Let $n \geq 2, n \in \mathbb{N},$ find the least positive real number $\lambda$ such that for arbitrary $a_i \in \mathbb{R}$ with $i = 1, 2, \ldots, n$ and $b_i \in \left[0, \frac{1}{2}\right]$ with $i = 1, 2, \ldots, n$, the following holds: \[\sum^n_{i=1} a_i = \sum^n_{i=1} b_i = 1 \Rightarrow \prod^n_{i=1} a_i \leq \lambda \sum^n_{i=1} a_i b_i.\]

Square $ABCD$ has side length $68$. Let $E$ be the midpoint of segment $\overline{CD}$, and let $F$ be the point on segment $\overline{AB}$ a distance $17$ from point $A$. Point $G$ is on segment $\overline{EF}$ so that $\overline{EF}$ is perpendicular to segment $\overline{GD}$. The length of segment $\overline{BG}$ can be written as $m\sqrt{n}$ where $m$ and $n$ are positive integers, and $n$ is not divisible by the square of any prime. Find $m+n$.

[b]p1.[/b] What is $(2 + 4 + ... + 20) - (1 + 3 + ...+ 19)$? [b]p2.[/b] Two ants start on opposite vertices of a dodecagon ($12$-gon). Each second, they randomly move to an adjacent vertex. What is the probability they meet after four moves? [b]p3.[/b] How many distinct $8$-letter strings can be made using $8$ of the $9$ letters from the words $FORK$ and $KNIFE$ (e.g., $FORKNIFE$)? [b]p4.[/b] Let $ABC$ be an equilateral triangle with side length $8$ and let $D$ be a point on segment $BC$ such that $BD = 2$. Given that $E$ is the midpoint of $AD$, what is the value of $CE^2 - BE^2$? [b][color=#f00](mistyped p4)[/color][/b] Let $ABC$ be an equilateral triangle with side length $8$ and let $D$ be a point on segment $BC$ such that $BD = 2$. Given that $E$ is the midpoint of $AD$, what is the value of $CE^2 + BE^2$? [b]p5.[/b] You have two fair six-sided dice, one labeled $1$ to $6$, and for the other one, each face is labeled $1$, $2$, $3$, or $4$ (not necessarily all numbers are used). Let $p$ be the probability that when the two dice are rolled, the number on the special die is smaller than the number on the normal die. Given that $p = 1/2$, how many distinct combinations of $1$, $2$, $3$, $4$ can appear on the special die? The arrangement of the numbers on the die does not matter. [b]p6.[/b] Let $\omega_1$ and $\omega_2$ be two circles with centers $A$ and $B$ and radii $3$ and $13$, respectively. Suppose $AB = 10$ and that $C$ is the midpoint of $AB$. Let $\ell$ be a line that passes through $C$ and is tangent to $\omega_1$ at $P$. Given that $\ell$ intersects $\omega_2$ at $X$ and $Y$ such that $XP < Y P$, what is $XP$? [b]p7.[/b] Let $f(x)$ be a cubic polynomial. Given that $f(1) = 13$, $f(4) = 19$, $f(7) = 7$, and $f(10) = 13$, find $f(13)$. [b]p8.[/b] For all integers $0 \le n \le 202$ not divisible by seven, define $f(n) = \{\sqrt{7n}\}$. For what value $n$ does $f(n)$ take its minimum value? (Note: $\{x\} = x - \lfloor x \rfloor$, where $\lfloor x \rfloor$ is the greatest integer less than or equal to $x$.) [b]p9.[/b] Let $ABC$ be a triangle with $AB = 14$ and $AC = 25$. Let the incenter of $ABC$ be $I$. Let line $AI$ intersect the circumcircle of $BIC$ at $D$ (different from $I$). Given that line $DC$ is tangent to the circumcircle of $ABC$, find the area of triangle $BCD$. [b]p10.[/b] Evaluate the infinite sum $$\frac{4^2 + 3}{1 \cdot 3 \cdot 5 \cdot 7} +\frac{6^2 + 3}{3 \cdot 5 \cdot 7 \cdot 9}+\frac{8^2 + 3}{5 \cdot 7 \cdot 9 \cdot 11}+ ...$$ PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

$n$ students take a test with $m$ questions, where $m,n\ge 2$ are integers. The score given to every question is as such: for a certain question, if $x$ students fails to answer it correctly, then those who answer it correctly scores $x$ points, while those who answer it wrongly scores $0$. The score of a student is the sum of his scores for the $m$ questions. Arrange the scores in descending order $p_1\ge p_2\ge \ldots \ge p_n$. Find the maximum value of $p_1+p_n$.

