Find all functions $f : \mathbb{Z}\rightarrow \mathbb{Z}$ satisfying \[f^{a^{2} + b^{2}}(a+b) = af(a) +bf(b)\] for all integers $a$ and $b$

How many positive integer $n$ are there satisfying the inequality $1+\sqrt{n^2-9n+20} > \sqrt{n^2-7n+12}$ ? $\textbf{(A)}\ 1 \qquad\textbf{(B)}\ 2 \qquad\textbf{(C)}\ 3 \qquad\textbf{(D)}\ 4 \qquad\textbf{(E)}\ \text{None}$

A robot moves in the plane in a straight line, but every one meter it turns $90^{\circ}$ to the right or to the left. At some point it reaches its starting point without having visited any other point more than once, and stops immediately. What are the possible path lengths of the robot?

Consider a $N\times N$ square board where $N\geq 3$ is an odd integer. The caterpillar Carl sits at the center of the square; all other cells contain distinct positive integers. An integer $n$ weights $1\slash n$ kilograms. Carl wants to leave the board but can eat at most $2$ kilograms. Determine whether Carl can always find a way out when a) $N=2003.$ b) $N$ is an arbitrary odd integer.

Determine the smallest possible real constant $C$ such that the inequality $$|x^3 + y^3 + z^3 + 1| \leq C|x^5 + y^5 + z^5 + 1|$$ holds for all real numbers $x, y, z$ satisfying $x + y + z = -1$.

An acute-angled triangle $ABC$ has two altitudes $BE$ and $CF$. The circle with diameter $AC$ intersects the segment $BE$ at point $P$. A circle with diameter $AB$ intersects the segment $CF$ at point $Q$ and the extension of this altitude at point $Q'$. Prove that $\angle PQ'Q = \angle PQB$.

Lines $l$ and $m$ are perpendicular. Line $l$ partitions a convex polygon into two parts of equal area, and partitions the projection of the polygon onto $m$ into two line segments of length $a$ and $b$ respectively. Determine the maximum value of $\left\lfloor \frac{1000a}{b} \right\rfloor$. (The floor notation $\lfloor x \rfloor$ denotes largest integer not exceeding $x$)

$a;b\in\mathbb{R^+}$. Prove the following inequality: $$\sqrt[3]{\frac{a}{b}}+\sqrt[3]{\frac{b}{a}}\leq\sqrt[3]{2(a+b)(\frac1{a}+\frac1{b})}$$

There are 300 apples, any two of which differ in weight by no more than twice. Prove that they can be arranged in packages of two apples so that any two packages differ in weight by no more than one and a half times.

Let $c>0$ be a given positive real and $\mathbb{R}_{>0}$ be the set of all positive reals. Find all functions $f \colon \mathbb{R}_{>0} \to \mathbb{R}_{>0}$ such that \[f((c+1)x+f(y))=f(x+2y)+2cx \quad \textrm{for all } x,y \in \mathbb{R}_{>0}.\]

What is the sum of the digits of the first $2010$ positive integers? $ \textbf{(A)}\ 30516 \qquad\textbf{(B)}\ 28068 \qquad\textbf{(C)}\ 25020 \qquad\textbf{(D)}\ 20100 \qquad\textbf{(E)}\ \text{None} $

