Assume $x_{1},x_{2},\dots,x_{n}\in\mathbb R^{+}$, $\sum_{i=1}^{n}x_{i}^{2}=n$, $\sum_{i=1}^{n}x_{i}\geq s>0$ and $0\leq\lambda\leq1$. Prove that at least $\left\lceil\frac{s^{2}(1-\lambda)^{2}}n\right\rceil$ of these numbers are larger than $\frac{\lambda s}{n}$.

Let $f(x)$ be a differentiable function such that $f'(x)+f(x)=4xe^{-x}\sin 2x,\ \ f(0)=0.$ Find $\lim_{n\to\infty}\sum_{k=1}^{n}f(k\pi).$

A positive integer $ n$ has 60 divisors and $ 7n$ has 80 divisors. What is the greatest integer $ k$ such that $ 7^k$ divides $ n$? $ \textbf{(A)}\ 0\qquad \textbf{(B)}\ 1\qquad \textbf{(C)}\ 2\qquad \textbf{(D)}\ 3\qquad \textbf{(E)}\ 4$

[u]Set 1[/u] [b]B1.[/b] Evaluate $2 + 0 - 2 \times 0$. [b]B2.[/b] It takes four painters four hours to paint four houses. How many hours does it take forty painters to paint forty houses? [b]B3.[/b] Let $a$ be the answer to this question. What is $\frac{1}{2-a}$? [u]Set 2[/u] [b]B4.[/b] Every day at Andover is either sunny or rainy. If today is sunny, there is a $60\%$ chance that tomorrow is sunny and a $40\%$ chance that tomorrow is rainy. On the other hand, if today is rainy, there is a $60\%$ chance that tomorrow is rainy and a $40\%$ chance that tomorrow is sunny. Given that today is sunny, the probability that the day after tomorrow is sunny can be expressed as n%, where n is a positive integer. What is $n$? [b]B5.[/b] In the diagram below, what is the value of $\angle DD'Y$ in degrees? [img]https://cdn.artofproblemsolving.com/attachments/0/8/6c966b13c840fa1885948d0e4ad598f36bee9d.png[/img] [b]B6.[/b] Christina, Jeremy, Will, and Nathan are standing in a line. In how many ways can they be arranged such that Christina is to the left of Will and Jeremy is to the left of Nathan? Note: Christina does not have to be next to Will and Jeremy does not have to be next to Nathan. For example, arranging them as Christina, Jeremy, Will, Nathan would be valid. [u]Set 3[/u] [b]B7.[/b] Let $P$ be a point on side $AB$ of square $ABCD$ with side length $8$ such that $PA = 3$. Let $Q$ be a point on side $AD$ such that $P Q \perp P C$. The area of quadrilateral $PQDB$ can be expressed in the form $m/n$ for relatively prime positive integers $m$ and $n$. Compute $m + n$. [b]B8.[/b] Jessica and Jeffrey each pick a number uniformly at random from the set $\{1, 2, 3, 4, 5\}$ (they could pick the same number). If Jessica’s number is $x$ and Jeffrey’s number is $y$, the probability that $x^y$ has a units digit of $1$ can be expressed as $m/n$ , where $m$ and $n$ are relatively prime positive integers. Find $m + n$. [b]B9.[/b] For two points $(x_1, y_1)$ and $(x_2, y_2)$ in the plane, we define the taxicab distance between them as $|x_1 - x_2| + |y_1 - y_2|$. For example, the taxicab distance between $(-1, 2)$ and $(3,\sqrt2)$ is $6-\sqrt2$. What is the largest number of points Nathan can find in the plane such that the taxicab distance between any two of the points is the same? [u]Set 4[/u] [b]B10.[/b] Will wants to insert some × symbols between the following numbers: $$1\,\,\,2\,\,\,3\,\,\,4\,\,\,6$$ to see what kinds of answers he can get. For example, here is one way he can insert $\times$ symbols: $$1 \times 23 \times 4 \times 6 = 552.$$ Will discovers that he can obtain the number $276$. What is the sum of the numbers that he multiplied together to get $276$? [b]B11.[/b] Let $ABCD$ be a parallelogram with $AB = 5$, $BC = 3$, and $\angle BAD = 60^o$ . Let the angle bisector of $\angle ADC$ meet $AC$ at $E$ and $AB$ at $F$. The length $EF$ can be expressed as $m/n$, where $m$ and $n$ are relatively prime positive integers. What is $m + n$? [b]B12.[/b] Find the sum of all positive integers $n$ such that $\lfloor \sqrt{n^2 - 2n + 19} \rfloor = n$. Note: $\lfloor x \rfloor$ denotes the greatest integer less than or equal to $x$. [u]Set 5[/u] [b]B13.[/b] This year, February $29$ fell on a Saturday. What is the next year in which February $29$ will be a Saturday? [b]B14.[/b] Let $f(x) = \frac{1}{x} - 1$. Evaluate $$f\left( \frac{1}{2020}\right) \times f\left( \frac{2}{2020}\right) \times f\left( \frac{3}{2020}\right) \times \times ... \times f\left( \frac{2019}{2020}\right) .$$ [b]B15.[/b] Square $WXYZ$ is inscribed in square $ABCD$ with side length $1$ such that $W$ is on $AB$, $X$ is on $BC$, $Y$ is on $CD$, and $Z$ is on $DA$. Line $W Y$ hits $AD$ and $BC$ at points $P$ and $R$ respectively, and line $XZ$ hits $AB$ and $CD$ at points $Q$ and $S$ respectively. If the area of $WXYZ$ is $\frac{13}{18}$ , then the area of $PQRS$ can be expressed as $m/n$ for relatively prime positive integers $m$ and $n$. What is $m + n$? PS. You had better use hide for answers. Last sets have been posted [url=https://artofproblemsolving.com/community/c4h2777424p24371574]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Define $\text{sgn}(x)$ to be $1$ when $x$ is positive, $-1$ when $x$ is negative, and $0$ when $x$ is $0.$ Compute $$\sum_{n=1}^{\infty} \frac{\text{sgn}(\sin(2^n))}{2^n}.$$ (The arguments to $\sin$ are in radians.)