Solve in positive real numbers the following equation $x^{-y} + y^{-x} = 4 - x - y$.

A function $f$ is given by $f(x) = x^2 - 2x$ . Prove that there exists a number a which satisfies $f(f(a)) = a$ without satisfying $f(a) = a$ .

Let $n$ be an integer greater than or equal to $1$. Find, as a function of $n$, the smallest integer $k\ge 2$ such that, among any $k$ real numbers, there are necessarily two of which the difference, in absolute value, is either strictly less than $1 / n$, either strictly greater than $n$.

Prove that in any polygon, there exist two sides whose radio is less than $2$.(Essentialy if $a_1\geq a_2\geq...\geq a_n$ are the sides of a polygon prove that there exist $i,j\in\{1,2,..,n\}$ so that $i<j$ and $\frac {a_i}{a_j}<2$).

Find all real solutions of the system of equations: \[\sum^n_{k=1} x^i_k = a^i\] for $i = 1,2, \ldots, n.$

Nine quadratics, $x^2+a_1x+b_1, x^2+a_2x+b_2,...,x^2+a_9x+b_9$ are written on the board. The sequences $a_1, a_2,...,a_9$ and $b_1, b_2,...,b_9$ are arithmetic. The sum of all nine quadratics has at least one real root. What is the the greatest possible number of original quadratics that can have no real roots?

An accurate 12-hour analog clock has an hour hand, a minute hand, and a second hand that are aligned at 12:00 o’clock and make one revolution in 12 hours, 1 hour, and 1 minute, respectively. It is well known, and not difficult to prove, that there is no time when the three hands are equally spaced around the clock, with each separating angle $ \frac{2 \cdot \pi}{3}.$ Let $ f(t), g(t), h(t)$ be the respective absolute deviations of the separating angles from \frac{2 \cdot \pi}{3} at $ t$ hours after 12:00 o’clock. What is the minimum value of $ max\{f(t), g(t), h(t)\}?$

Given a nonzero natural number $n$, let $f_n$ be the function of the closed interval $[0, 1]$ in $R$ defined like this: $$f_n(x) = \begin{cases}n^2x, \,\,\, if \,\,\, 0 \le x < 1/n\\ 3/n, \,\,\,if \,\,\,1/n \le x \le 1 \end{cases}$$ a) Represent the function graphically. b) Calculate $A_n =\int_0^1 f_n(x) dx$. c) Find, if it exists, $\lim_{n\to \infty} A_n$ .

The roots of the polynomial $f(x) = x^8 +x^7 -x^5 -x^4 -x^3 +x+ 1 $ are all roots of unity. We say that a real number $r \in [0, 1)$ is nice if $e^{2i \pi r} = \cos 2\pi r + i \sin 2\pi r$ is a root of the polynomial $f$ and if $e^{2i \pi r}$ has positive imaginary part. Let $S$ be the sum of the values of nice real numbers $r$. If $S =\frac{p}{q}$ for relatively prime positive integers $p, q$, find $p + q$.

Let $M, \alpha, \beta \in \mathbb{R} $ with $M > 0$ and $\alpha, \beta \in (0,1)$. If $R>1$ is a real number, we say that a sequence of positive real numbers $\{ C_n \}_{n\geq 0}$ is $R$-[i]inoceronte[/i] if $ \sum_{i=1}^n R^{n-i}C_i \leq R^n \cdot M$ for all $n \geq 1$. Determine the smallest real $R>1$ for which exists a $R$-[i]inoceronte[/i] sequence $ \{ C_n \}_{n\geq 0}$ such that $\sum_{n=1}^{\infty} \beta ^n C_n^{\alpha}$ diverges.

