For any odd prime $p$ and any integer $n,$ let $d_p (n) \in \{ 0,1, \dots, p-1 \}$ denote the remainder when $n$ is divided by $p.$ We say that $(a_0, a_1, a_2, \dots)$ is a [i]p-sequence[/i], if $a_0$ is a positive integer coprime to $p,$ and $a_{n+1} =a_n + d_p (a_n)$ for $n \geqslant 0.$ (a) Do there exist infinitely many primes $p$ for which there exist $p$-sequences $(a_0, a_1, a_2, \dots)$ and $(b_0, b_1, b_2, \dots)$ such that $a_n >b_n$ for infinitely many $n,$ and $b_n > a_n$ for infinitely many $n?$ (b) Do there exist infinitely many primes $p$ for which there exist $p$-sequences $(a_0, a_1, a_2, \dots)$ and $(b_0, b_1, b_2, \dots)$ such that $a_0 <b_0,$ but $a_n >b_n$ for all $n \geqslant 1?$ [I]United Kingdom[/i]

Consider $n$ points $A_1, A_2, \ldots, A_n$ on a circle. How many ways are there if we want to color these points by $p$ colors, so that each two neighbors points are colored with two different colors?

Which of the following is the correct order of the fractions $\frac{15}{11}, \frac{19}{15}$, and $\frac{17}{13}$, from least to greatest? $\textbf{(A) } \frac{15}{11} < \frac{17}{13} < \frac{19}{15} \qquad\textbf{(B) } \frac{15}{11} < \frac{19}{15} < \frac{17}{13} \qquad\textbf{(C) } \frac{17}{13} < \frac{19}{15} < \frac{15}{11} \newline\newline \qquad\textbf{(D) } \frac{19}{15} < \frac{15}{11} < \frac{17}{13} \qquad\textbf{(E) } \frac{19}{15} < \frac{17}{13} < \frac{15}{11}$

Consider a $2 \times 2$ grid of squares. David writes a positive integer in each of the squares. Next to each row, he writes the product of the numbers in the row, and next to each column, he writes the product of the numbers in each column. If the sum of the eight numbers he writes down is 2015, what is the minimum possible sum of the four numbers he writes in the grid?

Let $ABCD$ be a cyclic quadrilateral satysfing $AB=AD$ and $AB+BC=CD$. Determine $\measuredangle CDA$.

The sum of all real $x$ such that $(2^{x} - 4)^{3} + (4^{x} - 2)^{3} = (4^{x} + 2^{x} - 6)^{3}$ is $ \textbf{(A)}\ 3/2\qquad\textbf{(B)}\ 2\qquad\textbf{(C)}\ 5/2\qquad\textbf{(D)}\ 3\qquad\textbf{(E)}\ 7/2 $

Any two members of a club with $3n+1$ people plays ping-pong, tennis or chess with each other. Everyone has exactly $n$ partners who plays ping-pong, $n$ who play tennis and $n$ who play chess. Prove that we can choose three members of the club who play three different games amongst each other.

Two players by turn paint the vertices of triangles on the given picture each with his colour. At the end, each of small triangles is painted by the colour of the majority of its vertices. The winner is one who gets at least 6 triangles of his colour. If both players get at most 5, then it is a draw. Does any of them have winning strategy? If yes, then who wins? \[ \begin{picture}(40,50) \put(2,2){\put(0,0){\line(6,0){42}} \put(7,14){\line(6,0){28}} \put(14,28){\line(6,0){14}} \put(0,0){\line(1,2){21}} \put(14,0){\line(1,2){14}} \put(28,0){\line(1,2){7}} \put(14,28){\line(1,2){7}} \put(14,0){\line( \minus{} 1,2){7}} \put(28,0){\line( \minus{} 1,2){14}} \put(42,0){\line( \minus{} 1,2){21}} \put(0,0){\circle*{3}} \put(14,0){\circle*{3}} \put(28,0){\circle*{3}} \put(42,0){\circle*{3}} \put(7,14){\circle*{3}} \put(21,14){\circle*{3}} \put(35,14){\circle*{3}} \put(14,28){\circle*{3}} \put(28,28){\circle*{3}} \put(21,42){\circle*{3}}} \end{picture}\]

The function $f$ is increasing and convex (i.e. every straight line between two points on the graph of $f$ lies above the graph) and satisfies $f(f(x))=3^x$ for all $x\in\mathbb{R}$. If $f(0)=0.5$ determine $f(0.75)$ with an error of at most $0.025$. The following are corrent to the number of digits given: \[3^{0.25}=1.31607,\quad 3^{0.50}=1.73205,\quad 3^{0.75}=2.27951.\]

Let $ABCDE$ be a convex pentagon such that $\angle ABC = \angle AED = 90^\circ$. Suppose that the midpoint of $CD$ is the circumcenter of triangle $ABE$. Let $O$ be the circumcenter of triangle $ACD$. Prove that line $AO$ passes through the midpoint of segment $BE$.

Let $ p_1,\dots,p_k$ be prime numbers, and let $ S$ be the set of those integers whose all prime divisors are among $ p_1,\dots,p_k$. For a finite subset $ A$ of the integers let us denote by $ \mathcal G(A)$ the graph whose vertices are the elements of $ A$, and the edges are those pairs $ a,b\in A$ for which $ a \minus{} b\in S$. Does there exist for all $ m\geq 3$ an $ m$-element subset $ A$ of the integers such that (i) $ \mathcal G(A)$ is complete? (ii) $ \mathcal G(A)$ is connected, but all vertices have degree at most 2?

