Let $C$ be the set of points $(x,y)$ with integer coordinates in the plane where $1\leq x\leq 900$ and $1\leq y\leq 1000$. A polygon $P$ with vertices in $C$ is called [i]emerald[/i] if $P$ has exactly zero or two vertices in each row and each column and all the internal angles of $P$ are $90^\circ$ or $270^\circ$. Find the greatest value of $k$ such that we can color $k$ points in $C$ such that any subset of these $k$ points is not the set of vertices of an [i]emerald[/i] polygon. [img]https://cdn.discordapp.com/attachments/954427908359876608/1299737432010395678/image.png?ex=671e4a4f&is=671cf8cf&hm=ce008541975226a0e9ea53a93592a7469d8569baca945c1c207d4a722126bb60&[/img] On the left, an example of an emerald polygon; on the right, an example of a non-emerald polygon.

[b]p1.[/b] To convert between Fahrenheit, $F$, and Celsius, $C$, the formula is $F = \frac95 C + 32$. Jennifer, having no time to be this precise, instead approximates the temperature of Fahrenheit, $\widehat F$, as $\widehat F = 2C + 30$. There is a range of temperatures $C_1 \le C \le C_2$ such that for any $C$ in this range, $| \widehat F - F| \le 5$. Compute the ordered pair $(C_1,C_2)$. [b]p2.[/b] Compute integer $x$ such that $x^{23} = 27368747340080916343$. [b]p3.[/b] The number of ways to flip $n$ fair coins such that there are no three heads in a row can be expressed with the recurrence relation $$ S(n + 1) = a_0 S(n) + a_1 S(n - 1) + ... + a_k S(n - k) $$ for sufficiently large $n$ and $k$ where $S(n)$ is the number of valid sequences of length $n$. What is $\sum^k_{n=0}|a_n|$? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Given a polynomial$$P(x)=x^n+a_{n-1}x^{n-1}+...+a_1x+a_0\in \mathbb{Z}[x]$$ with degree $n\ge 2$ and $a_o\ne 0.$ Prove that if $|a_{n-1}|>1+|a_{n-2}|+...+|a_1|+|a_0|$, then $P(x)$ is irreducible in $\mathbb{Z}[x].$

Thomas and Nils are playing a game. They have a number of cards, numbered $1, 2, 3$, et cetera. At the start, all cards are lying face up on the table. They take alternate turns. The person whose turn it is, chooses a card that is still lying on the table and decides to either keep the card himself or to give it to the other player. When all cards are gone, each of them calculates the sum of the numbers on his own cards. If the difference between these two outcomes is divisible by $3$, then Thomas wins. If not, then Nils wins. (a) Suppose they are playing with $2018$ cards (numbered from $1$ to $2018$) and that Thomas starts. Prove that Nils can play in such a way that he will win the game with certainty. (b) Suppose they are playing with $2020 $cards (numbered from $1$ to $2020$) and that Nils starts. Which of the two players can play in such a way that he wins with certainty?

There are 22 black and 3 blue balls in a bag. Ahmet chooses an integer $ n$ in between 1 and 25. Betül draws $ n$ balls from the bag one by one such that no ball is put back to the bag after it is drawn. If exactly 2 of the $ n$ balls are blue and the second blue ball is drawn at $ n^{th}$ order, Ahmet wins, otherwise Betül wins. To increase the possibility to win, Ahmet must choose $\textbf{(A)}\ 2 \qquad\textbf{(B)}\ 11 \qquad\textbf{(C)}\ 12 \qquad\textbf{(D)}\ 13 \qquad\textbf{(E)}\ 23$

Let be given $n$ positive integer. Lets write with $a_n$ the number of positive integer pairs $(x,y)$ such that $x+y$ is even and $1\leq x\leq y\leq n$. Lets write with $b_n$ the number of positive integer pairs $(x,y)$ such that $x+y\leq n+1$ and $1\leq x\leq y\leq n$.

