Let $N>1$ and let $a_1,a_2,\dots,a_N$ be nonnegative reals with sum at most $500$. Prove that there exist integers $k\ge 1$ and $1=n_0<n_1<\dots<n_k=N$ such that \[\sum_{i=1}^k n_ia_{n_{i-1}}<2005.\]

[u]Round 1 [/u] [b]p1.[/b] A small pizza costs $\$4$ and has $6$ slices. A large pizza costs $\$9$ and has $14$ slices. If the MMATHS organizers got at least $400$ slices of pizza (having extra is okay) as cheaply as possible, how many large pizzas did they buy? [b]p2.[/b] Rachel flips a fair coin until she gets a tails. What is the probability that she gets an even number of heads before the tails? [b]p3.[/b] Find the unique positive integer $n$ that satisfies $n! \cdot (n + 1)! = (n + 4)!$. [u]Round 2 [/u] [b]p4.[/b] The Portland Malt Shoppe stocks $10$ ice cream flavors and $8$ mix-ins. A milkshake consists of exactly $1$ flavor of ice cream and between $1$ and $3$ mix-ins. (Mix-ins can be repeated, the number of each mix-in matters, and the order of the mix-ins doesn’t matter.) How many different milkshakes can be ordered? [b]p5.[/b] Find the minimum possible value of the expression $(x)^2 + (x + 3)^4 + (x + 4)^4 + (x + 7)^2$, where $x$ is a real number. [b]p6.[/b] Ralph has a cylinder with height $15$ and volume $\frac{960}{\pi}$ . What is the longest distance (staying on the surface) between two points of the cylinder? [u]Round 3 [/u] [b]p7.[/b] If there are exactly $3$ pairs $(x, y)$ satisfying $x^2 + y^2 = 8$ and $x + y = (x - y)^2 + a$, what is the value of $a$? [b]p8.[/b] If $n$ is an integer between $4$ and $1000$, what is the largest possible power of $2$ that $n^4 - 13n^2 + 36$ could be divisible by? (Your answer should be this power of $2$, not just the exponent.) [b]p9.[/b] Find the sum of all positive integers $n \ge 2$ for which the following statement is true: “for any arrangement of $n$ points in three-dimensional space where the points are not all collinear, you can always find one of the points such that the $n - 1$ rays from this point through the other points are all distinct.” [u]Round 4 [/u] [b]p10.[/b] Donald writes the number $12121213131415$ on a piece of paper. How many ways can he rearrange these fourteen digits to make another number where the digit in every place value is different from what was there before? [b]p11.[/b] A question on Joe’s math test asked him to compute $\frac{a}{b} +\frac34$ , where $a$ and $b$ were both integers. Because he didn’t know how to add fractions, he submitted $\frac{a+3}{b+4}$ as his answer. But it turns out that he was right for these particular values of $a$ and $b$! What is the largest possible value that a could have been? [b]p12.[/b] Christopher has a globe with radius $r$ inches. He puts his finger on a point on the equator. He moves his finger $5\pi$ inches North, then $\pi$ inches East, then $5\pi$ inches South, then $2\pi$ inches West. If he ended where he started, what is the largest possible value of $r$? PS. You should use hide for answers. Rounds 5-7 have be posted [url=https://artofproblemsolving.com/community/c4h2789002p24519497]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Find all fuctions $f,g:\mathbb{R}\rightarrow \mathbb{R}$ such that: $f(x-3f(y))=xf(y)-yf(x)+g(x) \forall x,y\in\mathbb{R}$ and $g(1)=-8$

Let $A=\sqrt{7+2\sqrt{10}} - \sqrt{7-2\sqrt{10}}.$ We can express $A$ as $a\sqrt{b},$ where $a,b$ are integers and $b$ is square-free. Compute $a+b.$

Let $P(x)$ be a monic polynomial of degree $100$ with $100$ distinct noninteger real roots. Suppose that each of polynomials $P(2x^2 - 4x)$ and $P(4x - 2x^2)$ has exactly $130$ distinct real roots. Prove that there exist non constant polynomials $A(x),B(x)$ such that $A(x)B(x) = P(x)$ and $A(x) = B(x)$ has no root in $(-1.1)$

Let $n$ be a fixed positive integer. Determine the smallest possible rank of an $n\times n$ matrix that has zeros along the main diagonal and strictly positive real numbers off the main diagonal. [i]Proposed by Ilya Bogdanov and Grigoriy Chelnokov, MIPT, Moscow.[/i]

Let $g(x)$ be a fixed polynomial with real coefficients and define $f(x)$ by $f(x) =x^2 + xg(x^3)$. Show that $f(x)$ is not divisible by $x^2 - x + 1$.

