There is a board with the shape of an equilateral triangle with side $n$ divided into triangular cells with the shape of equilateral triangles with side $ 1$ (the figure below shows the board when $n = 4$). Each and every triangular cell is colored either red or blue. What is the least number of cells that can be colored blue without two red cells sharing one side? [img]https://cdn.artofproblemsolving.com/attachments/0/1/d1f034258966b319dc87297bdb311f134497b5.png[/img]

Cassidy has string of $n$ bits, where $n$ is a positive integer, which initially are all $0$s or $1$s. Every second, Cassidy may choose to do one of two things: 1. Change the first bit (so the first bit changes from a $0$ to a $1$, or vice versa) 2. Change the first bit after the first $1$. Let $M$ be the minimum number of such moves it takes to get from $1\dots 1$ to $0 \dots 0$ (both of length $12$), and $N$ the number of starting sequences with $12$ bits that Cassidy can turn into all $0$s. Find $M + N$.

Find all functions $f:\mathbb{N}\setminus\{1\} \rightarrow\mathbb{N}$ such that for all distinct $x,y\in \mathbb{N}$ with $y\ge 2018$, $$\gcd(f(x),y)\cdot \mathrm{lcm}(x,f(y))=f(x)f(y).$$

The Young Scientist and the Old Scientist play the following game, taking turns in an alternating fashion, with the Young Scientist starting first. The player on turn fills in one of the stars in the equation \[ x^4 + *x^3 + *x^2 + *x + * = 0 \] with a positive real number. Who has a winning strategy if the goals of the players are: a) the Young Scientist - to make the resulting equation have no real roots, and the Old Scientist -- to make it have real roots? b) the Young Scientist - to make the resulting equation have real roots, and the Old Scientist -- to make it have none?

Consider a polinomial $p \in \mathbb{R}[x]$ of degree $n$ and with no real roots. Prove that $$\int_{-\infty}^{\infty}\frac{(p'(x))^2}{(p(x))^2+(p'(x))^2}dx$$ converges, and is less or equal than $n^{3/2}\pi.$

The acute angles of a right triangle are $a^{\circ}$ and $b^{\circ}$, where $a>b$ and both $a$ and $b$ are prime numbers. What is the least possible value of $b$? $\textbf{(A) }2\qquad\textbf{(B) }3\qquad\textbf{(C) }5\qquad\textbf{(D) }7\qquad\textbf{(E) }11$

Four positive integers $x,y,z$ and $t$ satisfy the relations \[ xy - zt = x + y = z + t. \] Is it possible that both $xy$ and $zt$ are perfect squares?

Let $E$ be a finite set of points in space such that $E$ is not contained in a plane and no three points of $E$ are collinear. Show that $E$ contains the vertices of a tetrahedron $T = ABCD$ such that $T \cap E = \{A,B,C,D\}$ (including interior points of $T$ ) and such that the projection of $A$ onto the plane $BCD$ is inside a triangle that is similar to the triangle $BCD$ and whose sides have midpoints $B,C,D.$

Let $A=(a_{jk})$ be a $10\times 10$ array of positive real numbers such that the sum of numbers in row as well as in each column is $1$. Show that there exists $j<k$ and $l<m$ such that \[a_{jl}a_{km}+a_{jm}a_{kl}\ge \frac{1}{50}\]

Let $n\ge 1$ be a positive integer. A square of side length $n$ is divided by lines parallel to each side into $n^2$ squares of side length $1$. Find the number of parallelograms which have vertices among the vertices of the $n^2$ squares of side length $1$, with both sides smaller or equal to $2$, and which have tha area equal to $2$. (Greece)

Let $a,b,c,d$ be odd integers such that $0<a<b<c<d$ and $ad=bc$. Prove that if $a+d=2^k$ and $b+c=2^m$ for some integers $k$ and $m$, then $a=1$.

Does there exist a surjective function $f:R \to R$ such that the expression $f(x + y) - f(x) - f(y)$ takes exactly two values $0$ and $1$ for various real $x$ and $y$ ? (E. Barabanov)

Let $ABC$ be an acute triangle ($AB < BC < AC$) with circumcircle $\Gamma$. Assume there exists $X \in AC$ satisfying $AB=BX$ and $AX=BC$. Points $D, E \in \Gamma$ are taken such that $\angle ADB<90^{\circ}$, $DA=DB$ and $BC=CE$. Let $P$ be the intersection point of $AE$ with the tangent line to $\Gamma$ at $B$, and let $Q$ be the intersection point of $AB$ with tangent line to $\Gamma$ at $C$. Show that the projection of $D$ onto $PQ$ lies on the circumcircle of $\triangle PAB$.