a) Two positive integers are chosen. The sum is revealed to logician $A$, and the sum of squares is revealed to logician $B$. Both $A$ and $B$ are given this information and the information contained in this sentence. The conversation between $A$ and $B$ goes as follows: $B$ starts B: ` I can't tell what they are.' A: ` I can't tell what they are.' B: ` I can't tell what they are.' A: ` I can't tell what they are.' B: ` I can't tell what they are.' A: ` I can't tell what they are.' B: ` Now I can tell what they are.' What are the two numbers? b) When $B$ first says that he cannot tell what the two numbers are, $A$ receives a large amount of information. But when $A$ first says that he cannot tell what the two numbers are, $B$ already knows that $A$ cannot tell what the two numbers are. What good does it do $B$ to listen to $A$?

Given that $ABC$ is a triangle, points $A_i, B_i, C_i \hspace{0.15cm} (i \in \{1,2,3\})$ and $O_A, O_B, O_C$ satisfy the following criteria: a) $ABB_1A_2, BCC_1B_2, CAA_1C_2$ are rectangles not containing any interior points of the triangle $ABC$, b) $\displaystyle \frac{AB}{BB_1} = \frac{BC}{CC_1} = \frac{CA}{AA_1}$, c) $AA_1A_3A_2, BB_1B_3B_2, CC_1C_3C_2$ are parallelograms, and d) $O_A$ is the centroid of rectangle $BCC_1B_2$, $O_B$ is the centroid of rectangle $CAA_1C_2$, and $O_C$ is the centroid of rectangle $ABB_1A_2$. Prove that $A_3O_A, B_3O_B,$ and $C_3O_C$ concur at a point. [i]Proposed by Farras Mohammad Hibban Faddila[/i]

Maria and Skyler have a square-shaped cookie with a side length of $1$ inch. They split the cookie by choosing two points on distinct sides of the cookie uniformly at random and cutting across the line segment formed by connecting the two points. If Maria always gets the larger piece, what is the expected amount of extra cookie in Maria’s piece compared to Skyler’s, in square inches?

The segment of length $\ell$ with the ends on the border of a triangle divides the area of that triangle in half. Prove that $\ell >r\sqrt2$, where $r$ is the radius of the inscribed circle of the triangle.

Find the maximum possible number of edges of a simple graph with $8$ vertices and without any quadrilateral. (a simple graph is an undirected graph that has no loops (edges connected at both ends to the same vertex) and no more than one edge between any two different vertices.)

