A straight one-mile stretch of highway, $40$ feet wide, is closed. Robert rides his bike on a path composed of semicircles as shown. If he rides at $5$ miles per hour, how many hours will it take to cover the one-mile stretch? Note: $1$ mile= $5280$ feet [asy]size(10cm); pathpen=black; pointpen=black; D(arc((-2,0),1,300,360)); D(arc((0,0),1,0,180)); D(arc((2,0),1,180,360)); D(arc((4,0),1,0,180)); D(arc((6,0),1,180,240)); D((-1.5,1)--(5.5,1)); D((-1.5,0)--(5.5,0),dashed); D((-1.5,-1)--(5.5,-1)); [/asy] $\textbf{(A) }\frac{\pi}{11}\qquad\textbf{(B) }\frac{\pi}{10}\qquad\textbf{(C) }\frac{\pi}{5}\qquad\textbf{(D) }\frac{2\pi}{5}\qquad \textbf{(E) }\frac{2\pi}{3}$

Let $ABC$ be an equilateral triangle. Suppose $D$ is the point on $BC$ such that $BD+DC = 1:3$. Let the perpendicular bisector of $AD$ intersect $AB,AC$ at $E,F$ respectively. Prove that $49 \cdot [BDE] = 25 \cdot [CDF]$, where $[XYZ]$ denotes the area of the triangle $XYZ$.

In quadrilateral $ABCD$, $\angle B = \angle D = 90$ and $AC = BC + DC$. Point $P$ of ray $BD$ is such that $BP = AD$. Prove that line $CP$ is parallel to the bisector of angle $ABD$. [i](Proposed by A.Trigub)[/i]

A cube with sides 1m in length is filled with water, and has a tiny hole through which the water drains into a cylinder of radius $1\text{ m}$. If the water level in the cube is falling at a rate of $1 \text{ cm/s}$, at what rate is the water level in the cylinder rising?

Olga and Sasha play a game on an infinite hexagonal grid. They take turns in placing a stone on a free hexagon of their choice. Olga starts the game. Just before the $2018$th stone is placed, a new rule comes into play. A stone may now be placed only on those free hexagons having at least two occupied neighbors. A player loses when she or he either is unable to make a move, or makes a move such that a pattern of the rhomboid shape as shown (rotated in any possible way) appears. Determine which player, if any, possesses a winning strategy.

[b]p1.[/b] Find the sum of the seventeenth powers of the seventeen roots of the seventeeth degree polynomial equation $x^{17} - 17x + 17 = 0$. [b]p2.[/b] Four identical spherical cows, each of radius $17$ meters, are arranged in a tetrahedral pyramid (their centers are the vertices of a regular tetrahedron, and each one is tangent to the other three). The pyramid of cows is put on the ground, with three of them laying on it. What is the distance between the ground and the top of the topmost cow? [b]p3.[/b] If $a_n$ is the last digit of $\sum^{n}_{i=1} i$, what would the value of $\sum^{1000}_{i=1}a_i$ be? [b]p4.[/b] If there are $15$ teams to play in a tournament, $2$ teams per game, in how many ways can the tournament be organized if each team is to participate in exactly $5$ games against dierent opponents? [b]p5.[/b] For $n = 20$ and $k = 6$, calculate $$2^k {n \choose 0}{n \choose k}- 2^{k-1}{n \choose 1}{{n - 1} \choose {k - 1}} + 2^{k-2}{n \choose 2}{{n - 2} \choose {k - 2}} +...+ (-1)^k {n \choose k}{{n - k} \choose 0}$$ where ${n \choose k}$ is the number of ways to choose $k$ things from a set of $n$. [b]p6.[/b] Given a function $f(x) = ax^2 + b$, with a$, b$ real numbers such that $$f(f(f(x))) = -128x^8 + \frac{128}{3}x^6 - \frac{16}{22}x^2 +\frac{23}{102}$$ , find $b^a$. [b]p7.[/b] Simplify the following fraction $$\frac{(2^3-1)(3^3-1)...(100^3-1)}{(2^3+1)(3^3+1)...(100^3+1)}$$ [b]p8.[/b] Simplify the following expression $$\frac{\sqrt{3 + \sqrt5} + \sqrt{3 - \sqrt5}}{\sqrt{3 - \sqrt8}} -\frac{4}{ \sqrt{8 - 2\sqrt{15}}}$$ [b]p9.[/b] Suppose that $p(x)$ is a polynomial of degree $100$ such that $p(k) = k2^{k-1}$ , $k =1, 2, 3 ,... , 100$. What is the value of $p(101)$ ? [b]p10. [/b] Find all $17$ real solutions $(w, x, y, z)$ to the following system of equalities: $$ 2w + w^2x = x$$ $$ 2x + x^2y=y $$ $$ 2y + y^2z=z $$ $$ -2z+z^2w=w $$ PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