Consider the integral lattice $\mathbb{Z}^n$, $n \geq 2$, in the Euclidean $n$-space. Define a [i]line[/i] in $\mathbb{Z}^n$ to be a set of the form $a_1 \times \cdots \times a_{k-1} \times \mathbb{Z} \times a_{k+1} \times \cdots \times a_n$ where $k$ is an integer in the range $1,2,\ldots,n$, and the $a_i$ are arbitrary integers. A subset $A$ of $\mathbb{Z}^n$ is called [i]admissible[/i] if it is non-empty, finite, and every [i]line[/i] in $\mathbb{Z}^{n}$ which intersects $A$ contains at least two points from $A$. A subset $N$ of $\mathbb{Z}^n$ is called [i]null[/i] if it is non-empty, and every [i]line[/i] in $\mathbb{Z}^n$ intersects $N$ in an even number of points (possibly zero). [b](a)[/b] Prove that every [i]admissible[/i] set in $\mathbb{Z}^2$ contains a [i]null[/i] set. [b](b)[/b] Exhibit an [i]admissible[/i] set in $\mathbb{Z}^3$ no subset of which is a [i]null[/i] set .

Two circles touch in $M$, and lie inside a rectangle $ABCD$. One of them touches the sides $AB$ and $AD$, and the other one touches $AD,BC,CD$. The radius of the second circle is four times that of the first circle. Find the ratio in which the common tangent of the circles in $M$ divides $AB$ and $CD$.

Let $a,b,c$ be the positive real numbers satisfying $a^2+b^2+c^2=3$. Prove that: $$\frac{a}{b(a+c)}+\frac{b}{c(b+a)}+\frac{c}{a(c+b)}\geq \frac{3}{2}.$$

In a triangle $ABC$ points $A_1,B_1,C_1$ are the midpoints of $BC,CA,AB$ respectively,and $A_2,B_2,C_2$ are the midpoints of the altitudes from $A,B,C$ respectively. Show that the lines $A_1A_2,B_1B_2,C_1,C_2$ are concurrent.

$P\in A_n(\mathbb R)=\{M_{n\times n}|M^2=M\}$. Which of the following are true? $\textbf{(A)}~P^T=P,\forall P\in A_n(\mathbb R)$ $\textbf{(B)}~\exists P\ne0,P\in A_n(\mathbb R)\text{ with }\operatorname{tr}(P)=0$ $\textbf{(C)}~\exists X_{n\times r}\text{ such that }Px=X\text{ for }r=\operatorname{rank}(P)$

[b]Q10.[/b] Solve the following equation \[\frac{1}{(x+29)^2}+ \frac{1}{(x+30)^2}= \frac{13}{36}.\]

$300$ people took part in the drawing for the main prize of the television lottery. They lined up in a circle, then, starting with someone who received number $1$, they began to count them. Moreover, every third person dropped out every time. (So, in the first round, everyone with numbers divisible by $3$ dropped out). The counting continued until there was only one person left. (It is clear that more than one circle was made). This person received the main prize. (It “accidentally” turned out to be the TV director’s mother-in-law). What number did this person have in the initial lineup?

A man has a seven-candle chandellier. The first evening he lighted one candle for one hour, the second evening he lighted two candles, also for one hour, and so on. After one hour the seventh evening, all seven candles simultaneously finished. How did the man choose the candles to light every evening?

What is the last digit of ${17^{17^{17^{17}}}}$?

Given the tetrahedron $ABCD$, whose opposite edges are equal, that is, $AB=CD, AC=BD$ and $BC=AD$. Prove that exist exactly $6$ planes intersecting the triangular angles of the tetrahedron and dividing the total surface and volume of this tetrahedron in half.

Let $f(x,y)$ and $g(x,y)$ be real valued functions defined for every $x,y \in \{1,2,..,2000\}$. If there exist $X,Y \subset \{1,2,..,2000\}$ such that $s(X)=s(Y)=1000$ and $x\notin X$ and $y\notin Y$ implies that $f(x,y)=g(x,y)$ than, what is the maximum number of $(x,y)$ couples where $f(x,y)\neq g(x,y)$.

