Let $f (n)$ be a function satisfying the following three conditions for all positive integers $n$: (a) $f (n)$ is a positive integer, (b) $f (n + 1) > f (n)$, (c) $f ( f (n)) = 3n$. Find $f (2001)$.

For each integer $n\ge 1,$ compute the smallest possible value of \[\sum_{k=1}^{n}\left\lfloor\frac{a_k}{k}\right\rfloor\] over all permutations $(a_1,\dots,a_n)$ of $\{1,\dots,n\}.$ [i]Proposed by Shahjalal Shohag, Bangladesh[/i]

The integers $a_0, a_1, a_2, a_3,\ldots$ are defined as follows: $a_0 = 1$, $a_1 = 3$, and $a_{n+1} = a_n + a_{n-1}$ for all $n \ge 1$. Find all integers $n \ge 1$ for which $na_{n+1} + a_n$ and $na_n + a_{n-1}$ share a common factor greater than $1$.

[u]Round 5[/u] [b]p13.[/b] Sally is at the special glasses shop, where there are many different optical lenses that distort what she sees and cause her to see things strangely. Whenever she looks at a shape through lens $A$, she sees a shape with $2$ more sides than the original (so a square would look like a hexagon). When she looks through lens $B$, she sees the shape with $3$ fewer sides (so a hexagon would look like a triangle). How many sides are in the shape that has $200$ more diagonals when looked at from lense $A$ than from lense $B$? [b]p14.[/b] How many ways can you choose $2$ cells of a $5$ by $5$ grid such that they aren't in the same row or column? [b]p15.[/b] If $a + \frac{1}{b} = (2015)^{-1}$ and $b + \frac{1}{a} = (2016)^2$ then what are all the possible values of $b$? [u]Round 6[/u] [b]p16.[/b] In Canadian football, linebackers must wear jersey numbers from $30 -35$ while defensive linemen must wear numbers from $33 -38$ (both intervals are inclusive). If a team has $5$ linebackers and $4$ defensive linemen, how many ways can it assign jersey numbers to the $9$ players such that no two people have the same jersey number? [b]p17.[/b] What is the maximum possible area of a right triangle with hypotenuse $8$? [b]p18.[/b] $9$ people are to play touch football. One will be designated the quarterback, while the other eight will be divided into two (indistinct) teams of $4$. How many ways are there for this to be done? [u]Round 7[/u] [b]p19.[/b] Express the decimal $0.3$ in base $7$. [b]p20.[/b] $2015$ people throw their hats in a pile. One at a time, they each take one hat out of the pile so that each has a random hat. What is the expected number of people who get their own hat? [b]p21.[/b] What is the area of the largest possible trapezoid that can be inscribed in a semicircle of radius $4$? [u]Round 8[/u] [b]p22.[/b] What is the base $7$ expression of $1211_3 \cdot 1110_2 \cdot 292_{11} \cdot 20_3$ ? [b]p23.[/b] Let $f(x)$ equal the ratio of the surface area of a sphere of radius $x$ to the volume of that same sphere. Let $g(x)$ be a quadratic polynomial in the form $x^2 + bx + c$ with $g(6) = 0$ and the minimum value of $g(x)$ equal to $c$. Express $g(x)$ as a function of $f(x)$ (e.g. in terms of $f(x)$). [b]p24.[/b] In the country of Tahksess, the income tax code is very complicated. Citizens are taxed $40\%$ on their first $\$20, 000$ and $45\%$ on their next $\$40, 000$ and $50\%$ on their next $\$60, 000$ and so on, with each $5\%$ increase in tax rate aecting $\$20, 000$ more than the previous tax rate. The maximum tax rate, however, is $90\%$. What is the overall tax rate (percentage of money owed) on $1$ million dollars in income? PS. You should use hide for answers. Rounds 1-4 have been posted [url=https://artofproblemsolving.com/community/c3h3157009p28696627]here [/url] and 9-12 [url=https://artofproblemsolving.com/community/c3h3158564p28715928]here[/url]. .Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

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?

For a set $S$, let $|S|$ denote the number of elements in $S$. Let $A$ be a set of positive integers with $|A| = 2001$. Prove that there exists a set $B$ such that all of the following conditions are fulfilled: a) $B \subseteq A$; b) $|B| \ge 668$; c) for any $x, y \in B$ we have $x + y \notin B$.

