Real numbers $x_1, x_2, x_3, x_4$ are roots of the fourth degree polynomial $W (x)$ with integer coefficients. Prove that if $x_3 + x_4$ is a rational number and $x_3x_4$ is a irrational number, then $x_1 + x_2 = x_3 + x_4$.

$p(x) $ is a real polynomial of degree $3$. Find necessary and sufficient conditions on its coefficients in order that $p(n)$ is integral for every integer $n$.

Prove that in any set of $2000$ distinct real numbers there exist two pairs $a>b$ and $c>d$ with $a \neq c$ or $b \neq d $, such that \[ \left| \frac{a-b}{c-d} - 1 \right|< \frac{1}{100000}. \]

Let $a, b, c$ be positive reals such that their sum is $1$. Prove that \[\frac{1}{ab+2c^{2}+2c}+\frac{1}{bc+2a^{2}+2a}+\frac{1}{ac+2b^{2}+2b}\geq \frac{1}{ab+bc+ac}.\]

Let $ f(x) \equal{} 1 \plus{} x \plus{} x^2 \plus{} \cdots \plus{} x^{100}$. Find $ f'(1)$.

Given finitely many open half planes on the Euclidean plane. The boundary lines of these half planes divide the plane into convex domains. Find a polynomial $C(q)$ of degree two so that the following holds: for any $q\ge 1$ integer, if the half planes cover each point of the plane at least $q$ times, then the set of points covered exactly $q$ times is the union of at most $C(q)$ domains. (translated by L. Erdős)

For any system $ x_1,x_2,...,x_n$ of positive real numbers, let $ f_1(x_1,x_2,...,x_n) \equal{} x_1$, and $ f_{\nu} \equal{} \frac {x_1 \plus{} 2x_2 \plus{} \cdots \plus{} \nu x_{\nu}}{\nu \plus{} (\nu \minus{} 1)x_1 \plus{} (\nu \minus{} 2)x_2 \plus{} \cdots \plus{} 1\cdot x_{\nu \minus{} 1}}$ for $ \nu \equal{} 2,3,...,n$. Show that for any $ \epsilon > 0$, a positive integer $ n_0 < n_0(\epsilon)$ can be found such that for every $ n > n_0$ there exists a system $ x_1',x_2',...,x_n'$ of positive real numbers with $ x_1' \plus{} x_2' \plus{} \cdots \plus{} x_n' \equal{} 1$ and $ f_{\nu}(x_1',x_2',...,x_n')\le \epsilon$ for $ \nu \equal{} 1,2,...,n$ .

Prove the inequality $x_1y_1+x_2y_2+x_2y_1+2x_2y_2\le 1996$ if $x_1^2+2x_1x_2+2x_2^2\le 998$ and $y_1^2+2y_1y_2+2y_2^2\le 3992$.

A real number $x$ satisfies $9^x + 3^x = 6$. Compute the value of $16^{1/x} + 4^{1/x} $.

Suppose that $4^{x_1}= 5, 5^{x_2}= 6,6^{x_3} = 7,..., 126^{x_{123}} = 127,127^{x_{124}} = 128$. What is the value of the product $X_1X_2... X_{124}$?

For positive real numbers $x$ and $y$, define their special mean to be average of their arithmetic and geometric means. Find the total number of pairs of integers $(x, y)$, with $x \le y$, from the set of numbers $\{1,2,...,2016\}$, such that the special mean of $x$ and $y$ is a perfect square.

Find all polynomials $P$ such that $P(x)+3P(x+2)=3P(x+1)+P(x+3)$ for all real numbers $x$.

If the product of $(\sqrt2 +\sqrt3+\sqrt5) (\sqrt2 +\sqrt3-\sqrt5) (\sqrt2 -\sqrt3+\sqrt5) (-\sqrt2 +\sqrt3+\sqrt5)$ is $12\sqrt6+ 6\sqrt{x}$ , find $x$. ([i]0 points[/i] - [b]THROWN OUT[/b])

suppose that $A$ is the set of all Closed intervals $[a,b] \subset \mathbb{R}$. Find all functions $f:\mathbb{R} \rightarrow A$ such that $\bullet$ $x \in f(y) \Leftrightarrow y \in f(x)$ $\bullet$ $|x-y|>2 \Leftrightarrow f(x) \cap f(y)=\varnothing$ $\bullet$ For all real numbers $0\leq r\leq 1$, $f(r)=[r^2-1,r^2+1]$ Proposed by Matin Yousefi

Find all functions $f: \mathbb{R} \to \mathbb{R}$ such that \[ f(xy) = yf(x) + x + f(f(y) - f(x)) \] for all $x,y \in \mathbb{R}$.