[b]p1.[/b] What is the area of a triangle with side lengths $ 6$, $ 8$, and $10$? [b]p2.[/b] Let $f(n) = \sqrt{n}$. If $f(f(f(n))) = 2$, compute $n$. [b]p3.[/b] Anton is buying AguaFina water bottles. Each bottle costs $14 $dollars, and Anton buys at least one water bottle. The number of dollars that Anton spends on AguaFina water bottles is a multiple of $10$. What is the least number of water bottles he can buy? [b]p4.[/b] Alex flips $3$ fair coins in a row. The probability that the first and last flips are the same can be expressed in the form $m/n$ for relatively prime positive integers $m$ and $n$. Compute $m + n$. [b]p5.[/b] How many prime numbers $p$ satisfy the property that $p^2 - 1$ is not a multiple of $6$? [b]p6.[/b] In right triangle $\vartriangle ABC$ with $AB = 5$, $BC = 12$, and $CA = 13$, point $D$ lies on $\overline{CA}$ such that $AD = BD$. The length of $CD$ can then be expressed in the form $m/n$ for relatively prime positive integers $m$ and $n$. Compute $m + n$. [b]p7.[/b] Vivienne is deciding on what courses to take for Spring $2021$, and she must choose from four math courses, three computer science courses, and five English courses. Vivienne decides that she will take one English course and two additional courses that are either computer science or math. How many choices does Vivienne have? [b]p8.[/b] Square $ABCD$ has side length $2$. Square $ACEF$ is drawn such that $B$ lies inside square $ACEF$. Compute the area of pentagon $AFECD$. [b]p9.[/b] At the Boba Math Tournament, the Blackberry Milk Team has answered $4$ out of the first $10$ questions on the Boba Round correctly. If they answer all $p$ remaining questions correctly, they will have answered exactly $\frac{9p}{5}\%$ of the questions correctly in total. How many questions are on the Boba Round? [b]p10.[/b] The sum of two positive integers is $2021$ less than their product. If one of them is a perfect square, compute the sum of the two numbers. [b]p11.[/b] Points $E$ and $F$ lie on edges $\overline{BC}$ and $\overline{DA}$ of unit square $ABCD$, respectively, such that $BE =\frac13$ and $DF =\frac13$ . Line segments $\overline{AE}$ and $\overline{BF}$ intersect at point $G$. The area of triangle $EFG$ can be written in the form $m/n$ , where $m$ and $n$ are relatively prime positive integers. Compute $m+n$. [b]p12.[/b] Compute the number of positive integers $n \le 2020$ for which $n^{k+1}$ is a factor of $(1+2+3+· · ·+n)^k$ for some positive integer $k$. [b]p13.[/b] How many permutations of $123456$ are divisible by their last digit? For instance, $123456$ is divisible by $6$, but $561234$ is not divisible by $4$. [b]p14.[/b] Compute the sum of all possible integer values for $n$ such that $n^2 - 2n - 120$ is a positive prime number. [b]p15. [/b]Triangle $\vartriangle ABC$ has $AB =\sqrt{10}$, $BC =\sqrt{17}$, and $CA =\sqrt{41}$. The area of $\vartriangle ABC$ can be expressed in the form $m/n$ for relatively prime positive integers $m$ and $n$. Compute $m + n$. [b]p16.[/b] Let $f(x) = \frac{1 + x^3 + x^{10}}{1 + x^{10}}$ . Compute $f(-20) + f(-19) + f(-18) + ...+ f(20)$. [b]p17.[/b] Leanne and Jing Jing are walking around the $xy$-plane. In one step, Leanne can move from any point $(x, y)$ to $(x + 1, y)$ or $(x, y + 1)$ and Jing Jing can move from $(x, y)$ to $(x - 2, y + 5)$ or $(x + 3, y - 1)$. The number of ways that Leanne can move from $(0, 0)$ to $(20, 20)$ is equal to the number of ways that Jing Jing can move from $(0, 0)$ to $(a, b)$, where a and b are positive integers. Compute the minimum possible value of $a + b$. [b]p18.[/b] Compute the number positive integers $1 < k < 2021$ such that the equation $x +\sqrt{kx} = kx +\sqrt{x}$ has a positive rational solution for $x$. [b]p19.[/b] In triangle $\vartriangle ABC$, point $D$ lies on $\overline{BC}$ with $\overline{AD} \perp \overline{BC}$. If $BD = 3AD$, and the area of $\vartriangle ABC$ is $15$, then the minimum value of $AC^2$ is of the form $p\sqrt{q} - r$, where $p, q$, and $r$ are positive integers and $q$ is not divisible by the square of any prime number. Compute $p + q + r$. [b]p20. [/b]Suppose the decimal representation of $\frac{1}{n}$ is in the form $0.p_1p_2...p_j\overline{d_1d_2...d_k}$, where $p_1, ... , p_j$ , $d_1,... , d_k$ are decimal digits, and $j$ and $k$ are the smallest possible nonnegative integers (i.e. it’s possible for $j = 0$ or $k = 0$). We define the [i]preperiod [/i]of $\frac{1}{n}$ to be $j$ and the [i]period [/i]of $\frac{1}{n}$ to be $k$. For example, $\frac16 = 0.16666...$ has preperiod $1$ and period $1$, $\frac17 = 0.\overline{142857}$ has preperiod $0$ and period $6$, and $\frac14 = 0.25$ has preperiod $2$ and period $0$. What is the smallest positive integer $n$ such that the sum of the preperiod and period of $\frac{1}{n}$ is $ 8$? PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $AC$ be a fixed chord of a circle $\omega$ with center $O$. Point $B$ moves along the arc $AC$. A fixed point $P$ lies on $AC$. The line passing through $P$ and parallel to $AO$ meets $BA$ at point $A_1$, the line passing through $P$ and parallel to $CO$ meets $BC$ at point $C_1$. Prove that the circumcenter of triangle $A_1BC_1$ moves along a straight line.

