Reduce the number $\sqrt[3]{2 +\sqrt5} + \sqrt[3]{2 -\sqrt5}$.

Define the function $f:(0,1)\to (0,1)$ by \[\displaystyle f(x) = \left\{ \begin{array}{lr} x+\frac 12 & \text{if}\ \ x < \frac 12\\ x^2 & \text{if}\ \ x \ge \frac 12 \end{array} \right.\] Let $a$ and $b$ be two real numbers such that $0 < a < b < 1$. We define the sequences $a_n$ and $b_n$ by $a_0 = a, b_0 = b$, and $a_n = f( a_{n -1})$, $b_n = f (b_{n -1} )$ for $n > 0$. Show that there exists a positive integer $n$ such that \[(a_n - a_{n-1})(b_n-b_{n-1})<0.\] [i]Proposed by Denmark[/i]

Real numbers $a \neq 0, b, c$ are given. Prove that there is a polynomial $P(x)$ with real coefficients such that the polynomial $x^2+1$ divides the polynomial $aP(x)^2+bP(x)+c$. [i]Proposed by A. Golovanov[/i]

Prove that in every triangle with sides $a, b, c$ and opposite angles $A, B, C$, is fulfilled (measuring the angles in radians) $$\frac{a A+bB+cC}{a+b+c} \ge \frac{\pi}{3}$$ Hint: Use $a \ge b \ge c \Rightarrow A \ge B \ge C$.