Find all real $x$ such that $ \sqrt{x - \frac{1}{x}} + \sqrt{1 - \frac{1}{x}}> \frac{x - 1}{x}$

Given positive integers $k, m, n$ such that $1 \leq k \leq m \leq n$. Evaluate \[\sum^{n}_{i=0} \frac{(-1)^i}{n+k+i} \cdot \frac{(m+n+i)!}{i!(n-i)!(m+i)!}.\]

Prove that every real function, defined on all of $\mathbb R$, can be represented as a sum of two functions whose graphs both have an axis of symmetry. [i]D. Tereshin[/i]

Given that $a$ is a real solution to the polynomial equation $$nx^n-x^{n-1}-x^{n-2}-\cdots-x-1=0$$ where $n$ is a positive integer, show that $a=1$ or $-1<a<0$.

Sequence ${u_n}$ is defined with $u_0=0,u_1=\frac{1}{3}$ and $$\frac{2}{3}u_n=\frac{1}{2}(u_{n+1}+u_{n-1})$$ $\forall n=1,2,...$ Show that $|u_n|\leq1$ $\forall n\in\mathbb{N}.$

We say that a number $n \ge 2$ has the property $(P)$ if, in its prime factorization, at least one of the factors has an exponent $3$. a) Determine the smallest number $N$ with the property that, no matter how we choose $N$ consecutive natural numbers, at least one of them has the property $(P).$ b) Determine the smallest $15$ consecutive numbers $a_1, a_2, \ldots, a_{15}$ that do not have the property $(P),$ such that the sum of the numbers $5 a_1, 5 a_2, \ldots, 5 a_{15}$ is a number with the property $(P).$

Let $a,b,c>0.$ Prove that$$\frac{a^3+b^2c+bc^2}{bc}+\frac{b^3+c^2a+ca^2}{ca}+\frac{c^3+a^2b+ab^2}{ab}\geq 3(a+b+c)$$ $$\frac{bc}{a^3+b^2c+bc^2}+\frac{ca}{b^3+c^2a+ca^2}+\frac{ab}{c^3+a^2b+ab^2}\leq \frac{1}{3}(\frac{1}{a}+\frac{1}{b}+\frac{1}{c})$$

$1600$ bananas are distributed among $100$ monkeys (it is possible that some monkeys do not receive bananas). Prvove that at least four monkeys receive the same amount of bananas.

If positive reals $ x_1,x_2,\cdots,x_n $ satisfy $\sum_{i=1}^{n}x_i=1.$ Prove that$$\sum_{i=1}^{n}\frac{1}{1+\sum_{j=1}^{i}x_j}<\sqrt{\frac{2}{3}\sum_{i=1}^{n}\frac{1}{x_i}} $$

Suppose that $k,m,n$ are positive integers with $k \le n$. Prove that: \[\sum_{r=0}^m \dfrac{k \binom{m}{r} \binom{n}{k}}{(r+k) \binom{m+n}{r+k}} = 1\]

For each $a$, $1<a<2$, the graphs of functions $y=1-|x-1|$ and $y=|2x-a|$ determine a figure. Prove that the area of this figure is less than $\frac13$.

Find all functions $f:\mathbb{R}\rightarrow\mathbb{R}$ such that $f(0)\neq 0$ and for all $x,y\in\mathbb{R}$, \[ f(x+y)^2 = 2f(x)f(y) + \max \left\{ f(x^2+y^2), f(x^2)+f(y^2) \right\}. \]

Let $P$ be a cubic monic polynomial with roots $a$, $b$, and $c$. If $P(1)=91$ and $P(-1)=-121$, compute the maximum possible value of \[\dfrac{ab+bc+ca}{abc+a+b+c}.\] [i]Proposed by David Altizio[/i]

Let $f = f(x,y,z)$ be a polynomial in three variables $x$, $y$, $z$ such that $f(w,w,w) = 0$ for all $w \in \mathbb{R}$. Show that there exist three polynomials $A$, $B$, $C$ in these same three variables such that $A + B + C = 0$ and \[ f(x,y,z) = A(x,y,z) \cdot (x-y) + B(x,y,z) \cdot (y-z) + C(x,y,z) \cdot (z-x). \] Is there any polynomial $f$ for which these $A$, $B$, $C$ are uniquely determined?