It is known that a polynomial $P$ with integer coefficients has degree $2022$. What is the maximum $n$ such that there exist integers $a_1, a_2, \cdots a_n$ with $P(a_i)=i$ for all $1\le i\le n$? [Extra: What happens if $P \in \mathbb{Q}[X]$ and $a_i\in \mathbb{Q}$ instead?]

Let $ u_1, u_2, \ldots, u_m$ be $ m$ vectors in the plane, each of length $ \leq 1,$ with zero sum. Show that one can arrange $ u_1, u_2, \ldots, u_m$ as a sequence $ v_1, v_2, \ldots, v_m$ such that each partial sum $ v_1, v_1 \plus{} v_2, v_1 \plus{} v_2 \plus{} v_3, \ldots, v_1, v_2, \ldots, v_m$ has length less than or equal to $ \sqrt {5}.$

Find all natural values of $k$ for which the system $$\begin{cases}x_1+x_2+\ldots+x_k=9\\\frac1{x_1}+\frac1{x_2}+\ldots+\frac1{x_k}=1\end{cases}$$ has solutions in positive numbers. Find these solutions. [i]I. Dimovski[/i]

Show that there is no integer-valued function on the integers such that $f(m+f(n))=f(m)-n$ for all $m,n$.

Evaluate $$\frac{(2 + 2)^2}{2^2} \cdot \frac{(3 + 3 + 3 + 3)^3}{(3 + 3 + 3)^3} \cdot \frac{(6 + 6 + 6 + 6 + 6 + 6)^6}{(6 + 6 + 6 + 6)^6}$$

Determine all triples of positive integers $(A,B,C)$ for which some function $f \colon \mathbb Z_{\geq 0} \to \mathbb Z_{\geq 0}$ satisfies \[ f^{f(y)} (y + f(2x)) + f^{f(y)} (2y) = (Ax+By)^{C} \] for all nonnegative integers $x$ and $y$, where $f^k$ as usual denotes $f$ composed $k$ times. Proposed by [i]Benny Wang[/i]

Derek and Jacob have a cake in the shape a rectangle with dimensions $14$ inches by $9$ inches. They make a deal to split it: Derek takes home the portion of the cake that is less than one inch from the border, while Jacob takes home the remainder of the cake. Let $D : J$ be the ratio of the amount of cake Derek took to the amount of cake Jacob took, where $D$ and $J$ are relatively prime positive integers. Find $D + J$.

Let $n$ be a positive integer and $u_1,u_2,\cdots ,u_n$ be positive integers not larger than $2^k, $ for some integer $k\geq 3.$ A representation of a non-negative integer $t$ is a sequence of non-negative integers $a_1,a_2,\cdots ,a_n$ such that $t=a_1u_1+a_2u_2+\cdots +a_nu_n.$ Prove that if a non-negative integer $t$ has a representation,then it also has a representation where less than $2k$ of numbers $a_1,a_2,\cdots ,a_n$ are non-zero.

Let $n$ be an integer satisfying $n\geq2$. Find the greatest integer not exceeding the expression: $E=1+\sqrt{1+\frac{2^2}{3!}}+\sqrt[3]{1+\frac{3^2}{4!}}+\dots+\+\sqrt[n]{1+\frac{n^2}{(n+1)!}}$

