Let's define [i]almost mean[/i] of numbers $a_1, a_2, \ldots, a_n$ as $\frac{a_1 + a_2 + \ldots + a_n}{n+1}$. Oleksiy has positive real numbers $b_1, b_2, \ldots, b_{2023}$, not necessarily distinct. For each pair $(i, j)$ with $1 \leq i, j \leq 2023$, Oleksiy wrote on a board [i]almost mean[/i] of numbers $b_i, b_{i+1}, \ldots, b_j$. Prove that there are at least $45$ distinct numbers on the board. [i]Proposed by Anton Trygub[/i]

Find the differentiable functions $ f:\mathbb{R}\longrightarrow (-\infty ,1) $ with the property $ f(1)=-1 $ and $$ f(x+y)=f(x)+f(y)-f(x)f(y) , $$ for any reals $ x,y. $ [i]Vasile Pop[/i]

Three circles of radius $ 1$ are externally tangent to each other and internally tangent to a larger circle. What is the radius of the large circle? [asy]unitsize(0.8cm); defaultpen(linewidth(.8pt)+fontsize(8pt)); real r = 1 + (2/3)*(sqrt(3)); pair A = dir(47.5)*(r - 1); pair B = dir(167.5)*(r - 1); pair C = dir(-72.5)*(r - 1); draw(Circle(A,1)); draw(Circle(B,1)); draw(Circle(C,1)); draw(Circle(origin,r));[/asy] $ \textbf{(A)}\ \frac{2 \plus{} \sqrt{6}}{3}\qquad \textbf{(B)}\ 2\qquad \textbf{(C)}\ \frac{2 \plus{} 3\sqrt{2}}{3}\qquad \textbf{(D)}\ \frac{3 \plus{} 2\sqrt{3}}{3}\qquad \textbf{(E)}\ \frac{3 \plus{} \sqrt{3}}{2}$

We are given an infinite set of rectangles in the plane, each with vertices of the form $(0, 0)$, $(0,m)$, $(n, 0)$ and $ (n,m)$, where $m$ and $n$ are positive integers. Prove that there exist two rectangles in the set such that one contains the other.

Find $\sum_{k=1}^\infty\frac{k^2-2}{(k+2)!}$.

Given an equilateral triangle with sidelength $k$ cm. With lines parallel to it's sides, we split it into $k^2$ small equilateral triangles with sidelength $1$ cm. This way, a triangular grid is created. In every small triangle of sidelength $1$ cm, we place exactly one integer from $1$ to $k^2$ (included), such that there are no such triangles having the same numbers. With vertices the points of the grid, regular hexagons are defined of sidelengths $1$ cm. We shall name as [i]value [/i] of the hexagon, the sum of the numbers that lie on the $6$ small equilateral triangles that the hexagon consists of . Find (in terms of the integer $k>4$) the maximum and the minimum value of the sum of the values of all hexagons .

[list=a][*] Find infinitely many pairs of integers $a$ and $b$ with $1<a<b$, so that $ab$ exactly divides $a^{2}+b^{2}-1$. [*] With $a$ and $b$ as above, what are the possible values of \[\frac{a^{2}+b^{2}-1}{ab}?\] [/list]

Cyclic quadrilateral $ABCD$ has positive integer side lengths $AB$, $BC$, $CA$, $AD$. It is known that $AD=2005$, $\angle{ABC}=\angle{ADC} = 90^o$, and $\max \{ AB,BC,CD \} < 2005$. Determine the maximum and minimum possible values for the perimeter of $ABCD$.

