In how many ways can you choose several numbers from the numbers $1,2,3,..., 11$ so that among the selected numbers there are not three consecutive numbers?

Find, and write out explicitly, a permutation $\{p(1),p(2),\ldots,p(20)\}$ of $\{1,2,\ldots,20\}$ such that $k+p(k)$ is a power of $2$ for $k=1,2,\ldots,20$, and prove that only one such permutation exists.

Let $I$ and $I_a$ be the incenter and excenter (opposite vertex $A$) of a triangle $ABC$, respectively. Let $A'$ be the point on its circumcircle opposite to $A$, and $A_1$ be the foot of the altitude from $A$. Prove that $\angle IA_1I_a=\angle IA'I_a$. [i](Proposed by Pavel Kozhevnikov)[/i]

In a country, some pairs of cities are connected by one-way roads. It turns out that every city has at least two out-going and two in-coming roads assigned to it, and from every city one can travel to any other city by a sequence of roads. Prove that it is possible to delete a cyclic route so that it is still possible to travel from any city to any other city.

[b]p1.[/b] Suppose $a$, $b$ and $c$ are integers such that $c$ divides $a^n + b^n$ for all integers, $n \ge 1$. If the greatest common divisor of $a$ and $b$ is $7$, what is the largest possible value of $c$? [b]p2.[/b] Consider a sequence of points $\{P_1, P_2,...\}$ on a circle w with the property that $\overline{P_{i+1}P_{i+2}}$ is parallel to the tangent line through $P_i$ for each $i \ge 1$. If $P_5 = P_1$, what is the largest possible angle formed by $P_1P_3P_2$? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Find all functions $f: \mathbb{R} \to \mathbb{R}$ such that $f(x+y) = \max(f(x),y) + \min(f(y),x)$. [i]George Xing.[/i]

If a rectangle’s length is increased by $30\%$ and its width is decreased by $30\%$, by what percentage does its area change? State whether the area increases or decreases.

$99$ cards each have a label chosen from $1,2,\dots,99$, such that no (non-empty) subset of the cards has labels with total divisible by $100$. Show that the labels must all be equal.

A function $f:[0,\infty)\to\mathbb R\setminus\{0\}$ is called [i]slowly changing[/i] if for any $t>1$ the limit $\lim_{x\to\infty}\frac{f(tx)}{f(x)}$ exists and is equal to $1$. Is it true that every slowly changing function has for sufficiently large $x$ a constant sign (i.e., is it true that for every slowly changing $f$ there exists an $N$ such that for every $x,y>N$ we have $f(x)f(y)>0$?)

Positive integers a, b, c, d, and e satisfy the equations $$(a + 1)(3bc + 1) = d + 3e + 1$$ $$(b + 1)(3ca + 1) = 3d + e + 13$$ $$(c + 1)(3ab + 1) = 4(26-d- e) - 1$$ Find $d^2+e^2$.

Fix an integer $n \ge 3$ and let $a_0 = n$. Does there exist a permutation $a_1, a_2,..., a_{n-1}$ of the first $n-1$ positive integers such that $\Sigma_{j=0}^{k-1} a_j$ is divisible by $a_k$ for all indices $k < n$?

In $\vartriangle PQR$, $\angle Q+10 = \angle R$. Let $M$ be the midpoint of $\overline{QR}$. If $m\angle PMQ = 100^o$, then find the measure of $\angle Q$ in degrees.

Determine all pairs $(n,m)$ of positive integers for which there exists an infinite sequence $\{x_k\}$ of $0$'s and $1$'s with the properties that if $x_i=0$ then $x_{i+m}=1$ and if $x_i = 1$ then $x_{i+n} = 0.$

Let $d>1$ be integer and $0<r<\frac12$. Show that there exist finitely many (depending only on $d,r$) nonzero vectors in $\mathbb{R}^d$ such that if the distance of a straight line in $\mathbb{R}^d$ from the integer lattice $\mathbb{Z}^d$ is at least $r$, then this line is orthogonal to one of these finitely many vectors. (translated by L. Erdős)

In the figure, it is given that angle $ C \equal{} 90^{\circ}, \overline{AD} \equal{} \overline{DB}, DE \perp AB, \overline{AB} \equal{} 20$, and $ \overline{AC} \equal{} 12$. The area of quadrilateral $ ADEC$ is: [asy]unitsize(7); defaultpen(linewidth(.8pt)+fontsize(10pt)); pair A,B,C,D,E; A=(0,0); B=(20,0); C=(36/5,48/5); D=(10,0); E=(10,75/10); draw(A--B--C--cycle); draw(D--E); label("$A$",A,SW); label("$B$",B,SE); label("$C$",C,N); label("$D$",D,S); label("$E$",E,NE); draw(rightanglemark(B,D,E,30));[/asy]$ \textbf{(A)}\ 75 \qquad\textbf{(B)}\ 58\frac {1}{2} \qquad\textbf{(C)}\ 48 \qquad\textbf{(D)}\ 37\frac {1}{2} \qquad\textbf{(E)}\ \text{none of these}$