The numbers $1, 2, ..., 2020$ and $2021$ are written on a blackboard. The following operation is executed: Two numbers are chosen, both are erased and replaced by the absolute value of their difference. This operation is repeated until there is only one number left on the blackboard. (a) Show that $2021$ can be the final number on the blackboard. (b) Show that $2020$ cannot be the final number on the blackboard. (Karl Czakler)

Points $A,M,B,C,D$ are given on a circle in this order such that $A$ and $B$ are equidistant from $M$. Lines $MD$ and $AC$ intersect at $E$ and lines $MC$ and $BD$ intersect at $F$. Prove that the quadrilateral $CDEF$ is inscridable in a circle.

Determine the smallest positive integer $n$ whose prime factors are all greater than $18$, and that can be expressed as $n = a^3 + b^3$ with positive integers $a$ and $b$.

Is it possible to fill a rectangular table with black and white squares (only) so, that the number of black squares will equal to the number of white squares, and each row and each column will have more than $75\%$ squares of the same colour?

A student will take an exam in which they have to solve three chosen problems by chance of a list of $10$ possible problems. It will be approved if it correctly resolves two problems. Considering that the student can solve five of the problems on the list and not know how to solve others, how likely is he to pass the exam?

Find all pairs $(x, y)$ of rational numbers such that $y^2 =x^3 -3x+2$.

Let $a, b$, and $c$ be positive real numbers. Prove: $\frac{a}{b}+ \frac{b}{c}+ \frac{c}{a}\le \frac{a^2}{b^2} + \frac{b^2}{c^2} + \frac{c^2}{a^2}$ .

A sequence of primes $p_1, p_2, \dots$ is given by two initial primes $p_1$ and $p_2$, and $p_{n+2}$ being the greatest prime divisor of $p_n + p_{n+1} + 2018$ for all $n \ge 1$. Prove that the sequence only contains finitely many primes for all possible values of $p_1$ and $p_2$.

Given a positive integer $n$. (a) If $P$ is a polynomial of degree $n$ where $P(x) \in \mathbb{Z}$ for every $x \in \mathbb{Z}$, prove that for every $a,b \in \mathbb{Z}$ where $P(a) \neq P(b)$, \[\text{lcm}(1, 2, \ldots, n) \ge \left| \dfrac{a-b}{P(a) - P(b)} \right|\] (b) Find one $P$ (for each $n$) such that the equality case above is achieved for some $a,b \in \mathbb{Z}$.

