Problem 3. Let $f:\left[ 0,\frac{\pi }{2} \right]\to \left[ 0,\infty \right)$ an increasing function .Prove that: (a) $\int_{0}^{\frac{\pi }{2}}{\left( f\left( x \right)-f\left( \frac{\pi }{4} \right) \right)}\left( \sin x-\cos x \right)dx\ge 0.$ (b) Exist $a\in \left[ \frac{\pi }{4},\frac{\pi }{2} \right]$ such that $\int_{0}^{a}{f\left( x \right)\sin x\ dx=}\int_{0}^{a}{f\left( x \right)\cos x\ dx}.$

Let $ABCD$ be a cyclic quadrilateral. Prove that there exists a point $X$ on segment $\overline{BD}$ such that $\angle BAC=\angle XAD$ and $\angle BCA=\angle XCD$ if and only if there exists a point $Y$ on segment $\overline{AC}$ such that $\angle CBD=\angle YBA$ and $\angle CDB=\angle YDA$.

Find all solutions of $a^2 + b^2 = n!$ for positive integers $a$, $b$, $n$ with $a \le b$ and $n < 14$.

In $\triangle ABC$, let $D$ be the point on side $BC$ such that $AB+BD=DC+CA.$ The line $AD$ intersects the circumcircle of $\triangle ABC$ again at point $X \neq A$. Prove that one of the common tangents of the circumcircles of $\triangle BDX$ and $\triangle CDX$ is parallel to $BC$.

Prove that for positive integers $x_{1},...,x_{n}$, we have $\prod_{1\leq i<j\leq n}(j-i)|\prod_{1\leq i<j\leq n}(x_{j}-x_{i})$

Determine the largest integer $n$ such that $2^n$ divides the decimal representation given by some permutation of the digits $2$, $0$, $1$, and $5$. (For example, $2^1$ divides $2150$. It may start with $0$.)

Find all non-negative integers $a, b, c$ such that the roots of equations: $\begin{cases}x^2 - 2ax + b = 0 \\ x^2- 2bx + c = 0 \\ x^2 - 2cx + a = 0 \end{cases}$ are non-negative integers.

Let $ABC$ be a triangle. consider an arbitrary point $P$ on the plain of $\triangle ABC$. Let $R,Q$ be the reflections of $P$ wrt $AB,AC$ respectively. Let $RQ\cap BC=T$. Prove that $\angle APB=\angle APC$ if and if only $\angle APT=90^{\circ}$.

Let $S$ be the set of positive integers $k$ such that the two parabolas$$y=x^2-k~~\text{and}~~x=2(y-20)^2-k$$intersect in four distinct points, and these four points lie on a circle with radius at most $21$. Find the sum of the least element of $S$ and the greatest element of $S$.

Let $p$ be an odd prime. Show that for at least $(p+1)/2$ values of $n$ in $\{0,1,2,\dots,p-1\},$ \[\sum_{k=0}^{p-1}k!n^k \quad \text{is not divisible by }p.\]

Distinct pebbles are placed on a $1001 \times 1001$ board consisting of $1001^2$ unit tiles, such that every unit tile consists of at most one pebble. The [i]pebble set[/i] of a unit tile is the set of all pebbles situated in the same row or column with said unit tile. Determine the minimum amount of pebbles that must be placed on the board so that no two distinct tiles have the same [i]pebble set[/i]. [hide=Where's the Algebra Problem?]It's already posted [url=https://artofproblemsolving.com/community/c6h2742895_simple_inequality]here[/url].[/hide]

Find the largest integer such that every number after the first is one less than the previous one and is divisible by each of its own numbers.

Ben works quickly on his homework, but tires quickly. The first problem takes him $1$ minute to solve, and the second problem takes him $2$ minutes to solve. It takes him $N$ minutes to solve problem $N$ on his homework. If he works for an hour on his homework, compute the maximum number of problems he can solve.

Points $A_1$, $B_1$, $C_1$ are midpoints of sides $BC$, $AC$, $AB$ of triangle $ABC$. On midlines $C_1B_1$ and $A_1B_1$ points $E$ and $F$ are chosen such that $BE$ is the angle bisector of $AEB_1$ and $BF$ is the angle bisector of $CFB_1$. Prove that bisectors of angles $ABC$ and $FBE$ coincide. [I]Proposed by F. Baharev[/i]

A polynomial $p(x) = \sum_{j=1}^{2n-1} a_j x^j$ with real coefficients is called [i]mountainous[/i] if $n \ge 2$ and there exists a real number such that the polynomial's coefficients satisfy $a_1=1, a_{j+1}-a_j=k$ for $1 \le j \le n-1,$ and $a_{j+1}-a_j=-k$ for $n \le j \le 2n-2;$ we call $k$ the [i]step size[/i] of $p(x).$ A real number $k$ is called [i]good[/i] if there exists a mountainous polynomial $p(x)$ with step size $k$ such that $p(-3)=0.$ Let $S$ be the sum of all good numbers $k$ satisfying $k \ge 5$ or $k \le 3.$ If $S=\tfrac{b}{c}$ for relatively prime positive integers $b,c,$ find $b+c.$

Assume the $m$ is a given integer greater than $ 1$. Find the largest number $C$ such that for all $n \in N$ we have $$\sum_{1\le k \le m ,\,\, (k,m)=1}\frac{1}{k}\ge C \sum_{k=1}^{m}\frac{1}{k}$$