Set $A $ has 20 elements, and set $B $ has 15 elements. What is the smallest possible number of elements in $A \cup B $, the union of $A $ and $B $? $ \textbf{(A)}\ 5 \qquad\textbf{(B)}\ 15 \qquad\textbf{(C)}\ 20\qquad\textbf{(D)}\ 35\qquad\textbf{(E)}\ 300 $

Consider the integer $30x070y03$ where $x, y$ are unknown digits. Find all possible values of $x, y$ so that the given integer is a multiple of $37$.

Once upon a time, three boys visited a library for the first time. The first decided to visit the library every second day. The second decided to visit the library every third day. The third decided to visit the library every fourth day. The librarian noticed, that the library doesn't work on Wednesdays. The boys decided to visit library on Thursdays, if they have to do it on Wednesdays, but to restart the day counting in these cases. They strictly obeyed these rules. Some Monday later I met them all in that library. What day of week was when they visited a library for the first time?

Let $A = \{1,2,...,m + n\}$, where $m,n$ are positive integers, and let the function f : $A \to A$ be defined by: $f(m) = 1$, $f(m+n) = m+1$ and $f(i) = i+1$ for all the other $i$. (a) Prove that if $m$ and $n$ are odd, then there exists a function $g : A \to A$ such that $g(g(a)) = f(a)$ for all $a \in A$. (b) Prove that if $m$ is even, then there is a function $g : A\to A$ such that $g(g(a))=f(a)$ for all $a \in A$ is and only if $n = m$.

Let $ABC$ be an acute triangle with side lengths $AB = 7$, $ BC = 12$, $AC = 10$, and let $\omega$ be its incircle. If $\omega$ is touching $AB$, $AC$ at $F, E$, respectively, and if $EF$ intersects $BC$ at $X$, suppose that the ratio in which the angle bisector of $\angle BAC$ divides the segment connecting midpoint of $EX$ and $C$ is $\frac{a}{b}$ , where $a, b$ are relatively prime integers. Find $a + b$.

Calculate $\displaystyle \sum_{n=1}^\infty \ln \left(1+\frac{1}{n}\right) \ln\left( 1+\frac{1}{2n}\right)\ln\left( 1+\frac{1}{2n+1}\right)$.

a) One has a set of stones with weights $1, 2, \ldots, 20$ grams. Find all $k$ for which it is possible to place $k$ and the rest $20-k$ stones from the set respectively on the two pans of a balance so that equilibrium is achieved. b) One has a set of stones with weights $1, 2, \ldots, 51$ grams. Find all $k$ for which it is possible to place $k$ and the rest $51-k$ stones from the set respectively on the two pans of a balance so that equilibrium is achieved. c) One has a set of stones with weights $1, 2, \ldots, n$ grams ($n\in\mathbb{N}$). Find all $n$ and $k$ for which it is possible to place $k$ and the rest $n-k$ stones from the set respectively on the two pans of a balance so that equilibrium is achieved. [size=75] a) and b) were proposed at the festival, c) is a generalization[/size]

In triangle $ABC$, let $I, O, H$ be the incenter, circumcenter and orthocenter, respectively. Suppose that $AI = 11$ and $AO = AH = 13$. Find $OH$. [i]Proposed by Kevin You[/i]

Let $B = (-1, 0)$ and $C = (1, 0)$ be fixed points on the coordinate plane. A nonempty, bounded subset $S$ of the plane is said to be [i]nice[/i] if $\text{(i)}$ there is a point $T$ in $S$ such that for every point $Q$ in $S$, the segment $TQ$ lies entirely in $S$; and $\text{(ii)}$ for any triangle $P_1P_2P_3$, there exists a unique point $A$ in $S$ and a permutation $\sigma$ of the indices $\{1, 2, 3\}$ for which triangles $ABC$ and $P_{\sigma(1)}P_{\sigma(2)}P_{\sigma(3)}$ are similar. Prove that there exist two distinct nice subsets $S$ and $S'$ of the set $\{(x, y) : x \geq 0, y \geq 0\}$ such that if $A \in S$ and $A' \in S'$ are the unique choices of points in $\text{(ii)}$, then the product $BA \cdot BA'$ is a constant independent of the triangle $P_1P_2P_3$.

Let $a_1, a_2, ..., a_9$ be non-negative real numbers such that $a_1 = a_9 = 0$ and at least one of the remaining terms is different from $0$. a) Prove that for some $i$ $(i = 2, ..., 8$) ,holds that $a_{i-1} + a_{i+1} < 2a_i.$ b) Will the previous statement be true, if we change the number $2$ for $1.9$ in the inequality?