Let $(x_1, y_1, z_1), (x_2, y_2, z_2), \cdots, (x_{19}, y_{19}, z_{19})$ be integers. Prove that there exist pairwise distinct subscripts $i, j, k$ such that $x_i+x_j+x_k$, $y_i+y_j+y_k$, $z_i+z_j+z_k$ are all multiples of $3$.

Let $a$ be the largest root of the equation $x^3 - 3x^2 + 1 = 0$. Find the first $200$ decimal digits for the number $a^{2000}$.

Let $P$ be the set of points in the plane, and let $f : P \to P$ be a function such that the image through $f$ of any triangle is a square (any polygon is considered to be formed by the reunion of the points on its sides). Prove that $f(P)$ is a square.

Let $ b, m, n$ be positive integers such that $ b > 1$ and $ m \neq n.$ Prove that if $ b^m \minus{} 1$ and $ b^n \minus{} 1$ have the same prime divisors, then $ b \plus{} 1$ is a power of 2.

[b]p1.[/b] Find the sum of the seventeenth powers of the seventeen roots of the seventeeth degree polynomial equation $x^{17} - 17x + 17 = 0$. [b]p2.[/b] Four identical spherical cows, each of radius $17$ meters, are arranged in a tetrahedral pyramid (their centers are the vertices of a regular tetrahedron, and each one is tangent to the other three). The pyramid of cows is put on the ground, with three of them laying on it. What is the distance between the ground and the top of the topmost cow? [b]p3.[/b] If $a_n$ is the last digit of $\sum^{n}_{i=1} i$, what would the value of $\sum^{1000}_{i=1}a_i$ be? [b]p4.[/b] If there are $15$ teams to play in a tournament, $2$ teams per game, in how many ways can the tournament be organized if each team is to participate in exactly $5$ games against dierent opponents? [b]p5.[/b] For $n = 20$ and $k = 6$, calculate $$2^k {n \choose 0}{n \choose k}- 2^{k-1}{n \choose 1}{{n - 1} \choose {k - 1}} + 2^{k-2}{n \choose 2}{{n - 2} \choose {k - 2}} +...+ (-1)^k {n \choose k}{{n - k} \choose 0}$$ where ${n \choose k}$ is the number of ways to choose $k$ things from a set of $n$. [b]p6.[/b] Given a function $f(x) = ax^2 + b$, with a$, b$ real numbers such that $$f(f(f(x))) = -128x^8 + \frac{128}{3}x^6 - \frac{16}{22}x^2 +\frac{23}{102}$$ , find $b^a$. [b]p7.[/b] Simplify the following fraction $$\frac{(2^3-1)(3^3-1)...(100^3-1)}{(2^3+1)(3^3+1)...(100^3+1)}$$ [b]p8.[/b] Simplify the following expression $$\frac{\sqrt{3 + \sqrt5} + \sqrt{3 - \sqrt5}}{\sqrt{3 - \sqrt8}} -\frac{4}{ \sqrt{8 - 2\sqrt{15}}}$$ [b]p9.[/b] Suppose that $p(x)$ is a polynomial of degree $100$ such that $p(k) = k2^{k-1}$ , $k =1, 2, 3 ,... , 100$. What is the value of $p(101)$ ? [b]p10. [/b] Find all $17$ real solutions $(w, x, y, z)$ to the following system of equalities: $$ 2w + w^2x = x$$ $$ 2x + x^2y=y $$ $$ 2y + y^2z=z $$ $$ -2z+z^2w=w $$ PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

[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].

Find all non-negative integer numbers $n$ for which there exists integers $a$ and $b$ such that $n^2=a+b$ and $n^3=a^2+b^2.$

Let $a$ and $ b$ be positive integers (of one or more digits) such that $ b$ is divisible by $a$, and if we write $a$ and $ b$, one after the other in this order, we get the number $(a + b)^2$. Prove that $\frac{b}{a}= 6$.

Determine all natural numbers $n$ of the form $n=[a,b]+[b,c]+[c,a]$ where $a,b,c$ are positive integers and $[u,v]$ is the least common multiple of the integers $u$ and $v$.

You have a set of five weights, together with a balance that allows you to compare the weight of two things. The weights are known to be $10$, $20$,$30$,$40$ and $50$ grams, but are otherwise identical except for their labels. The $10$ and $50$ gram weights are clearly labelled, but the labels have been erased on the remaining weights. Using the balance exactly once, is it possible to determine what one of the three unlabelled weights is? If so, explain how, and if not, explain why not.

Determine $ \gcd (n^2\plus{}2,n^3\plus{}1)$ for $ n\equal{}9^{753}$.

Prove that if $n$ is a natural number such that $1 + 2^n + 4^n$ is prime then $n = 3^k$ for some $k \in N_0$.

How many ways are there to choose three digits $A,B,C$ with $1 \le A \le 9$ and $0 \le B,C \le 9$ such that $\overline{ABC}_b$ is even for all choices of base $b$ with $b \ge 10$?

Find all triples $(a,b,c)$ of positive integers such that if $n$ is not divisible by any prime less than $2014$, then $n+c$ divides $a^n+b^n+n$. [i]Proposed by Evan Chen[/i]

Proove that in every five positive numbers there is a pair, say $a,b$, for which $$\left| \frac{1}{a+25}- \frac{1}{b+25}\right| <\frac{1}{100}.$$

a) Prove that $x^2 + y^2 + z^2 = 2xyz$ for integer $x, y, z$ only if $x = y = z = 0$. b) Find integers $x, y, z, u$ such that $x^2 + y^2 + z^2 + u^2 = 2xyzu$.

The radius $r$ of a circle with center at the origin is an odd integer. There is a point ($p^m, q^n$) on the circle, with $p,q$ prime numbers and $m,n$ positive integers. Determine $r$.

Find the number of positive integers less than $2018$ that are divisible by $6$ but are not divisible by at least one of the numbers $4$ or $9$.

Let $p$ be a prime number congruent to $3$ modulo $4$, and consider the equation $(p+2)x^{2}-(p+1)y^{2}+px+(p+2)y=1$. Prove that this equation has infinitely many solutions in positive integers, and show that if $(x,y) = (x_{0}, y_{0})$ is a solution of the equation in positive integers, then $p | x_{0}$.

Prove that there are at most three primes between $10$ and $10^{10}$ all of whose decimal digits are $1$.

The positive integers \( a \), \( b \), \( c \), \( d \) are such that \( (a+c)(b+d) = (ab-cd)^2 \). Prove that \( 4ad + 1 \) and \( 4bc + 1 \) are perfect squares of natural numbers.

Let $ \mathbb{Z}$ denote the set of all integers. Prove that for any integers $ A$ and $ B,$ one can find an integer $ C$ for which $ M_1 \equal{} \{x^2 \plus{} Ax \plus{} B : x \in \mathbb{Z}\}$ and $ M_2 \equal{} {2x^2 \plus{} 2x \plus{} C : x \in \mathbb{Z}}$ do not intersect.

Find the smallest $x \in\mathbb{N}$ for which $\frac{7x^{25}-10}{83}$ is an integer.

Let $T_n$ denotes the least natural such that $$n\mid 1+2+3+\cdots +T_n=\sum_{i=1}^{T_n} i$$ Find all naturals $m$ such that $m\ge T_m$. [i]Proposed by Nicolás De la Hoz [/i]

Find all positive integers $k$, for which the product of some consecutive $k$ positive integers ends with $k$. [i]Proposed by Oleksiy Masalitin[/i]