In $\triangle ABC$, $M$ is the midpoint of side $BC$, $AN$ bisects $\angle BAC$, $BN\perp AN$ and $\theta$ is the measure of $\angle BAC$. If sides $AB$ and $AC$ have lengths $14$ and $19$, respectively, then length $MN$ equals [asy] size(230); defaultpen(linewidth(0.7)+fontsize(10)); pair B=origin, A=14*dir(36), C=intersectionpoint(B--(9001,0), Circle(A,19)), M=midpoint(B--C), b=A+14*dir(A--C), N=foot(A, B, b); draw(N--B--A--N--M--C--A^^B--M); markscalefactor=0.1; draw(rightanglemark(B,N,A)); pair point=N; label("$A$", A, dir(point--A)); label("$B$", B, dir(point--B)); label("$C$", C, dir(point--C)); label("$M$", M, S); label("$N$", N, dir(30)); label("$19$", (A+C)/2, dir(A--C)*dir(90)); label("$14$", (A+B)/2, dir(A--B)*dir(270)); [/asy] $\displaystyle \text{(A)} \ 2 \qquad \text{(B)} \ \frac{5}{2} \qquad \text{(C)} \ \frac{5}{2} - \sin \theta \qquad \text{(D)} \ \frac{5}{2} - \frac{1}{2} \sin \theta \qquad \text{(E)} \ \frac{5}{2} - \frac{1}{2} \sin \left(\frac{1}{2} \theta\right)$

In an acute-angled triangle $ABC$, point $I$ is the center of the inscribed circle, point $T$ is the midpoint of the arc $ABC$ of the circumcircle of triangle $ABC$. It turned out that $\angle AIT = 90^o$ . Prove that $AB + AC = 3BC$. (Matthew of Kursk)

Ivan has a parallelogram whose interior angles are $60^{\circ}, 120^{\circ}, 60^{\circ}, 120^{\circ}$ respectively, and all side lengths are integers. Is it possible that one of the diagonals has length $\sqrt{2024}$?

Let $ABC$ be a triangle inscribed in the circle $(O)$ and $P$ is a point inside the triangle $ABC$. Let $D$ be a point on $(O)$ such that $AD \perp AP$. The line $CD$ cuts the perpendicular bisector of $BC$ at $M$. The line $AD$ cuts the line passing through $B$ and is perpendicular to $BP$ at $Q$. Let $N$ be the reflection of $Q$ through $M$. Prove that $CN \perp CP$.

a) Let $a,b\in R$, $a <b$. Prove that $x \in (a,b)$ if and only if there exists $\lambda \in (0,1)$ such that $x=\lambda a +(1-\lambda)b$. b) If the function $f: R \to R$ has the property: $$f (\lambda x+(1-\lambda) y) < \lambda f(x) + (1-\lambda)f(y), \forall x,y \in R, x\ne y, \forall \lambda \in (0,1), $$ prove that one cannot find four points on the function’s graph that are the vertices of a parallelogram

Given a pyramid with square base $ ABCD $ and vertex $ S $. Find the shortest path whose starting and ending point is the point $ S $ and which passes through all the vertices of the base.

