Find the maximum constant $C$ such that, whenever $\{a_n \}_{n=1}^{\infty}$ is a sequence of positive real numbers satisfying $a_{n+1}-a_n=a_n(a_n+1)(a_n+2)$, we have $$\frac{a_{2023}-a_{2020}}{a_{2022}-a_{2021}}>C.$$

Suppose $a,b,$ and $c$ are real numbers such that \begin{align*} a^2-bc &= \ 14, \\ b^2-ca &= \ 14, \text{ and} \\ c^2-ab &=-3. \end{align*} Compute $|a+b+c|.$

For every positive integer $n$, the symbol $a_n/b_n$ is the simplest form of the fraction $1+1/2+...+1/n$. Prove that for every pair of positive integers $(M, N)$ we can always find a positive integer $m$ where $(a_n, N) = 1$ for all $n = m, m + 1, ...,m + M$.

Let be three discs $ D_1,D_2,D_3. $ For each $ i,j\in\{1,2,3\} , $ denote $ a_{ij} $ as being the area of $ D_i\cap D_j. $ If $ x_1,x_2,x_3\in\mathbb{R} $ such that $ x_1x_2x_3\neq 0, $ then $$ a_{11} x_1^2+a_{22} x_2^2+a_{33} x_3^2+2a_{12} x_1x_2+2a_{23 }x_2x_3+2a_{31} x_3x_1>0. $$ [i]Vasile Pop[/i]

Find $$\sum^{100}_{i=1}i \gcd(i ,100).$$

Let $a, b, c$ be side lengths of a triangle, and define $s =\frac{a+b+c}{2}$. Prove that $$\frac{2a(2a-s)}{b + c}+\frac{2b(2b - s)}{c + a}+\frac{2c(2c - s)}{a + b}\ge s.$$

There are $ n \plus{} 1$ cells in a row labeled from $ 0$ to $ n$ and $ n \plus{} 1$ cards labeled from $ 0$ to $ n$. The cards are arbitrarily placed in the cells, one per cell. The objective is to get card $ i$ into cell $ i$ for each $ i$. The allowed move is to find the smallest $ h$ such that cell $ h$ has a card with a label $ k > h$, pick up that card, slide the cards in cells $ h \plus{} 1$, $ h \plus{} 2$, ... , $ k$ one cell to the left and to place card $ k$ in cell $ k$. Show that at most $ 2^n \minus{} 1$ moves are required to get every card into the correct cell and that there is a unique starting position which requires $ 2^n \minus{} 1$ moves. [For example, if $ n \equal{} 2$ and the initial position is 210, then we get 102, then 012, a total of 2 moves.]

Given the lengths of two sides of a triangle and that of the bisector of the angle between these sides, construct the triangle.

For positive integers $m, n$ ($m>n$), $a_{n+1}, a_{n+2}, ..., a_m$ are non-negative integers that satisfy the following inequality. $$ 2> \frac{a_{n+1}}{n+1} \ge \frac{a_{n+2}}{n+2} \ge \cdots \ge \frac{a_m}{m}$$ Find the number of pair $(a_{n+1}, a_{n+2}, \cdots, a_m)$.

Solve in $\mathbb{R}$ the equation : $(x+1)^5 + (x+1)^4(x-1) + (x+1)^3(x-1)^2 +$ $ (x+1)^2(x-1)^3 + (x+1)(x-1)^4 + (x-1)^5 =$ $ 0$.

Given a prime number $p>5$. It is known that the length of the smallest period of the fraction $1/p$ is a multiple of three. This period (including possible leading zeros) was written on a strip of paper and cut into three equal-length parts $a$, $b$, $c$ (they may also have leading zeros). What could be the sum of the three periodic fractions: $0.(a)$, $0.(b)$, and $0.(c)$? [i]Proposed by A. Khrabrov[/i]

The product $a_1 \cdot a_2 \cdot ... \cdot a_{100}$ is written on the board , where $a_1$, $a_2$, $ ... $, $a_{100}$, are natural numbers. Let's consider $99$ expressions, each of which is obtained by replacing one of the multiplication signs with an addition sign. It is known that the values of exactly $32$ of these expressions are even. What is the largest number of even numbers among $a_1$, $a_2$, $ ... $, $a_{100}$ could it be?

Let $P$ be an arbitrary point on the incircle $k$ of triangle $ABC$ with center $I$, different from the points of tangency with its sides. The tangent to $k$ at $P$ intersects the lines $BC$, $AC$, $AB$ at points $A_0$, $B_0$, $C_0$, respectively. The lines through $A_0$, $B_0$, $C_0$, parallel to the bisectors of the angles $\angle BAC$, $\angle ABC$, $\angle ACB$, form a triangle $\Delta$. Prove that the line $PI$ is tangent to the circumcircle of $\Delta$.

Given a prime $p$, prove that the sum $\sum_{k=1}^{\lfloor \frac{q}{p} \rfloor}{k^{p-1}}$ is not divisible by $q$ for all but finitely many primes $q$.

Let $p$ be an odd prime.Positive integers $a,b,c,d$ are less than $p$,and satisfy $p|a^2+b^2$ and $p|c^2+d^2$.Prove that exactly one of $ac+bd$ and $ad+bc$ is divisible by $p$

There are $n=1681$ children, $a_1,a_2,...,a_{n}$ seated clockwise in a circle on the floor. The teacher walks behind the children in the clockwise direction with a box of $1000$ candies. She drops a candy behind the first child $a_1$. She then skips one child and drops a candy behind the third child, $a_3$. Now she skips two children and drops a candy behind the next child, $a_6$. She continues this way, at each stage skipping one child more than at the preceding stage before dropping a candy behind the next child. How many children will never receive a candy? Justify your answer.

Let $M$ be any point on the circumcircle of triangle $ABC$. Suppose the tangents from $M$ to the incircle meet $BC$ at two points $X_1$ and $X_2$. Prove that the circumcircle of triangle $MX_1X_2$ intersects the circumcircle of $ABC$ again at the tangency point of the $A$-mixtilinear incircle.