A quadrilateral $ABCD$ is inscribed, as shown, in a square of area one unit. Prove that $$2\le |AB|^2+|BC|^2+|CD|^2+|DA|^2\le 4$$ [asy] size(6cm); draw((0,0)--(10,0)); draw((10,0)--(10,10)); draw((0,10)--(10,10)); draw((0,0)--(0,10)); dot((0,8.5)); dot((3.5,10)); dot((10,3.5)); dot((3.5,0)); label("$D$",(0,8.5),W); label("$A$",(3.5,10),NE); label("$B$",(10,3.5),E); label("$C$",(3.5,0),S); draw((0,8.5)--(3.5,10)); draw((3.5,10)--(10,3.5)); draw((10,3.5)--(3.5,0)); draw((3.5,0)--(0,8.5)); [/asy]

Let $ABC$ be a triangle with $\angle ABC=90^{\circ}$. Let $D$ and $E$ be the feet from $B$ and $C$ to the median from $A$, respectively. Suppose $DE=4$ and $CD=5$. Find the area of $ABC.$

What is the minimum possible perimeter of a triangle two of whose sides are along the x- and y-axes and such that the third contains the point $(1,2)$?

Rectangle $ABCD$ and right triangle $DCE$ have the same area. They are joined to form a trapezoid, as shown. What is $DE$? [asy] size(250); defaultpen(linewidth(0.8)); pair A=(0,5),B=origin,C=(6,0),D=(6,5),E=(18,0); draw(A--B--E--D--cycle^^C--D); draw(rightanglemark(D,C,E,30)); label("$A$",A,NW); label("$B$",B,SW); label("$C$",C,S); label("$D$",D,N); label("$E$",E,S); label("$5$",A/2,W); label("$6$",(A+D)/2,N); [/asy] $\textbf{(A) }12\qquad\textbf{(B) }13\qquad\textbf{(C) }14\qquad\textbf{(D) }15\qquad \textbf{(E) }16$