[b]p1.[/b] Evaluate $1^2 - 2^2 + 3^2 - 4^2 + ...+ 19^2 - 20^2 + 21^2$. [b]p2.[/b] Kevin is playing in a table-tennis championship against Vincent. Kevin wins the championship if he wins two matches against Vincent, while Vincent must win three matches to win the championship. Given that both players have a $50\%$ chance of winning each match and there are no ties, the probability that Vincent loses the championship can be written in the form $\frac{a}{b}$ , where $a$ and $b$ are relatively prime positive integers. Find $a + b$. [b]p3.[/b] For how many positive integers $n$ less than $2000$ is $n^{3n}$ a perfect fourth power? [b]p4.[/b] Given that a coin of radius $\sqrt{3}$ cm is tossed randomly onto a plane tiled by regular hexagons of side length $14$ cm, the chance that it lands strictly inside of a hexagon can be written in the form $\frac{p}{q}$ , where $p$ and $q$ are relatively prime positive integers. Find $p + q$. [b]p5.[/b] Given that $A,C,E,I, P,$ and $M$ are distinct nonzero digits such that $$EPIC + EMCC + AMC = PEACE,$$ what is the least possible value of $PEACE$? [b]p6.[/b] A palindrome is a number that reads the same forwards and backwards. Call a number palindrome-ish if it is not a palindrome but we can make it a palindrome by changing one digit (we cannot change the first digit to zero). For instance, $4009$ is palindrome-ish because we can change the $4$ to a $9$. How many palindrome-ish four-digit numbers are there? [b]p7.[/b] Given that the heights of triangle $ABC$ have lengths $\frac{15}{7}$ , $5$, and $3$, what is the square of the area of $ABC$? [b]p8.[/b] Suppose that cubic polynomial $P(x)$ has leading coecient $1$ and three distinct real roots in the interval $[-20, 2]$. Given that the equation $P\left(x + \frac{1}{x} \right) = 0$ has exactly two distinct real solutions, the range of values that $P(3)$ can take is the open interval $(a, b)$. Compute $b - a$. [b]p9.[/b] Vincent the Bug has $17$ students in his class lined up in a row. Every day, starting on January $1$, $2021$, he performs the same series of swaps between adjacent students. One example of a series of swaps is: swap the $4$th and the $5$th students, then swap the $2$nd and the $3$rd, then the $3$rd and the $4$th. He repeats this series of swaps every day until the students are in the same arrangement as on January $1$. What is the greatest number of days this process could take? [b]p10.[/b] The summation $$\sum^{18}_{i=1}\frac{1}{i}$$ can be written in the form $\frac{a}{b}$ , where $a$ and $b$ are relatively prime positive integers. Compute the number of divisors of $b$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

$ABC$ is a right triangle with $\angle A = 30^o$ and circumcircle $O$. Circles $\omega_1$, $\omega_2$, and $\omega_3$ lie outside $ABC$ and are tangent to $O$ at $T_1$, $T_2$, and $T_3$ respectively and to $AB$, $BC$, and $CA$ at $S_1$, $S_2$, and $S_3$, respectively. Lines $T_1S_1$, $T_2S_2$, and $T_3S_3$ intersect $O$ again at $A'$, $B'$, and $C'$, respectively. What is the ratio of the area of $A'B'C'$ to the area of $ABC$?

Find the sum of all positive integers $n$ for which $\mid 2^n + 5^n - 65 \mid$ is a perfect square.

Suppose $r \ge 2$ is an integer, and let $m_1, n_1, m_2, n_2, \dots, m_r, n_r$ be $2r$ integers such that $$\left|m_in_j-m_jn_i\right|=1$$ for any two integers $i$ and $j$ satisfying $1 \le i<j \le r$. Determine the maximum possible value of $r$. [i]Proposed by B Sury[/i]

Four real numbers $x_0>x_1>x_2>x_3>0$, if $\log_{\frac{x_0}{x_1}}1993+\log_{\frac{x_1}{x_2}}1993+\log_{\frac{x_2}{x_3}}1993\geq k\cdot\log_{\frac{x_0}{x_3}}1993$ for all $x_0,x_1,x_2,x_3$, then the maximum value of $k$ is________.

Let $ABC$ be a triangle. Let its altitudes $AD$, $BE$ and $CF$ concur at $H$. Let $K, L$ and $M$ be the midpoints of $BC$, $CA$ and $AB$, respectively. Prove that, if $\angle BAC = 60^o$, then the midpoints of the segments $AH$, $DK$, $EL$, $FM$ are concyclic.

Sea $f: R^+ \to R$ defined as $$f (x) = \frac{1}{\sqrt[3]{x^2 + 6x + 9} + \sqrt[3]{x^2 + 4x + 3} + \sqrt[3]{x^2 + 2x + 1}}$$ Calculate $$f (1) + f (2) + f (3) + ... + f (2016).$$

Function $f(x)$ satisfies that $f(3+x)=f(3-x)$. Also, equation $f(x)=0$ has six different real roots, then the sum of these roots is $\text{(A)}18\qquad\text{(B)}12\qquad\text{(C)}9\qquad\text{(D)}0$

Jackson's paintbrush makes a narrow strip that is $6.5$ mm wide. Jackson has enough paint to make a strip of 25 meters. How much can he paint, in $\text{cm}^2$? $\textbf{(A) }162{,}500\qquad\textbf{(B) }162.5\qquad\textbf{(C) }1{,}625\qquad\textbf{(D) }1{,}625{,}000\qquad\textbf{(E) }16{,}250$