Let $\triangle ABC$ be a right triangle with right angle $\angle B$. Suppose the angle bisector $l$ of $B$ divides the hypotenuse $AC$ into two segments of length $\sqrt{3}-1$ and $\sqrt{3}+1$. What is the measure of the smaller angle between $l$ and $AC$, in radians?

Let $\triangle ABC$ be a right-angled triangle with $\angle BAC = 90^{\circ}$, and let $E$ be the foot of the perpendicular from $A$ to $BC$. Let $Z \neq A$ be a point on the line $AB$ with $AB = BZ$. Let $(c)$ and $(c_1)$ be the circumcircles of the triangles $\triangle AEZ$ and $\triangle BEZ$, respectively. Let $(c_2)$ be an arbitrary circle passing through the points $A$ and $E$. Suppose $(c_1)$ meets the line $CZ$ again at the point $F$, and meets $(c_2)$ again at the point $N$. If $P$ is the other point of intersection of $(c_2)$ with $AF$, prove that the points $N$, $B$, $P$ are collinear.

Prove that all quadrilaterals $ABCD$ where $\angle B = \angle D = 90^o$, $|AB| = |BC|$ and $|AD| + |DC| = 1$, have the same area. [img]https://1.bp.blogspot.com/-55lHuAKYEtI/XzRzDdRGDPI/AAAAAAAAMUk/n8lYt3fzFaAB410PQI4nMEz7cSSrfHEgQCLcBGAsYHQ/s0/2016%2Bmohr%2Bp3.png[/img]

Let $f$ and $g$ be two nonzero polynomials with integer coefficients and $\deg f>\deg g$. Suppose that for infinitely many primes $p$ the polynomial $pf+g$ has a rational root. Prove that $f$ has a rational root.

Find all pairs $(p,n)$ so that $p$ is a prime number, $n$ is a positive integer and \[p^3-2p^2+p+1=3^n \] holds.

The length of the common chord of two intersecting circles is $16$ feet. If the radii are $10$ feet and $17$ feet, a possible value for the distance between the centers of teh circles, expressed in feet, is: $\text{(A)} \ 27 \qquad \text{(B)} \ 21 \qquad \text{(C)} \ \sqrt{389} \qquad \text{(D)} \ 15 \qquad \text{(E)} \ \text{undetermined}$

The tourist on an island can play the "getting treasure" game. He has to open a series of doors, each door is colored with one of n colors, according to the following rules: [i](i)[/i] The tourist has n keys, each key with a different color. [i](ii)[/i] Once a key is used, it is not permitted to change until it is destroyed. [i](iii)[/i] Each key can open any door, and keeps intact when it opens the door having different color with it, but is destroyed when it opens the door having the same color with it. Find the least number of doors to ensure that no tourist, no matter how he choose the order of the keys to use, can get the treasure.

A dart board is a regular octagon divided into regions as shown. Suppose that a dart thrown at the board is equally likely to land anywhere on the board. What is probability that the dart lands within the center square? [asy] unitsize(10mm); defaultpen(linewidth(.8pt)+fontsize(10pt)); dotfactor=4; pair A=(0,1), B=(1,0), C=(1+sqrt(2),0), D=(2+sqrt(2),1), E=(2+sqrt(2),1+sqrt(2)), F=(1+sqrt(2),2+sqrt(2)), G=(1,2+sqrt(2)), H=(0,1+sqrt(2)); draw(A--B--C--D--E--F--G--H--cycle); draw(A--D); draw(B--G); draw(C--F); draw(E--H); [/asy] $ \textbf{(A)}\ \frac{\sqrt{2} - 1}{2} \qquad\textbf{(B)}\ \frac{1}{4} \qquad\textbf{(C)}\ \frac{2 - \sqrt{2}}{2} \qquad\textbf{(D)}\ \frac{\sqrt{2}}{4} \qquad\textbf{(E)}\ 2 - \sqrt{2}$

Let $a$, $b$ and $c$ be any positive real numbers. Prove that $$\dfrac{a^2+b^2}{2c}+\dfrac{a^2+c^2}{2b}+\dfrac{b^2+c^2}{2a} \geqslant a+b+c.$$