Find the area of the smallest region bounded by the graphs of $y=|x|$ and $x^2+y^2=4$. $\textbf{(A) }\frac{\pi}{4}\qquad\textbf{(B) }\frac{3\pi}{4}\qquad\textbf{(C) }\pi\qquad\textbf{(D) }\frac{3\pi}{2}\qquad\textbf{(E) }2\pi$

Triangle $ABC$ satisfies $\overline{CA} > \overline{AB}$. Let the incenter of triangle $ABC$ be $\omega$, which touches $BC, CA, AB$ at $D, E, F$, respectively. Let $M$ be the midpoint of $BC$. Let the circle centered at $M$ passing through $D$ intersect $DE, DF$ at $P(\neq D), Q(\neq D)$, respecively. Let line $AP$ meet $BC$ at $N$, line $BP$ meet $CA$ at $L$. Prove that the three lines $EQ, FP, NL$ are concurrent.

In an acute triangle $ABC$,$O$ is the center of the circumscribed circle $\omega$, $P$ is the point of intersection of the tangents to $\omega$ through the points $B$ and $C$, the median AM intersects the circle $\omega$ at point $D$. Prove that points $A, D, P$ and $O$ lie on the same circle. (D. Prokopenko)

The letter [b]M[/b] in the figure below is first reflected over the line $q$ and then reflected over the line $p$. What is the resulting image? [asy] // pog diagram usepackage("newtxtext"); size(3cm); draw((-1,0)--(1,0)); draw((0,-1)--(0,1)); label("$\textbf{\textsf{M}}$",(0.25,0.6)); draw((-0.8,-0.8)--(0.8,0.8),linewidth(1.1)); label("$p$", (-1,0),NE); label("$q$", (-0.75,-0.75), N*1.5); [/asy] [asy] // pog diagram usepackage("newtxtext"); size(12.5cm); draw((-1,0)--(1,0)); draw((0,-1)--(0,1)); label(rotate(90)*"$\textbf{\textsf{M}}$",(0.6,-0.25)); draw((-0.8,-0.8)--(0.8,0.8),linewidth(1.1)); label("$\textbf{(A)}$",(-1,1),W); draw((2,0)--(4,0)); draw((3,-1)--(3,1)); label(rotate(270)*"$\textbf{\textsf{M}}$",(2.8,0.7)); draw((2.2,-0.8)--(3.8,0.8),linewidth(1.1)); label("$\textbf{(B)}$",(2,1),W); draw((5,0)--(7,0)); draw((6,-1)--(6,1)); label(rotate(90)*"$\textbf{\textsf{M}}$",(5.4,0.2)); draw((5.2,-0.8)--(6.8,0.8),linewidth(1.1)); label("$\textbf{(C)}$",(5,1),W); draw((-1,-2.5)--(1,-2.5)); draw((0,-3.5)--(0,-1.5)); label(rotate(180)*"$\textbf{\textsf{M}}$",(-0.25,-3.1)); draw((-0.8,-3.3)--(0.8,-1.7),linewidth(1.1)); label("$\textbf{(D)}$",(-1,-1.5),W); draw((2,-2.5)--(4,-2.5)); draw((3,-3.5)--(3,-1.5)); label(rotate(270)*"$\textbf{\textsf{M}}$",(3.6,-2.75)); draw((2.2,-3.3)--(3.8,-1.7),linewidth(1.1)); label("$\textbf{(E)}$",(2,-1.5),W); [/asy]

