Prove for any positives $a,b,c$ the inequality $$ \sqrt[3]{\dfrac{a}{b}}+\sqrt[5]{\dfrac{b}{c}}+\sqrt[7]{\dfrac{c}{a}}>\dfrac{5}{2}$$

There are five positive integers written in a row. Each one except for the first one is the minimal positive integer that is not a divisor of the previous one. Can all these five numbers be distinct? Boris Frenkin

For every positive integer $k$ let $a_{k,1},a_{k,2},\ldots$ be a sequence of positive integers. For every positive integer $k$ let sequence $\{a_{k+1,i}\}$ be the difference sequence of $\{a_{k,i}\}$, i.e. for all positive integers $k$ and $i$ the following holds: $a_{k,i+1}-a_{k,i}=a_{k+1,i}$. Is it possible that every positive integer appears exactly once among numbers $a_{k,i}$? [i]Proposed by Dávid Matolcsi, Berkeley[/i]

Let $ABC$ be a triangle such that $\angle BAC = 90^\circ$ and $AB < AC$. We divide the interior of the triangle into the following six regions: \begin{align*} S_1=\text{set of all points }\mathit{P}\text{ inside }\triangle ABC\text{ such that }PA<PB<PC \\ S_2=\text{set of all points }\mathit{P}\text{ inside }\triangle ABC\text{ such that }PA<PC<PB \\ S_3=\text{set of all points }\mathit{P}\text{ inside }\triangle ABC\text{ such that }PB<PA<PC \\ S_4=\text{set of all points }\mathit{P}\text{ inside }\triangle ABC\text{ such that }PB<PC<PA \\ S_5=\text{set of all points }\mathit{P}\text{ inside }\triangle ABC\text{ such that }PC<PA<PB \\ S_6=\text{set of all points }\mathit{P}\text{ inside }\triangle ABC\text{ such that }PC<PB<PA\end{align*} Suppose that the ratio of the area of the largest region to the area of the smallest non-empty region is $49 : 1$. Determine the ratio $AC : AB$.

Let $ ABC$ be a right triangle with $ \angle A \equal{} 90^\circ$. Let $ D$ be the midpoint of $ AB$ and let $ E$ be a point on segment $ AC$ such that $ AD \equal{} AE$. Let $ BE$ meet $ CD$ at $ F$. If $ \angle BFC \equal{} 135^\circ$, determine $ BC / AB$.

Let $ABC$ be triangle with circumcircle $(O)$. Let $AO$ cut $(O)$ again at $A'$. Perpendicular bisector of $OA'$ cut $BC$ at $P_A$. $P_B,P_C$ define similarly. Prove that : I) Point $P_A,P_B,P_C$ are collinear. II ) Prove that the distance of $O$ from this line is equal to $\frac {R}{2}$ where $R$ is the radius of the circumcircle.

Function $f(x)=ax^2+8x+3(a<0)$. For any given nerative number $a$, define the largest positive number $l(a)$: $|f(x)|\leq5$ for all $x\in[0,l(a)]$. Find the largest $l(a)$, and $a$ when $l(a)$ takes its maximum value.

Let $ABCD$ be a rectangle and $P$ a point outside of it such that $\angle{BPC} = 90^{\circ}$ and the area of the pentagon $ABPCD$ is equal to $AB^{2}$. Show that $ABPCD$ can be divided in 3 pieces with straight cuts in such a way that a square can be built using those 3 pieces, without leaving any holes or placing pieces on top of each other. Note: the pieces can be rotated and flipped over.

Let $ABC$ be an acute triangle, with $AB \ne AC$. Let $D$ be the midpoint of the line segment $BC$, and let $E$ and $F$ be the projections of $D$ onto the sides $AB$ and $AC$, respectively. If $M$ is the midpoint of the line segment $EF$, and $O$ is the circumcenter of triangle $ABC$, prove that the lines $DM$ and $AO$ are parallel. [hide=PS] As source was given [url=https://artofproblemsolving.com/community/c629086_caucasus_mathematical_olympiad]Caucasus MO[/url], but I was unable to find this problem in the contest collections [/hide]

A [i] magic square[/i] is a $3\times 3$ grid of numbers in which the sums of the numbers in each row, column, and long diagonal are all equal. How many magic squares exist where each of the integers from $11$ to $19$ inclusive is used exactly once and two of the numbers are already placed as shown below? $\begin{tabular}{|l|l|l|l|} \hline & & 18 \\ \hline & 15 & \\ \hline & & \\ \hline \end{tabular}$

Let $A=\{a_1,a_2,\cdots,a_n,b_1,b_2,\cdots,b_n\}$ be a set with $2n$ distinct elements, and $B_i\subseteq A$ for any $i=1,2,\cdots,m.$ If $\bigcup_{i=1}^m B_i=A,$ we say that the ordered $m-$tuple $(B_1,B_2,\cdots,B_m)$ is an ordered $m-$covering of $A.$ If $(B_1,B_2,\cdots,B_m)$ is an ordered $m-$covering of $A,$ and for any $i=1,2,\cdots,m$ and any $j=1,2,\cdots,n,$ $\{a_j,b_j\}$ is not a subset of $B_i,$ then we say that ordered $m-$tuple $(B_1,B_2,\cdots,B_m)$ is an ordered $m-$covering of $A$ without pairs. Define $a(m,n)$ as the number of the ordered $m-$coverings of $A,$ and $b(m,n)$ as the number of the ordered $m-$coverings of $A$ without pairs. $(1)$ Calculate $a(m,n)$ and $b(m,n).$ $(2)$ Let $m\geq2,$ and there is at least one positive integer $n,$ such that $\dfrac{a(m,n)}{b(m,n)}\leq2021,$ Determine the greatest possible values of $m.$

There is a square table with side $n$. Is it possible to enter the numbers $0$, $1$ or $2$ into the cells of this table so that all sums of numbers in rows and columns are different and take values from $1$ to $2n$, if: a) $n = 7$ ? b) $n = 8$ ?

