Found problems: 85335
Given that $x, y$, and $z$ are a combination of positive integers such that $xyz = 2(x + y + z)$, compute the sum of all possible values of $x + y + z$.
Prove that diagonals of a convex quadrilateral are perpendicular if and only if inside of the quadrilateral there is a point, whose orthogonal projections on sides of the quadrilateral are vertices of a rectangle.
Determine all strictly increasing functions $f : R \to R$ such that for all $x \ne y$ to hold $$\frac{2\left[f(y)-f\left(\frac{x+y}{2}\right) \right]}{f(x)-f(y)}=\frac{f(x)-f(y)}{2\left[f\left(\frac{x+y}{2}\right)-f(x) \right]}$$
The sidelines $AB$ and $CD$ of a trapezoid meet at point $P$, and the diagonals of this trapezoid meet at point $Q$. Point $M$ on the smallest base $BC$ is such that $AM=MD$. Prove that $\angle PMB=\angle QMB$.
Two positive integers $r$ and $k$ are given as is an infinite sequence of positive integers $a_1 \le a_2 \le a_3 \le ..$ such that $\frac{r}{a_r}= k + 1$. Prove that there is a positive integer $t$ such that $\frac{t}{a_t}= k$.
Prove that there are infinitely many natural numbers $a,b,c,u$ and $v$ with greatest common divisor $1$ satisfying the system of equations: $a+b+c=u+v$ and $a^2+b^2+c^2=u^2+v^2$
Set positive integer $m=2^k\cdot t$, where $k$ is a non-negative integer, $t$ is an odd number, and let $f(m)=t^{1-k}$. Prove that for any positive integer $n$ and for any positive odd number $a\le n$, $\prod_{m=1}^n f(m)$ is a multiple of $a$.
Consider a function $f:\mathbb{R}\rightarrow \mathbb{R}$ such that:
1. $f$ has one-side limits in any $a\in \mathbb{R}$ and $f(a-0)\le f(a)\le f(a+0)$.
2. for any $a,b\in \mathbb{R},\ a<b$, we have $f(a-0)<f(b-0)$.
Prove that $f$ is strictly increasing.
[i]Mihai Piticari & Sorin Radulescu[/i]
There are five dots arranged in a line from left to right. Each of the dots is colored from one of five colors so that no $3$ consecutive dots are all the same color. How many ways are there to color the dots?
Determine all integers $x$ satisfying
\[ \left[\frac{x}{2}\right] \left[\frac{x}{3}\right] \left[\frac{x}{4}\right] = x^2. \]
($[y]$ is the largest integer which is not larger than $y.$)
Consider $A$, the set of natural numbers with exactly $2019$ natural divisors , and for each $n \in A$, denote $$S_n=\frac{1}{d_1+\sqrt{n}}+\frac{1}{d_2+\sqrt{n}}+...+\frac{1}{d_{2019}+\sqrt{n}}$$
where $d_1,d_2, .., d_{2019}$ are the natural divisors of $n$.
Determine the maximum value of $S_n$ when $n$ goes through the set $ A$.
Assume the adjoining chart shows the $1980$ U.S. population, in millions, for each region by ethnic group. To the nearest percent, what percent of the U.S. Black population lived in the South?
\[\begin{tabular}[t]{c|cccc}
& NE & MW & South & West \\ \hline
White & 42 & 52 & 57 & 35 \\
Black & 5 & 5 & 15 & 2 \\
Asian & 1 & 1 & 1 & 3 \\
Other & 1 & 1 & 2 & 4
\end{tabular}\]
$\text{(A)}\ 20\% \qquad \text{(B)}\ 25\% \qquad \text{(C)}\ 40\% \qquad \text{(D)}\ 56\% \qquad \text{(E)}\ 80\% $
Does there exist a convex polyhedron such that it has diagonals and each of them is shorter than each of its edges?
Given a finite binary string $S$ of symbols $X,O$ we define $\Delta(S)=n(X)-n(O)$ where $n(X),n(O)$ respectively denote number of $X$'s and $O$'s in a string. For example $\Delta(XOOXOOX)=3-4=-1$. We call a string $S$ $\emph{balanced}$ if every substring $T$ of $S$ has $-2\le \Delta(T)\le 2$. Find number of balanced strings of length $n$.
A row of 2021 balls is given. Pasha and Vova play a game, taking turns to perform moves; Pasha begins. On each turn a boy should paint a non-painted ball in one of the three available colors: red, yellow, or green (initially all balls are non-painted). When all the balls are colored, Pasha wins, if there are three consecutive balls of different colors; otherwise Vova wins. Who has a winning strategy?
Suppose $(a,b)$ is an ordered pair of integers such that the three numbers $a$, $b$, and $ab$ form an arithmetic progression, in that order. Find the sum of all possible values of $a$.
[i]Proposed by Nathan Xiong[/i]
We are given two mutually tangent circles in the plane, with radii $r_1, r_2$. A line intersects these circles in four points, determining three segments of equal length. Find this length as a function of $r_1$ and $r_2$ and the condition for the solvability of the problem.
Let $0 < a < b$. Prove that
$\int_a^b (x^2+1)e^{-x^2} dx \geq e^{-a^2} - e^{-b^2}$.
Consider all lines which meet the graph of $$y=2x^4+7x^3+3x-5$$ in four distinct points, say $(x_i,y_i), i=1,2,3,4.$ Show that $$\frac{x_1+x_2+x_3+x_4}{4}$$ is independent of the line and find its value.
Can the numbers from 1 to 81 be written in a 9×9 board, so that the sum of numbers in each 3×3 square is the same?
[i]S. Tokarev[/i]
Let $ABCD$ be a cyclic quadrilateral. Point $P$ is on line $CB$ such that $CP=CA$and $B$ lies between $C$ and $P$. Point $Q$ is on line $CD$ such that $CQ=CA$ and $D$ lies between $C$ and $Q$. Prove that the incentre of triangle $ABD$ lies on line $PQ.$
Let $ n \in \mathbb{Z}^\plus{}$ and let $ a, b \in \mathbb{R}.$ Determine the range of $ x_0$ for which
\[ \sum^n_{i\equal{}0} x_i \equal{} a \text{ and } \sum^n_{i\equal{}0} x^2_i \equal{} b,\]
where $ x_0, x_1, \ldots , x_n$ are real variables.
An isosceles triangle $ABC$ is inscribed in a circle with $\angle ACB = 90^o$ and $EF$ is a chord of the circle such that neither E nor $F$ coincide with $C$. Lines $CE$ and $CF$ meet $AB$ at $D$ and $G$ respectively. Prove that $|CE|\cdot |DG| = |EF| \cdot |CG|$.
Given prime numbers $p$ and $q$ with $p<q$, determine all pairs $(x,y)$ of positive integers such that
$$\frac1x+\frac1y=\frac1p-\frac1q.$$
Let $G$ be a finite group with the following property:
If $f$ is an automorphism of $G$, then there exists $m\in\mathbb{N^\star}$, so that $f(x)=x^{m} $ for all $x\in G$.
Prove that G is commutative.
[i]Marian Andronache[/i]