[b]p4[/b] Define the sequence ${a_n}$ as follows: 1) $a_1 = -1$, and 2) for all $n \ge 2$, $a_n = 1 + 2 + . . . + n - (n + 1)$. For example, $a_3 = 1+2+3-4 = 2$. Find the largest possible value of $k$ such that $a_k+a_{k+1} = a_{k+2}$. [b]p5[/b] The taxicab distance between two points $(a, b)$ and $(c, d)$ on the coordinate plane is $|c-a|+|d-b|$. Given that the taxicab distance between points $A$ and $B$ is $8$ and that the length of $AB$ is $k$, find the minimum possible value of $k^2$. [b]p6[/b] For any two-digit positive integer $\overline{AB}$, let $f(\overline{AB}) = \overline{AB}-A\cdot B$, or in other words, the result of subtracting the product of its digits from the integer itself. For example, $f(\overline{72}) = 72-7\cdot 2 = 58$. Find the maximum possible $n$ such that there exist distinct two-digit integers$ \overline{XY}$ and $\overline{WZ}$ such that $f(\overline{XY} ) = f(\overline{WZ}) = n$.

Let $a_1, a_2, a_3, \ldots$ be a sequence of positive integers satisfying the following property: for all positive integers $k < \ell$, for all distinct integers $m_1, m_2, \ldots, m_k$ and for all distinct integers $n_1, n_2, \ldots, n_\ell$, \[ a_{m_1} + a_{m_2} + \cdots + a_{m_k} \leqslant a_{n_1} + a_{n_2} + \cdots + a_{n_\ell}. \] Prove that there exist two integers $N$ and $b$ such that $a_n = b$ for all $n \geqslant N$.

On an auto trip, the distance read from the instrument panel was $450$ miles. With snow tires on for the return trip over the same route, the reading was $440$ miles. Find, to the nearest hundredth of an inch, the increase in radius of the wheels if the original radius was $15$ inches. $\textbf{(A) }.33\qquad\textbf{(B) }.34\qquad\textbf{(C) }.35\qquad\textbf{(D) }.38\qquad \textbf{(E) }.66$

Let $S$ be a set of $6$ positive real numbers such that $\left(a,b \in S \right) \left(a>b \right) \Rightarrow a+b \in S$ or $a-b \in S$ Prove that if we sort these numbers in ascending order, then they form an arithmetic progression

Una rolls $6$ standard $6$-sided dice simultaneously and calculates the product of the $6{ }$ numbers obtained. What is the probability that the product is divisible by $4?$ $\textbf{(A)}\: \frac34\qquad\textbf{(B)} \: \frac{57}{64}\qquad\textbf{(C)} \: \frac{59}{64}\qquad\textbf{(D)} \: \frac{187}{192}\qquad\textbf{(E)} \: \frac{63}{64}$

Let $k$ be a positive integer and let $S$ be a finite set of odd prime numbers. Prove that there is at most one way (up to rotation and reflection) to place the elements of $S$ around the circle such that the product of any two neighbors is of the form $x^2+x+k$ for some positive integer $x$.

Jamie accidentally misinterprets the rules of the order of operations, and adds or subtracts before multiplying or dividing. What would be her result for the equation $4 + 3 \times 1 - 2$? $\textbf{(A) }{-}7\qquad\textbf{(B) }{-}5\qquad\textbf{(C) }5\qquad\textbf{(D) }7\qquad\textbf{(E) }9$

All of the polygons in the figure below are regular. Prove that $ABCD$ is an isosceles trapezoid. [img]https://cdn.artofproblemsolving.com/attachments/e/a/3f4de32becf4a90bf0f0b002fb4d8e724e8844.png[/img] [i]Proposed by Mahdi Etesamifard - Iran[/i]

Submit a positive integer $x$ between $1$ and $10$ inclusive. Your score on the problem will be proportional to \[ \frac{11-x}{n} \] where $n$ is the number of teams that also submit the number $x$. [i]2015 CCA Math Bonanza Lightning Round #5.4[/i]

How many natural numbers are there whose square is a thirty-digit number which has the following curious property: If that thirty-digit number is divided from left to right into three groups of ten digits, then the numbers given by the middle group and the right group formed numbers are both four times the number formed by the left group?

1-Let $A$ and $B$ be two diametrically opposite points on a circle with radius $1$. Points $P_1,P_2,...,P_n$ are arbitrarily chosen on the circle. Let a and b be the geometric means of the distances of $P_1,P_2,...,P_n$ from $A$ and $B$, respectively. Show that at least one of the numbers $a$ and $b$ does not exceed $\sqrt{2}$

In triangle $ABC$, let $D$ lie on $AB$ such that $AD = AC$ and $\angle ADC = 20^{\circ}$. Let $l$ be a line through $B$ parallel to $CD$. Let $E$ lie on $l$ with $BE = AD$ so that $AE$ intersects segment $BC$ at $F$. If $\angle ABC = 10^{\circ}$, find the degree measure of $\angle FDC$.