Let $P_1P_2\ldots P_{24}$ be a regular $24$-sided polygon inscribed in a circle $\omega$ with circumference $24$. Determine the number of ways to choose sets of eight distinct vertices from these $24$ such that none of the arcs has length $3$ or $8$.

For a positive integer $n \ge 2$, define a sequence $a_1, a_2, \cdots ,a_n$ as the following. $$ a_1 = \frac{n(2n-1)(2n+1)}{3}$$ $$a_k = \frac{(n+k-1)(n-k+1)}{2(k-1)(2k+1)}a_{k-1}, \text{ } (k=2,3, \cdots n)$$ (a) Show that $a_1, a_2, \cdots a_n$ are all integers. (b) Prove that there are exactly one number out of $a_1, a_2, \cdots a_n$ which is not a multiple of $2n-1$ and exactly one number out of $a_1, a_2, \cdots a_n$ which is not a multiple of $2n+1$ if and only if $2n-1$ and $2n+1$ are all primes.

Alice is thinking of a positive real number $x$, and Bob is thinking of a positive real number $y$. Given that $x^{\sqrt{y}}=27$ and $(\sqrt{x})^y=9$, compute $xy$.

If $x, y$, and $z$ are real numbers with $\frac{x - y}{z}+\frac{y - z}{x}+\frac{z - x}{y}= 36$, find $2012 +\frac{x - y}{z}\cdot \frac{y - z}{x}\cdot\frac{z - x}{y}$ .

Three fair dice are tossed at random (i.e., all faces have the same probability of coming up). What is the probability that the three numbers turned up can be arranged to form an arithmetic progression with common difference one? $\textbf{(A) }\frac{1}{6}\qquad\textbf{(B) }\frac{1}{9}\qquad\textbf{(C) }\frac{1}{27}\qquad\textbf{(D) }\frac{1}{54}\qquad \textbf{(E) }\frac{7}{36}$

Show that if the point $ M $ is situated in the interior of a square $ ABCD, $ then, among the segments $ MA,MB,MC,MD, $ [b]a)[/b] at most one of them is greater with a factor of $ \sqrt 5/2 $ than the side of the square. [b]b)[/b] at most two of them are greater than the side of the square. [b]c)[/b] at most three of them are greater with a factor of $ \sqrt 2/2 $ than the side of the square.

We assume a truck as a $1 \times (k + 1)$ tile. Our parking is a $(2k + 1) \times (2k + 1)$ table and there are $t$ trucks parked in it. Some trucks are parked horizontally and some trucks are parked vertically in the parking. The vertical trucks can only move vertically (in their column) and the horizontal trucks can only move horizontally (in their row). Another truck is willing to enter the parking lot (it can only enter from somewhere on the boundary). For $3k + 1 < t < 4k$, prove that we can move other trucks forward or backward in such a way that the new truck would be able to enter the lot. Prove that the statement is not necessarily true for $t = 3k + 1$.

For nonnegative integers $m$ and $n$, define the sequence $a(m,n)$ of real numbers as follows. Set $a(0,0)=2$ and for every natural number $n$, set $a(0,n)=1$ and $a(n,0)=2$. Then for $m,n\geq1$, define \[ a(m,n)=a(m-1,n)+a(m,n-1). \] Prove that for every natural number $k$, all the roots of the polynomial $P_{k}(x)=\sum_{i=0}^{k}a(i,2k+1-2i)x^{i}$ are real.

Given natural number n. Suppose that $N$ is the maximum number of elephants that can be placed on a chessboard measuring $2 \times n$ so that no two elephants are mutually under attack. Determine the number of ways to put $N$ elephants on a chessboard sized $2 \times n$ so that no two elephants attack each other. Alternative Formulation: Determine the number of ways to put $2015$ elephants on a chessboard measuring $2 \times 2015$ so there are no two elephants attacking each othe PS. Elephant = Bishop

Define the modulo $2553$ distance $d(x, y)$ between two integers $x, y$ to be the smallest nonnegative integer $d$ equivalent to either $x - y$ or $y - x$ modulo $2553$. Show that, given a set S of integers such that $|S| \ge 70$, there must be $m, n \in S$ with $d(m, n) \le 36$.

Let $n$ be a positive integer. Prove that the number of lines which go through the origin and precisely one other point with integer coordinates $(x,y),0\le x,y\le n$, is at least $\frac{n^2}{4}$.

Evaluate $ \int_1^{e^2} \frac{(e^{\sqrt{x}}\minus{}e^{\minus{}\sqrt{x}})\cos \left(e^{\sqrt{x}}\plus{}e^{\minus{}\sqrt{x}}\plus{}\frac{\pi}{4}\right)\plus{}(e^{\sqrt{x}}\plus{}e^{\minus{}\sqrt{x}})\cos \left(e^{\sqrt{x}}\minus{}e^{\minus{}\sqrt{x}}\plus{}\frac{\pi}{4}\right)}{\sqrt{x}}\ dx.$

Let $\sum_{i=1}^n a_i x_i =0$, $a_i\in \mathbb{Z}$. It is known that however we color $\mathbb{N}$ with finite number of colors, then the upper equation has a solution $x_1,x_2,...,x_n$ in one color. Prove that there is some non-empty sum of its coefficients equal to 0.