Let $n$ be a positive integer. A sequence of $n$ positive integers (not necessarily distinct) is called [b]full[/b] if it satisfies the following condition: for each positive integer $k\geq2$, if the number $k$ appears in the sequence then so does the number $k-1$, and moreover the first occurrence of $k-1$ comes before the last occurrence of $k$. For each $n$, how many full sequences are there ?

Let $ABCD$ be a cyclic quadrilateral. Prove that $AC > BD$ if and only if $(AD-BC)(AB- CD) > 0$. (V. Yasinsky)

In an acute non-isosceles triangle $ABC$, the inscribed circle touches side $BC$ at point $T, Q$ is the midpoint of altitude $AK$, $P$ is the orthocenter of the triangle formed by the bisectors of angles $B$ and $C$ and line $AK$. Prove that the points $P, Q$ and $T$ lie on the same line. (D. Prokopenko)

Let $ f:\mathbb{R}\longrightarrow\mathbb{R} $ be a bounded function in some neighbourhood of $ 0, $ such that there are three real numbers $ a>0, b>1, c $ with the property that $$ f(ax)=bf(x)+c,\quad\forall x\in\mathbb{R} . $$ Show that $ f $ is continuous at $ 0 $ if and only if $ c=0. $

Let $n$ be a positive integer and $A$ a set containing $8n + 1$ positive integers co-prime with $6$ and less than $30n$. Prove that there exist $a, b \in A$ two different numbers such that $a$ divides $b$.

A function $f:\mathbb{R} \to \mathbb{R}$ has the following property: $$\text{For every } x,y \in \mathbb{R} \text{ such that }(f(x)+y)(f(y)+x) > 0, \text{ we have } f(x)+y = f(y)+x.$$ Prove that $f(x)+y \leq f(y)+x$ whenever $x>y$.

Let $ n \geq 3$ be an odd integer. Determine the maximum value of \[ \sqrt{|x_{1}\minus{}x_{2}|}\plus{}\sqrt{|x_{2}\minus{}x_{3}|}\plus{}\ldots\plus{}\sqrt{|x_{n\minus{}1}\minus{}x_{n}|}\plus{}\sqrt{|x_{n}\minus{}x_{1}|},\] where $ x_{i}$ are positive real numbers from the interval $ [0,1]$.

Prove that for each $k$ points in the plane, no three collinear and having integral distances from each other. If we have an infinite set of points with integral distances from each other, then all points are collinear. [i](Anonymous314)[/i] PS. wording needs to be fixed , [url=http://www.mathcenter.net/forum/showthread.php?t=7288]source[/url]

Let $$ A = \log (1) + \log 2 + \log(3) + \cdots + \log(2023) $$ and $$ B = \log(1/1) + \log(1/2) + \log(1/3) + \cdots + \log(1/2023). $$ What is the value of $A + B\ ?$ $($logs are logs base $10)$ $$ \mathrm a. ~ 0\qquad \mathrm b.~1\qquad \mathrm c. ~{-\log(2023!)} \qquad \mathrm d. ~\log(2023!) \qquad \mathrm e. ~{-2023} $$

Find the largest integer $N \in \{1, 2, \ldots , 2019 \}$ such that there exists a polynomial $P(x)$ with integer coefficients satisfying the following property: for each positive integer $k$, $P^k(0)$ is divisible by $2020$ if and only if $k$ is divisible by $N$. Here $P^k$ means $P$ applied $k$ times, so $P^1(0)=P(0), P^2(0)=P(P(0)),$ etc.

Alice has 24 apples. In how many ways can she share them with Becky and Chris so that each of the people has at least 2 apples? $\textbf{(A) }105\qquad\textbf{(B) }114\qquad\textbf{(C) }190\qquad\textbf{(D) }210\qquad\textbf{(E) }380$

Two projectiles are launched from a $35$ meter ledge as shown in the diagram. One is launched from a $37$ degree angle above the horizontal and the other is launched from $37$ degrees below the horizontal. Both of the launches are given the same initial speed of $v_\text{0} = \text{50 m/s}$. [asy] size(300); import graph; draw((-8,0)--(0,0)--(0,-11)--(30,-11)); draw((0,-11)--(-4.5,-11),dashdotted); draw((0,0)--(12,0),dashdotted); label(scale(0.75)*"35 m",(0,-5.5),5*W); draw((-4,-4.5)--(-4,-0.5),EndArrow(size=5)); draw((-4,-6)--(-4,-10.5),EndArrow(size=5)); // Projectiles real f(real x){ return -11x^2/49; } draw(graph(f,0,7),dashed+linewidth(1.5)); real g(real x){ return -6x^2/145+119x/145; } draw(graph(g,0,29),dashed+linewidth(1.5)); // Labels label(scale(0.75)*"Projectile 1",(20,2),E); label(scale(0.75)*"Projectile 2",(6,-7),E); [/asy] The difference in the times of flight for these two projectiles, $t_1-t_2$, is closest to (A) $\text{3 s}$ (B) $\text{5 s}$ (C) $\text{6 s}$ (D) $\text{8 s}$ (E) $\text{10 s}$

Given are $4004$ distinct points, which lie in the interior of a convex polygon of area $1$. Prove that there exists a convex polygon of area $\frac{1}{2003}$, included in the given polygon, such that it does not contain any of the given points in its interior.