Let $\mathcal{A}$ denote the set of all polynomials in three variables $x, y, z$ with integer coefficients. Let $\mathcal{B}$ denote the subset of $\mathcal{A}$ formed by all polynomials which can be expressed as \begin{align*} (x + y + z)P(x, y, z) + (xy + yz + zx)Q(x, y, z) + xyzR(x, y, z) \end{align*} with $P, Q, R \in \mathcal{A}$. Find the smallest non-negative integer $n$ such that $x^i y^j z^k \in \mathcal{B}$ for all non-negative integers $i, j, k$ satisfying $i + j + k \geq n$.

In the adjoining figure the five circles are tangent to one another consecutively and to the lines $L_1$ and $L_2$ ($L_1$ is the line that is above the circles and $L_2$ is the line that goes under the circles). If the radius of the largest circle is 18 and that of the smallest one is 8, then the radius of the middle circle is [asy] size(250);defaultpen(linewidth(0.7)); real alpha=5.797939254, x=71.191836; int i; for(i=0; i<5; i=i+1) { real r=8*(sqrt(6)/2)^i; draw(Circle((x+r)*dir(alpha), r)); x=x+2r; } real x=71.191836+40+20*sqrt(6), r=18; pair A=tangent(origin, (x+r)*dir(alpha), r, 1), B=tangent(origin, (x+r)*dir(alpha), r, 2); pair A1=300*dir(origin--A), B1=300*dir(origin--B); draw(B1--origin--A1); pair X=(69,-5), X1=reflect(origin, (x+r)*dir(alpha))*X, Y=(200,-5), Y1=reflect(origin, (x+r)*dir(alpha))*Y, Z=(130,0), Z1=reflect(origin, (x+r)*dir(alpha))*Z; clip(X--Y--Y1--X1--cycle); label("$L_2$", Z, S); label("$L_1$", Z1, dir(2*alpha)*dir(90));[/asy] $\text{(A)} \ 12 \qquad \text{(B)} \ 12.5 \qquad \text{(C)} \ 13 \qquad \text{(D)} \ 13.5 \qquad \text{(E)} \ 14$

Let $a, b, c,x, y, z$ be positive reals such that $\frac{1}{x}+\frac{1}{y}+\frac{1}{z}=1$. Prove that \[a^x+b^y+c^z\ge \frac{4abcxyz}{(x+y+z-3)^2}.\] [i]Proposed by Daniel Liu[/i]

In chess, a knight placed on a chess board can move by jumping to an adjacent square in one direction (up, down, left, or right) then jumping to the next two squares in a perpendicular direction. We then say that a square in a chess board [i]can be attacked[/i] by a knight if the knight can end up on that square after a move. Thus, depending on where a knight is placed, it can attack as many as eight squares, or maybe even less. In a $10 \times 10$ chess board, what is the maximum number of knights that can be placed such that each square on the board can be attacked by at most one knight?

Does there exist a convex pentagon, all of whose vertices are lattice points in the plane, with no lattice point in the interior?

Let $0 \leq \alpha , \beta , \gamma \leq \frac{\pi}{2}$, such that $\cos ^{2} \alpha + \cos ^{2} \beta + \cos ^{2} \gamma = 1$. Prove that $2 \leq (1 + \cos ^{2} \alpha ) ^{2} \sin^{4} \alpha + (1 + \cos ^{2} \beta ) ^{2} \sin ^{4} \beta + (1 + \cos ^{2} \gamma ) ^{2} \sin ^{4} \gamma \leq (1 + \cos ^{2} \alpha )(1 + \cos ^{2} \beta)(1 + \cos ^{2} \gamma ).$

We are given a supply of $a \times b$ tiles with $a$ and $b$ distinct positive integers. The tiles are to be used to tile a $28 \times 48$ rectangle. Find $a, b$ such that the tile has the smallest possible area and there is only one possible tiling. (If there are two distinct tilings, one of which is a reflection of the other, then we treat that as more than one possible tiling. Similarly for other symmetries.) Find $a, b$ such that the tile has the largest possible area and there is more than one possible tiling.

