Nine distinct positive integers summing to $74$ are put into a $3 \times 3$ grid. Simultaneously, the number in each cell is replaced with the sum of the numbers in its adjacent cells. (Two cells are adjacent if they share an edge.) After this, exactly four of the numbers in the grid are $23$. Determine, with proof, all possible numbers that could have been originally in the center of the grid.

For a given positive integer $ k$ denote the square of the sum of its digits by $ f_1(k)$ and let $ f_{n\plus{}1}(k) \equal{} f_1(f_n(k)).$ Determine the value of $ f_{1991}(2^{1990}).$

There are $n>1$ cities in Jansonland, with two-way roads joining certain pairs of cities. Janson will send a few robots one-by-one to build more roads. The robots operate as such: 1. Janson first selects an integer $k$ and a list of cities $a_0, a_1, \dots, a_k$ (cities can repeat). 2. The robot begins at $a_0$ and goes to $a_1$, then $a_2$, and so on until $a_k$. 3. When the robot goes from $a_i$ to $a_{i+1}$, if there is no road then the robot builds a road, but if there is a road then the robot destroys the road. In terms of $n$, determine the smallest constant $k$ such that Janson can always achieve a configuration such that every pair of cities has a road connecting them using no more than $k$ robots. [i](Proposed by Ho Janson)[/i]

Let $a,b,c,d$ be real positive numbers such that $a^2+b^2+c^2+d^2 = 1$. Prove that $(1-a)(1-b)(1-c)(1-d) \geq abcd$.

$a, b, c$ are real positive numbers for which $a+b+c=3$. Prove that $a^{12}+b^{12}+c^{12}+8(ab+bc+ca) \geq 27$

A one-player game is played as follows: There is a bowl at each integer on the $Ox$-axis. All the bowls are initially empty, except for that at the origin, which contains $n \geq 2$ stones. A move is either (A) to remove two stones from a bowl and place one in each of the two adjacent bowls, or (B) to remove a stone from each of two adjacent bowls and to add one stone to the bowl immediately to their left. Show that only a finite number of moves can be made and that the final position (when no more moves are possible) is independent of the moves made (for a given $n$).

Show that the equation $x^4 +y^4 +4z^4 =1$ has infinitely many rational solutions.

Determine all prime numbers $p, q$ and $r$ with $p + q^2 = r^4$. [i](Karl Czakler)[/i]

To any triangle with side lengths $a,b,c$ and the corresponding angles $\alpha, \beta, \gamma$ (measured in radians), the 6-tuple $(a,b,c,\alpha, \beta, \gamma)$ is assigned. Find the minimum possible number $n$ of distinct terms in the 6-tuple assigned to a scalene triangle.

A palindrome in base $b$ is a positive integer whose base-$b$ digits read the same backwards and forwards; for example, $2002$ is a $4$-digit palindrome in base $10$. Note that $200$ is not a palindrome in base $10$, but it is a $3$-digit palindrome: $242$ in base $9$, and $404$ in base $7$. Prove that there is an integer which is a $3$-digit palindrome in base $b$ for at least $2002$ different values of $b$.

In quadrilateral $ABCD$, $AB \parallel CD$ and $BC \perp AB$. Lines $AC$ and $BD$ intersect at $E$. If $AB = 20$, $BC = 2016$, and $CD = 16$, find the area of $\triangle BCE$. [i]Proposed by Harrison Wang[/i]

As shown in the figure, given $\vartriangle ABC$ with $\angle B$, $\angle C$ acute angles, $AD \perp BC$, $DE \perp AC$, $M$ midpoint of $DE$, $AM \perp BE$. Prove that $\vartriangle ABC$ is isosceles. [img]https://cdn.artofproblemsolving.com/attachments/a/8/f553c33557979f6f7b799935c3bde743edcc3c.png[/img]

Consider a triangle $\vartriangle ABC$ with $AB = 5$ and $BC = 4$. Let $G$ be the centroid of the triangle, and let $P$ lie on line $AG$ such that $AG = GP$ and $P\ne A$. Suppose that $P$ lies on the circumcircle of $\vartriangle ABC$. Compute $CA$.

