The function $f$ is increasing and convex (i.e. every straight line between two points on the graph of $f$ lies above the graph) and satisfies $f(f(x))=3^x$ for all $x\in\mathbb{R}$. If $f(0)=0.5$ determine $f(0.75)$ with an error of at most $0.025$. The following are corrent to the number of digits given: \[3^{0.25}=1.31607,\quad 3^{0.50}=1.73205,\quad 3^{0.75}=2.27951.\]

[b]p1.[/b] Ana, Bob, Cho, Dan, and Eve want to use a microwave. In order to be fair, they choose a random order to heat their food in (all orders have equal probability). Ana's food needs $5$ minutes to cook, Bob's food needs $7$ minutes, Cho's needs $1$ minute, Dan's needs $12$ minutes, and Eve's needs $5$ minutes. What is the expected number of minutes Bob has to wait for his food to be done? [b]p2.[/b] $ABC$ is an equilateral triangle. $H$ lies in the interior of $ABC$, and points $X$, $Y$, $Z$ lie on sides $AB, BC, CA$, respectively, such that $HX\perp AB$, $HY \perp BC$, $HZ\perp CA$. Furthermore, $HX =2$, $HY = 3$, $HZ = 4$. Find the area of triangle $ABC$. [b]p3.[/b] Amy, Ben, and Chime play a dice game. They each take turns rolling a die such that the $first$ person to roll one of his favorite numbers wins. Amy's favorite number is $1$, Ben's favorite numbers are $2$ and $3$, and Chime's are $4$, $5$, and $6$. Amy rolls first, Ben rolls second, and Chime rolls third. If no one has won after Chime's turn, they repeat the sequence until someone has won. What's the probability that Chime wins the game? [b]p4.[/b] A point $P$ is chosen randomly in the interior of a square $ABCD$. What is the probability that the angle $\angle APB$ is obtuse? [b]p5.[/b] Let $ABCD$ be the quadrilateral with vertices $A = (3, 9)$, $B = (1, 1)$, $C = (5, 3)$, and $D = (a, b)$, all of which lie in the first quadrant. Let $M$ be the midpoint of $AB$, $N$ the midpoint of $BC$, $O$ the midpoint of $CD$, and $P$ the midpoint of $AD$. If $MNOP$ is a square, find $(a, b)$. [b]p6.[/b] Let $M$ be the number of positive perfect cubes that divide $60^{60}$. What is the prime factorization of $M$? [b]p7.[/b] Given that $x$, $y$, and $z$ are complex numbers with $|x|=|y| =|z|= 1$, $x + y + z = 1$ and $xyz = 1$, find $|(x + 2)(y + 2)(z + 2)|$. [b]p8.[/b] If $f(x)$ is a polynomial of degree $2008$ such that $f(m) = \frac{1}{m}$ for $m = 1, 2, ..., 2009$, find $f(2010)$. [b]p9.[/b] A drunkard is randomly walking through a city when he stumbles upon a $2 \times 2$ sliding tile puzzle. The puzzle consists of a $2 \times 2$ grid filled with a blank square, as well as $3$ square tiles, labeled $1$, $2$, and $3$. During each turn you may fill the empty square by sliding one of the adjacent tiles into it. The following image shows the puzzle's correct state, as well as two possible moves you can make: [img]https://cdn.artofproblemsolving.com/attachments/c/6/7ddd9305885523deeee2a530dc90505875d1cc.png[/img] Assuming that the puzzle is initially in an incorrect (but solvable) state, and that the drunkard will make completely random moves to try and solve it, how many moves is he expected to make before he restores the puzzle to its correct state? [b]p10.[/b] How many polynomials $p(x)$ exist such that the coeffients of $p(x)$ are a rearrangement of $\{0, 1, 2, .., deg \, p(x)\}$ and all of the roots of $p(x)$ are rational? (Note that the leading coefficient of $p(x)$ must be nonzero.) PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $a_1,a_2,\ldots ,a_9$ be any non-negative numbers such that $a_1=a_9=0$ and at least one of the numbers is non-zero. Prove that for some $i$, $2\le i\le 8$, the inequality $a_{i-1}+a_{i+1}<2a_i$ holds. Will the statement remain true if we change the number $2$ in the last inequality to $1.9$?

The equation $ x^2 \plus{} ax \plus{} b \equal{} 0$ has two distinct real roots. Prove that the equation $ x^4 \plus{} ax^3 \plus{} (b \minus{} 2)x^2 \minus{} ax \plus{} 1 \equal{} 0$ has four distinct real roots.

Find all functions $f:\mathbb{Z}\to \mathbb{Z}$ satisfying the condition $f(n)-f(n+f(m))=m$ for all $m,n\in \mathbb{Z}$

The sequence $a_i$ is defined as $a_1 = 2, a_2 = 3$, and $a_{n+1} = 2a_{n-1}$ or $a_{n+1} = 3a_n - 2a_{n-1}$ for all integers $n \ge 2$. Prove that no term in $a_i$ is in the range $[1612, 2012]$.

