Each white point in the figure below has to be completed with one of the integers $1, 2, ..., 9$, without repetitions, such that the sum of the three numbers in the external circle is equal to the sum of the four numbers in each internal circle that don't belong to the external circle. $(a)$ Show a solution. $(b)$ Prove that, in any solution, the number $9$ must belong to the external circle.

What is the tens digit in the sum $ 7! \plus{} 8! \plus{} 9! \plus{} \cdots \plus{} 2006!$? $ \textbf{(A) } 1 \qquad \textbf{(B) } 3 \qquad \textbf{(C) } 4 \qquad \textbf{(D) } 6 \qquad \textbf{(E) } 9$

Let $ABC$ be an isosceles triangle $ABC$ with base $BC$ insribed in a circle. The segment $AD$ is the diameter of the circle, and point $P$ lies on the smaller arc $BD$. Line $DP$ intersects rays $AB$ and $AC$ at points $M$ and $N$, and the lines $BP$ and $CP$ intersects the line $AD$ at points $Q$ and $R$. Prove that the midpoint of the segment $MN$ lies on the circumscribed circle of triangle $PQR$.

Kevin is trying to solve an economics question which has six steps. At each step, he has a probability $p$ of making a sign error. Let $q$ be the probability that Kevin makes an even number of sign errors (thus answering the question correctly!). For how many values of $0 \le p \le 1$ is it true that $p+q=1$? [i]Proposed by Evan Chen[/i]

On the circle $K$ there are given three distinct points $A,B,C$. Construct (using only a straightedge and a compass) a fourth point $D$ on $K$ such that a circle can be inscribed in the quadrilateral thus obtained.

Given a triangle $ABC$, point $D$ on $[AB], E$ on $[AC]$, $|AD| = |DE| = |AC| , |BD| = |AE| , DE$ is parallel to $BC$. Prove that the length $|BD|$ equals to the side of a regular decagon inscribed in a circle with the radius $R=|AC|$.

Find all pairs of integers $(m,n)$ such that the numbers $A=n^2+2mn+3m^2+2$, $B=2n^2+3mn+m^2+2$, $C=3n^2+mn+2m^2+1$ have a common divisor greater than $1$.

Let the function $f$ be defined on the set $\mathbb{N}$ such that $\text{(i)}\ \ \quad f(1)=1$ $\text{(ii)}\ \quad f(2n+1)=f(2n)+1$ $\text{(iii)}\quad f(2n)=3f(n)$ Determine the set of values taken $f$.

Prove or disprove that there exists an integer which is doubled when the initial digit is transferred to the end.

What is the largest integer less than or equal to $4x^3 - 3x$, where $x=\frac{\sqrt[3]{2+\sqrt3}+\sqrt[3]{2-\sqrt3}}{2}$ ? (A) $1$, (B) $2$, (C) $3$, (D) $4$, (E) None of the above.

Let $ABCD$ be a cyclic quadrilateral, so that $|AB| + |CD| = |BC|$. Show that the intersection of the bisector of $\angle DAB$ and $\angle CDA$ lies on the side $BC$.

[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 $n\ge3$ be a positive integer. Alex and Lizzie play a game. Alex chooses $n$ positive integers (not necessarily distinct), writes them on a blackboard, and does nothing further. Then, Lizzie is allowed to pick some of the numbers―but not all of them―and replace them each by their average. For example, if $n=7$ and the numbers Alex writes on the blackboard to start are $1,2,4,5,9,4,11$, then on her first turn Lizzie could pick $1,4,9$, erase them, and replace them each with the number $\tfrac{1+4+9}{3}$, leaving on the blackboard the numbers $\tfrac{14}{3},2,\tfrac{14}{3},5,\tfrac{14}{3},4,11$. Lizzie can repeat this process of selecting and averaging some numbers as often as she wants. Lizzie wins the game if eventually all of the numbers written on the blackboard are equal. Find all positive integers $n\ge3$ such that no matter what numbers Alex picks, Lizzie can win the game.