[b]p1.[/b] BmMT is in a week, and we don’t have any problems! Let’s write $1$ on the first day, $2$ on the second day, $4$ on the third, $ 8$ on the fourth, $16$ on the fifth, $32$ on the sixth, and $64$ on the seventh. After seven days, how many problems will we have written in total? [b]p2.[/b] $100$ students are taking a ten-point exam. $50$ students scored $8$ points, $30$ students scored $7$ points, and the rest scored $9$ points. What is the average score for the exam? [b]p3.[/b] Rebecca has four pairs of shoes. Rebecca may or may not wear matching shoes. However, she will always use a left-shoe for her left foot and a right-shoe for her right foot. How many ways can Rebecca wear shoes? [b]p4.[/b] A council of $111$ mathematicians voted on whether to hold their conference in Beijing or Shanghai. The outcome of an initial vote was $70$ votes in favor of Beijing, and 41 votes in favor of Shanghai. If the vote were to be held again, what is the minimum number of mathematicians that would have to change their votes in order for Shanghai to win a majority of votes? [b]p5.[/b] What is the area of the triangle bounded by the line $20x + 16y = 160$, the $x$-axis, and the $y$-axis? [b]p6.[/b] Suppose that $3$ runners start running from the start line around a circular $800$-meter track and that their speeds are $100$, $160$, and $200$ meters per minute, respectively. How many minutes will they run before all three are next at the start line at the same time? [b]p7.[/b] Brian’s lawn is in the shape of a circle, with radius $10$ meters. Brian can throw a frisbee up to $50$ meters from where he stands. What is the area of the region (in square meters) in which the frisbee can land, if Brian can stand anywhere on his lawn? [b]p8.[/b] A seven digit number is called “bad” if exactly four of its digits are $0$ and the rest are odd. How many seven digit numbers are bad? [b]p9.[/b] Suppose you have a $3$-digit number with only even digits. What is the probability that twice that number also has only even digits? [b]p10.[/b] You have a flight on Air China from Beijing to New York. The flight will depart any time between $ 1$ p.m. and $6$ p.m., uniformly at random. Your friend, Henry, is flying American Airlines, also from Beijing to New York. Henry’s flight will depart any time between $3$ p.m. and $5$ p.m., uniformly at random. What is the probability that Henry’s flight departs before your flight? [b]p11.[/b] In the figure below, three semicircles are drawn outside the given right triangle. Given the areas $A_1 = 17$ and $A_2 = 14$, find the area $A_3$. [img]https://cdn.artofproblemsolving.com/attachments/4/4/28393acb3eba83a5a489e14b30a3e84ffa60fb.png[/img] [b]p12.[/b] Consider a circle of radius $ 1$ drawn tangent to the positive $x$ and $y$ axes. Now consider another smaller circle tangent to that circle and also tangent to the positive $x$ and $y$ axes. Find the radius of the smaller circle. [img]https://cdn.artofproblemsolving.com/attachments/7/4/99b613d6d570db7ee0b969f57103d352118112.png[/img] [b]p13.[/b] The following expression is an integer. Find this integer: $\frac{\sqrt{20 + 16\frac{\sqrt{20+ 16\frac{20 + 16...}{2}}}{2}}}{2}$ [b]p14.[/b] Let $2016 = a_1 \times a_2 \times ... \times a_n$ for some positive integers $a_1, a_2, ... , a_n$. Compute the smallest possible value of $a_1 + a_2 + ...+ a_n$. [b]p15.[/b] The tetranacci numbers are defined by the recurrence $T_n = T_{n-1} + T_{n-2} + T_{n-3} + T_{n-4}$ and $T_0 = T_1 = T_2 = 0$ and $T_3 = 1$. Given that $T_9 = 29$ and $T_{14} = 773$, calculate $T_{15}$. [b]p16.[/b] Find the number of zeros at the end of $(2016!)^{2016}$. Your answer should be an integer, not its prime factorization. [b]p17.[/b] A DJ has $7$ songs named $1, 2, 3, 4, 5, 6$, and $7$. He decides that no two even-numbered songs can be played one after the other. In how many different orders can the DJ play the $7$ songs? [b]p18.[/b] Given a cube, how many distinct ways are there (using $6$ colors) to color each face a distinct color? Colorings are distinct if they cannot be transformed into one another by a sequence of rotations. [b]p19. [/b]Suppose you have a triangle with side lengths $3, 4$, and $5$. For each of the triangle’s sides, draw a square on its outside. Connect the adjacent vertices in order, forming $3$ new triangles (as in the diagram). What is the area of this convex region? [img]https://cdn.artofproblemsolving.com/attachments/4/c/ac4dfb91cd055badc07caface93761453049fa.png[/img] [b]p20.[/b] Find $x$ such that $\sqrt{c +\sqrt{c - x}} = x$ when $c = 4$. PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].