[i]25 problems for 30 minutes[/i] [b]p1.[/b] Somya has a football game $4$ days from today. If the day before yesterday was Wednesday, what day of the week is the game? [b]p2.[/b] Sammy writes the following equation: $$\frac{2 + 2}{8 + 8}=\frac{x}{8}.$$ What is the value of $x$ in Sammy's equation? [b]p3.[/b] On $\pi$ day, Peter buys $7$ pies. The pies costed $\$3$, $\$1$, $\$4$, $\$1$, $\$5$, $\$9$, and $\$2$. What was the median price of Peter's $7$ pies in dollars? [b]p4.[/b] Antonio draws a line on the coordinate plane. If the line passes through the points ($1, 3$) and ($-1,-1$), what is slope of the line? [b]p5.[/b] Professor Varun has $25$ students in his science class. He divides his students into the maximum possible number of groups of $4$, but $x$ students are left over. What is $x$? [b]p6.[/b] Evaluate the following: $$4 \times 5 \div 6 \times 3 \div \frac47$$ [b]p7.[/b] Jonny, a geometry expert, draws many rectangles with perimeter $16$. What is the area of the largest possible rectangle he can draw? [b]p8.[/b] David always drives at $60$ miles per hour. Today, he begins his trip to MIT by driving $60$ miles. He stops to take a $20$ minute lunch break and then drives for another $30$ miles to reach the campus. What is the total time in minutes he spends getting to MIT? [b]p9.[/b] Richard has $5$ hats: blue, green, orange, red, and purple. Richard also has 5 shirts of the same colors: blue, green, orange, red, and purple. If Richard needs a shirt and a hat of different colors, how many outts can he wear? [b]p10.[/b] Poonam has $9$ numbers in her bag: $1, 2, 3, 4, 5, 6, 7, 8, 9$. Eric runs by with the number $36$. How many of Poonam's numbers evenly divide Eric's number? [b]p11.[/b] Serena drives at $45$ miles per hour. If her car runs at $6$ miles per gallon, and each gallon of gas costs $2$ dollars, how many dollars does she spend on gas for a $135$ mile trip? [b]p12.[/b] Grace is thinking of two integers. Emmie observes that the sum of the two numbers is $56$ but the difference of the two numbers is $30$. What is the sum of the squares of Grace's two numbers? [b]p13.[/b] Chang stands at the point ($3,-3$). Fang stands at ($-3, 3$). Wang stands in-between Chang and Fang; Wang is twice as close to Fang as to Chang. What is the ordered pair that Wang stands at? [b]p14.[/b] Nithin has a right triangle. The longest side has length $37$ inches. If one of the shorter sides has length $12$ inches, what is the perimeter of the triangle in inches? [b]p15.[/b] Dora has $2$ red socks, $2$ blue socks, $2$ green socks, $2$ purple socks, $3$ black socks, and $4$ gray socks. After a long snowstorm, her family loses electricity. She picks socks one-by-one from the drawer in the dark. How many socks does she have to pick to guarantee a pair of socks that are the same color? [b]p16.[/b] Justin selects a random positive $2$-digit integer. What is the probability that the sum of the two digits of Justin's number equals $11$? [b]p17.[/b] Eddie correctly computes $1! + 2! + .. + 9! + 10!$. What is the remainder when Eddie's sum is divided by $80$? [b]p18.[/b] $\vartriangle PQR$ is drawn such that the distance from $P$ to $\overline{QR}$ is $3$, the distance from $Q$ to $\overline{PR}$ is $4$, and the distance from $R$ to $\overline{PQ}$ is $5$. The angle bisector of $\angle PQR$ and the angle bisector of $\angle PRQ$ intersect at $I$. What is the distance from $I$ to $\overline{PR}$? [b]p19.[/b] Maxwell graphs the quadrilateral $|x - 2| + |y + 2| = 6$. What is the area of the quadrilateral? [b]p20.[/b] Uncle Gowri hits a speed bump on his way to the hospital. At the hospital, patients who get a rare disease are given the option to choose treatment $A$ or treatment $B$. Treatment $A$ will cure the disease $\frac34$ of the time, but since the treatment is more expensive, only $\frac{8}{25}$ of the patients will choose this treatment. Treatment $B$ will only cure the disease $\frac{1}{2}$ of the time, but since it is much more aordable, $\frac{17}{25}$ of the patients will end up selecting this treatment. Given that a patient was cured, what is the probability that the patient selected treatment $A$? [b]p21.[/b] In convex quadrilateral $ABCD$, $AC = 28$ and $BD = 15$. Let $P, Q, R, S$ be the midpoints of $AB$, $BC$, $CD$ and $AD$ respectively. Compute $PR^2 + QS^2$. [b]p22.[/b] Charlotte writes the polynomial $p(x) = x^{24} - 6x + 5$. Let its roots be $r_1$, $r_2$, $...$, $r_{24}$. Compute $r^{24}_1 +r^{24}_2 + r^{24}_3 + ... + r^{24}_24$. [b]p23.[/b] In rectangle $ABCD$, $AB = 6$ and $BC = 4$. Let $E$ be a point on $CD$, and let $F$ be the point on $AB$ which lies on the bisector of $\angle BED$. If $FD^2 + EF^2 = 52$, what is the length of $BE$? [b]p24.[/b] In $\vartriangle ABC$, the measure of $\angle A$ is $60^o$ and the measure of $\angle B$ is $45^o$. Let $O$ be the center of the circle that circumscribes $\vartriangle ABC$. Let $I$ be the center of the circle that is inscribed in $\vartriangle ABC$. Finally, let $H$ be the intersection of the $3$ altitudes of the triangle. What is the angle measure of $\angle OIH$ in degrees? [b]p25.[/b] Kaitlyn fully expands the polynomial $(x^2 + x + 1)^{2018}$. How many of the coecients are not divisible by $3$? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $ABC$ be a triangle such that $AB=2$, $CA=3$, and $BC=4$. A semicircle with its diameter on $BC$ is tangent to $AB$ and $AC$. Compute the area of the semicircle.

Jeffrey stands on a straight horizontal bridge that measures $20000$ meters across. He wishes to place a pole vertically at the center of the bridge so that the sum of the distances from the top of the pole to the two ends of the bridge is $20001$ meters. To the nearest meter, how long of a pole does Jeffrey need?

The triangular plot of ACD lies between Aspen Road, Brown Road and a railroad. Main Street runs east and west, and the railroad runs north and south. The numbers in the diagram indicate distances in miles. The width of the railroad track can be ignored. How many square miles are in the plot of land ACD? [asy] size(250); defaultpen(linewidth(0.55)); pair A=(-6,0), B=origin, C=(0,6), D=(0,12); pair ac=C+2.828*dir(45), ca=A+2.828*dir(225), ad=D+2.828*dir(A--D), da=A+2.828*dir(D--A), ab=(2.828,0), ba=(-6-2.828, 0); fill(A--C--D--cycle, gray); draw(ba--ab); draw(ac--ca); draw(ad--da); draw((0,-1)--(0,15)); draw((1/3, -1)--(1/3, 15)); int i; for(i=1; i<15; i=i+1) { draw((-1/10, i)--(13/30, i)); } label("$A$", A, SE); label("$B$", B, SE); label("$C$", C, SE); label("$D$", D, SE); label("$3$", (1/3,3), E); label("$3$", (1/3,9), E); label("$3$", (-3,0), S); label("Main", (-3,0), N); label(rotate(45)*"Aspen", A--C, SE); label(rotate(63.43494882)*"Brown", A--D, NW); [/asy] $\textbf{(A)}\ 2\qquad \textbf{(B)}\ 3 \qquad \textbf{(C)}\ 4.5 \qquad \textbf{(D)}\ 6 \qquad \textbf{(E)}\ 9$