[b]p1.[/b] Given that $7 \times 22 \times 13 = 2002$, compute $14 \times 11 \times 39$. [b]p2.[/b] Ariel the frog is on the top left square of a $8 \times 10$ grid of squares. Ariel can jump from any square on the grid to any adjacent square, including diagonally adjacent squares. What is the minimum number of jumps required so that Ariel reaches the bottom right corner? [b]p3.[/b] The distance between two floors in a building is the vertical distance from the bottom of one floor to the bottom of the other. In Evans hall, the distance from floor $7$ to floor $5$ is $30$ meters. There are $12$ floors on Evans hall and the distance between any two consecutive floors is the same. What is the distance, in meters, from the first floor of Evans hall to the $12$th floor of Evans hall? [b]p4.[/b] A circle of nonzero radius $ r$ has a circumference numerically equal to $\frac13$ of its area. What is its area? [b]p5.[/b] As an afternoon activity, Emilia will either play exactly two of four games (TwoWeeks, DigBuild, BelowSaga, and FlameSymbol) or work on homework for exactly one of three classes (CS61A, Math 1B, Anthro 3AC). How many choices of afternoon activities does Emilia have? [b]p6.[/b] Matthew wants to buy merchandise of his favorite show, Fortune Concave Decagon. He wants to buy figurines of the characters in the show, but he only has $30$ dollars to spend. If he can buy $2$ figurines for $4$ dollars and $5$ figurines for $8$ dollars, what is the maximum number of figurines that Matthew can buy? [b]p7.[/b] When Dylan is one mile from his house, a robber steals his wallet and starts to ride his motorcycle in the direction opposite from Dylan’s house at $40$ miles per hour. Dylan dashes home at $10$ miles per hour and, upon reaching his house, begins driving his car at $60$ miles per hour in the direction of the robber’s motorcycle. How long, starting from when the robber steals the wallet, does it take for Dylan to catch the robber? Express your answer in minutes. [b]p8.[/b] Deepak the Dog is tied with a leash of $7$ meters to a corner of his $4$ meter by $6$ meter rectangular shed such that Deepak is outside the shed. Deepak cannot go inside the shed, and the leash cannot go through the shed. Compute the area of the region that Deepak can travel to. [img]https://cdn.artofproblemsolving.com/attachments/f/8/1b9563776325e4e200c3a6d31886f4020b63fa.png[/img] [b]p9.[/b] The quadratic equation $a^2x^2 + 2ax -3 = 0$ has two solutions for x that differ by $a$, where $a > 0$. What is the value of $a$? [b]p10.[/b] Find the number of ways to color a $2 \times 2$ grid of squares with $4$ colors such that no two (nondiagonally) adjacent squares have the same color. Each square should be colored entirely with one color. Colorings that are rotations or reflections of each other should be considered different. [b]p11[/b]. Given that $\frac{1}{y^2+5} - \frac{3}{y^4-39} = 0$, and $y \ge 0$, compute $y$. [b]p12.[/b] Right triangle $ABC$ has $AB = 5$, $BC = 12$, and $CA = 13$. Point $D$ lies on the angle bisector of $\angle BAC$ such that $CD$ is parallel to $AB$. Compute the length of $BD$. [img]https://cdn.artofproblemsolving.com/attachments/c/3/d5cddb0e8ac43c35ddfc94b2a74b8d022292f2.png[/img] [b]p13.[/b] Let $x$ and $y$ be real numbers such that $xy = 4$ and $x^2y + xy^2 = 25$. Find the value of $x^3y +x^2y^2 + xy^3$. [b]p14.[/b] Shivani is planning a road trip in a car with special new tires made of solid rubber. Her tires are cylinders that are $6$ inches in width and have diameter $26$ inches, but need to be replaced when the diameter is less than $22$ inches. The tire manufacturer says that $0.12\pi$ cubic inches will wear away with every single rotation. Assuming that the tire manufacturer is correct about the wear rate of their tires, and that the tire maintains its cylindrical shape and width (losing volume by reducing radius), how many revolutions can each tire make before she needs to replace it? [b]p15.[/b] What’s the maximum number of circles of radius $4$ that fit into a $24 \times 15$ rectangle without overlap? [b]p16.[/b] Let $a_i$ for $1 \le i \le 10$ be a finite sequence of $10$ integers such that for all odd $i$, $a_i = 1$ or $-1$, and for all even $i$, $a_i = 1$, $-1$, or $0$. How many sequences a_i exist such that $a_1+a_2+a_3+...+a_{10} = 0$? [b]p17.[/b] Let $\vartriangle ABC$ be a right triangle with $\angle B = 90^o$ such that $AB$ and $BC$ have integer side lengths. Squares $ABDE$ and $BCFG$ lie outside $\vartriangle ABC$. If the area of $\vartriangle ABC$ is $12$, and the area of quadrilateral $DEFG$ is $38$, compute the perimeter of $\vartriangle ABC$. [img]https://cdn.artofproblemsolving.com/attachments/b/6/980d3ba7d0b43507856e581476e8ad91886656.png[/img] [b]p18.[/b] What is the smallest positive integer $x$ such that there exists an integer $y$ with $\sqrt{x} +\sqrt{y} = \sqrt{1025}$ ? [b]p19. [/b]Let $a =\underbrace{19191919...1919}_{19\,\, is\,\,repeated\,\, 3838\,\, times}$. What is the remainder when $a$ is divided by $13$? [b]p20.[/b] James is watching a movie at the cinema. The screen is on a wall and is $5$ meters tall with the bottom edge of the screen $1.5$ meters above the floor. The floor is sloped downwards at $15$ degrees towards the screen. James wants to find a seat which maximizes his vertical viewing angle (depicted below as $\theta$ in a two dimensional cross section), which is the angle subtended by the top and bottom edges of the screen. How far back from the screen in meters (measured along the floor) should he sit in order to maximize his vertical viewing angle? [img]https://cdn.artofproblemsolving.com/attachments/1/5/1555fb2432ee4fe4903accc3b74ea7215bc007.png[/img] PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Maximum value of $\sin^4\theta +\cos^6\theta $ will be ? [list=1] [*] $\frac{1}{2\sqrt{2}}$ [*] $\frac{1}{2}$ [*] $\frac{1}{\sqrt{2}}$ [*] 1 [/list]

Given the sequence $(u_n)_{n=1}^{\infty}$, where $u_1 = 1, u_2 = 2$, and $u_{n + 2} = u_{n + 1} +u_ n+ \frac{(-1)^n-1}{2}$ for any positive integers $n$. Prove that every positive integers can be expressed as the sum of some distinguished numbers of the sequence of numbers $(u_n)_{n=1}^{\infty}$ Nguyen Duy Thai Son, The University of Danang, Da Nang.

The function $f(x)=x^2+ \sin x$ and the sequence of positive numbers $\{ a_n \}$ satisfy $a_1=1$, $f(a_n)=a_{n-1}$, where $n \geq 2$. Prove that there exists a positive integer $n$ such that $a_1+a_2+ \dots + a_n > 2020$.

