Determine all positive real $M$ such that for any positive reals $a,b,c$, at least one of $a + \dfrac{M}{ab}, b + \dfrac{M}{bc}, c + \dfrac{M}{ca}$ is greater than or equal to $1+M$.

Sir Alex plays the following game on a row of 9 cells. Initially, all cells are empty. In each move, Sir Alex is allowed to perform exactly one of the following two operations: [list=1] [*] Choose any number of the form $2^j$, where $j$ is a non-negative integer, and put it into an empty cell. [*] Choose two (not necessarily adjacent) cells with the same number in them; denote that number by $2^j$. Replace the number in one of the cells with $2^{j+1}$ and erase the number in the other cell. [/list] At the end of the game, one cell contains $2^n$, where $n$ is a given positive integer, while the other cells are empty. Determine the maximum number of moves that Sir Alex could have made, in terms of $n$. [i]Proposed by Warut Suksompong, Thailand[/i]

A function $f: Z \to Z$ has the following properties: $f (92 + x) = f (92 - x)$ $f (19 \cdot 92 + x) = f (19 \cdot 92 - x)$ ($19 \cdot 92 = 1748$) $f (1992 + x) = f (1992 - x)$ for all integers $x$. Can all positive divisors of $92$ occur as values of f?

$a_{n}$ ($n$ is integer) is a sequence from positive reals that \[a_{n}\geq \frac{a_{n+2}+a_{n+1}+a_{n-1}+a_{n-2}}4\] Prove $a_{n}$ is constant.

One point of the plane is called $rational$ if both coordinates are rational and $irrational$ if both coordinates are irrational. Check whether the following statements are true or false: [b]a)[/b] Every point of the plane is in a line that can be defined by $2$ rational points. [b]b)[/b] Every point of the plane is in a line that can be defined by $2$ irrational points. This maybe is not algebra so sorry if I putted it in the wrong category!

The numbers from $1$ to $2025$ are arranged in some order in the cells of the $1 \times 2025$ strip. Let's call a [i]flip[/i] an operation that takes two arbitrary cells of a strip and swaps the numbers written in them, but only if the larger of these numbers is located to the left of the smaller one. A [i]flop[/i] is a set of several flips that do not contain common cells that are executed simultaneously. (For example, a simultaneous flip between the 2nd and 8th cells and a flip between the 5th and 101st cells.) Prove that there exists a sequence of $66$ flops such that for any initial arrangement, applying this sequence of flops to it will result in the numbers being ordered from left to right in ascending order.

$77$ stones weighing $1,2,\dots, 77$ grams are divided into $k$ groups such that total weights of each group are different from each other and each group contains less stones than groups with smaller total weights. For how many $k\in \{9,10,11,12\}$, is such a division possible? $ \textbf{(A)}\ 4 \qquad\textbf{(B)}\ 3 \qquad\textbf{(C)}\ 2 \qquad\textbf{(D)}\ 1 \qquad\textbf{(E)}\ \text{None of above} $

Find all primes $p$ for which $p^2 - p + 1$ is a perfect cube.

A rectangular field is half as wide as it is long and is completely enclosed by $ x$ yards of fencing. The area in terms of $ x$ is: $ \textbf{(A)}\ \frac {x^2}{2} \qquad\textbf{(B)}\ 2x^2 \qquad\textbf{(C)}\ \frac {2x^2}{9} \qquad\textbf{(D)}\ \frac {x^2}{18} \qquad\textbf{(E)}\ \frac {x^2}{72}$

The exam has $25$ topics, each of which has $8$ questions. On a test, there are $4$ questions of different topics. Is it possible to make $50$ tests so that each question was asked exactly once, and for any two topics there is a test where are questions of both topics?

There are $ n \geq 3$ job openings at a factory, ranked $1$ to $ n$ in order of increasing pay. There are $ n$ job applicants, ranked from $1$ to $ n$ in order of increasing ability. Applicant $ i$ is qualified for job $ j$ if and only if $ i \geq j.$ The applicants arrive one at a time in random order. Each in turn is hired to the highest-ranking job for which he or she is qualified AND which is lower in rank than any job already filled. (Under these rules, job $1$ is always filled, and hiring terminates thereafter.) Show that applicants $ n$ and $ n \minus{} 1$ have the same probability of being hired.

