Found problems: 85335
Solve the following equation in positive integers $x, y$: $x^{2017} - 1 = (x - 1)(y^{2015}- 1)$
A function $f(x)$ defined for $x\ge 0$ satisfies the following conditions:
i. for $x,y\ge 0$, $f(x)f(y)\le x^2f(y/2)+y^2f(x/2)$;
ii. there exists a constant $M$($M>0$), such that $|f(x)|\le M$ when $0\le x\le 1$.
Prove that $f(x)\le x^2$.
A right triangle with perpendicular sides $a$ and $b$ and hypotenuse $c$ has the following properties:
$a = p^m$ and $b = q^n$ with $p$ and $q$ prime numbers and $m$ and $n$ positive integers, $c = 2k +1$ with $k$ a positive integer.
Determine all possible values of $c$ and the associated values of $a$ and $b$.
Consider the real numbers $a_1,a_2,...,a_{2n}$ whose sum is equal to $0$. Prove that among pairs $(a_i,a_j) , i<j$ where $ i,j \in \{1,2,...,2n\} $ .there are at least $2n-1$ pairs with the property that $a_i+a_j\ge 0$.
The two figures depicted below consisting of $6$ and $10$ unit squares, respectively, are called staircases.
Consider a $2018\times 2018$ board consisting of $2018^2$ cells, each being a unit square. Two arbitrary
cells were removed from the same row of the board. Prove that the rest of the board cannot be cut (along the cell borders) into staircases (possibly rotated).
Let $n>1$ be a positive integer. Show that the number of residues modulo $n^2$ of the elements of the set $\{ x^n + y^n : x,y \in \mathbb{N} \}$ is at most $\frac{n(n+1)}{2}$.
[I]Proposed by N. Safaei (Iran)[/i]
Georg chooses three distinct digits among $1, 2, . . . , 9$ and writes them down on three cards. When the cards are laid down next to each other, a three-digit number is formed. Georg tells his mother that the sum of the largest and the second-largest number that can be formed in this manner is $1732$. Can she figure out which three digits Georg has chosen?
Find the intersection of all sets of consecutive positive integers having at least four elements and the sum of elements equal to $2001$.
Let $ABC$ be a triangle in which $\angle BAC = 60^{\circ}$ . Let $P$ (similarly $Q$) be the point of intersection of the bisector of $\angle ABC$(similarly of $\angle ACB$) and the side $AC$(similarly $AB$). Let $r_1$ and $r_2$ be the in-radii of the triangles $ABC$ and $AP Q$, respectively. Determine the circum-radius of $APQ$ in terms of $r_1$ and $r_2$.
Two straight pipes (circular cylinders), with radii $1$ and $\frac{1}{4}$, lie parallel and in contact on a flat floor. The figure below shows a head-on view. What is the sum of the possible radii of a third parallel pipe lying on the same floor and in contact with both?
[asy]
size(6cm);
draw(circle((0,1),1), linewidth(1.2));
draw((-1,0)--(1.25,0), linewidth(1.2));
draw(circle((1,1/4),1/4), linewidth(1.2));
[/asy]
$\textbf{(A)}~\displaystyle\frac{1}{9}
\qquad\textbf{(B)}~1
\qquad\textbf{(C)}~\displaystyle\frac{10}{9}
\qquad\textbf{(D)}~\displaystyle\frac{11}{9}
\qquad\textbf{(E)}~\displaystyle\frac{19}{9}$
Given is a right triangle $ABC$ with perimeter $2$, with $\angle B=90^o$ . Point $S$ is the center of the excircle to the side $AB$ of the triangle and $H$ is the intersection of the heights of the triangle $ABS$ . Determine the smallest possible length of the segment $HS $.
