Let $ \theta$ be an angle in the interval $ (0,\pi/2)$. Given that $ \cos \theta$ is irrational, and that $ \cos k \theta$ and $ \cos[(k \plus{} 1)\theta ]$ are both rational for some positive integer $ k$, show that $ \theta \equal{} \pi/6$.

Let $m,n\geq 2$ be integers. Let $f(x_1,\dots, x_n)$ be a polynomial with real coefficients such that $$f(x_1,\dots, x_n)=\left\lfloor \frac{x_1+\dots + x_n}{m} \right\rfloor\text{ for every } x_1,\dots, x_n\in \{0,1,\dots, m-1\}.$$ Prove that the total degree of $f$ is at least $n$.

Let $ a,b \in [0,1]$. Prove that \[ \frac 1{1\plus{}a\plus{}b} \leq 1 \minus{} \frac {a\plus{}b}2 \plus{} \frac {ab}3.\]

For an integer $n\ge 2$ find all $a_1,a_2,\cdots ,a_n, b_1,b_2,\cdots , b_n$ so that (a) $0\le a_1\le a_2\le \cdots \le a_n\le 1\le b_1\le b_2\le \cdots \le b_n;$ (b) $\sum_{k=1}^n (a_k+b_k)=2n;$ (c) $\sum_{k=1}^n (a_k^2+b_k^2)=n^2+3n.$

Solve the system of equations:$$\left\{ \begin{array}{l} ({x^2} + y)\sqrt {y - 2x} - 4 = 2{x^2} + 2x + y\\ {x^3} - {x^2} - y + 6 = 4\sqrt {x + 1} + 2\sqrt {y - 1} \end{array} \right.(x,y \in \mathbb{R}).$$

Reduce $$\frac{2}{\sqrt{4-3\sqrt[4]{5} + 2\sqrt[4]{25}-\sqrt[4]{125}}}$$ to its lowest form. Then generalize this result and show that it holds for any positive $n$.

Real numbers $a,b$ and $c$ satisfy the equalities $2015 (a + b + c) =1$ and $ab+bc+ca=2015 abc$. Determine the numeric value of the expression $E=a^{2015}+b^{2015}+c^{2015}.$

Let $(m,n)$ be the pairs of integers satisfying $2(8n^3+m^3)+6(m^2-6n^2)+3(2m+9n)=437$. Find the sum of all possible values of $mn$.

Find all pairs of real numbers $(x,y)$ satisfying $\left\{\begin{array}{rl} 2x^2+xy &=1 \\ \frac{9x^2}{2(1-x)^4}&=1+\frac{3xy}{2(1-x)^2} \end{array}\right.$

Let $u_1, u_2, \dots, u_{2019}$ be real numbers satisfying \[u_{1}+u_{2}+\cdots+u_{2019}=0 \quad \text { and } \quad u_{1}^{2}+u_{2}^{2}+\cdots+u_{2019}^{2}=1.\] Let $a=\min \left(u_{1}, u_{2}, \ldots, u_{2019}\right)$ and $b=\max \left(u_{1}, u_{2}, \ldots, u_{2019}\right)$. Prove that \[ a b \leqslant-\frac{1}{2019}. \]

Prove that the number $x^8+\frac{1}{x^8}$ is an integer if $x+\frac{1}{x }$ is an integer.

Let $n\geqslant 3$ be a fixed integer and $\omega=\cos\dfrac{2\pi}n+i\sin\dfrac{2\pi}n$. Show that for every $a\in\mathbb{C}$ and $r>0$, the number \[\sum\limits_{k=1}^n \dfrac{|a-r\omega^k|^2}{|a|^2+r^2}\] is an integer. Interpet this result geometrically. [i]Octavian Stănășilă[/i]

A train accelerates at $10$ mph/min, and decelerates at $20$ mph/min. The train’s maximum speed is $300$ mph. What’s the shortest amount of the time that the train could take to travel $500$ miles, if it has to be stationary at both the start and end of its trip? Please give your answer in minutes.

Find all pariwise distinct real numbers $x,y,z$ such that $\left\{\frac{x-y}{y-z},\frac{y-z}{z-x},\frac{z-x}{x-y} \right\} = \{x,y,z\}$. (It means, those three fractions make a permutation of $x, y$, and $z$.)

Let $a$ and $b$ be two positive real numbers with $a \le 2b \le 4a$. Prove that $4ab \le2 (a^2+ b^2) \le 5 ab$.