In a five term arithmetic sequence, the first term is $2020$ and the last term is $4040.$ Find the second term of the sequence. [i]Proposed by Ada Tsui[/i]

Let $a, b, c$ be three real numbers which are roots of a cubic polynomial, and satisfy $a+b+c=6$ and $ab+bc+ca=9$. Suppose $a<b<c$. Show that $$0<a<1<b<3<c<4.$$

Let $n$ be an odd natural number and $A,B \in \mathcal{M}_n(\mathbb{C})$ be two matrices such that $(A-B)^2=O_n.$ Prove that $\det(AB-BA)=0.$

Find, with proof, all real numbers $ x \in \lbrack 0, \frac {\pi}{2} \rbrack$, such that $ (2 \minus{} \sin 2x)\sin (x \plus{} \frac {\pi}{4}) \equal{} 1$.

Find all square roots of integers, namely $ p, $ such that $ \left(\frac{p}{2}\right)^2 <3<\left(\frac{p+1}{2}\right)^2. $

Consider a regular $2n + 1$-gon $P$ in the plane, where n is a positive integer. We say that a point $S$ on one of the sides of $P$ can be seen from a point $E$ that is external to $P$, if the line segment $SE$ contains no other points that lie on the sides of $P$ except $S$. We want to color the sides of $P$ in $3$ colors, such that every side is colored in exactly one color, and each color must be used at least once. Moreover, from every point in the plane external to $P$, at most $2$ different colors on $P$ can be seen (ignore the vertices of $P$, we consider them colorless). Find the largest positive integer for which such a coloring is possible.

Isosceles $ \triangle ABC$ has a right angle at $ C$. Point $ P$ is inside $ \triangle ABC$, such that $ PA \equal{} 11, PB \equal{} 7,$ and $ PC \equal{} 6$. Legs $ \overline{AC}$ and $ \overline{BC}$ have length $ s \equal{} \sqrt {a \plus{} b\sqrt {2}}$, where $ a$ and $ b$ are positive integers. What is $ a \plus{} b$? [asy]pointpen = black; pathpen = linewidth(0.7); pen f = fontsize(10); size(5cm); pair B = (0,sqrt(85+42*sqrt(2))); pair A = (B.y,0); pair C = (0,0); pair P = IP(arc(B,7,180,360),arc(C,6,0,90)); D(A--B--C--cycle); D(P--A); D(P--B); D(P--C); MP("A",D(A),plain.E,f); MP("B",D(B),plain.N,f); MP("C",D(C),plain.SW,f); MP("P",D(P),plain.NE,f);[/asy] $ \textbf{(A) } 85 \qquad \textbf{(B) } 91 \qquad \textbf{(C) } 108 \qquad \textbf{(D) } 121 \qquad \textbf{(E) } 127$

Let $A$ and $B$ be two fixed points of a given circle and $XY$ a diameter of this circle. Find the locus of the intersection points of lines $AX$ and $BY$ . ($BY$ is not a diameter of the circle). Albanian National Mathematical Olympiad 2010---12 GRADE Question 1.

Three spheres are tangent to one plane, to a straight line perpendicular to this plane, and in pairs to each other. The radius of the largest sphere is $1$. Within what limits can the radius of the smallest sphere vary?

Define a sequence by $a_1=1,a_2=\frac12$, and $a_{n+2}=a_{n+1}-\frac{a_na_{n+1}}2$ for $n$ a positive integer. Find $\lim_{n\to\infty}na_n$.

In rectangle $ABCD$, the points $M,N,P, Q$ lie on $AB$, $BC$, $CD$, $DA$ respectively such that the area of triangles $AQM$, $BMN$, $CNP$, $DPQ$ are equal. Prove that the quadrilateral $MNPQ$ is parallelogram. by Mahdi Etesami Fard

Let M be a subst of {1,2,...,2006} with the following property: For any three elements x,y and z (x<y<z) of M, x+y does not divide z. Determine the largest possible size of M. Justify your claim.