A function $f : R -\{0\} \to R$ has the following properties: (i) $f(x)- f(y) = f(x)f\left(\frac{1}{y}\right)- f(y)f\left(\frac{1}{x}\right)$ for all $x,y \ne 0$, (ii) $f$ takes the value $\frac12$ at least once. Determine $f(-1)$. Prove that $f$ is a periodic function

Let $f:[0,1]\to\mathbb R$ be a function satisfying $$|f(x)-f(y)|\le|x-y|$$for every $x,y\in[0,1]$. Show that for every $\varepsilon>0$ there exists a countable family of rectangles $(R_i)$ of dimensions $a_i\times b_i$, $a_i\le b_i$ in the plane such that $$\{(x,f(x)):x\in[0,1]\}\subset\bigcup_iR_i\text{ and }\sum_ia_i<\varepsilon.$$(The edges of the rectangles are not necessarily parallel to the coordinate axes.)

What is the value of $ x$ if $ |x \minus{} 1| \equal{} |x \minus{} 2|$? $ \textbf{(A)}\ \minus{}\!\frac {1}{2}\qquad \textbf{(B)}\ \frac {1}{2}\qquad \textbf{(C)}\ 1\qquad \textbf{(D)}\ \frac {3}{2}\qquad \textbf{(E)}\ 2$

Cat and Claire are having a discussion about their favorite positive two-digit numbers. [b]Cat:[/b] My number has a $1$ in its tens digit. Is it possible that your number is a multiple of my number? [b]Claire:[/b] No, however, my number is not prime. Additionally, if I told you the two digits of my number, you still wouldn't know my number. [b]Cat:[/b] Aha, my number and your number aren't relatively prime! [b]Claire:[/b] Then our numbers must share the same ones digit! What is the product of Cat and Claire's numbers?

a) Let $ n_{1},n_{2},\dots$ be a sequence of natural number such that $ n_{i}\geq2$ and $ \epsilon_{1},\epsilon_{2},\dots$ be a sequence such that $ \epsilon_{i}\in\{1,2\}$. Prove that the sequence: \[ \sqrt[n_{1}]{\epsilon_{1}\plus{}\sqrt[n_{2}]{\epsilon_{2}\plus{}\dots\plus{}\sqrt[n_{k}]{\epsilon_{k}}}}\]is convergent and its limit is in $ (1,2]$. Define $ \sqrt[n_{1}]{\epsilon_{1}\plus{}\sqrt[n_{2}]{\epsilon_{2}\plus{}\dots}}$ to be this limit. b) Prove that for each $ x\in(1,2]$ there exist sequences $ n_{1},n_{2},\dots\in\mathbb N$ and $ n_{i}\geq2$ and $ \epsilon_{1},\epsilon_{2},\dots$, such that $ n_{i}\geq2$ and $ \epsilon_{i}\in\{1,2\}$, and $ x\equal{}\sqrt[n_{1}]{\epsilon_{1}\plus{}\sqrt[n_{2}]{\epsilon_{2}\plus{}\dots}}$

Let $\ a,x,y,z>0$. Prove that: $\frac{a+y}{a+z}x+\frac{a+z}{a+x}y+\frac{a+x}{a+y}z\geq{x+y+z}\geq\frac{a+z}{a+x}x+\frac{a+x}{a+y}y+\frac{a+y}{a+z}z$

In parallelogram $ABCD$ with acute angle $A$ a point $N$ is chosen on the segment $AD$, and a point $M$ on the segment $CN$ so that $AB = BM = CM$. Point $K$ is the reflection of $N$ in line $MD$. The line $MK$ meets the segment $AD$ at point $L$. Let $P$ be the common point of the circumcircles of $AMD$ and $CNK$ such that $A$ and $P$ share the same side of the line $MK$. Prove that $\angle CPM = \angle DPL$.

Find all real solutions $(x,y)$ to the equation $y^4+2y^2+8x^2+16x^4 = 24xy-8$.

Let $f(x) = x^2 - x + 1$. Prove that for every natural $m>1$ the numbers $$m, f(m), f(f(m)), ...$$ are relatively prime.

There exists a quadrilateral $\mathcal{Q} _{1}$ such that the midpoints of its sides lie on a circle. Prove that there exists a cyclic quadrilateral $\mathcal{Q} _{2}$ with the same sides as $\mathcal{Q} _{1}$ with two of the same angles.

Find all functions $f:\mathbb{N}\to\mathbb{N}$ such that $$|xf(y)-yf(x)|$$ is a perfect square for every $x,y \in \mathbb{N}$

Suppose $n$ houses are to be assigned to $n$ people. Each person ranks the houses in the order of preference, with no ties. After the assignment is made, it is observed that every other assignment would assign to at least one person a less preferred house. Prove that there is at least one person who received the house he/she preferred most under this assignment.