Five people are standing in a straight line, and the distance between any two people is a unique positive integer number of units. Find the least possible distance between the leftmost and rightmost people in the line in units.

[b]p1.[/b] There are $32$ balls in a box: $6$ are blue, $8$ are red, $4$ are yellow, and $14$ are brown. If I pull out three balls at once, what is the probability that none of them are brown? [b]p2.[/b] Circles $A$ and $B$ are concentric, and the area of circle $A$ is exactly $20\%$ of the area of circle $B$. The circumference of circle $B$ is $10$. A square is inscribed in circle $A$. What is the area of that square? [b]p3.[/b] If $x^2 +y^2 = 1$ and $x, y \in R$, let $q$ be the largest possible value of $x+y$ and $p$ be the smallest possible value of $x + y$. Compute $pq$. [b]p4.[/b] Yizheng and Jennifer are playing a game of ping-pong. Ping-pong is played in a series of consecutive matches, where the winner of a match is given one point. In the scoring system that Yizheng and Jennifer use, if one person reaches $11$ points before the other person can reach $10$ points, then the person who reached $11$ points wins. If instead the score ends up being tied $10$-to-$10$, then the game will continue indefinitely until one person’s score is two more than the other person’s score, at which point the person with the higher score wins. The probability that Jennifer wins any one match is $70\%$ and the score is currently at $9$-to-$9$. What is the probability that Yizheng wins the game? [b]p5.[/b] The squares on an $8\times 8$ chessboard are numbered left-to-right and then from top-to-bottom (so that the top-left square is $\#1$, the top-right square is $\#8$, and the bottom-right square is $\#64$). $1$ grain of wheat is placed on square $\#1$, $2$ grains on square $\#2$, $4$ grains on square $\#3$, and so on, doubling each time until every square of the chessboard has some number of grains of wheat on it. What fraction of the grains of wheat on the chessboard are on the rightmost column? [b]p6.[/b] Let $f$ be any function that has the following property: For all real numbers $x$ other than $0$ and $1$, $$f \left( 1 - \frac{1}{x} \right) + 2f \left( \frac{1}{1 - x}\right)+ 3f(x) = x^2.$$ Compute $f(2)$. [b]p7.[/b] Find all solutions of: $$(x^2 + 7x + 6)^2 + 7(x^2 + 7x + 6)+ 6 = x.$$ [b]p8.[/b] Let $\vartriangle ABC$ be a triangle where $AB = 25$ and $AC = 29$. $C_1$ is a circle that has $AB$ as a diameter and $C_2$ is a circle that has $BC$ as a diameter. $D$ is a point on $C_1$ so that $BD = 15$ and $CD = 21$. $C_1$ and $C_2$ clearly intersect at $B$; let $E$ be the other point where $C_1$ and $C_2$ intersect. Find all possible values of $ED$. PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Find all functions $f:\mathbb{R}\to \mathbb{R}$ such that for all real numbers $a,b,$ and $c$: (i) If $a+b+c\ge 0$ then $f(a^3)+f(b^3)+f(c^3)\ge 3f(abc).$ (ii) If $a+b+c\le 0$ then $f(a^3)+f(b^3)+f(c^3)\le 3f(abc).$ [i]Proposed by Ashwin Sah[/i]

There are $n$ points in the plane so that no two pairs are equidistant. Each point is connected to the nearest point by a segment. Show that no point is connected to more than five points.

Solve the Dirichlet problem \[\Delta u=0\text{ for }x^2+y^2<1\] \[u=1\text{ for }x^2+y^2=1,\;y>0\] \[u=-1\text{ for }x^2+y^2=1,\;y<0\]

Let $a$ be an odd positive integer. Prove that if $d$ divides $a^2+2$, then $d\equiv1\pmod8$ or $d\equiv3\pmod8$.

We say that a positive integer $n$ is [i]good[/i] if the result of multiplying the first $n$ positive odd integers consists only of the digits $1$, $3$, $5$ and $9$. For example, $n = 3$ is good because $1 \times 3 \times 5 = 15$, but $n = 4$ is not good because $1 \times 3 \times 5 \times 7 = 105$. Determine all the good numbers.

Two externally tangent circles have radius $2$ and radius $3$. Two lines are drawn, each tangent to both circles, but not at the point where the circles are tangent to each other. What is the area of the quadrilateral whose vertices are the four points of tangency between the circles and the lines?

Compute the remainder when $2^{30}$ is divided by $1000$.

Let $a$ be a natural number, $a>3$. Prove there is an infinity of numbers n, for which $a+n|a^{n}+1$

Given $0 < a \le x_1\le x_2\le ... \le x_n \le b$. Prove that $$(x_1+x_2+...+x_n)\left ( \frac{1}{x_1}+ \frac{1}{x_2}+...+ \frac{1}{x_n}\right)\le \frac{(a+b)^2}{4ab}n^2$$

Consider the system \[x+y=z+u,\] \[2xy=zu.\] Find the greatest value of the real constant $m$ such that $m \le \frac{x}{y}$ for any positive integer solution $(x, y, z, u)$ of the system, with $x \ge y$.