Let $a$ and $b$ be arbitrary distinct numbers. Prove that the equation $(x +a) (x+b)=2x+a+b$ has two different roots.

Prove that the equation $x^n - a_1x^{n-1} - a_2x^{n-2} - ... -a_{n-1}x - a_n = 0$, where $a_1 \ge 0, a_2 \ge 0, . . . , a_n \ge 0$, cannot have two positive roots.

The real numbers $a$ and $b$ satisfy the relation $a + b \ge 1$. Show that $8 (a^4 + b^4) \ge 1$.

Let $a,b,c$ be the lengths of the sides of a given triangle.If positive reals $x,y,z$ satisfy $x+y+z=1,$ find the maximum of $axy+byz+czx.$

Let $\mathbb R$ be the set of real numbers. Determine all functions $f:\mathbb R\to\mathbb R$ that satisfy the equation $$\sum_{i=1}^{2015} f(x_i + x_{i+1}) + f\left( \sum_{i=1}^{2016} x_i \right) \le \sum_{i=1}^{2016} f(2x_i)$$ for all real numbers $x_1, x_2, ... , x_{2016}.$

a) Find the minimal value of the polynomial $$P(x,y) = 4 + x^2y^4 + x^4y^2 - 3x^2y^2$$ b) Prove that it cannot be represented as a sum of the squares of some polynomials of $x,y$.

Solve in real numbers the system of equations: \begin{align*} \frac{1}{xy}&=\frac{x}{z}+1 \\ \frac{1}{yz}&=\frac{y}{x}+1 \\ \frac{1}{zx}&=\frac{z}{y}+1 \\ \end{align*}

Find the smallest positive real $k$ satisfying the following condition: for any given four DIFFERENT real numbers $a,b,c,d$, which are not less than $k$, there exists a permutation $(p,q,r,s)$ of $(a,b,c,d)$, such that the equation $(x^{2}+px+q)(x^{2}+rx+s)=0$ has four different real roots.

Find all real constants c for which there exist strictly increasing sequence $a$ of positive integers such that $(a_{2n-1}+a_{2n})/{a_n}=c$ for all positive intеgers n.

[b]p1A[/b] Positive reals $x$, $y$, and $z$ are such that $x/y +y/x = 7$ and $y/z +z/y = 7$. There are two possible values for $z/x + x/z;$ find the greater value. [b]p1B[/b] Real values $x$ and $y$ are such that $x+y = 2$ and $x^3+y^3 = 3$. Find $x^2+y^2$. [b]p2[/b] Set $A = \{5, 6, 8, 13, 20, 22, 33, 42\}$. Let $\sum S$ denote the sum of the members of $S$; then $\sum A = 149$. Find the number of (not necessarily proper) subsets $B$ of $A$ for which $\sum B \ge 75$. [b]p3[/b] $99$ dots are evenly spaced around a circle. Call two of these dots ”close” if they have $0$, $1$, or $2$ dots between them on the circle. We wish to color all $99$ dots so that any two dots which are close are colored differently. How many such colorings are possible using no more than $4$ different colors? [b]p4[/b] Given a $9 \times 9$ grid of points, count the number of nondegenerate squares that can be drawn whose vertices are in the grid and whose center is the middle point of the grid. PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

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

Let $k$ be an integer. If the equation $(x-1)|x+1|=x+\frac{k}{2020}$ has three distinct real roots, how many different possible values of $k$ are there?

Let $x$ be a real number such that $x^2 -x+1 = 7$ and $x^2 +x+1 = 13$. Compute the value of $x^4$.

Prove that for any integer $a$ the polynomial $3x^{2n}+ax^n+2$ cannot be divided by $2x^{2m}+ax^m+3$ without a remainder.

Find largest possible constant $M$ such that, for any sequence $a_n$, $n=0,1,2,...$ of real numbers, that satisfies the conditions : i) $a_0=1$, $a_1=3$ ii) $a_0+a_1+...+a_{n-1} \ge 3 a_n - a_{n+1}$ for any integer $n\ge 1$ to be true that $$\frac{a_{n+1}}{a_n} >M$$ for any integer $n\ge 0$.

Find all pairs of positive integers $ a,b$ such that \begin{align*} b^2 + b+ 1 & \equiv 0 \pmod a \\ a^2+a+1 &\equiv 0 \pmod b . \end{align*}

Determine all functions $f:(0,\infty)\to\mathbb{R}$ satisfying $$\left(x+\frac{1}{x}\right)f(y)=f(xy)+f\left(\frac{y}{x}\right)$$ for all $x,y>0$.

Given a positive integer $k > 1$, find all positive integers $n$ such that the polynomial $$P(z) = z^n + \sum_{j=0}^{2^k-2} z^j = 1 +z +z^2 + \cdots +z^{2^k-2} + z^n$$ has a complex root $w$ such that $|w| = 1$.

Compute $$ 2022^{\left(2022^{\cdot ^ {\cdot ^{\cdot ^{\left(2022^{2022}\right)}}}}\right)} \pmod{111} $$ where there are $2022$ $2022$s. (Give the answer as an integer from $0$ to $110$). [i]Proposed by David Tang[/i]