Let $ABC$ be an acute triangle. Let $\omega$ be a circle whose centre $L$ lies on the side $BC$. Suppose that $\omega$ is tangent to $AB$ at $B'$ and $AC$ at $C'$. Suppose also that the circumcentre $O$ of triangle $ABC$ lies on the shorter arc $B'C'$ of $\omega$. Prove that the circumcircle of $ABC$ and $\omega$ meet at two points. [i]Proposed by Härmel Nestra, Estonia[/i]

Let $C_1$ and $C_2$ be two tangent unit circles inside a circle $C$ of radius $2$. Circle $C_3$ inside $C$ is tangent to the circles $C,C_1,C_2$, and circle $C_4$ inside $C$ is tangent to $C,C_1,C_3$. Prove that the centers of $C,C_1,C_3$ and $C_4$ are vertices of a rectangle.

Let $\triangle ABC$ have circumcenter $O$ and incenter $I$ with $\overline{IA}\perp\overline{OI}$, circumradius $13$, and inradius $6$. Find $AB\cdot AC$.

Ralph is standing along a road which heads straight east. If you go nine miles east, make a left turn, and travel seven miles north, you will find Pamela with her mountain bike. At exactly the same time that Ralph begins running eastward along the road at 6 miles per hour, Pamela begins biking in a straight line at 10 miles per hour. Pamela’s direction is chosen so that she will reach a point on the road where Ralph is running at exactly the same time Ralph reaches that same point. Let $M$ and $N$ be relatively prime positive integers such that $\frac{M}{N}$ is the number of hours that it takes Pamela and Ralph to meet. Find $M+N$.

Let $ I$ be the incenter of triangle $ ABC.$ Let $ M,N$ be the midpoints of $ AB,AC,$ respectively. Points $ D,E$ lie on $ AB,AC$ respectively such that $ BD\equal{}CE\equal{}BC.$ The line perpendicular to $ IM$ through $ D$ intersects the line perpendicular to $ IN$ through $ E$ at $ P.$ Prove that $ AP\perp BC.$

