ across $\overline{AC},\overline{AB}$ by $D,E$, respectively, and let $O$ be the circumcenter of triangle $ABC$. Let $P$ be a point such that $\triangle DPO\sim\triangle PEO$, and let $X$ and $Y$ be the midpoints of the major and minor arcs $\widehat{BC}$ of the circumcircle of triangle $ABC$. Find $PX \cdot PY$.
[i]Proposed by Michael Kural[/i]
There are several triangles. From them a new triangle is obtained according to the following rule. The largest side of the new triangle is equal to the sum of the large sides of the data, the middle one is equal to the sum of the middle sides, and the smallest one is the sum of the smaller ones. Prove that if all the angles of these triangles were less than $a$, and $\phi$, where $\phi$ is the largest angle of the resulting triangle, then $\cos \phi \ge 1-\sin (a/2)$.
In a basketball tournament any two of the $n$ teams $S_1,S_2,...,S_n$ play one match (no draws).
Denote by $v_i$ and $p_i$ the number of victories and defeats of team $S_i$ ($i = 1,2,...,n$), respectively.
Prove that $v^2_1 +v^2_2 +...+v^2_n = p^2_1 +p^2_2 +...+p^2_n$
What is the largest integer $n$ with no repeated digits that is relatively prime to $6$? Note that two numbers are considered relatively prime if they share no common factors besides $1$.
[i]Proposed by Jacob Stavrianos[/i]