Suppose $ A_1,\dots, A_6$ are six sets each with four elements and $ B_1,\dots,B_n$ are $ n$ sets each with two elements, Let $ S \equal{} A_1 \cup A_2 \cup \cdots \cup A_6 \equal{} B_1 \cup \cdots \cup B_n$. Given that each elements of $ S$ belogs to exactly four of the $ A$'s and to exactly three of the $ B$'s, find $ n$.

Consider $n \geq 2 $ positive numbers $0<x_1 \leq x_2 \leq ... \leq x_n$, such that $x_1 + x_2 + ... + x_n = 1$. Prove that if $x_n \leq \dfrac{2}{3}$, then there exists a positive integer $1 \leq k \leq n$ such that $\dfrac{1}{3} \leq x_1+x_2+...+x_k < \dfrac{2}{3}$.

Let $a,b,c>0$ be real numbers. Prove that $$\frac{a+b}{\sqrt{b+c}}+\frac{b+c}{\sqrt{c+a}}+\frac{c+a}{\sqrt{a+b}}\geq \sqrt{2a}+ \sqrt{2b}+ \sqrt{2c}$$ Б. Кайрат (Казахстан), А. Храбров

Find all real $a$ such that there exists a function $f: R \to R$ satisfying the equation $f(\sin x )+ a f(\cos x) = \cos 2x$ for all real $x$. I.Voronovich

[b]p1.[/b] Two players play the following game. On the lowest left square of an $8\times 8$ chessboard there is a rook. The first player is allowed to move the rook up or to the right by an arbitrary number of squares. The second player is also allowed to move the rook up or to the right by an arbitrary number of squares. Then the first player is allowed to do this again, and so on. The one who moves the rook to the upper right square wins. Who has a winning strategy? [b]p2.[/b] In Crocodile Country there are banknotes of $1$ dollar, $10$ dollars, $100$ dollars, and $1,000$ dollars. Is it possible to get 1,000,000 dollars by using $250,000$ banknotes? [b]p3.[/b] Fifteen positive numbers (not necessarily whole numbers) are placed around the circle. It is known that the sum of every four consecutive numbers is $30$. Prove that each number is less than $15$. [b]p4.[/b] Donald Duck has $100$ sticks, each of which has length $1$ cm or $3$ cm. Prove that he can break into $2$ pieces no more than one stick, after which he can compose a rectangle using all sticks. [b]p5.[/b] Three consecutive $2$ digit numbers are written next to each other. It turns out that the resulting $6$ digit number is divisible by $17$. Find all such numbers. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

For any real number $a$, find all real numbers $x$ that satisfy the following equation. $$(2x + 1)^4 + ax(x + 1) - \frac{x}{2}= 0$$

A line segment is divided so that the lesser part is to the greater part as the greater part is to the whole. If $ R$ is the ratio of the lesser part to the greater part, then the value of \[ R^{[R^{(R^2\plus{}R^{\minus{}1})}\plus{}R^{\minus{}1}]}\plus{}R^{\minus{}1}\] is $ \textbf{(A)}\ 2 \qquad \textbf{(B)}\ 2R \qquad \textbf{(C)}\ R^{\minus{}1} \qquad \textbf{(D)}\ 2\plus{}R^{\minus{}1} \qquad \textbf{(E)}\ 2\plus{}R$

Consider the sequence $z_n = (1+i)(2+i)...(n+i)$. Prove that the sequence $Im$ $z_n$ contains infinitely many positive and infinitely many negative numbers.

If the coefficient of the third term in the binomial expansion of $(1 - 3x)^{1/4}$ is $-a/b$, where $ a$ and $b$ are relatively prime integers, find $a+b$.

Find a polynomial $ p\left(x\right)$ with real coefficients such that $ \left(x\plus{}10\right)p\left(2x\right)\equal{}\left(8x\minus{}32\right)p\left(x\plus{}6\right)$ for all real $ x$ and $ p\left(1\right)\equal{}210$.