Represent the number $ 2008$ as a sum of natural number such that the addition of the reciprocals of the summands yield 1.

Leonard wrote three 3-digit numbers on the board whose sum is $1000$. All of the nine digits are different. Determine which digit does not appear on the board. [i]Proposed by Giorgi Arabidze, Georgia[/i]

Let $a_1 \ge a_2 \ge ... \ge a_n > 0$ be real numbers. Prove that $$a_1a_2(a_1 - a_2) + a_2a_3(a_2 - a_3) +...+ a_{n-1}a_n(a_{n-1} - a_n) \ge a_1a_n(a_1 - a_n)$$

Let $k\ge2$ be an integer. Find the smallest integer $n \ge k+1$ with the property that there exists a set of $n$ distinct real numbers such that each of its elements can be written as a sum of $k$ other distinct elements of the set.

Show that a polynomial of odd degree $2m+1$ over $\mathbb{Z}$, \[f(x)=c_{2m+1}x^{2m+1}+\cdots+c_{1}x+c_{0},\] is irreducible if there exists a prime $p$ such that \[p \not\vert c_{2m+1}, p \vert c_{m+1}, c_{m+2}, \cdots, c_{2m}, p^{2}\vert c_{0}, c_{1}, \cdots, c_{m}, \; \text{and}\; p^{3}\not\vert c_{0}.\]

Find the least positive integer $k$ for which the equation $\lfloor \frac{2002}{n}\rfloor = k$ has no integer solutions for $n.$ (The notation $\lfloor x \rfloor$ means the greatest integer less than or equal to $x.$)

Let $P(x, y)$ be a non-constant homogeneous polynomial with real coefficients such that $P(\sin t, \cos t) = 1$ for every real number $t$. Prove that there exists a positive integer $k$ such that $P(x, y) = (x^2 + y^2)^k$.

Let $a_0,$ $a_1,$ $\ldots,$ $a_n$ be complex numbers such that [center]$|a_nz^n+a_{n-1}z^{n-1}+\ldots+a_1z+a_0| \le 1,$ for any $z \in \mathbb{C}$ with $|z|=1.$[/center] Prove that $|a_k| \le 1$ and $|a_0+a_1+\ldots+a_n-(n+1)a_k| \le n,$ for any $k=\overline{0,n}.$

Define a sequence $a_0$, $a_1$, $a_2$, $,...$ recursively by $a_0 = 0$, $a_1 = 1$, and $a_{n+2} = a_{n+1} + xa_n$ for each $n \ge 0$ and some real number $x$. The infinite series $$ \sum^{\infty}_{n=0}\frac{a_n}{10^n} = 1.$$ Compute $x$.

Real numbers $a,b,c$ with $a\neq b$ verify $$a^2(b+c)=b^2(c+a)=2023.$$ Find the numerical value of $E=c^2(a+b)$.

Let $K$ be a field having $q=p^n$ elements, where $p$ is a prime and $n\ge 2$ is an arbitrary integer number. For any $a\in K$, one defines the polynomial $f_a=X^q-X+a$. Show that: $a)$ $f=(X^q-X)^q-(X^q-X)$ is divisible by $f_1$; $b)$ $f_a$ has at least $p^{n-1}$ essentially different irreducible factors $K[X]$.

Let $a\le 1$ be a real number. Sequence $\{x_n\}$ satisfies $x_0=0, x_{n+1}= 1-a\cdot e^{x_n}$, for all $n\ge 1$, where $e$ is the natural logarithm. Prove that for any natural $n$, $x_n\ge 0$.

Find all functions $f : R \to R$ such that for all $x, y \in R$ holds $$yf(2x) - xf(2y) = 8xy(x^2 - y^2).$$

Find all functions $f : R \to R$ which satisfy for all $x, y \in R$ the relation $f(f(f(x) + y) + y) = x + y + f(y)$

Find all function $f:\mathbb{Z}\rightarrow\mathbb{Z}$ such that $f(f(x)+f(y))+f(x)f(y)=f(x+y)f(x-y)$ for all integer $x,y$

Calculate $1^2-2^2+3^2-4^2+...-2018^2+2019^2$.

How many 7-element subsets of $\{1, 2, 3,\ldots , 14\}$ are there, the sum of whose elements is divisible by $14$?