Call a set of positive integers $\textit{good}$ if there is a partition of it into two sets $S$ and $T$, such that there do not exist three elements $a, b, c \in S$ such that $a^b = c$ and such that there do not exist three elements $a, b, c \in T$ such that $a^b = c$ ($a$ and $b$ need not be distinct). Find the smallest positive integer $n$ such that the set $\{2, 3, 4, \dots, n\}$ is \textit{not} good.

Let $ABCD$ be a rhombus. Eight additional points $X_1$, $X_2$, $Y_1$, $Y_2$, $Z_1$, $Z_2$, $W_1$, $W_2$ were chosen so that the quadrilaterals $AX_1BX_2$, $BY_1CY_2$, $CZ_1DZ_2$, $DW_1AW_2$ are squares. Prove that the eight new points lie on two straight lines.

Let $n$ be a positive integer. Define a chameleon to be any sequence of $3n$ letters, with exactly $n$ occurrences of each of the letters $a, b,$ and $c$. Define a swap to be the transposition of two adjacent letters in a chameleon. Prove that for any chameleon $X$ , there exists a chameleon $Y$ such that $X$ cannot be changed to $Y$ using fewer than $3n^2/2$ swaps.

One base of a trapezoid is 100 units longer than the other base. The segment that joins the midpoints of the legs divides the trapezoid into two regions whose areas are in the ratio $2: 3.$ Let $x$ be the length of the segment joining the legs of the trapezoid that is parallel to the bases and that divides the trapezoid into two regions of equal area. Find the greatest integer that does not exceed $x^2/100.$

What is the sum of all $x$ that satisfy $|2x-4| = 2$? $\textbf{(A) } 1\qquad\textbf{(B) } 2\qquad\textbf{(C) } 3\qquad\textbf{(D) } 4\qquad\textbf{(E) } 5$

Let $ABCD$ be an isosceles trapezoid with $AB=5$, $CD = 8$, and $BC = DA = 6$. There exists an angle $\theta$ such that there is only one point $X$ satisfying $\angle AXD = 180^{\circ} - \angle BXC = \theta$. Find $\sin(\theta)^2$.

Given \(n \in \mathbb{N}\), let \(\sigma (n)\) denote the sum of the divisors of \(n\) and \(\phi (n)\) denote the number of integers \(n \geq m\) for which \(\gcd(m,n) = 1\). Show that for all \(n \in \mathbb{N}\), \[\large \frac{1}{\sigma (n)} + \frac{1}{\phi (n)} \geq \frac{2}{n}\] and determine when equality holds.

Juan and Pedro play alternately on the given grid. Each one in turn traces $1$ to $5$ routes different from the ones outlined above, that join $A$ with $B$, moving only to the right and upwards on the grid lines. Juan starts playing. The one who traces a route that passes through $C$ or $D$ loses. Prove that one of them can win regardless of how the other plays. [img]https://cdn.artofproblemsolving.com/attachments/2/7/6a24ca9c4c1c710bd41e44bfcab3d3b61b6d4f.png[/img]

What is the smallest possible perimeter of a triangle with integer coordinate vertices, area $\frac12$, and no side parallel to an axis?

A math contest consists of $9$ objective type questions and $6$ fill in the blanks questions. From a school some number of students took the test and it was noticed that all students had attempted exactly $14$ out of $15$ questions. Let $O_1, O_2, \dots , O_9$ be the nine objective questions and $F_1, F_2, \dots , F_6$ be the six fill inthe blanks questions. Let $a_{ij}$ be the number of students who attemoted both questions $O_i$ and $F_j$. If the sum of all the $a_{ij}$ for $i=1, 2,\dots , 9$ and $j=1, 2,\dots , 6$ is $972$, then find the number of students who took the test in the school.

Find all functions such that $ f: \mathbb{R}^\plus{} \rightarrow \mathbb{R}^\plus{}$ and $ f(x\plus{}f(y))\equal{}yf(xy\plus{}1)$ for every $ x,y\in \mathbb{R}^\plus{}$.

[i]Nyihaha[/i] and [i]Bruhaha[/i] are two neighbouring islands, both having $n$ inhabitants. On island [i]Nyihaha[/i] every inhabitant is either a Knight or a Knave. Knights always tell the truth and Knaves always lie. The inhabitants of island [i]Bruhaha[/i] are normal people, who can choose to tell the truth or lie. When a visitor arrives on any of the two islands, the following ritual is performed: every inhabitant points randomly to another inhabitant (indepently from each other with uniform distribution), and tells "He is a Knight" or "He is a Knave'". On sland [i]Nyihaha[/i], Knights have to tell the truth and Knaves have to lie. On island [i]Bruhaha[/i] every inhabitant tells the truth with probability $1/2$ independently from each other. Sinbad arrives on island [i]Bruhaha[/i], but he does not know whether he is on island [i]Nyihaha[/i] or island [i]Bruhaha[/i]. Let $p_n$ denote the probability that after observing the ritual he can rule out being on island [i]Nyihaha[/i]. Is it true that $p_n\to 1$ if $n\to\infty$? [i]Proposed by Dávid Matolcsi, Berkeley[/i]

Find the largest possible real part of \[(75+117i)z+\frac{96+144i}{z}\] where $z$ is a complex number with $|z|=4$.

(a) Find all positive integers $ n$ for which $ 2^n\minus{}1$ is divisible by $ 7$. (b) Prove that there is no positive integer $ n$ for which $ 2^n\plus{}1$ is divisible by $ 7$.