[b]p1.[/b] What is $6\times 7 + 4 \times 7 + 6\times 3 + 4\times 3$? [b]p2.[/b] How many integers $n$ have exactly $\sqrt{n}$ factors? [b]p3.[/b] A triangle has distinct angles $3x+10$, $2x+20$, and $x+30$. What is the value of $x$? [b]p4.[/b] If $4$ people of the Math Club are randomly chosen to be captains, and Henry is one of the $30$ people eligible to be chosen, what is the probability that he is not chosen to be captain? [b]p5.[/b] $a, b, c, d$ is an arithmetic sequence with difference $x$ such that $a, c, d$ is a geometric sequence. If $b$ is $12$, what is $x$? (Note: the difference of an aritmetic sequence can be positive or negative, but not $0$) [b]p6.[/b] What is the smallest positive integer that contains only $0$s and $5$s that is a multiple of $24$. [b]p7.[/b] If $ABC$ is a triangle with side lengths $13$, $14$, and $15$, what is the area of the triangle made by connecting the points at the midpoints of its sides? [b]p8.[/b] How many ways are there to order the numbers $1,2,3,4,5,6,7,8$ such that $1$ and $8$ are not adjacent? [b]p9.[/b] Find all ordered triples of nonnegative integers $(x, y, z)$ such that $x + y + z = xyz$. [b]p10.[/b] Noah inscribes equilateral triangle $ABC$ with area $\sqrt3$ in a cricle. If $BR$ is a diameter of the circle, then what is the arc length of Noah's $ARC$? [b]p11.[/b] Today, $4/12/14$, is a palindromic date, because the number without slashes $41214$ is a palindrome. What is the last palindromic date before the year $3000$? [b]p12.[/b] Every other vertex of a regular hexagon is connected to form an equilateral triangle. What is the ratio of the area of the triangle to that of the hexagon? [b]p13.[/b] How many ways are there to pick four cards from a deck, none of which are the same suit or number as another, if order is not important? [b]p14.[/b] Find all functions $f$ from $R \to R$ such that $f(x + y) + f(x - y) = x^2 + y^2$. [b]p15.[/b] What are the last four digits of $1(1!) + 2(2!) + 3(3!) + ... + 2013(2013!)$/ [b]p16.[/b] In how many distinct ways can a regular octagon be divided up into $6$ non-overlapping triangles? [b]p17.[/b] Find the sum of the solutions to the equation $\frac{1}{x-3} + \frac{1}{x-5} + \frac{1}{x-7} + \frac{1}{x-9} = 2014$ . [b]p18.[/b] How many integers $n$ have the property that $(n+1)(n+2)(n+3)(n+4)$ is a perfect square of an integer? [b]p19.[/b] A quadrilateral is inscribed in a unit circle, and another one is circumscribed. What is the minimum possible area in between the two quadrilaterals? [b]p20.[/b] In blindfolded solitary tic-tac-toe, a player starts with a blank $3$-by-$3$ tic-tac-toe board. On each turn, he randomly places an "$X$" in one of the open spaces on the board. The game ends when the player gets $3$ $X$s in a row, in a column, or in a diagonal as per normal tic-tac-toe rules. (Note that only $X$s are used, not $O$s). What fraction of games will run the maximum $7$ amount of moves? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Prove that for every pair of positive integers $k$ and $n$, there exists integer $x_1$, $x_2$,$...$, $x_k$ with $0 \le x_j \le 2^{k-1}\cdot \sqrt[k]{n}$ for $j = 1$, $2$, $...$, $k$, and such that $$x_1 + x^2_2+ x^3_3+...+ x^k_k= n.$$

Two circles $\omega_1$ and $\omega_2$, centered at $O_1$ and $O_2$, respectively, meet at points $A$ and $B$. A line through $B$ intersects $\omega_1$ again at $C$ and $\omega_2$ again at $D$. The tangents to $\omega_1$ and $\omega_2$ at $C$ and $D$, respectively, meet at $E$, and the line $AE$ intersects the circle $\omega$ through $AO_1O_2$ at $F$. Prove that the length of segment $EF$ is equal to the diameter of $\omega$.

A finite sequence of decimal digits from $\{0,1,\cdots, 9\}$ is said to be [i]common[/i] if for each sufficiently large positive integer $n$, there exists a positive integer $m$ such that the expansion of $n$ in base $m$ ends with this sequence of digits. For example, $0$ is common because for any large $n$, the expansion of $n$ in base $n$ is $10$, whereas $00$ is not common because for any squarefree $n$, the expansion of $n$ in any base cannot end with $00$. Determine all common sequences. [i]Proposed by Wong Jer Ren[/i]

Piglet added together three consecutive whole numbers, then the next three numbers, and multiplied one sum by the other. Could the product be equal to $111,111,111$?

