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 $S$ be the set of all polygonal areas in a plane. Prove that there is a function $f : S \to (0,1)$ which satisfies $f(S_1 \cup S_2) = f(S_1)+ f(S_2)$ for any $S_1,S_2 \in S$ which have common points only on their borders

Let \( \omega \) be the circumcircle of triangle \( ABC \), where \( AB > AC \). Let \( N \) be the midpoint of arc \( \smile\!BAC \), and \( D \) a point on the circle \( \omega \) such that \( ND \perp AB \). Let \( I \) be the incenter of triangle \( ABC \). Reconstruct triangle \( ABC \), given the marked points \( A, D, \) and \( I \). Proposed by Oleksii Karlyuchenko and Hryhorii Filippovskyi

Let $H$ be the orthocenter of acute-angled triangle $ABC$ and $M$ be the midpoint of $AH$. Line $BH$ cuts $AC$ at $D$. Consider point $E$ such that $BC$ is the perpendicular bisector of $DE$. Segments $CM$ and $AE$ intersect at $F$. Show that $BF$ is perpendicular to $CM$. [i]Proposed by Germán Puga[/i]

A convex pentagon $ABCDE$ satisfies $AB=BC=CA$ and $CD=DE=EC$. Let $S$ be the center of the equilateral triangle $ABC$ and $M$ and $N$ be the midpoints of $BD$ and $AE$, respectively. Prove that the triangles $SME$ and $SND$ are similar.

Let $N$ be a convex polygon with 1415 vertices and perimeter 2001. Prove that we can find 3 vertices of $N$ which form a triangle of area smaller than 1.

Let $ABC$ be a triangle whose incircle $(I)$ touches $BC, CA, AB$ at $D, E, F$, respectively. The line passing through $A$ and parallel to $BC$ cuts $DE, DF$ at $M, N$, respectively. The circumcircle of triangle $DMN$ cuts $(I)$ again at $L$. a) Let $K$ be the intersection of $N E$ and $M F$. Prove that $K$ is the orthocenter of the triangle $DMN$. b) Prove that $A, K, L$ are collinear.

In a scalene triangle $ABC$ with $\angle A = 90^\circ,$ the tangent line at $A$ to its circumcircle meets line $BC$ at $M$ and the incircle touches $AC$ at $S$ and $AB$ at $R.$ The lines $RS$ and $BC$ intersect at $N,$ while the lines $AM$ and $SR$ intersect at $U.$ Prove that the triangle $UMN$ is isosceles.

The angles $A$ and $B$ of base of the isosceles triangle $ABC$ are equal to $40^o$. Inside $\vartriangle ABC$, $P$ is such that $\angle PAB = 30^o$ and $\angle PBA = 20^o$. Calculate, without table, $\angle PCA$.

A point $M$ lies on the side $BC$ of square $ABCD$. Let $X$, $Y$ , and $Z$ be the incenters of triangles $ABM$, $CMD$, and $AMD$ respectively. Let $H_x$, $H_y$, and $H_z$ be the orthocenters of triangles $AXB$, $CY D$, and $AZD$. Prove that $H_x$, $H_y$, and $H_z$ are collinear.

Let $ABC$ be an acute triangle.The lines $l_1$ and $l_2$ are perpendicular to $AB$ at the points $A$ and $B$, respectively.The perpendicular lines from the midpoint $M$ of $AB$ to the lines $AC$ and $BC$ intersect $l_1$ and $l_2$ at the points $E$ and $F$, respectively.If $D$ is the intersection point of the lines $EF$ and $MC$, prove that \[\angle ADB = \angle EMF.\]

In the plane of a given triangle $A_1A_2A_3$ determine (with proof) a straight line $l$ such that the sum of the distances from $A_1, A_2$, and $A_3$ to $l$ is the least possible.

Let $A, B, C$, and $D$ be four different points in the plane. Three of the line segments $AB, AC, AD, BC, BD$, and $CD$ have length $a$. The other three have length $b$, where $b > a$. Determine all possible values of the quotient $\frac{b}{a}$. .