Let $ABCD$ be a rectangle, and let $\omega$ be a circular arc passing through the points $A$ and $C$. Let $\omega_{1}$ be the circle tangent to the lines $CD$ and $DA$ and to the circle $\omega$, and lying completely inside the rectangle $ABCD$. Similiarly let $\omega_{2}$ be the circle tangent to the lines $AB$ and $BC$ and to the circle $\omega$, and lying completely inside the rectangle $ABCD$. Denote by $r_{1}$ and $r_{2}$ the radii of the circles $\omega_{1}$ and $\omega_{2}$, respectively, and by $r$ the inradius of triangle $ABC$. [b](a)[/b] Prove that $r_{1}+r_{2}=2r$. [b](b)[/b] Prove that one of the two common internal tangents of the two circles $\omega_{1}$ and $\omega_{2}$ is parallel to the line $AC$ and has the length $\left|AB-AC\right|$.

A white cylindrical silo has a diameter of 30 feet and a height of 80 feet. A red stripe with a horizontal width of 3 feet is painted on the silo, as shown, making two complete revolutions around it. What is the area of the stripe in square feet? [asy] size(250);defaultpen(linewidth(0.8)); draw(ellipse(origin, 3, 1)); fill((3,0)--(3,2)--(-3,2)--(-3,0)--cycle, white); draw((3,0)--(3,16)^^(-3,0)--(-3,16)); draw((0, 15)--(3, 12)^^(0, 16)--(3, 13)); filldraw(ellipse((0, 16), 3, 1), white, black); draw((-3,11)--(3, 5)^^(-3,10)--(3, 4)); draw((-3,2)--(0,-1)^^(-3,1)--(-1,-0.89)); draw((0,-1)--(0,15), dashed); draw((3,-2)--(3,-4)^^(-3,-2)--(-3,-4)); draw((-7,0)--(-5,0)^^(-7,16)--(-5,16)); draw((3,-3)--(-3,-3), Arrows(6)); draw((-6,0)--(-6,16), Arrows(6)); draw((-2,9)--(-1,9), Arrows(3)); label("$3$", (-1.375,9.05), dir(260), fontsize(7)); label("$A$", (0,15), N); label("$B$", (0,-1), NE); label("$30$", (0, -3), S); label("$80$", (-6, 8), W);[/asy] $ \textbf{(A)}\; 120\qquad \textbf{(B)}\; 180\qquad \textbf{(C)}\; 240\qquad \textbf{(D)}\; 360\qquad \textbf{(E)}\; 480$

Seven children, each with the same birthday, were born in seven consecutive years. The sum of the ages of the youngest three children in $42$. What is the sum of the ages of the oldest three? $ \mathrm{(A) \ } 51 \qquad \mathrm{(B) \ } 54 \qquad \mathrm {(C) \ } 57 \qquad \mathrm{(D) \ } 60 \qquad \mathrm{(E) \ } 63$

Jacob and Laban take turns playing a game. Each of them starts with the list of square numbers $1, 4, 9, \dots, 2021^2$, and there is a whiteboard in front of them with the number $0$ on it. Jacob chooses a number $x^2$ from his list, removes it from his list, and replaces the number $W$ on the whiteboard with $W + x^2$. Laban then does the same with a number from his list, and the repeat back and forth until both of them have no more numbers in their list. Now every time that the number on the whiteboard is divisible by $4$ after a player has taken his turn, Jacob gets a sheep. Jacob wants to have as many sheep as possible. What is the greatest number $K$ such that Jacob can guarantee to get at least $K$ sheep by the end of the game, no matter how Laban plays?

Let $~$ $ a,\,b,\,c,\,d\, >\, 0$ $~$ and $~$ $a+b+c+d\, =\, 1\, .$ $~$ Prove the inequality \[ \frac{1}{4a+3b+c}+\frac{1}{3a+b+4d}+\frac{1}{a+4c+3d}+\frac{1}{4b+3c+d}\; \ge\; 2\, . \]

Let $S$ be the set of positive integers $n$ for which $\tfrac{1}{n}$ has the repeating decimal representation $0.\overline{ab} = 0.ababab\cdots,$ with $a$ and $b$ different digits. What is the sum of the elements of $S$? $ \textbf{(A)}\ 11\qquad\textbf{(B)}\ 44\qquad\textbf{(C)}\ 110\qquad\textbf{(D)}\ 143\qquad\textbf{(E)}\ 155\qquad $