A circle centered at point $O$ is separated by points $A_1,A_2,...,A_n$ on $n$ equal parts (points are listed sequentially clockwise) and the rays $OA_1,OA_2,...,OA_n$ are constructed. The angle $A_2OA_3$ is divided by rays into two equal angles at vertex $O$, the angle $A_3OA_4$ is divided into three equal angles, and so on, finally, the angle $A_nOA_1$ divided into $n$ equal angles at vertex $O$. A point belonging to the ray other than $OA_1$, is connected by a segment with its orthogonal projection $B_0$ on the neighboring (clockwise) arrow) with ray $OA_1$, point$ B_1$ is connected by a segment with its orthogonal projection on the next (clockwise) ray, etc. As a result of such process it turns out the broken line $B_0B_1B_2B_3...$ infinitely "twists". Consider the question of giving the thus obtained broken numerical value of "length" $L (n)$ and explore the value of $L(n)$ depending on $n$.

Let $ABC$ be an acute scalene triangle, with the feet of $A,B,C$ onto $BC,CA,AB$ being $D,E,F$ respectively. Let $W$ be a point inside $ABC$ whose reflections over $BC,CA,AB$ are $W_a,W_b,W_c$ respectively. Finally, let $N$ and $I$ be the circumcenter and the incenter of $W_aW_bW_c$ respectively. Prove that, if $N$ coincides with the nine-point center of $DEF$, the line $WI$ is parallel to the Euler line of $ABC$. [i]Proposed by Navneel Singhal, India and Massimiliano Foschi, Italy[/i]

A convex quadrilateral $ABCD$ is circumscribed about circle $\omega$. A tangent to $\omega$ parallel to $AC$ intersects $BD$ at a point $P$ outside of $\omega$. The second tangent from $P$ to $\omega$ touches $\omega$ at a point $T$. Prove that $\omega$ and circumcircle of $ATC$ are tangent. [i]Proposed by Nikolai Beluhov, Bulgaria[/i]

The sum of the squares of three positive numbers is $160$. One of the numbers is equal to the sum of the other two. The difference between the smaller two numbers is $4.$ What is the difference between the cubes of the smaller two numbers? [i]Author: Ray Li[/i] [hide="Clarification"]The problem should ask for the positive difference.[/hide]

The incircle of triangle $ABC$, is tangent to sides $BC,CA$ and $AB$ in $D,E$ and $F$ respectively. The reflection of $F$ with respect to $B$ and the reflection of $E$ with respect to $C$ are $T$ and $S$ respectively. Prove that the incenter of triangle $AST$ is inside or on the incircle of triangle $ABC$. [i]Proposed by Mehdi E'tesami Fard[/i]

[list=a] [*]A convex pentagon is partitioned into three triangles by nonintersecting diagonals. Is it possible for centroids of these triangles to lie on a common straight line? [*]The same question for a non-convex pentagon. [/list] [i]Alexandr Gribalko[/i]

Let $A,B$ be two distinct points on a given circle $O$ and let $P$ be the midpoint of the line segment AB. Let $O_1$ be the circle tangent to the line $AB$ at $P$ and tangent to the circle $O$. Let $l$ be the tangent line, different from the line $AB$, to $O_1$ passing through $A$. Let $C$ be the intersection point, different from $A$, of $l$ and $O$. Let $Q$ be the midpoint of the line segment $BC$ and $O_2$ be the circle tangent to the line $BC$ at $Q$ and tangent to the line segment $AC$. Prove that the circle $O_2$ is tangent to the circle $O$.

From a point $P$ outside of a circle with centre $O$, tangent segments $PA$ and $PB$ are drawn. $\frac{1}{OA^2}+\frac{1}{PA^2}=\frac{1}{16}$ then $AB=$ [list=1] [*] 4 [*] 6 [*] 8 [*] 10 [/list]

We are given an equilateral triangle $ABC$, of side $a$, inside its circumscribed circle. We consider the smallest of the two portions of circle limited between $AB$ and the circumference. If we consider parallel lines to $BC$, some of them cut the portion of circle in a segment. Which is the maximum possible length for one of the segments?