If $k$ is the number of distinct prime divisors of a natural number $n$, prove that log $n \geq k$ log $2$.

George walks $1$ mile to school. He leaves home at the same time each day, walks at a steady speed of $3$ miles per hour, and arrives just as school begins. Today he was distracted by the pleasant weather and walked the first $\frac{1}{2}$ mile at a speed of only $2$ miles per hour. At how many miles per hour must George run the last $\frac{1}{2}$ mile in order to arrive just as school begins today? $\textbf{(A) }4\qquad\textbf{(B) }6\qquad\textbf{(C) }8\qquad\textbf{(D) }10\qquad \textbf{(E) }12$

For a real coefficient cubic polynomial $f(x)=ax^3+bx^2+cx+d$, denote three roots of the equation $f(x)=0$ by $\alpha,\beta,\gamma$. Prove that the three roots $\alpha,\beta,\gamma$ are distinct real numbers iff the real symmetric matrix $$\begin{pmatrix} 3 & p_1 & p_2 \\ p_1 & p_2 & p_3 \\ p_2 & p_3 & p_4 \end{pmatrix},\quad p_i = \alpha^i + \beta^i + \gamma^i$$ is positive definite.

Let $n$ be a fixed positive integer. Which is greater?[list=1] [*]The number of $n$-tuples of integers whose largest value is $7$ and whose smallest value is $0$; or [*]The number of ordered triples $(A, B, C)$ that satisfy the following property: $A$, $B$, $C$ are subsets of $\{1, 2, 3, \dots , n\}$, and neither $C\subseteq A\cup B$, nor $B\subseteq A\cup C$. [/list] Your answer can be: $(1)$, $(2)$, the two counts are equal, or it depends on $n$.

The sidelines $AB$ and $CD$ of a trapezoid meet at point $P$, and the diagonals of this trapezoid meet at point $Q$. Point $M$ on the smallest base $BC$ is such that $AM=MD$. Prove that $\angle PMB=\angle QMB$.

The side $AC$ of an acute triangle $ABC$ is the diameter of the circle $c_1$ and side $BC$ is the diameter of the circle $c_2$. Let $E$ be the foot of the altitude drawn from the vertex $B$ of the triangle and $F$ the foot of the altitude drawn from the vertex $A$. In addition, let $L$ and $N$ be the points of intersection of the line $BE$ with the circle $c_1$ (the point $L$ lies on the segment $BE$) and the points of intersection of $K$ and $M$ of line $AF$ with circle $c_2$ (point $K$ is in section $AF$). Prove that $K LM N$ is a cyclic quadrilateral.

If $ a^x \equal{} c^q \equal{} b$ and $ c^y \equal{} a^z \equal{} d$, then $ \textbf{(A)}\ xy \equal{} qz \qquad\textbf{(B)}\ \frac {x}{y} \equal{} \frac {q}{z} \qquad\textbf{(C)}\ x \plus{} y \equal{} q \plus{} z \qquad\textbf{(D)}\ x \minus{} y \equal{} q \minus{} z$ $ \textbf{(E)}\ x^y \equal{} q^z$

Prove that, for any positive integer $n$, there exists a polynomial $p(x)$ of degree at most $n$ whose coefficients are all integers such that, $p(k)$ is divisible by $2^n$ for every even integer $k$, and $p(k) -1$ is divisible by $2^n$ for every odd integer $k$.

Let $ABC$ be an acute triangle with orthocentre $H$, and let $M$ be the midpoint of $AC$. The point $C_1$ on $AB$ is such that $CC_1$ is an altitude of the triangle $ABC$. Let $H_1$ be the reflection of $H$ in $AB$. The orthogonal projections of $C_1$ onto the lines $AH_1$, $AC$ and $BC$ are $P$, $Q$ and $R$, respectively. Let $M_1$ be the point such that the circumcentre of triangle $PQR$ is the midpoint of the segment $MM_1$. Prove that $M_1$ lies on the segment $BH_1$.

Given an integer $h > 1$. Let's call a positive common fraction (not necessarily irreducible) [i]good[/i] if the sum of its numerator and denominator is equal to $h$. Let's say that a number $h$ is [i]remarkable[/i] if every positive common fraction whose denominator is less than $h$ can be expressed in terms of good fractions (not necessarily various) using the operations of addition and subtraction. Prove that $h$ is remarkable if and only if it is prime. (Recall that an common fraction has an integer numerator and a natural denominator.)

Let $AH$ be the altitude of a given triangle $ABC.$ The points $I_b$ and $I_c$ are the incenters of the triangles $ABH$ and $ACH$ respectively. $BC$ touches the incircle of the triangle $ABC$ at a point $L.$ Find $\angle LI_bI_c.$

All vertices of a polygon $P$ lie at points with integer coordinates in the plane, and all sides of $P$ have integer lengths. Prove that the perimeter of $P$ must be an even number.

Let $n, k$ be positive integers with $k \geq 3$. The edges of of a complete graph $K_n$ are colored in $k$ colors, such that for any color $i$ and any two vertices, there exists a path between them, consisting only of edges in color $i$. Prove that there exist three vertices $A, B, C$ of $K_n$, such that $AB, BC$ and $CA$ are all distinctly colored.