Every cell of a $20 \times 20$ table has to be coloured black or white (there are $2^{400}$ such colourings in total). Given any colouring $P$, we consider division of the table into rectangles with sides in the grid lines where no rectangle contains more than two black cells and where the number of rectangles containing at most one black cell is the least possible. We denote this smallest possible number of rectangles containing at most one black cell by $f(P)$. Determine the maximum value of $f(P)$ as $P$ ranges over all colourings.

Find all pairs of integers $(a,b)$ such that $(b^2+7(a-b))^2=a^{3}b$.

The line segments $AB$ and $CD$ are perpendicular and intersect at point $X$. Additionally, the following equalities hold: $AC=BD$, $AD=BX$, and $DX=1$. Determine the length of segment $CX$.

In the following $6\times 6$ matrix, one can choose any $k\times k$ submatrix, with $1<k\leq6 $ and add $1$ to all its entries. Is it possible to perform the operation a finite number of times so that all the entries in the $6\times 6$ matrix are multiples of $3$? $ \begin{pmatrix} 2 & 0 & 1 & 0 & 2 & 0 \\ 0 & 2 & 0 & 1 & 2 & 0 \\ 1 & 0 & 2 & 0 & 2 & 0 \\ 0 & 1 & 0 & 2 & 2 & 0 \\ 1 & 1 & 1 & 1 & 2 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \end{pmatrix} $ Note: A $p\times q$ submatrix of a $m\times n$ matrix (with $p\leq m$, $q\leq n$) is a $p\times q$ matrix formed by taking a block of the entries of this size from the original matrix.

Let $ABC$ be an acute angled triangle with orthocenter $H$. centroid $G$ and circumcircle $\omega$. Let $D$ and $M$ respectively be the intersection of lines $AH$ and $AG$ with side $BC$. Rays $MH$ and $DG$ interect $ \omega$ again at $P$ and $Q$ respectively. Prove that $PD$ and $QM$ intersect on $\omega$.

[b]p1.[/b] What is the units digit of $1 + 9 + 9^2 +... + 9^{2015}$ ? [b]p2.[/b] In Fourtown, every person must have a car and therefore a license plate. Every license plate must be a $4$-digit number where each digit is a value between $0$ and $9$ inclusive. However $0000$ is not a valid license plate. What is the minimum population of Fourtown to guarantee that at least two people who have the same license plate? [b]p3.[/b] Two sides of an isosceles triangle $\vartriangle ABC$ have lengths $9$ and $4$. What is the area of $\vartriangle ABC$? [b]p4.[/b] Let $x$ be a real number such that $10^{\frac{1}{x}} = x$. Find $(x^3)^{2x}$. [b]p5.[/b] A Berkeley student and a Stanford student are going to visit each others campus and go back to their own campuses immediately after they arrive by riding bikes. Each of them rides at a constant speed. They first meet at a place $17.5$ miles away from Berkeley, and secondly $10$ miles away from Stanford. How far is Berkeley away from Stanford in miles? [b]p6.[/b] Let $ABCDEF$ be a regular hexagon. Find the number of subsets $S$ of $\{A,B,C,D,E, F\}$ such that every edge of the hexagon has at least one of its endpoints in $S$. [b]p7.[/b] A three digit number is a multiple of $35$ and the sum of its digits is $15$. Find this number. [b]p8.[/b] Thomas, Olga, Ken, and Edward are playing the card game SAND. Each draws a card from a $52$ card deck. What is the probability that each player gets a dierent rank and a different suit from the others? [b]p9.[/b] An isosceles triangle has two vertices at $(1, 4)$ and $(3, 6)$. Find the $x$-coordinate of the third vertex assuming it lies on the $x$-axis. [b]p10.[/b] Find the number of functions from the set $\{1, 2,..., 8\}$ to itself such that $f(f(x)) = x$ for all $1 \le x \le 8$. [b]p11.[/b] The circle has the property that, no matter how it's rotated, the distance between the highest and the lowest point is constant. However, surprisingly, the circle is not the only shape with that property. A Reuleaux Triangle, which also has this constant diameter property, is constructed as follows. First, start with an equilateral triangle. Then, between every pair of vertices of the triangle, draw a circular arc whose center is the $3$rd vertex of the triangle. Find the ratio between the areas of a Reuleaux Triangle and of a circle whose diameters are equal. [b]p12.[/b] Let $a$, $b$, $c$ be positive integers such that gcd $(a, b) = 2$, gcd $(b, c) = 3$, lcm $(a, c) = 42$, and lcm $(a, b) = 30$. Find $abc$. [b]p13.[/b] A point $P$ is inside the square $ABCD$. If $PA = 5$, $PB = 1$, $PD = 7$, then what is $PC$? [b]p14.[/b] Find all positive integers $n$ such that, for every positive integer $x$ relatively prime to $n$, we have that $n$ divides $x^2 - 1$. You may assume that if $n = 2^km$, where $m$ is odd, then $n$ has this property if and only if both $2^k$ and $m$ do. [b]p15.[/b] Given integers $a, b, c$ satisfying $$abc + a + c = 12$$ $$bc + ac = 8$$ $$b - ac = -2,$$ what is the value of $a$? [b]p16.[/b] Two sides of a triangle have lengths $20$ and $30$. The length of the altitude to the third side is the average of the lengths of the altitudes to the two given sides. How long is the third side? [b]p17.[/b] Find the number of non-negative integer solutions $(x, y, z)$ of the equation $$xyz + xy + yz + zx + x + y + z = 2014.$$ [b]p18.[/b] Assume that $A$, $B$, $C$, $D$, $E$, $F$ are equally spaced on a circle of radius $1$, as in the figure below. Find the area of the kite bounded by the lines $EA$, $AC$, $FC$, $BE$. [img]https://cdn.artofproblemsolving.com/attachments/7/7/57e6e1c4ef17f84a7a66a65e2aa2ab9c7fd05d.png[/img] [b]p19.[/b] A positive integer is called cyclic if it is not divisible by the square of any prime, and whenever $p < q$ are primes that divide it, $q$ does not leave a remainder of $1$ when divided by $p$. Compute the number of cyclic numbers less than or equal to $100$. [b]p20.[/b] On an $8\times 8$ chess board, a queen can move horizontally, vertically, and diagonally in any direction for as many squares as she wishes. Find the average (over all $64$ possible positions of the queen) of the number of squares the queen can reach from a particular square (do not count the square she stands on). PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

