Prove that every bijective function $ f: \mathbb{Z}\rightarrow\mathbb{Z}$ can be written in the way $ f\equal{}u\plus{}v$ where $ u,v: \mathbb{Z}\rightarrow\mathbb{Z}$ are bijective functions.

Four runners are preparing to begin a $1$-mile race from the same starting line. When the race starts, runners Alice, Bob, and Charlie all travel at constant speeds of $8$ mph, $4$ mph, and $2$ mph, respectively. The fourth runner, Dave, is initially half as slow as Charlie, but Dave has a superpower where he suddenly doubles his running speed every time a runner finishes the race. How many hours does it take for Dave to finish the race?

Prove that the affixes of three pairwise distinct complex numbers $ z_0,z_1,z_2 $ represent an isosceles triangle with right angle at $ z_0 $ if and only if $ \left( z_1-z_0 \right)^2 =-\left( z_2-z_0 \right)^2. $

There're $n$ students whose names are different from each other. Everyone has $n-1$ envelopes initially with the others' name and address written on them respectively. Everyone also has at least one greeting card with her name signed on it. Everyday precisely a student encloses a greeting card (which can be the one received before) with an envelope (the name on the card and the name on envelope cannot be the same) and post it to the appointed student by a same day delivery. Prove that when no one can post the greeting cards in this way any more: (i) Everyone still has at least one card; (ii) If there exist $k$ students $p_1, p_2, \cdots, p_k$ so that $p_i$ never post a card to $p_{i+1}$, where $i = 1,2, \cdots, k$ and $p_{k+1} = p_1$, then these $k$ students have prepared the same number of greeting cards initially.

Find all positive integer solutions for the equation $(3x+1)(3y+1)(3z+1)=34xyz$ Thx

A watch loses $2\frac{1}{2}$ minutes per day. It is set right at $1$ P.M. on March 15. Let $n$ be the positive correction, in minutes, to be added to the time shown by the watch at a given time. When the watch shows $9$ A.M. on March 21, $n$ equals: $\textbf{(A) }14\frac{14}{23}\qquad\textbf{(B) }14\frac{1}{14}\qquad\textbf{(C) }13\frac{101}{115}\qquad\textbf{(D) }13\frac{83}{115}\qquad \textbf{(E) }13\frac{13}{23}$

Given triangle $ABC$. Let $BPCQ$ be a parallelogram ($P$ is not on $BC$). Let $U$ be the intersection of $CA$ and $BP$, $V$ be the intersection of $AB$ and $CP$, $X$ be the intersection of $CA$ and the circumcircle of triangle $ABQ$ distinct from $A$, and $Y$ be the intersection of $AB$ and the circumcircle of triangle $ACQ$ distinct from $A$. Prove that $\overline{BU} = \overline{CV}$ if and only if the lines $AQ$, $BX$, and $CY$ are concurrent. [i]Proposed by Li4.[/i]

$4^4 \cdot 9^4 \cdot 4^9 \cdot 9^9=$ $ \textbf{(A)}\ 13^{13} \qquad\textbf{(B)}\ 13^{36} \qquad\textbf{(C)}\ 36^{13} \qquad\textbf{(D)}\ 36^{36} \qquad\textbf{(E)}\ 1296^{26} $

Let $A_M$ be the midpoint of side $BC$ of an acute-angled $\Delta ABC$, and $A_H$ be the foot of the altitude to this side. Points $B_M, B_H, C_M, C_H$ are defined similarly. Prove that one of the ratios $A_MA_H : A_HA, B_MB_H : B_HB, C_MC_H : C_HC$ is equal to the sum of two remaining ratios

Let $ n$ be a positive integer, $ X \equal{} \{1, 2, \ldots , n\},$ and $ k$ a positive integer such that $ \frac{n}{2} \leq k \leq n.$ Determine, with proof, the number of all functions $ f : X \mapsto X$ that satisfy the following conditions: [b](i)[/b] $ f^2 \equal{} f;$ [b](ii)[/b] the number of elements in the image of $ f$ is $ k;$ [b](iii)[/b] for each $ y$ in the image of $ f,$ the number of all points $ x \in X$ such that $ f(x)\equal{}y$ is at most $ 2.$

Let $f(n)$ be the smallest number of 1s needed to represent the positive integer $n$ using only 1s, $+$ signs, $\times$ signs and brackets $(,)$. For example, you could represent 80 with 13 1s as follows: $(1+1+1+1+1)(1+1+1+1)(1+1+1+1)$. Show that $3 \log(n) \leq \log(3)f(n) \leq 5 \log(n)$ for $n > 1$.

Let $n$ be a positive integer, and let $p$ be a prime, such that $n>p$. Prove that : \[ \displaystyle \binom np \equiv \left\lfloor\frac{n}{p}\right\rfloor \ \pmod p. \]

Ten coins of $1$ cm radius are placed around a circle as indicated in the figure. Each coin is tangent to the circle and its two neighboring coins. Prove that the sum of the areas of the ten coins is twice the area of the circle. [img]https://cdn.artofproblemsolving.com/attachments/5/e/edf7a7d39d749748f4ae818853cb3f8b2b35b5.gif[/img]

Let $f_1 = (a_1, a_2, \dots , a_n) , n > 2$, be a sequence of integers. From $f_1$ one constructs a sequence $f_k$ of sequences as follows: if $f_k = (c_1, c_2, \dots, cn)$, then $f_{k+1} = (c_{i_{1}}, c_{i_{2}}, c_{i_{3}} + 1, c_{i_{4}} + 1, . . . , c_{i_{n}} + 1)$, where $(c_{i_{1}}, c_{i_{2}},\dots , c_{i_{n}})$ is a permutation of $(c_1, c_2, \dots, c_n)$. Give a necessary and sufficient condition for $f_1$ under which it is possible for $f_k$ to be a constant sequence $(b_1, b_2,\dots , b_n), b_1 = b_2 =\cdots = b_n$, for some $k.$