Let $P \in Q[x]$ be a polynomial of degree $2016$ whose leading coefficient is $1$. A positive integer $m$ is [i]nice [/i] if there exists some positive integer $n$ such that $m = n^3 + 3n + 1$. Suppose that there exist infinitely many positive integers $n$ such that $P(n)$ are nice. Prove that there exists an arithmetic sequence $(n_k)$ of arbitrary length such that $P(n_k)$ are all nice for $k = 1,2, 3$,

A polynomial $P(x)$ with real coefficients and degree $2021$ is given. For any real $a$ polynomial $x^{2022}+aP(x)$ has at least one real root. Find all possible values of $P(0)$

Determine all polynomials $P$ with real coefficients satisfying the following condition: whenever $x$ and $y$ are real numbers such that $P(x)$ and $P(y)$ are both rational, so is $P(x + y)$.

Let $0<k<\frac{1}{2}$ be a real number and let $a_0, b_0$ be arbitrary real numbers in $(0,1)$. The sequences $(a_n)_{n\ge 0}$ and $(b_n)_{n\ge 0}$ are then defined recursively by $$a_{n+1} = \dfrac{a_n+1}{2} \text{ and } b_{n+1} = b_n^k$$ for $n\ge 0$. Prove that $a_n<b_n$ for all sufficiently large $n$. [i]Proposed by Michael Ma

Let $A,B$ be matrices of dimension $2010\times2010$ which commute and have real entries, such that $A^{2010}=B^{2010}=I$, where $I$ is the identity matrix. Prove that if $\operatorname{tr}(AB)=2010$, then $\operatorname{tr}(A)=\operatorname{tr}(B)$.

Find all prime number $p$ such that there exists an integer-coefficient polynomial $f(x)=x^{p-1}+a_{p-2}x^{p-2}+…+a_1x+a_0$ that has $p-1$ consecutive positive integer roots and $p^2\mid f(i)f(-i)$, where $i$ is the imaginary unit.