[hide=R stands for Ramanujan , P stands for Pascal]they had two problem sets under those two names[/hide] [u] Set 1[/u] [b]R1.1 / P1.1[/b] Find $291 + 503 - 91 + 492 - 103 - 392$. [b]R1.2[/b] Let the operation $a$ & $b$ be defined to be $\frac{a-b}{a+b}$. What is $3$ & $-2$? [b]R1.3[/b]. Joe can trade $5$ apples for $3$ oranges, and trade $6$ oranges for $5$ bananas. If he has $20$ apples, what is the largest number of bananas he can trade for? [b]R1.4[/b] A cone has a base with radius $3$ and a height of $5$. What is its volume? Express your answer in terms of $\pi$. [b]R1.5[/b] Guang brought dumplings to school for lunch, but by the time his lunch period comes around, he only has two dumplings left! He tries to remember what happened to the dumplings. He first traded $\frac34$ of his dumplings for Arman’s samosas, then he gave $3$ dumplings to Anish, and lastly he gave David $\frac12$ of the dumplings he had left. How many dumplings did Guang bring to school? [u]Set 2[/u] [b]R2.6 / P1.3[/b] In the recording studio, Kanye has $10$ different beats, $9$ different manuscripts, and 8 different samples. If he must choose $1$ beat, $1$ manuscript, and $1$ sample for his new song, how many selections can he make? [b]R2.7[/b] How many lines of symmetry does a regular dodecagon (a polygon with $12$ sides) have? [b]R2.8[/b] Let there be numbers $a, b, c$ such that $ab = 3$ and $abc = 9$. What is the value of $c$? [b]R2.9[/b] How many odd composite numbers are there between $1$ and $20$? [b]R2.10[/b] Consider the line given by the equation $3x - 5y = 2$. David is looking at another line of the form ax - 15y = 5, where a is a real number. What is the value of a such that the two lines do not intersect at any point? [u]Set 3[/u] [b]R3.11[/b] Let $ABCD$ be a rectangle such that $AB = 4$ and $BC = 3$. What is the length of BD? [b]R3.12[/b] Daniel is walking at a constant rate on a $100$-meter long moving walkway. The walkway moves at $3$ m/s. If it takes Daniel $20$ seconds to traverse the walkway, find his walking speed (excluding the speed of the walkway) in m/s. [b]R3.13 / P1.3[/b] Pratik has a $6$ sided die with the numbers $1, 2, 3, 4, 6$, and $12$ on the faces. He rolls the die twice and records the two numbers that turn up on top. What is the probability that the product of the two numbers is less than or equal to $12$? [b]R3.14 / P1.5[/b] Find the two-digit number such that the sum of its digits is twice the product of its digits. [b]R3.15[/b] If $a^2 + 2a = 120$, what is the value of $2a^2 + 4a + 1$? PS. You should use hide for answers. R16-30 /P6-10/ P26-30 have been posted [url=https://artofproblemsolving.com/community/c3h2786837p24497019]here[/url], and P11-25 [url=https://artofproblemsolving.com/community/c3h2786880p24497350]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

A vertex of a tetrahedron is called perfect if the three edges at this vertex are sides of a certain triangle. How many perfect vertices can a tetrahedron have?

Point $K$ is marked on the diagonal $AC$ in rectangle $ABCD$ so that $CK = BC$. On the side $BC$, point $M$ is marked so that $KM = CM$. Prove that $AK + BM = CM$.

Let ${c\equiv c\left(O, R\right)}$ be a circle with center ${O}$ and radius ${R}$ and ${A, B}$ be two points on it, not belonging to the same diameter. The bisector of angle${\angle{ABO}}$ intersects the circle ${c}$ at point ${C}$, the circumcircle of the triangle $AOB$ , say ${c_1}$ at point ${K}$ and the circumcircle of the triangle $AOC$ , say ${{c}_{2}}$ at point ${L}$. Prove that point ${K}$ is the circumcircle of the triangle $AOC$ and that point ${L}$ is the incenter of the triangle $AOB$. Evangelos Psychas (Greece)

In triangle $ABC$ with $\overline{AB}=\overline{AC}=3.6$, a point $D$ is taken on $AB$ at a distance $1.2$ from $A$. Point $D$ is joined to $E$ in the prolongation of $AC$ so that triangle $AED$ is equal in area to $ABC$. Then $\overline{AE}$ is: $ \textbf{(A)}\ 4.8 \qquad\textbf{(B)}\ 5.4\qquad\textbf{(C)}\ 7.2\qquad\textbf{(D)}\ 10.8\qquad\textbf{(E)}\ 12.6 $

Prove that every triangle can be cut into three pieces so that every piece has axis of symmetry. Show how to cut it (a) using three line segments, (b) using two line segments.