Find all functions $f: \mathbb{Z}\rightarrow\mathbb{Z}$ such that for all $x,y \in \mathbb{Z}$: \[f(x-y+f(y))=f(x)+f(y).\]

There is a lamp on each cell of a $2017 \times 2017$ board. Each lamp is either on or off. A lamp is called [i]bad[/i] if it has an even number of neighbours that are on. What is the smallest possible number of bad lamps on such a board? (Two lamps are neighbours if their respective cells share a side.)

Find the sum of all possible $n$ such that $n$ is a positive integer and there exist $a, b, c$ real numbers such that for every integer $m$, the quantity $\frac{2013m^3 + am^2 + bm + c}{n}$ is an integer.

Solve the system of equations $$\begin{cases} \sin^3 x+\sin^4 y=1 \\ \cos^4 x+\cos^5 y =1\end{cases}$$

Let a board game has $10$ cards: $3$ [b]skull[/b] cards, $5$ [b]coin[/b] cards and $2$ [b]blank[/b] cards. We put these $10$ cards downward and shuffle them and take cards one by one from the top. Once $3$ [b]skull[/b] cards or [b]coin[/b] cards appears we stop. What is the possibility of it stops because there appears $3$ [b]skull[/b] cards?

$M$ is any point on the side $AB$ of the triangle $ABC$. $r,r_1,r_2$ are the radii of the circles inscribed in $ABC,AMC,BMC$. $q$ is the radius of the circle on the opposite side of $AB$ to $C$, touching the three sides of $AB$ and the extensions of $CA$ and $CB$. Similarly, $q_1$ and $q_2$. Prove that $r_1r_2q=rq_1q_2$.

Find all solutions of the following system of $n$ equations in $n$ variables: \[\begin{array}{c}\ x_1|x_1| - (x_1 - a)|x_1 - a| = x_2|x_2|,x_2|x_2| - (x_2 - a)|x_2 - a| = x_3|x_3|,\ \vdots \ x_n|x_n| - (x_n - a)|x_n - a| = x_1|x_1|\end{array}\] where $a$ is a given number.

A sequence $\{a_n\}_{n\geq 1}$ of positive reals is defined by the rule $a_{n+1}a_{n-1}^5=a_n^4a_{n-2}^2$ for integers $n>2$ together with the initial values $a_1=8$ and $a_2=64$ and $a_3=1024$. Compute \[\sqrt{a_1+\sqrt{a_2+\sqrt{a_3+\cdots}}}\]

The altitudes through the vertices $A,B,C$ of an acute-angled triangle $ABC$ meet the opposite sides at $D,E,F,$ respectively. The line through $D$ parallel to $EF$ meets the lines $AC$ and $AB$ at $Q$ and $R$, respectively. The line $EF$ meets $BC$ at $P$. Prove that the circumcircle of the triangle $PQR$ passes through the midpoint of $BC$.

Let $a,b,c $be positive real numbers.Prove that $\frac{8}{(a+b)^2 + 4abc} + \frac{8}{(b+c)^2 + 4abc} + \frac{8}{(a+c)^2 + 4abc} + a^2 + b^2 + c ^2 \ge \frac{8}{a+3} + \frac{8}{b+3} + \frac{8}{c+3}$.

At a party there are $n$ women and $n$ men. Each woman likes $r$ of the men, and each man likes $s$ of then women. For which $r$ and $s$ must there be a man and a woman who like each other?

Let $ m, n $ natural numbers with $ m\ge 2,n\ge 3. $ Prove that there exist $ m $ distinct multiples of $ n-1, $ namely, $ a_1,a_2,a_3,...,a_m, $ such that: $$ \frac{1}{n} =\sum_{i=1}^m \frac{(-1)^{i-1}}{a_i} . $$