Use the following description of a machine to solve the first 4 problems in the round. A machine displays four digits: $0000$. There are two buttons: button $A$ moves all digits one position to the left and fills the rightmost position with $0$ (for example, it changes $1234$ to $2340$), and button $B$ adds $11$ to the current number, displaying only the last four digits if the sum is greater than $9999$ (for example, it changes $1234$ to $1245$, and changes $9998$ to $0009$). We can denote a sequence of moves by writing down the buttons pushed from left to right. A sequence of moves that outputs $2100$, for example, is $BABAA$. [b]p1[/b]. Give a sequence of $17$ or less moves so that the machine displays $2020$. [b]p2.[/b] Using the same machine, how many outputs are possible if you make at most three moves? [b]p3.[/b] Button $ B$ now adds n to the four digit display, while button $ A$ remains the same. For how many positive integers $n \le 20$ (including $11$) can every possible four-digit output be reached? [b]p4.[/b] Suppose the function of button $ A$ changes to: move all digits one position to the right and fill the leftmost position with $2$. Then, what is the minimum number of moves required for the machine to display $2020$, if it initially displays $0000$? [b]p5.[/b] In the figure below, every inscribed triangle has vertices that are on the midpoints of its circumscribed triangle’s sides. If the area of the largest triangle is $64$, what is the area of the shaded region? [img]https://cdn.artofproblemsolving.com/attachments/6/f/fe17b6a6d0037163f0980a5a5297c1493cc5bb.png[/img] [b]p6.[/b] A bee flies $10\sqrt2$ meters in the direction $45^o$ clockwise of North (that is, in the NE direction). Then, the bee turns $135^o$ clockwise, and flies $20$ forward meters. It continues by turning $60^o$ counterclockwise, and flies forward $14$ meters. Finally, the bee turns $120^o$ clockwise and flies another $14$ meters forward before finally finding a flower to pollinate. How far is the bee from its starting location in meters? [b]p7.[/b] All the digits of a $15$-digit number are either $p$ or $c$. $p$ shows up $3$ more times than $c$ does, and the average of the digits is $c - p$. What is $p + c$? [b]p8.[/b] Let $m$ be the sum of the factors of $75$ (including $1$ and $75$ itself). What is the ones digit of $m^{75}$ ? [b]p9.[/b] John flips a coin twice. For each flip, if it lands tails, he does nothing. If it lands heads, he rolls a fair $4$-sided die with sides labeled 1 through $4$. Let $a/b$ be the probability of never rolling a $3$, in simplest terms. What is $a + b$? [b]p10.[/b] Let $\vartriangle ABC$ have coordinates $(0, 0)$, $(0, 3)$,$(18, 0)$. Find the number of integer coordinates interior (excluding the vertices and edges) of the triangle. [b]p11.[/b] What is the greatest integer $k$ such that $2^k$ divides the value $20! \times 20^{20}$? [b]p12.[/b] David has $n$ pennies, where $n$ is a natural number. One apple costs $3$ pennies, one banana costs $5$ pennies, and one cranberry costs $7$ pennies. If David spends all his money on apples, he will have $2$ pennies left; if David spends all his money on bananas, he will have $4$ pennies left; is David spends all his money on cranberries, he will have $6$ pennies left. What is the second least possible amount of pennies that David can have? [b]p13.[/b] Elvin is currently at Hopperville which is $40$ miles from Waltimore and $50$ miles from Boshington DC. He takes a taxi back to Waltimore, but unfortunately the taxi gets lost. Elvin now finds himself at Kinsville, but he notices that he is still $40$ miles from Waltimore and $50$ miles from Boshington $DC$. If Waltimore and Boshington DC are $30$ miles apart, What is the maximum possible distance between Hopperville and Kinsville? [b]p14.[/b] After dinner, Rick asks his father for $1000$ scoops of ice cream as dessert. Rick’s father responds, “I will give you $2$ scoops of ice cream, plus $ 1$ additional scoop for every ordered pair $(a, b)$ of real numbers satisfying $\frac{1}{a + b}= \frac{1}{a}+ \frac{1}{b}$ you can find.” If Rick finds every solution to the equation, how many scoops of ice cream will he receive? [b]p15.[/b] Esther decides to hold a rock-paper-scissors tournament for the $56$ students at her school. As a rule, competitors must lose twice before they are eliminated. Each round, all remaining competitors are matched together in best-of-1 rock-paper-scissors duels. If there is an odd number of competitors in a round, one random competitor will not compete that round. What is the maximum number of matches needed to determine the rock-paper-scissors champion? [b]p16.[/b] $ABCD$ is a rectangle. $X$ is a point on $\overline{AD}$, $Y$ is a point on $\overline{AB}$, and $N$ is a point outside $ABCD$ such that $XYNC$ is also a rectangle and $YN$ intersects $\overline{BC}$ at its midpoint $M$. $ \angle BYM = 45^o$. If $MN = 5$, what is the sum of the areas of $ABCD$ and $XYNC$? [b]p17. [/b] Mr. Brown has $10$ identical chocolate donuts and $15$ identical glazed donuts. He knows that Amar wants $6$ donuts, Benny wants $9$ donuts, and Callie wants $9$ donuts. How many ways can he distribute out his $25$ donuts? [b]p18.[/b] When Eric gets on the bus home, he notices his $ 12$-hour watch reads $03: 30$, but it isn’t working as expected. The second hand makes a full rotation in $4$ seconds, then makes another in $8$ seconds, then another in $ 12$ seconds, and so on until it makes a full rotation in $60$ seconds. Then it repeats this process, and again makes a full rotation in $4$ second, then $8$ seconds, etc. Meanwhile, the minute hand and hour hand continue to function as if every full rotation of the second hand represents $60$ seconds. When Eric gets off the bus $75$ minutes later, his watch reads $AB: CD$. What is $A + B + C + D$? [b]p19.[/b] Alex and Betty want to meet each other at the airport. Alex will arrive at the airport between $12: 00$ and $13: 15$, and will wait for Betty for $15$ minutes before he leaves. Betty will arrive at the airport between $12: 30$ and $13: 10$, and will wait for Alex for $10$ minutes before she leaves. The chance that they arrive at any time in their respective time intervals is equally likely. The probability that they will meet at the airport can be expressed as $a/b$ where $a/b$ is a fraction written in simplest form. What is $a + b$? [b]p20.[/b] Let there be $\vartriangle ABC$ such that $A = (0, 0)$, $B = (23, 0)$, $C = (a, b)$. Furthermore, $D$, the center of the circle that circumscribes $\vartriangle ABC$, lies on $\overline{AB}$. Let $\angle CDB = 150^o$. If the area of $\vartriangle ABC$ is $m/n$ where $m, n$ are in simplest integer form, find the value of $m \,\, \mod \,\,n$ (The remainder of $m$ divided by $n$). PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Triangle $ABC$ has $\angle BAC=90^\circ$. A semicircle with diameter $XY$ is inscribed inside $\triangle ABC$ such that it is tangent to a point $D$ on side $BC$, with $X$ on $AB$ and $Y$ on $AC$. Let $O$ be the midpoint of $XY$. Given that $AB=3$, $AC=4$, and $AX=\tfrac{9}{4}$, compute the length of $AO$.