[b]Q.[/b] Find the minimum value of the expression for $x,y,z\in \mathbb{R}^{+}$ $$\sum_{\text{cyc}}\frac{(x+1)^{4}+2(y+1)^{6}-(y+1)^{4}}{(y+1)^{6}}$$ [i]Proposed by Aritra12[/i]

a) Let $ x_1, x_2, ..., x_n $ ($ n> 2 $) be negative real numbers and $ x_1 + x_2 + ... + x_n = m. $ Determine the maximum value of the sum $ S = x_1x_2 + x_1x_3 + \dots + x_1x_n + x_2x_3 + x_2x_4 + \dots + x_2x_n + \dots + x_ {n-1} x_n. $ b) Let $ x_1, x_2, ..., x_n $ ($ n> 2 $) be nonnegative natural numbers and $ x_1 + x_2 + ... + x_n = m. $ Determine the maximum value of the sum $ S = x_1x_2 + x_1x_3 + \dots + x_1x_n + x_2x_3 + x_2x_4 + \dots + x_2x_n + \dots + x_ {n-1} x_n. $

Look at these fractions. At firs step we have $ \frac{0}{1}$ and $ \frac{1}{0}$, and at each step we write $ \frac{a\plus{}b}{c\plus{}d}$ between $ \frac{a}{b}$ and $ \frac{c}{d}$, and we do this forever \[ \begin{array}{ccccccccccccccccccccccccc}\frac{0}{1}&&&&&&&&\frac{1}{0}\\ \frac{0}{1}&&&&\frac{1}{1}&&&&\frac{1}{0}\\ \frac{0}{1}&&\frac{1}{2}&&\frac{1}{1}&&\frac{2}{1}&&\frac{1}{0}\\ \frac{0}{1}&\frac{1}{3}&\frac{1}{2}&\frac{2}{3}&\frac{1}{1}&\frac{3}{2}&\frac{2}{1}&\frac{3}{1}&\frac{1}{0}\\ &&&&\dots\end{array}\] a) Prove that each of these fractions is irreducible. b) In the plane we have put infinitely many circles of diameter 1, over each integer on the real line, one circle. The inductively we put circles that each circle is tangent to two adjacent circles and real line, and we do this forever. Prove that points of tangency of these circles are exactly all the numbers in part a(except $ \frac{1}{0}$). [img]http://i2.tinypic.com/4m8tmbq.png[/img] c) Prove that in these two parts all of positive rational numbers appear. If you don't understand the numbers, look at [url=http://upload.wikimedia.org/wikipedia/commons/2/21/Arabic_numerals-en.svg]here[/url].

Let $Q(x)$ be a non-zero polynomial and $k$ be a natural number. Prove that the polynomial $P(x) = (x-1)^kQ(x)$ has at least $k+1$ non-zero coefficients.