For any positive integer $n$ let $s(n)$ be the sum of its digits in decimal representation. Find all numbers $n$ for which $s(n)$ is the largest proper divisor of $n$.

All six sides of a rectangular solid were rectangles. A one-foot cube was cut out of the rectangular solid as shown. The total number of square feet in the surface of the new solid is how many more or less than that of the original solid? [asy] unitsize(20); draw((0,0)--(1,0)--(1,3)--(0,3)--cycle); draw((1,0)--(1+9*sqrt(3)/2,9/2)--(1+9*sqrt(3)/2,15/2)--(1+5*sqrt(3)/2,11/2)--(1+5*sqrt(3)/2,9/2)--(1+2*sqrt(3),4)--(1+2*sqrt(3),5)--(1,3)); draw((0,3)--(2*sqrt(3),5)--(1+2*sqrt(3),5)); draw((1+9*sqrt(3)/2,15/2)--(9*sqrt(3)/2,15/2)--(5*sqrt(3)/2,11/2)--(5*sqrt(3)/2,5)); draw((1+5*sqrt(3)/2,9/2)--(1+2*sqrt(3),9/2)); draw((1+5*sqrt(3)/2,11/2)--(5*sqrt(3)/2,11/2)); label("$1'$",(.5,0),S); label("$3'$",(1,1.5),E); label("$9'$",(1+9*sqrt(3)/4,9/4),S); label("$1'$",(1+9*sqrt(3)/4,17/4),S); label("$1'$",(1+5*sqrt(3)/2,5),E);label("$1'$",(1/2+5*sqrt(3)/2,11/2),S); [/asy] $\text{(A)}\ 2\text{ less} \qquad \text{(B)}\ 1\text{ less} \qquad \text{(C)}\ \text{the same} \qquad \text{(D)}\ 1\text{ more} \qquad \text{(E)}\ 2\text{ more}$

We have $n$ points in the plane such that they are not all collinear. We call a line $\ell$ a 'good' line if we can divide those $n$ points in two sets $A,B$ such that the sum of the distances of all points in $A$ to $\ell$ is equal to the sum of the distances of all points in $B$ to $\ell$. Prove that there exist infinitely many points in the plane such that for each of them we have at least $n+1$ good lines passing through them.

Let $f_1(x)=x,f_n(x)=x+\frac{1}{14}\int_0^\pi xf_{n-1}(t)\cos ^ 3 t\ dt\ (n\geq 2)$. Find $\lim_{n\to\infty} f_n(x)$

All the cells of a $10\times10$ board are colored white initially. Two players are playing a game with alternating moves. A move consists of coloring any un-colored cell black. A player is considered to loose, if after his move no white domino is left. Which of the players has a winning strategy? [I]Proposed by A. Khrabrov[/i]

In the figure below, $AB = 15$, $BD = 18$, $AF = 15$, $DF = 12$, $BE = 24$, and $CF = 17$. Find $BG : FG$. [img]https://cdn.artofproblemsolving.com/attachments/9/e/dc171c52961442f9846d2fce858937ff9fb7e8.png[/img]

Determine all positive integers relatively prime to all the terms of the infinite sequence \[ a_n=2^n+3^n+6^n -1,\ n\geq 1. \]

Circles ${\omega_1}$ , ${\omega_2}$ are externally tangent at point M and tangent internally with circle ${\omega_3}$ at points ${K}$ and $L$ respectively. Let ${A}$ and ${B}$ be the points that their common tangent at point ${M}$ of circles ${\omega_1}$ and ${\omega_2}$ intersect with circle ${\omega_3.}$ Prove that if ${\angle KAB=\angle LAB}$ then the segment ${AB}$ is diameter of circle ${\omega_3.}$ Theoklitos Paragyiou (Cyprus)

A quadratic trinomial $p(x)$ with real coefficients is given. Prove that there is a positive integer $n$ such that the equation $p(x) = \frac{1}{n}$ has no rational roots.

Given a square $ABCD$ of side $a$, we consider the circle $\omega$, tangent to side $BC$ and to the two semicircles of diameters $AB$ and $CD$. Calculate the radius of circle $\omega$,

Find three non-zero reals such that all quadratics with those numbers as coefficients have two distinct rational roots.

You have a paper pentagon, $ABCDE$, such that $AB = BC = 3$ cm, $CD = DE= 5$ cm, $EA = 4$ cm, $\angle ABC = 100^o$ ,$ \angle CDE = 80^o$. You have to divide the pentagon into four triangles, by three straight cuts, so that with the four triangles assemble a rectangle, without gaps or overlays. (The triangles can be rotated and / or turned around.)

Determine the roots of unity in the field of $ p$-adic numbers. [i]L. Fuchs[/i]