[b]p1.[/b] How many real roots does the equation $2x^7 + x^5 + 4x^3 + x + 2 = 0$ have? [b]p2.[/b] Given that $f(n) = 1 +\sum^n_{j=1}(1 +\sum^j_{i=1}(2i + 1))$, find the value of $f(99)-\sum^{99}_{i=1} i^2$. [b]p3.[/b] A rectangular prism with dimensions $1\times a \times b$, where $1 < a < b < 2$, is bisected by a plane bisecting the longest edges of the prism. One of the smaller prisms is bisected in the same way. If all three resulting prisms are similar to each other and to the original box, compute $ab$. Note: Two rectangular prisms of dimensions $p \times q\times r$ and$ x\times y\times z$ are similar if $\frac{p}{x} = \frac{q}{y} = \frac{r}{z}$ . [b]p4.[/b] For fixed real values of $p$, $q$, $r$ and $s$, the polynomial $x^4 + px^3 + qx^2 + rx + s$ has four non real roots. The sum of two of these roots is $4 + 7i$, and the product of the other two roots is $3 - 4i$. Compute $q$. [b]p5.[/b] There are $10$ seats in a row in a theater. Say we have an infinite supply of indistinguishable good kids and bad kids. How many ways can we seat $10$ kids such that no two bad kids are allowed to sit next to each other? [b]p6.[/b] There are an infinite number of people playing a game. They each pick a different positive integer $k$, and they each win the amount they chose with probability $\frac{1}{k^3}$ . What is the expected amount that all of the people win in total? [b]p7.[/b] There are $100$ donuts to be split among $4$ teams. Your team gets to propose a solution about how the donuts are divided amongst the teams. (Donuts may not be split.) After seeing the proposition, every team either votes in favor or against the propisition. The proposition is adopted with a majority vote or a tie. If the proposition is rejected, your team is eliminated and will never receive any donuts. Another remaining team is chosen at random to make a proposition, and the process is repeated until a proposition is adopted, or only one team is left. No promises or deals need to be kept among teams besides official propositions and votes. Given that all teams play optimally to maximize the expected value of the number of donuts they receive, are completely indifferent as to the success of the other teams, but they would rather not eliminate a team than eliminate one (if the number of donuts they receive is the same either way), then how much should your team propose to keep? [b]p8.[/b] Dominic, Mitchell, and Sitharthan are having an argument. Each of them is either credible or not credible – if they are credible then they are telling the truth. Otherwise, it is not known whether they are telling the truth. At least one of Dominic, Mitchell, and Sitharthan is credible. Tim knows whether Dominic is credible, and Ethan knows whether Sitharthan is credible. The following conversation occurs, and Tim and Ethan overhear: Dominic: “Sitharthan is not credible.” Mitchell: “Dominic is not credible.” Sitharthan: “At least one of Dominic or Mitchell is credible.” Then, at the same time, Tim and Ethan both simultaneously exclaim: “I can’t tell exactly who is credible!” They each then think for a moment, and they realize that they can. If Tim and Ethan always tell the truth, then write on your answer sheet exactly which of the other three are credible. [b]p9.[/b] Pick an integer $n$ between $1$ and $10$. If no other team picks the same number, we’ll give you $\frac{n}{10}$ points. [b]p10.[/b] Many quantities in high-school mathematics are left undefined. Propose a definition or value for the following expressions and justify your response for each. We’ll give you $\frac15$ points for each reasonable argument. $$(i) \,\,\,(.5)! \,\,\, \,\,\,(ii) \,\,\,\infty \cdot 0 \,\,\, \,\,\,(iii) \,\,\,0^0 \,\,\, \,\,\,(iv)\,\,\, \prod_{x\in \emptyset}x \,\,\, \,\,\,(v)\,\,\, 1 - 1 + 1 - 1 + ...$$ [b]p11.[/b] On the back of your answer sheet, write the “coolest” math question you know, and include the solution. If the graders like your question the most, then you’ll get a point. (With your permission, we might include your question on the Mixer next year!) PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Prove that there does not exist a function $f : \mathbb R^+\to\mathbb R^+$ such that \[f(f(x)+y)=f(x)+3x+yf(y)\] for all positive reals $x,y$.

Find all functions $f:\mathbb{R}\rightarrow \mathbb{R}$ such that for all $x,y\in \mathbb{R}$: $$f\left(f(x)^2-y^2\right)^2+f(2xy)^2=f\left(x^2+y^2\right)^2$$ [i]Proposed by Ali Behrouz - Mojtaba Zare Bidaki[/i]

Consider all quadratic trinomials $x^2+px+q$ with $p,q\in\{1,\ldots,2001\}$. Which of them has more elements: those having integer roots, or those having no real roots?

For how many ordered pairs $(x,y)$ of integers is it true that $0<x<y<10^{6}$ and that the arithmetic mean of $x$ and $y$ is exactly $2$ more than the geometric mean of $x$ and $y?$

The sequence of real numbers $a_1,a_2,\dots$ is defined as follows: $a_1=56$ and $a_{n+1}=a_n-\frac{1}{a_n}$ for $n\ge 1$. Show that there is an integer $1\leq{k}\leq2002$ such that $a_k<0$.

A two-pan balance is innacurate since its balance arms are of different lengths and its pans are of different weights. Three objects of different weights $A$, $B$, and $C$ are each weighed separately. When placed on the left-hand pan, they are balanced by weights $A_1$, $B_1$, and $C_1$, respectively. When $A$ and $B$ are placed on the right-hand pan, they are balanced by $A_2$ and $B_2$, respectively. Determine the true weight of $C$ in terms of $A_1, B_1, C_1, A_2$, and $B_2$.

$x^3+(\log_2{5}+\log_3{2}+\log_5{3})x=(\log_2{3}+\log_3{5}+\log_5{2})x^2+1$

Does there exist any function $f: \mathbb{R}^+ \to \mathbb{R}$ such that for every positive real number $x,y$ the following is true : $$f(xf(x)+yf(y)) = xy$$