What is the value of the improper integral $ \int_0 ^ {\pi} \log (\sin (x)) dx$?

In a given sequence $\{S_1,S_2,...,S_k\}$, for terms $n\ge3$, $S_n=\sum_{i=1}^{n-1} i\cdot S_{n-i}$. For example, if the first two elements are 2 and 3, respectively, the third entry would be $1\cdot3+2\cdot2=7$, and the fourth would be $1\cdot7+2\cdot3+3\cdot2=19$, and so on. Given that a sequence of integers having this form starts with 2, and the 7th element is 68, what is the second element?

Let $ p_i $ denote the $ i $-th prime number. Let $ n = \lfloor\alpha^{2015}\rfloor $, where $ \alpha $ is a positive real number such that $ 2 < \alpha < 2.7 $. Prove that $$ \displaystyle\sum_{2 \le p_i \le p_j \le n}\frac{1}{p_ip_j} < 2017 $$

For a multiple of $ kb$ of $ b$ let $ a \% kb$ be the greatest number such that $ a \% kb \equal{} a \bmod b$ which is smaller than $ kb$ and not greater than $ a$ itself. Let $ n \in \mathbb{Z}^ \plus{} .$ Determine all integer pairs $ (a,b)$ with: \[ a\%b \plus{} a\%2b \plus{} a\%3b \plus{} \ldots \plus{} a\%nb \equal{} a \plus{} b \]

A chess tournament had 10 participants. Each round, the participants split into pairs, and each pair played a game. In total, each participant played with every other participant exactly once, and in at least half of the games both the players were from the same town. Prove that during each round there was a game played by two participants from the same town. [i](Boris Frenkin)[/i]

There are forty weights: $1, 2, \cdots , 40$ grams. Ten weights with even masses were put on the left pan of a balance. Ten weights with odd masses were put on the right pan of the balance. The left and the right pans are balanced. Prove that one pan contains two weights whose masses differ by exactly $20$ grams. [i](4 points)[/i]

Three circles pass through a point $P$, and the second points of their intersection $A, B, C$ lie on a straight line. Let $A_1 B_1, C_1$ be the second meets of lines $AP, BP, CP$ with the corresponding circles. Let $C_2$ be the intersections of lines $AB_1$ and $BA_1$. Let $A_2, B_2$ be defined similarly. Prove that the triangles $A_1B_1C_1$ and $A_2B_2C_2$ are equal,

Let $ABC$ be an acute triangle with altitude $AD$ ($D \in BC$). The line through $C$ parallel to $AB$ meets the perpendicular bisector of $AD$ at $G$. Show that $AC = BC$ if and only if $\angle AGC = 90^{\circ}$.

Let $ f(x) \equal{} c_m x^m \plus{} c_{m\minus{}1} x^{m\minus{}1} \plus{}...\plus{} c_1 x \plus{} c_0$, where each $ c_i$ is a non-zero integer. Define a sequence $ \{ a_n \}$ by $ a_1 \equal{} 0$ and $ a_{n\plus{}1} \equal{} f(a_n)$ for all positive integers $ n$. (a) Let $ i$ and $ j$ be positive integers with $ i<j$. Show that $ a_{j\plus{}1} \minus{} a_j$ is a multiple of $ a_{i\plus{}1} \minus{} a_i$. (b) Show that $ a_{2008} \neq 0$

There are four points $A, B, C, D$ on a circle. Circles are drawn through each pair of neighboring points. Denote the intersection points of neighboring circles by $A_1, B_1, C_1, D_1$. (Some of these points may coincide with previously given ones.) Prove that points $A_1, B_1, C_1, D_1$ lie on one circle.

In quadrilateral $ABCD$, diagonals $AC$ and $BD$ intersect at $E$. If $AB=BE=5$, $EC=CD=7$, and $BC=11$, compute $AE$.

Let $a_1,a_2,\ldots,a_n,\ldots$ be any permutation of all positive integers. Prove that there exist infinitely many positive integers $i$ such that $\gcd(a_i,a_{i+1})\leq \frac{3}{4} i$.