[b]p1.[/b] What is the slope of the line perpendicular to the the graph $\frac{x}{4}+\frac{y}{9}= 1$ at $(0, 9)$? [b]p2.[/b] A boy is standing in the middle of a very very long staircase and he has two pogo sticks. One pogo stick allows him to jump $220$ steps up the staircase. The second pogo stick allows him to jump $125$ steps down the staircase. What is the smallest positive number of steps that he can reach from his original position by a series of jumps? [b]p3.[/b] If you roll three fair six-sided dice, what is the probability that the product of the results will be a multiple of $3$? [b]p4.[/b] Right triangle $ABC$ has squares $ABXY$ and $ACWZ$ drawn externally to its legs and a semicircle drawn externally to its hypotenuse $BC$. If the area of the semicircle is $18\pi$ and the area of triangle $ABC$ is $30$, what is the sum of the areas of squares $ABXY$ and $ACWZ$? [img]https://cdn.artofproblemsolving.com/attachments/5/1/c9717e7731af84e5286335420b73b974da2643.png[/img] [b]p5.[/b] You have a bag containing $3$ types of pens: red, green, and blue. $30\%$ of the pens are red pens, and $20\%$ are green pens. If, after you add $10$ blue pens, $60\%$ of the pens are blue pens, how many green pens did you start with? [b]p6.[/b] Canada gained partial independence from the United Kingdom in $1867$, beginning its long role as the headgear of the United States. It gained its full independence in $1982$. What is the last digit of $1867^{1982}$? [b]p7.[/b] Bacon, Meat, and Tomato are dealing with paperwork. Bacon can fill out $5$ forms in $3$ minutes, Meat can fill out $7$ forms in $5$ minutes, and Tomato can staple $3$ forms in $1$ minute. A form must be filled out and stapled together (in either order) to complete it. How long will it take them to complete $105$ forms? [b]p8.[/b] Nice numbers are defined to be $7$-digit palindromes that have no $3$ identical digits (e.g., $1234321$ or $5610165$ but not $7427247$). A pretty number is a nice number with a $7$ in its decimal representation (e.g., $3781873$). What is the $7^{th}$ pretty number? [b]p9.[/b] Let $O$ be the center of a semicircle with diameter $AD$ and area $2\pi$. Given square $ABCD$ drawn externally to the semicircle, construct a new circle with center $B$ and radius $BO$. If we extend $BC$, this new circle intersects $BC$ at $P$. What is the length of $CP$? [img]https://cdn.artofproblemsolving.com/attachments/b/1/be15e45cd6465c7d9b45529b6547c0010b8029.png[/img] [b]p10.[/b] Derek has $10$ American coins in his pocket, summing to a total of $53$ cents. If he randomly grabs $3$ coins from his pocket, what is the probability that they're all different? [b]p11.[/b] What is the sum of the whole numbers between $6\sqrt{10}$ and $7\pi$ ? [b]p12.[/b] What is the volume of a cylinder whose radius is equal to its height and whose surface area is numerically equal to its volume? [b]p13.[/b] $15$ people, including Luke and Matt, attend a Berkeley Math meeting. If Luke and Matt sit next to each other, a fight will break out. If they sit around a circular table, all positions equally likely, what is the probability that a fight doesn't break out? [b]p14.[/b] A non-degenerate square has sides of length $s$, and a circle has radius $r$. Let the area of the square be equal to that of the circle. If we have a rectangle with sides of lengths $r$, $s$, and its area has an integer value, what is the smallest possible value for $s$? [b]p15.[/b] How many ways can you arrange the letters of the word "$BERKELEY$" such that no two $E$'s are next to each other? [b]p16.[/b] Kim, who has a tragic allergy to cake, is having a birthday party. She invites $12$ people but isn't sure if $11$ or $12$ will show up. However, she needs to cut the cake before the party starts. What is the least number of slices (not necessarily of equal size) that she will need to cut to ensure that the cake can be equally split among either $11$ or $12$ guests with no excess? [b]p17.[/b] Tom has $2012$ blue cards, $2012$ red cards, and $2012$ boxes. He distributes the cards in such a way such that each box has at least $1$ card. Sam chooses a box randomly, then chooses a card randomly. Suppose that Tom arranges the cards such that the probability of Sam choosing a blue card is maximized. What is this maximum probability? [b]p18.[/b] Allison wants to bake a pie, so she goes to the discount market with a handful of dollar bills. The discount market sells different fruit each for some integer number of cents and does not add tax to purchases. She buys $22$ apples and $7$ boxes of blueberries, using all of her dollar bills. When she arrives back home, she realizes she needs more fruit, though, so she grabs her checkbook and heads back to the market. This time, she buys $31$ apples and $4$ boxes of blueberries, for a total of $60$ cents more than her last visit. Given she spent less than $100$ dollars over the two trips, how much (in dollars) did she spend on her first trip to the market? [b]p19.[/b] Consider a parallelogram $ABCD$. Let $k$ be the line passing through A and parallel to the bisector of $\angle ABC$, and let $\ell$ be the bisector of $\angle BAD$. Let $k$ intersect line $CD$ at $E$ and $\ell$ intersect line $CD$ at $F$. If $AB = 13$ and $BC = 37$, find the length $EF$. [b]p20.[/b] Given for some real $a, b, c, d,$ $$P(x) = ax^4 + bx^3 + cx^2 + dx$$ $$P(-5) = P(-2) = P(2) = P(5) = 1$$ Find $P(10).$ PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].