Two bodies simultaneously emerge from points $A$ and $B$ towards each other. Every time each body finds itself at point $A$ or point $B$, it changes the direction of its movement to the opposite (the absolute values of the velocities of both bodies are constant). A body that initially leaves point $A$ travels the path $AB$ in $101$ seconds. A body that initially left point $B$ travels the same path $AB$ in $201$ seconds. After $2 \cdot 101 \cdot 201$ seconds, each body will be at its starting point. How many times during this time will the first body overtake the second?

Consider a polynomial $P(x) = \prod^9_{j=1}(x+d_j),$ where $d_1, d_2, \ldots d_9$ are nine distinct integers. Prove that there exists an integer $N,$ such that for all integers $x \geq N$ the number $P(x)$ is divisible by a prime number greater than 20. [i]Proposed by Luxembourg[/i]

Solve the equation $$\frac{10}{x+10}+\frac{10\cdot 9}{(x+10)(x+9)}+\frac{10\cdot 9\cdot 8}{(x+10)(x+9)(x+8)}+ ...+\frac{10\cdot 9\cdot ... \cdot 2 \cdot 1}{(x+10)(x+9)\cdot ... \cdot(x+1)}=11$$

Suppose $S=\{1,2,3,...,2017\}$,for every subset $A$ of $S$,define a real number $f(A)\geq 0$ such that: $(1)$ For any $A,B\subset S$,$f(A\cup B)+f(A\cap B)\leq f(A)+f(B)$; $(2)$ For any $A\subset B\subset S$, $f(A)\leq f(B)$; $(3)$ For any $k,j\in S$,$$f(\{1,2,\ldots,k+1\})\geq f(\{1,2,\ldots,k\}\cup \{j\});$$ $(4)$ For the empty set $\varnothing$, $f(\varnothing)=0$. Confirm that for any three-element subset $T$ of $S$,the inequality $$f(T)\leq \frac{27}{19}f(\{1,2,3\})$$ holds.

We say that a doubly infinite sequence $. . . , s_{−2}, s_{−1}, s_{0}, s_1, s_2, . . .$ is subaveraging if $s_n = (s_{n−1} + s_{n+1})/4$ for all integers n. (a) Find a subaveraging sequence in which all entries are different from each other. Prove that all entries are indeed distinct. (b) Show that if $(s_n)$ is a subaveraging sequence such that there exist distinct integers m, n such that $s_m = s_n$, then there are infinitely many pairs of distinct integers i, j with $s_i = s_j$ .

Find all $f$ functions from real numbers to itself such that for all real numbers $x,y$ the equation \[f(f(y)+x^2+1)+2x=y+(f(x+1))^2\] holds.

Find all the functions $f: \mathbb{Z}\to \mathbb{Z}$ satisfying the following property: if $a$, $b$ and $c$ are integers such that $a+b+c=0$, then $$f(a)+f(b)+f(c)=a^2+b^2+c^2.$$

Let $S$ be the sum of all integers $b$ for which the polynomial $x^2+bx+2008b$ can be factored over the integers. Compute $|S|$.

Find all functions $f:\mathbb{R}^{+}\rightarrow \mathbb{R}^{+}$ such that $x,y\in \mathbb{R}^{+},$ \[ f\left(\frac{y}{f(x+1)}\right)+f\left(\frac{x+1}{xf(y)}\right)=f(y) \]

We say that a number is [i]angelic[/i] if it is greater than $10^{100}$ and all of its digits are elements of $\{1,3,5,7,8\}$. Suppose $P$ is a polynomial with nonnegative integer coefficients such that over all positive integers $n$, if $n$ is angelic, then the decimal representation of $P(s(n))$ contains the decimal representation of $s(P(n))$ as a contiguous substring, where $s(n)$ denotes the sum of digits of $n$. Prove that $P$ is linear and its leading coefficient is $1$ or a power of $10$. [i]Proposed by Grant Yu[/i]

Solve the following equation in positive numbers. $$(2a+1)(2a^2+2a+1)(2a^4+4a^3+6a^2+4a+1)=828567056280801$$