[u]Round 1 [/u] [b]p1.[/b] In the convex quadrilateral $ABCD$ with diagonals $AC$ and $BD$, you know that angle $BAC$ is congruent to angle $CBD$, and that angle $ACD$ is congruent to angle $ADB$. Show that angle $ABC$ is congruent to angle $ADC$. [img]https://cdn.artofproblemsolving.com/attachments/5/d/41cd120813d5541dc73c5d4a6c86cc82747fcc.png[/img] [b]p2.[/b] In how many different ways can you place $12$ chips in the squares of a $4 \times 4$ chessboard so that (a) there is at most one chip in each square, and (b) every row and every column contains exactly three chips. [b]p3.[/b] Students from Hufflepuff and Ravenclaw were split into pairs consisting of one student from each house. The pairs of students were sent to Honeydukes to get candy for Father's Day. For each pair of students, either the Hufflepuff student brought back twice as many pieces of candy as the Ravenclaw student or the Ravenclaw student brought back twice as many pieces of candy as the Hufflepuff student. When they returned, Professor Trelawney determined that the students had brought back a total of $1000$ pieces of candy. Could she have possibly been right? Why or why not? Assume that candy only comes in whole pieces (cannot be divided into parts). [b]p4.[/b] While you are on a hike across Deception Pass, you encounter an evil troll, who will not let you across the bridge until you solve the following puzzle. There are six stones, two colored red, two colored yellow, and two colored green. Aside from their colors, all six stones look and feel exactly the same. Unfortunately, in each colored pair, one stone is slightly heavier than the other. Each of the lighter stones has the same weight, and each of the heavier stones has the same weight. Using a balance scale to make TWO measurements, decide which stone of each color is the lighter one. [b]p5.[/b] Alex, Bob and Chad are playing a table tennis tournament. During each game, two boys are playing each other and one is resting. In the next game the boy who lost a game goes to rest, and the boy who was resting plays the winner. By the end of tournament, Alex played a total of $10$ games, Bob played $15$ games, and Chad played $17$ games. Who lost the second game? [u]Round 2 [/u] [b]p6.[/b] Consider a set of finitely many points on the plane such that if we choose any three points $A,B,C$ from the set, then the area of the triangle $ABC$ is less than $1$. Show that all of these points can be covered by a triangle whose area is less than $4$. [b]p7.[/b] A palindrome is a number that is the same when read forward and backward. For example, $1771$ and $23903030932$ are palindromes. Can the number obtained by writing the numbers from $1$ to $n$ in order be a palindrome for some $n > 1$ ? (For example, if $n = 11$, the number obtained is $1234567891011$, which is not a palindrome.) PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $n$ be a positive integer and $f(x)=a_mx^m+\ldots + a_1X+a_0$, with $m\ge 2$, a polynomial with integer coefficients such that: a) $a_2,a_3\ldots a_m$ are divisible by all prime factors of $n$, b) $a_1$ and $n$ are relatively prime. Prove that for any positive integer $k$, there exists a positive integer $c$, such that $f(c)$ is divisible by $n^k$.

Let $n \in N, n\ge 2$, and $a_1, a_2, ..., a_n, b_1, b_2, ..., b_n$ be real positive numbers such that $$\frac{a_1}{b_1} \le \frac{a_2}{b_2} \le ... \le\frac{a_n}{b_n}.$$ Find the largest real $c$ so that $$(a_1-b_1c)x_1+(a_2-b_2c)x_2+...+(a_n-b_nc)x_n \ge 0,$$ for every $x_1, x_2,..., x_n > 0$, with $x_1\le x_2\le ...\le x_n$.

The polynomial $x^8 +x^7$ is written on a blackboard. In a move, Peter can erase the polynomial $P(x)$ and write down $(x+1)P(x)$ or its derivative $P'(x).$ After a while, the linear polynomial $ax+b$ with $a\ne 0$ is written on the board. Prove that $a-b$ is divisible by $49.$

Suppose that $y$ is a positive integer written only with digit $1$, in base $9$ system. Prove that $y$ is a triangular number, that is, exists positive integer $n$ such that the number $y$ is the sum of the $n$ natural numbers from $1$ to $n$.