Let $ A\equal{}(a_{ij})_{1\leq i,j\leq n}$ be a real $ n\times n$ matrix, such that $ a_{ij} \plus{} a_{ji} \equal{} 0$, for all $ i,j$. Prove that for all non-negative real numbers $ x,y$ we have \[ \det(A\plus{}xI_n)\cdot \det(A\plus{}yI_n) \geq \det (A\plus{}\sqrt{xy}I_n)^2.\]
If a b c positive reals smaller than 1, prove:
a+b+c+2abc>ab+bc+ca+2(abc)^(1/2)
There are $2017$ frogs and $2017$ toads in a room. Each frog is friends with exactly $2$ distinct toads. Let $N$ be the number of ways to pair every frog with a toad who is its friend, so that no toad is paired with more than one frog. Let $D$ be the number of distinct possible values of $N$, and let $S$ be the sum of all possible value of $N$. Find the ordered pair $(D, S)$.
Let $n$ and $p$ be positive integers, with $p>3$ prime, such that:
i) $n\mid p-3;$
ii) $p\mid (n+1)^3-1.$
Show that $pn+1$ is the cube of an integer.
Let \[N= \sum_{k=1}^{1000}k(\lceil \log_{\sqrt{2}}k\rceil-\lfloor \log_{\sqrt{2}}k \rfloor).\] Find the remainder when N is divided by 1000. (Here $\lfloor x \rfloor$ denotes the greatest integer that is less than or equal to x, and $\lceil x \rceil$ denotes the least integer that is greater than or equal to x.)
Determine all positive integers $ n$, for which $ 2^{n\minus{}1}n\plus{}1$ is a perfect square.
Two distinct positive even integers sum to $8.$ Determine the larger of the $2$ integers.
Find all functions $f:\mathbb{Z} \rightarrow \mathbb{R}$ such that $f(1)=\tfrac{5}{2}$ and that \[f(x)f(y)=f(x+y)+f(x-y)\] for all integers $x$ and $y$.
Find all positive integers $m, n$ such that $\frac{117}{158} > \frac{m}{n} > \frac{97}{131}$ and $n \le 500$.
In a certain land, all Arogs are Brafs, all Crups are Brafs, all Dramps are Arogs, and all Crups are Dramps. Which of the following statements is implied by these facts?
$ \textbf{(A)}\ \text{All Dramps are Brafs and are Crups.}\qquad \\
\textbf{(B)}\ \text{All Brafs are Crups and are Dramps.}\qquad \\
\textbf{(C)}\ \text{All Arogs are Crups and are Dramps.}\qquad \\
\textbf{(D)}\ \text{All Crups are Arogs and are Brafs.}\qquad \\
\textbf{(E)}\ \text{All Arogs are Dramps and some Arogs may not be Crups.}$
Compute the number of ways to color the vertices of a regular heptagon red, green, or blue (with rotations and reflections distinct) such that no isosceles triangle whose vertices are vertices of the heptagon has all three vertices the same color.
Max flips $2020$ fair coins. Let the probability that there are at most $505$ heads be $p$. Estimate $-\log_2(p)$ to 5 decimal places, in the form $x.abcde$ where $x$ is a positive integer and $a, b, c, d, e$ are decimal digits.
Point $P$ is taken on the extension of side $AB$ of an equilateral triangle $ABC$ so that $A$ is between $B$ and $P$. Denote by $a$ the side length of triangle $ABC$, by $r_1$ the inradius of triangle $PAC$, and by $r_2$ the exradius of triangle $PBC$ opposite $P$. Find the sum $r_1+r_2$ as a function in $a$.
$u$ is a real parameter such that $0<u<1$.
For $0\le x \le u$, $f(x)=0$.
For $u\le x \le n$, $f(x)=1-\left(\sqrt{ux}+\sqrt{(1-u)(1-x)}\right)^2$.
The sequence $\{u_n\}$ is define recursively as follows: $u_1=f(1)$ and $u_n=f(u_{n-1})$ $\forall n\in \mathbb{N}, n\neq 1$.
Show that there exists a positive integer $k$ for which $u_k=0$.