Let $n$ be an integer greater than two, and let $A_1,A_2, \cdots , A_{2n}$ be pairwise distinct subsets of $\{1, 2, ,n\}$. Determine the maximum value of \[\sum_{i=1}^{2n} \dfrac{|A_i \cap A_{i+1}|}{|A_i| \cdot |A_{i+1}|}\] Where $A_{2n+1}=A_1$ and $|X|$ denote the number of elements in $X.$

Solve the system of equations: $$\begin{cases} x^3 + y^3= 7 \\ xy (x + y) = -2\end{cases}$$

A sequence $(a_{n})$ of positive integers is given by $a_{n}=[n+\sqrt{n}+\frac{1}{2}]$. Find all of positive integers which belong to the sequence.

Find all the finite sets $A$ of real positive numbers having at least two elements, with the property that $a^2 + b^2 \in A$ for every $a, b \in A$ with $a \ne b$

$ f: \mathbb N \times \mathbb Z \rightarrow \mathbb Z$ satisfy the given conditions $ a)$ $ f(0,0)\equal{}1$ , $ f(0,1)\equal{}1$ , $ b)$ $ \forall k \notin \left\{0,1\right\}$ $ f(0,k)\equal{}0$ and $ c)$ $ \forall n \geq 1$ and $ k$ , $ f(n,k)\equal{}f(n\minus{}1,k)\plus{}f(n\minus{}1,k\minus{}2n)$ find the sum $ \displaystyle\sum_{k\equal{}0}^{\binom{2009}{2}}f(2008,k)$

Let $a,b, c$ be real numbers with $a+b+c = 2018$. Suppose $x, y$, and $z$ are the distinct positive real numbers which are satisfied $a = x^2 - yz - 2018, b = y^2 - zx - 2018$ , and $c = z^2 - xy - 2018$. Compute the value of the following expression $P = \frac{\sqrt{a^3 + b^3 + c^3 - 3abc}}{x^3 + y^3 + z^3 - 3xyz}$

$1+2+3+4+5+6=6+7+8$. What is the smallest number $k$ greater than $6$ for which: $1 + 2 +...+ k = k + (k+1) +...+ n$, with $n$ an integer greater than $k$ ?

Solve in the set of real numbers the equation $$ 16\{ x \}^2-8x=-1, $$ where $ \{\} $ denotes the fractional part.

A polynomial $ P(x)$ with integer coefficients is called good,if it can be represented as a sum of cubes of several polynomials (in variable $ x$) with integer coefficients.For example,the polynomials $ x^3 \minus{} 1$ and $ 9x^3 \minus{} 3x^2 \plus{} 3x \plus{} 7 \equal{} (x \minus{} 1)^3 \plus{} (2x)^3 \plus{} 2^3$ are good. a)Is the polynomial $ P(x) \equal{} 3x \plus{} 3x^7$ good? b)Is the polynomial $ P(x) \equal{} 3x \plus{} 3x^7 \plus{} 3x^{2008}$ good? Justify your answers.

Find the smallest odd integer $ k$ such that: for every $ 3\minus{}$degree polynomials $ f$ with integer coefficients, if there exist $ k$ integer $ n$ such that $ |f(n)|$ is a prime number, then $ f$ is irreducible in $ \mathbb{Z}[n]$.

Find all $a\in\mathbb{N}$ such that exists a bijective function $g :\mathbb{N} \to \mathbb{N}$ and a function $f:\mathbb{N}\to\mathbb{N}$, such that for all $x\in\mathbb{N}$, $$f(f(f(...f(x)))...)=g(x)+a$$ where $f$ appears $2009$ times. [i](tatari/nightmare)[/i]

Let $a$ and $b$ be positive integers. Prove that the numbers $an^2+b$ and $a(n+1)^2+b$ are both perfect squares only for finitely many integers $n$.

Given complex numbers $a,b,c$, we have that $|az^2 + bz +c| \leq 1$ holds true for any complex number $z, |z| \leq 1$. Find the maximum value of $|bc|$.