Let $H$ be a regular hexagon with side length one. Peter picks a point $P$ uniformly and at random within $H$, then draws the largest circle with center $P$ that is contained in $H$. What is this probability that the radius of this circle is less than $1/2$ ?

An $n \times n$ table is called good if one can paint its cells with three colors so that, for any two different rows and two different columns, the four cells at their intersections are not all of the same color. a)Show, that exists good $9 \times 9$ good table. b)Prove, that fif $n \times n$ table is good, then $n<11$

Triangle $T_1$ has sides of length $a_1$, $b_1$, and $c_1$; its area is $K_1$. Triangle $T_2$ has sides of length $a_2$, $b_2$, and $c_2$; its area is $K_2$. Triangle $T_3$ has sides of length $a_1 + a_2$, $b_1 + b_2$, and $c_1 + c_2$; its area is $K_3$. (a) Prove that $K_1^2 + K_2^2 < K_3^2$. (b) Prove that $\sqrt{K_1} + \sqrt{K_2} \le \sqrt{K_3} \,$.

The distance between the city and the house of Borya is 2km. Once Borya went from the city to his house with speed 4km/h. Simultaneously with that a dog Sharik started running out of house in the direction to city, and whenever Sharik meets Borya or the house, it starts running back (so the dog runs between Borya and the house), and when the dog runs to the house, its speed is 8km/h, and when it runs from the house, its speed is 12km/h. What distance will Sharik run until Borya comes to the house? [i]Yauheni Barabanau[/i]

A parallelogram $ABCD$ is given. The perpendicular from $C$ to $CD$ meets the perpendicular from $A$ to $BD$ at point $F$, and the perpendicular from $B$ to $AB$ meets the perpendicular bisector to $AC$ at point $E$. Find the ratio in which side $BC$ divides segment $EF$.

The first four terms of an arithmetic sequence are $a, x, b, 2x$. The ratio of $a$ to $b$ is $ \textbf{(A)}\ \frac{1}{4} \qquad\textbf{(B)}\ \frac{1}{3} \qquad\textbf{(C)}\ \frac{1}{2} \qquad\textbf{(D)}\ \frac{2}{3} \qquad\textbf{(E)}\ 2 $

A chord $CD$ of a circle with center $O$ is perpendicular to a diameter $AB$. A chord $AE$ bisects the radius $OC$. Show that the line $DE$ bisects the chord $BC$ [i]V. Gordon[/i]

Find the remainder when \[9 \times 99 \times 999 \times \cdots \times \underbrace{99\cdots9}_{\text{999 9's}}\] is divided by $ 1000$.

Let $ a\equal{}\sqrt[1992]{1992}$. Which number is greater \[ \underbrace{a^{a^{a^{\ldots^{a}}}}}_{1992}\quad\text{or}\quad 1992? \]

Determine the minimum value of $\dfrac{m^m}{1\cdot 3\cdot 5\cdot \ldots \cdot(2m-1)}$ for positive integers $m$.

$\xi, \xi'$ are iid random variables. let F have the distribution function $\xi+\xi'$, and G have the uniform distribution over the interval [-1,1]. Prove that $\max | F ( x ) - G ( x ) | \geq 10^{-1994}$ .

Brownian motion (for example, pollen grains in water randomly pushed by collisions from water molecules) simplified to one dimension and beginning at the origin has several interesting properties. If $B(t)$ denotes the position of the particle at time $t$, the average of $B(t)$ is $x=0$, but the averate of $B(t)^{2}$ is $t$, and these properties of course still hold if we move the space and time origins ($x=0$ and $t=0$) to a later position and time of the particle (past and future are independent). What is the average of the product $B(t)B(s)$?