$\{a_{n}\}_{n\geq 0}$ and $\{b_{n}\}_{n\geq 0}$ are two sequences of positive integers that $a_{i},b_{i}\in \{0,1,2,\cdots,9\}$. There is an integer number $M$ such that $a_{n},b_{n}\neq 0$ for all $n\geq M$ and for each $n\geq 0$ $$(\overline{a_{n}\cdots a_{1}a_{0}})^{2}+999 \mid(\overline{b_{n}\cdots b_{1}b_{0}})^{2}+999 $$ prove that $a_{n}=b_{n}$ for $n\geq 0$.\\ (Note that $(\overline{x_nx_{n-1}\dots x_0}) = 10^n\times x_n + \dots + 10\times x_1 + x_0$.) [i]Proposed by Yahya Motevassel[/i]

Triangle $ABC$ is an equilateral triangle with side length $1$. Let $X_0,X_1,... $ be an infinite sequence of points such that the following conditions hold: $\bullet$ $X_0$ is the center of $ABC$ $\bullet$ For all $i \ge 0$, $X_{2i+1}$ lies on segment $AB$ and $X_{2i+2}$ lies on segment $AC$. $\bullet$ For all $i \ge 0$, $\angle X_iX_{i+1}X_{i+2} = 90^o.$ $\bullet$ For all $i \ge 1$, $X_{i+2}$ lies in triangle $AX_iX_{i+1}$. Find the maximum possible value of $\sum^{\infty}_{i=0}|X_iX_{i+1}|$, where $|PQ|$ is the length of line segment $PQ$.

Two positive real numbers $\alpha, \beta$ satisfies that for any positive integers $k_1,k_2$, it holds that $\lfloor k_1 \alpha \rfloor \neq \lfloor k_2 \beta \rfloor$, where $\lfloor x \rfloor$ denotes the largest integer less than or equal to $x$. Prove that there exist positive integers $m_1,m_2$ such that $\frac{m_1}{\alpha}+\frac{m_2}{\beta}=1$.

If $P_1P_2P_3P_4P_5P_6$ is a regular hexagon whose apothem (distance from the center to midpoint of a side) is $2$, and $Q_i$ is the midpoint of side $P_iP_{i+1}$ for $i=1,2,3,4$, then the area of quadrilateral $Q_1Q_2Q_3Q_4$ is $\textbf{(A) }6\qquad\textbf{(B) }2\sqrt{6}\qquad\textbf{(C) }\frac{8\sqrt{3}}{3}\qquad\textbf{(D) }3\sqrt{3}\qquad\textbf{(E) }4\sqrt{3}$

Acute triangle $ABC$ has $\angle BAC < 45^\circ$. Point $D$ lies in the interior of triangle $ABC$ so that $BD = CD$ and $\angle BDC = 4 \angle BAC$. Point $E$ is the reflection of $C$ across line $AB$, and point $F$ is the reflection of $B$ across line $AC$. Prove that lines $AD$ and $EF$ are perpendicular.

Let $0<a<b<1$ be reals numbers and \[g(x)=\left\{\begin{array}{cc}x+1-a,&\mbox{ if } 0<x<b\\b-a, & \mbox{ if } x=a \\x-a, & \mbox{ if } a<x<b\\1-a ,&\mbox{ if } x=b \\ x-a ,&\mbox{ if } b<x<1 \end{array}\right.\] Give that there exist $n+1$ reals numbers $0<x_0<x_1<...<x_n<1$, for which $g^{[n]}(x_i)=x_i \ (0 \leq i \leq n)$. Prove that there exists a positive integer $N$, such that $g^{[N]}(x)=x$ for all $0<x<1$. ($g^{[n]}(x)= \underbrace{g(g(....(g(x))....))}_{\text{n times}}$) Official solution [url=http://www.artofproblemsolving.com/Forum/viewtopic.php?f=125&t=365714&p=2011659#p2011659]here[/url]

Let $O$ be the circumcenter of an acute triangle $ABC$. Line $OA$ intersects the altitudes of $ABC$ through $B$ and $C$ at $P$ and $Q$, respectively. The altitudes meet at $H$. Prove that the circumcenter of triangle $PQH$ lies on a median of triangle $ABC$.

A regular tetrahedron with an edge of $30$ cm rests on one of its faces. Assuming it is hollow, $2$ liters of water are poured into it. Find the height of the ''upper'' liquid and the area of the ''free'' surface of the water.

The graph $G$ with 2014 vertices doesn’t contain any 3-cliques. If the set of the degrees of the vertices of $G$ is $\{1,2,...,k\}$, find the greatest possible value of $k$.

Let a$,b,c$ be sidelengths of a triangle and $m_a,m_b,m_c$ the medians at the corresponding sides. Prove that $$m_a\left(\frac{b}{a}-1\right)\left(\frac{c}{a}-1\right)+ m_b\left(\frac{a}{b}-1\right)\left(\frac{c}{b}-1\right) +m_c\left(\frac{a}{c}-1\right)\left(\frac{b}{c}-1\right)\geq 0.$$ (FYROM)

Let $f$ be a monic cubic polynomial satisfying $f(x) + f(-x) = 0$ for all real numbers $x$. For all real numbers $y$, define $g(y)$ to be the number of distinct real solutions $x$ to the equation $f(f(x)) = y$. Suppose that the set of possible values of $g(y)$ over all real numbers $y$ is exactly $\{1, 5, 9\}$. Compute the sum of all possible values of $f(10)$.