Let $OABC$ be a unit square in the $xy$-plane with $O(0,0),A(1,0),B(1,1)$ and $C(0,1)$. Let $u=x^2-y^2$ and $v=2xy$ be a transformation of the $xy$-plane into the $uv$-plane. The transform (or image) of the square is: [asy] size(150); defaultpen(linewidth(0.8)+fontsize(8)); draw((-2.5,0)--(2.5,0),EndArrow(size=7)); draw((0,-3)--(0,3),EndArrow(size=7)); label("$O$",(0,0),SW); label("$u$",(2.5,0),E); label("$v$",(0,3),N); draw((0,2)--(1,0)--(0,-2)--(-1,0)--cycle); label("$(0,2)$",(0,2),NE); label("$(1,0)$",(1,0),SE); label("$(0,-2)$",(0,-2),SE); label("$(-1,0)$",(-1,0),SW); label("$\textbf{(A)}$",(-2,1.5)); [/asy] [asy] size(150); defaultpen(linewidth(0.8)+fontsize(8)); draw((-2.5,0)--(2.5,0),EndArrow(size=7)); draw((0,-3)--(0,3),EndArrow(size=7)); label("$O$",(0,0),SW); label("$u$",(2.5,0),E); label("$v$",(0,3),N); draw((0,2)..(1,0)..(0,-2)^^(0,-2)..(-1,0)..(0,2)); label("$(0,2)$",(0,2),NE); label("$(1,0)$",(1,0),SE); label("$(0,-2)$",(0,-2),SE); label("$(-1,0)$",(-1,0),SW); label("$\textbf{(B)}$",(-2,1.5)); [/asy] [asy] size(150); defaultpen(linewidth(0.8)+fontsize(8)); draw((-2.5,0)--(2.5,0),EndArrow(size=7)); draw((0,-3)--(0,3),EndArrow(size=7)); label("$O$",(0,0),SW); label("$u$",(2.5,0),E); label("$v$",(0,3),N); draw((0,2)--(1,0)--(-1,0)--cycle); label("$(0,2)$",(0,2),NE); label("$(1,0)$",(1,0),S); label("$(-1,0)$",(-1,0),S); label("$\textbf{(C)}$",(-2,1.5)); [/asy] [asy] size(150); defaultpen(linewidth(0.8)+fontsize(8)); draw((-2.5,0)--(2.5,0),EndArrow(size=7)); draw((0,-3)--(0,3),EndArrow(size=7)); label("$O$",(0,0),SW); label("$u$",(2.5,0),E); label("$v$",(0,3),N); draw((0,2)..(1/2,3/2)..(1,0)--(-1,0)..(-1/2,3/2)..(0,2)); label("$(0,2)$",(0,2),NE); label("$(1,0)$",(1,0),S); label("$(-1,0)$",(-1,0),S); label("$\textbf{(D)}$",(-2,1.5)); [/asy] [asy] size(150); defaultpen(linewidth(0.8)+fontsize(8)); draw((-2.5,0)--(2.5,0),EndArrow(size=7)); draw((0,-3)--(0,3),EndArrow(size=7)); label("$O$",(0,0),SW); label("$u$",(2.5,0),E); label("$v$",(0,3),N); draw((0,1)--(1,0)--(0,-1)--(-1,0)--cycle); label("$(0,1)$",(0,1),NE); label("$(1,0)$",(1,0),SE); label("$(0,-1)$",(0,-1),SE); label("$(-1,0)$",(-1,0),SW); label("$\textbf{(E)}$",(-2,1.5)); [/asy]

Find the integer $x$ for which $135^3+138^3=x^3-1.$

Let the incircle of triangle $ABC$ touches the sides $AB,BC,CA$ in $C_1,A_1,B_1$ respectively. If $A_1C_1$ cuts the parallel to $BC$ from $A$ at $K$ prove that $\angle KB_1A_1=90.$