Prove that for any positive integer $n > 2$ we can find $n$ distinct positive integers, the sum of whose reciprocals is equal to $1$.

Let $X$ be a bounded, nonempty set of points in the Cartesian plane. Let $f(X)$ be the set of all points that are at a distance of at most $1$ from some point in $X$. Let $f_n(X) = f(f(\cdots(f(X))\cdots))$ ($n$ times). Show that $f_n(X)$ becomes “more circular” as $n$ gets larger. In other words, if $r_n = \sup\{\text{radii of circles contained in } f_n(X) \}$ and $R_n = \inf \{\text{radii of circles containing } f_n(X)\}$, then show that $R_n/r_n$ gets arbitrarily close to $1$ as $n$ becomes arbitrarily large. [hide]I'm not sure that I'm posting this in a right forum. If it's in a wrong forum, please mods move it.[/hide]

Find the sum of all prime numbers between $ 1$ and $ 100$ that are simultaneously $ 1$ greater than a multiple of $ 4$ and $ 1$ less than a multiple of $ 5$. $ \textbf{(A)}\ 118\qquad \textbf{(B)}\ 137\qquad \textbf{(C)}\ 158\qquad \textbf{(D)}\ 187 \qquad \textbf{(E)}\ 245$

A triangle $ABC$, its circumcircle, and its incenter $I$ are drawn on the plane. Construct the circumcenter of $ABC$ using only a ruler.

Evaluate the product $\prod_{k=0}^{2^{1999}}(4\sin^2 \frac{k\pi}{2^{2000}}-3)$

For all natural numbers $n$, define $f(n) = \tau (n!) - \tau ((n-1)!)$, where $\tau(a)$ denotes the number of positive divisors of $a$. Prove that there exist infinitely many composite $n$, such that for all naturals $m < n$, we have $f(m) < f(n)$.

Are there $1998$ different positive integers, the product of any two being divisible by the square of their difference?

Find all functions $f : R^+ \to R^+$ such that $f(3 (f (xy))^2 + (xy)^2) = (xf (y) + yf (x))^2$ for any $x, y > 0$.

Show that for each odd prime $p$, there is an integer $g$ such that $1<g<p$ and $g$ is a primitive root modulo $p^n$ for every positive integer $n$.

On Qingqing Grassland, there are 7 sheep numberd $1,2,3,4,5,6,7$ and 2017 wolves numberd $1,2,\cdots,2017$. We have such strange rules: (1) Define $P(n)$: the number of prime numbers that are smaller than $n$. Only when $P(i)\equiv j\pmod7$, wolf $i$ may eat sheep $j$ (he can also choose not to eat the sheep). (2) If wolf $i$ eat sheep $j$, he will immediately turn into sheep $j$. (3) If a wolf can make sure not to be eaten, he really wants to experience life as a sheep. Assume that all wolves are very smart, then how many wolves will remain in the end?

Find the smallest constant $C > 0$ for which the following statement holds: among any five positive real numbers $a_1,a_2,a_3,a_4,a_5$ (not necessarily distinct), one can always choose distinct subscripts $i,j,k,l$ such that \[ \left| \frac{a_i}{a_j} - \frac {a_k}{a_l} \right| \le C. \]

The sequences $(a_n), (b_n),$ and $(c_n)$ are defined by $a_0 = 1, b_0 = 0, c_0 = 0,$ and \[a_n = a_{n-1} + \frac{c_{n-1}}{n}, b_n = b_{n-1} +\frac{a_{n-1}}{n}, c_n = c_{n-1} +\frac{b_{n-1}}{n}\] for all $n \geq1$. Prove that \[\left|a_n -\frac{n + 1}{3}\right|<\frac{2}{\sqrt{3n}}\] for all $n \geq 1$.

Let \[ f(x_1,x_2,x_3) = -2 \cdot (x_1^3+x_2^3+x_3^3) + 3 \cdot (x_1^2(x_2+x_3) + x_2^2 \cdot (x_1+x_3) + x_3^2 \cdot ( x_1+x_2 ) - 12x_1x_2x_3. \] For any reals $r,s,t$, we denote \[ g(r,s,t)=\max_{t\leq x_3\leq t+2} |f(r,r+2,x_3)+s|. \] Find the minimum value of $g(r,s,t)$.

An international society has its members from six different countries. The list of members contain $1978$ names, numbered $1, 2, \dots, 1978$. Prove that there is at least one member whose number is the sum of the numbers of two members from his own country, or twice as large as the number of one member from his own country.

Find the integers $x, y, z$ for which $$\dfrac{1}{x+\dfrac{1}{y+\dfrac{1}{z}}}=\dfrac{7}{17}$$