In a circle, chord $AB$ has length $5$ and chord $AC$ has length $7$. Arc $AC$ is twice the length of arc $AB$, and both arcs have degree less than $180$. Compute the area of the circle.

What is the minimum number of cells that can be painted black in white square $300 \times 300$ so that no three black cells form a corner, and after painting any white cell this condition was it violated?

For positive integers $k$ and $n$, we know $k \geq n!$. Prove that $ \phi (k) \geq (n-1)!$

Let $I\subset \mathbb R$ be a non-empty open interval, $\varepsilon\geq 0$ and $f\colon I\rightarrow\mathbb R$ a function satisfying the $$f(tx+(1-t)y)\leq tf(x)+(1-t)f(y)+\varepsilon t(1-t)|x-y|$$ inequality for all $x,y\in I$ and $t\in [0,1]$. Prove that there exists a convex $g\colon I\rightarrow\mathbb R$ function, such that the function $l :=f-g$ has the $\varepsilon$-Lipschitz property, that is $$|l(x)-l(y)|\leq \varepsilon|x-y|\text{ for all }x,y\in I$$

For real numbers $a,b,c$ and positive number $\lambda$ such that three real roots $x_1,x_2,x_3$ of $f(x)=x^3+ax^2+bx+c$ satisfying: $(1) x_2-x_1=\lambda$; $(2) x_3>\frac{1}{2}(x_1+x_2)$. Find the maximum value of $\frac{2a^3+27c-9ab}{\lambda^3}$

Positive integers $1,2,...,n$ are written on а blackboard ($n >2$ ). Every minute two numbers are erased and the least prime divisor of their sum is written. In the end only the number 97 remains. Find the least $n$ for which it is possible.

In the country of Sikinia there are finitely many cities. From each city, exactly three roads go out and each road goes to another Sikinian city. A tourist starts a trip from city $A$ and drives according to the following rule: he turns left at the first city, then right at the next city, and so on, alternately. Show that he will eventually return to $A.$

Do there exist a positive integer $k$ and a non-constant sequence $a_1, a_2, a_3, ...$ of positive integers such that $a_n = gcd(a_{n+k}, a_{n+k+1})$ for all positive integers $n$?

Suppose that $P(x)$ is a polynomial with real coefficients, such that for some positive real numbers $c$ and $d$, and for all natural numbers $n$, we have $c|n|^3\leq |P(n)|\leq d|n|^3$. Prove that $P(x)$ has a real zero.

Let $ABCDEF$ be a convex hexagon. In the hexagon there is a point $K$, such that $ABCK,DEFK$ are both parallelograms. Prove that the three lines connecting $A,B,C$ to the midpoints of segments $CE,DF,EA$ meet at one point.

Let $(I),(O)$ be the incircle, and, respectiely, circumcircle of $ABC$. $(I)$ touches $BC,CA,AB$ in $D,E,F$ respectively. We are also given three circles $\omega_a,\omega_b,\omega_c$, tangent to $(I),(O)$ in $D,K$ (for $\omega_a$), $E,M$ (for $\omega_b$), and $F,N$ (for $\omega_c$). [b]a)[/b] Show that $DK,EM,FN$ are concurrent in a point $P$; [b]b)[/b] Show that the orthocenter of $DEF$ lies on $OP$.

Numbers $1,2,3,...,2n$ are divided onto two equal groups. Let $a_1,a_2,...,a_n$ be the first group numbers in the increasing order, and $b_1,b_2,...,b_n$ -- the second group numbers in the decreasing order. Prove that $$|a_1 - b_1| + |a_2 - b_2| + ... + |a_n - b_n| = n^2$$

Consider the system of congruences $$\begin{cases} xy \equiv - 1 \,\, (mod z) \\ yz \equiv 1 \, \, (mod x) \\zx \equiv 1 \, \, (mod y)\end {cases}$$ Find the number of triples $(x,y, z) $ of distinct positive integers satisfying this system such that one of the numbers $x,y, z$ equals $19$.

Anthony writes the $(n+1)^2$ distinct positive integer divisors of $10^n$, each once, on a whiteboard. On a move, he may choose any two distinct numbers $a$ and $b$ on the board, erase them both, and write $\gcd(a, b)$ twice. Anthony keeps making moves until all of the numbers on the board are the same. Find the minimum possible number of moves Anthony could have made. [i]Proposed by Andrew Wen[/i]