Solve the equation $a + b + 4 = 4\sqrt{a\sqrt{b}}$ in real numbers

Let $n > 1$ be an odd positive integer. Assume that positive integers $x_1, x_2,..., x_n \ge 0$ satisfy: $$\begin{cases} (x_2 - x_1)^2 + 2(x_2 +x_1) + 1 = n^2 \\ (x_3 -x_2)^2 + 2(x_3 +x_2) + 1 = n^2 \\ ...\\ (x_1 - x_n)^2 + 2(x_1 + x_n)+ 1 = n^2 \end {cases}$$ Show that there exists $j, 1 \le j \le n$, such that $x_j = x_{j+1}$. Here $x_{n+1} = x_1$.

Find all values of $a$ such that $x^6 - 6x^5 + 12x^4 + ax^3 + 12x^2 - 6x +1$ is nonnegative for all real $x$.

[u]Round 1[/u] [b]1.1.[/b] What is the area of a circle with diameter $2$? [b]1.2.[/b] What is the slope of the line through $(2, 1)$ and $(3, 4)$? [b]1.3.[/b] What is the units digit of $2^2 \cdot 4^4 \cdot 6^6$ ? [u]Round 2[/u] [b]2.1.[/b] Find the sum of the roots of $x^2 - 5x + 6$. [b]2. 2.[/b] Find the sum of the solutions to $|2 - x| = 1$. [b]2.3.[/b] On April $1$, $2018$, Mr. Dospinescu, Mr. Phaovibul and Mr. Pohoata all go swimming at the same pool. From then on, Mr. Dospinescu returns to the pool every 4th day, Mr. Phaovibul returns every $7$th day and Mr. Pohoata returns every $13$th day. What day will all three meet each other at the pool again? Give both the month and the day. [u]Round 3[/u] [b]3. 1.[/b] Kendall and Kylie are each selling t-shirts separately. Initially, they both sell t-shirts for $\$ 33$ each. A week later, Kendall marks up her t-shirt price by $30 \%$, but after seeing a drop in sales, she discounts her price by $30\%$ the following week. If Kim wants to buy $360$ t-shirts, how much money would she save by buying from Kendall instead of Kylie? Write your answer in dollars and cents. [b]3.2.[/b] Richard has English, Math, Science, Spanish, History, and Lunch. Each class is to be scheduled into one distinct block during the day. There are six blocks in a day. How many ways could he schedule his classes such that his lunch block is either the $3$rd or $4$th block of the day? [b]3.3.[/b] How many lattice points does $y = 1 + \frac{13}{17}x$ pass through for $x \in [-100, 100]$ ? (A lattice point is a point where both coordinates are integers.) [u]Round 4[/u] [b]4. 1.[/b] Unsurprisingly, Aaron is having trouble getting a girlfriend. Whenever he asks a girl out, there is an eighty percent chance she bursts out laughing in his face and walks away, and a twenty percent chance that she feels bad enough for him to go with him. However, Aaron is also a player, and continues asking girls out regardless of whether or not previous ones said yes. What is the minimum number of girls Aaron must ask out for there to be at least a fifty percent chance he gets at least one girl to say yes? [b]4.2.[/b] Nithin and Aaron are two waiters who are working at the local restaurant. On any given day, they may be fired for poor service. Since Aaron is a veteran who has learned his profession well, the chance of him being fired is only $\frac{2}{25}$ every day. On the other hand, Nithin (who never paid attention during job training) is very lazy and finds himself constantly making mistakes, and therefore the chance of him being fired is $\frac{2}{5}$. Given that after 1 day at least one of the waiters was fired, find the probability Nithin was fired. [b]4.3.[/b] In a right triangle, with both legs $4$, what is the sum of the areas of the smallest and largest squares that can be inscribed? An inscribed square is one whose four vertices are all on the sides of the triangle. PS. You should use hide for answers. Rounds 5-8 have been posted [url=https://artofproblemsolving.com/community/c3h2784569p24468582]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Prove that (a) for each $n \ge 1$ $$\sum_{k=0}^n C_{n}^{k} \left(\frac{k}{n}-\frac{1}{2} \right)^2 \frac{1}{2^n}=\frac{1}{4n}$$ (b) for every n \ge m \ge 2 $$\sum_{\ell=0}^n \sum_{k_1+...+k_n=\ell,k_i=0,...,m} \frac{\ell!}{k_1!...k_n!} \frac{1}{(m+1)^n} \left(\frac{\ell}{n}-\frac{m}{2} \right)^2= \left(\frac{m^3-3m^2}{12(m+1)}+\frac{m}{2}-\frac{m}{3(m+1)}\right)n$$