The incircle of a triangle $ABC$ is tangent to the sides $AB,AC$ at $M,N$ respectively. Suppose $P$ is the intersection between $MN$ and the bisector of $\angle ABC$. Prove that $BP$ and $CP$ are perpendicular.

On the sides of the triangle $ABC$, three similar isosceles triangles $ABP \ (AP = PB)$, $AQC \ (AQ = QC)$, and $BRC \ (BR = RC)$ are constructed. The first two are constructed externally to the triangle $ABC$, but the third is placed in the same half-plane determined by the line $BC$ as the triangle $ABC$. Prove that $APRQ$ is a parallelogram.

Let $ABC$ be a triangle and $D$ be the midpoint of the segment $BC$. The circle that passes through $D$ and tangent to $AB$ at $B$, and the circle that passes through $D$ and tangent to $AC$ at $C$ intersect at $M\neq D$. Let $M'$ be the reflection of $M$ with respect to $BC$. Prove that $M'$ is on $AD$.

[center][u]Guts Round[/u] / [u]Set 7[/u][/center] [b]p19.[/b] Let $a$ be the answer to Problem 19, $b$ be the answer to Problem 20, and $c$ be the answer to Problem 21. Compute the real value of $a$ such that $$\sqrt{a(101b + 1)} - 1 = \sqrt{b(c - 1)}+ 10\sqrt{(a - c)b}.$$ [b]p20.[/b] Let $a$ be the answer to Problem 19, $b$ be the answer to Problem 20, and $c$ be the answer to Problem 21. For some triangle $\vartriangle ABC$, let $\omega$ and $\omega_A$ be the incircle and $A$-excircle with centers $I$ and $I_A$, respectively. Suppose $AC$ is tangent to $\omega$ and $\omega_A$ at $E$ and $E'$, respectively, and $AB$ is tangent to $\omega$ and $\omega_A$ at $F$ and $F'$ respectively. Furthermore, let $P$ and $Q$ be the intersections of $BI$ with $EF$ and $CI$ with $EF$, respectively, and let $P'$ and $Q'$ be the intersections of $BI_A$ with $E'F'$ and $CI_A$ with $E'F'$, respectively. Given that the circumradius of $\vartriangle ABC$ is a, compute the maximum integer value of $BC$ such that the area $[P QP'Q']$ is less than or equal to $1$. [b]p21.[/b] Let $a$ be the answer to Problem 19, $b$ be the answer to Problem 20, and $c$ be the answer to Problem 21. Let $c$ be a positive integer such that $gcd(b, c) = 1$. From each ordered pair $(x, y)$ such that $x$ and $y$ are both integers, we draw two lines through that point in the $x-y$ plane, one with slope $\frac{b}{c}$ and one with slope $-\frac{c}{b}$ . Given that the number of intersections of these lines in $[0, 1)^2$ is a square number, what is the smallest possible value of $ c$? Note that $[0, 1)^2$ refers to all points $(x, y)$ such that $0 \le x < 1$ and $ 0 \le y < 1$.

In a triangle $ ABC$ with $ \angle A\equal{}90^{\circ}$, $ D$ is the midpoint of $ BC$, $ F$ that of $ AB$, $ E$ that of $ AF$ and $ G$ that of $ FB$. Segment $ AD$ intersects $ CE,CF$ and $ CG$ in $ P,Q$ and $ R$, respectively. Determine the ratio: $ \frac{PQ}{QR}$.

A certain state issues license plates consisting of six digits (from 0 to 9). The state requires that any two license plates differ in at least two places. (For instance, the numbers 027592 and 020592 cannot both be used.) Determine, with proof, the maximum number of distinct license plates that the state can use.