Let $A$ be a non-empty set of positive integers. Suppose that there are positive integers $b_1,\ldots b_n$ and $c_1,\ldots,c_n$ such that - for each $i$ the set $b_iA+c_i=\left\{b_ia+c_i\colon a\in A\right\}$ is a subset of $A$, and - the sets $b_iA+c_i$ and $b_jA+c_j$ are disjoint whenever $i\ne j$ Prove that \[{1\over b_1}+\,\ldots\,+{1\over b_n}\leq1.\]

A [i]magic square[/i] is a table [img]https://cdn.artofproblemsolving.com/attachments/7/9/3b1e2b2f5d2d4c486f57c4ad68b66f7d7e56dd.png[/img] in which all the natural numbers from $1$ to $16$ appear and such that: $\bullet$ all rows have the same sum $s$. $\bullet$ all columns have the same sum $s$. $\bullet$ both diagonals have the same sum $s$ . It is known that $a_{22} = 1$ and $a_{24} = 2$. Calculate $a_{44}$.

Find the least possible value of $(x + y)(y + z)$ for positive reals satisfying $(x + y + z) xyz = 1$.

Let the real numbers $a,b,c,d$ satisfy the relations $a+b+c+d=6$ and $a^2+b^2+c^2+d^2=12.$ Prove that \[36 \leq 4 \left(a^3+b^3+c^3+d^3\right) - \left(a^4+b^4+c^4+d^4 \right) \leq 48.\] [i]Proposed by Nazar Serdyuk, Ukraine[/i]

Let $P$ be the set of points in the Euclidean plane, and let $L$ be the set of lines in the same plane. Does there exist an one-to-one mapping (injective function) $f : L \to P$ such that for each $\ell \in L$ we have $f(\ell) \in \ell$?

Given a real number $\alpha$, a function $f$ is defined on pairs of nonnegative integers by $f(0,0) = 1, f(m,0) = f(0,m) = 0$ for $m > 0$, $f(m,n) = \alpha f(m,n-1)+(1- \alpha)f(m -1,n-1)$ for $m,n > 0$. Find the values of $\alpha$ such that $| f(m,n)| < 1989$ holds for any integers $m,n \ge 0$.

Determine all polynomials $P(x,y)$ with real coefficients such that \[P(x+y,x-y)=2P(x,y) \qquad \forall x,y\in\mathbb{R}.\]

Do positive reals $a, b, c, x$ such that $a^2+ b^2 = c^2$ and $(a + x)^2+ (b +x)^2 = (c + x)^2$ exist?

p1. A telephone number with $7$ digits is called a [i]Beautiful Number [/i]if the digits are which appears in the first three numbers (the three must be different) repeats on the next three digits or the last three digits. For example some beautiful numbers: $7133719$, $7131735$, $7130713$, $1739317$, $5433354$. If the numbers are taken from $0, 1, 2, 3, 4, 5, 6, 7, 8$ or $9$, but the number the first cannot be $0$, how many Beautiful Numbers can there be obtained? p2. Find the number of natural numbers $n$ such that $n^3 + 100$ is divisible by $n +10$ p3. A function $f$ is defined as in the following table. [img]https://cdn.artofproblemsolving.com/attachments/5/5/620d18d312c1709b00be74543b390bfb5a8edc.png[/img] Based on the definition of the function $f$ above, then a sequence is defined on the general formula for the terms is as follows: $U_1=2$ and $U_{n+1}=f(U_n)$ , for $n = 1, 2, 3, ...$ p4. In a triangle $ABC$, point $D$ lies on side $AB$ and point $E$ lies on side $AC$. Prove for the ratio of areas: $\frac{ADE }{ABC}=\frac{AD\times AE}{AB\times AC}$ p5. In a chess tournament, a player only plays once with another player. A player scores $1$ if he wins, $0$ if he loses, and $\frac12$ if it's a draw. After the competition ended, it was discovered that $\frac12$ of the total value that earned by each player is obtained from playing with 10 different players who got the lowest total points. Especially for those in rank bottom ten, $\frac12$ of the total score one gets is obtained from playing with $9$ other players. How many players are there in the competition?

Suppose $a,b,c$ are real numbers so that $a+b+c=15$ and $ab+ac+bc=27$. Find the range of values that may be obtained by the expression $abc$.