There are exactly two four-digit numbers that are multiples of three where their first digit is double their second digit, their third digit is three more than their fourth digit, and their second digit is $2$ less than their fourth digit. Find the difference of these two numbers.

Each of $2023$ balls is placed in on of $3$ bins. Which of the following is closest to the probability that each of the bins will contain an odd number of balls? $\textbf{(A) } \frac{2}{3} \qquad \textbf{(B) } \frac{3}{10} \qquad \textbf{(C) } \frac{1}{2} \qquad \textbf{(D) } \frac{1}{3} \qquad \textbf{(E) } \frac{1}{4}$

In each cell of a $4 \times 4$ grid, one of the two diagonals is drawn uniformly at random. Compute the probability that the resulting $32$ triangular regions can be colored red and blue so that any two regions sharing an edge have different colors.

Let $C_1$ and $C_2$ be two concentric circles with center $O$, in such a way that the radius of $C_1$ is smaller than the radius of $C_2$. Let $P$ be a point other than $O$ that is in the interior of $C_1$, and $L$ a line through $P$ and intersects $C_1$ at $A$ and $B$. Ray $\overrightarrow{OB}$ intersects $C_2$ at $C$. Determine the locus that determines the circumcenter of triangle $ABC$ as $L$ varies.

Define \[f(h,t) = \begin{cases} 8h & h = t \\ (h-t)^2 & h \neq t. \end{cases}\] Cody plays a game with a fair coin, where he begins by flipping it once. At each turn in the game, if he has flipped $h$ heads and $t$ tails and $h + t < 6$, he can choose either to stop and receive $f(h,t)$ dollars or he can flip the coin again; if $h + t = 6$ then the game ends and he receives $f(h,t)$ dollars. If Cody plays to maximize expectancy, how much money, in dollars, can he expect to win from this game?

Five points are chosen on the sphere, no three of them lying on a great circle (a great circle is the intersection of the sphere with some plane passing through the sphere’s centre). Two great circles not containing any of the chosen points are called equivalent if one of them can be moved to the other without passing through any chosen points. (a) How many nonequivalent great circles not containing any chosen points can be drawn on the sphere? (b) Answer the same problem, but with $n$ chosen points.

At the grocery store last week, small boxes of facial tissue were priced at 4 boxes for \$5. This week they are on sale at 5 boxes for \$4. The percent decrease in the price per box during the sale was closest to $\textbf{(A)}\ 30\% \qquad \textbf{(B)}\ 35\% \qquad \textbf{(C)}\ 40\% \qquad \textbf{(D)}\ 45\% \qquad \textbf{(E)}\ 65\%$

For every positive integer$ n$, let $P(n)$ be the greatest prime divisor of $n^2+1$. Show that there are infinitely many quadruples $(a, b, c, d)$ of positive integers that satisfy $a < b < c < d$ and $P(a) = P(b) = P(c) = P(d)$.

In each cell of the table $ 3 \times 3 $ there is one of the numbers $1, 2$ and $3$. Dima counted the sum of the numbers in each row and in each column. What is the greatest number of different sums he could get?

Define an [i]upno[/i] to be a positive integer of $2$ or more digits where the digits are strictly increasing moving left to right. Similarly, define a [i]downno[/i] to be a positive integer of $2$ or more digits where the digits are strictly decreasing moving left to right. For instance, the number $258$ is an [i]upno[/i] and $8620$ is a [i]downno[/i]. Let $U$ equal the total number of [i]upno[/i]s and let $D$ equal the total number of [i]downno[/i]s. What is $|U-D|$? $\textbf{(A)}~512\qquad\textbf{(B)}~10\qquad\textbf{(C)}~0\qquad\textbf{(D)}~9\qquad\textbf{(E)}~511$

Let $a$, $b$, $c$ be real numbers, not all of them are equal. Prove that $a+b+c=0$ if and only if $a^2+ab+b^2=b^2+bc+c^2=c^2+ca+a^2$.

The $48$ faces of $8$ unit cubes are painted white. What is the smallest number of these faces that can be repainted black so that it becomes impossible to arrange the $8$ unit cubes into a two by two by two cube, each of whose $6$ faces is totally white?

For how many postive integers $n$ less than $2013$, does $p^2+p+1$ divide $n$ where $p$ is the least prime divisor of $n$? $ \textbf{(A)}\ 212 \qquad\textbf{(B)}\ 206 \qquad\textbf{(C)}\ 191 \qquad\textbf{(D)}\ 185 \qquad\textbf{(E)}\ 173 $

Find all triples $(p, x, y)$ consisting of a prime number $p$ and two positive integers $x$ and $y$ such that $x^{p -1} + y$ and $x + y^ {p -1}$ are both powers of $p$. [i]Proposed by Belgium[/i]

In a planar coordinate system, the points have non-negative integer coordinates numbered according to the figure. E.g. the point $(3,1)$ has the number $12$. [img]https://cdn.artofproblemsolving.com/attachments/a/a/28725d75f281ac4618129067037d751c8d8f83.png[/img] What is the number of the point$(x,y)$?

Six tennis players gather to play in a tournament where each pair of persons play one game, with one person declared the winner and the other person the loser. A triplet of three players $\{\mathit{A}, \mathit{B}, \mathit{C}\}$ is said to be [i]cyclic[/i] if $\mathit{A}$ wins against $\mathit{B}$, $\mathit{B}$ wins against $\mathit{C}$ and $\mathit{C}$ wins against $\mathit{A}$. [list] [*] (a) After the tournament, the six people are to be separated in two rooms such that none of the two rooms contains a cyclic triplet. Prove that this is always possible. [*] (b) Suppose there are instead seven people in the tournament. Is it always possible that the seven people can be separated in two rooms such that none of the two rooms contains a cyclic triplet?[/list]

For which positive real numbers $a,b$ does the inequality \[x_1x_2+x_2x_3+\ldots x_{n-1}x_n+x_nx_1\ge x_1^ax_2^bx_3^a+ x_2^ax_3^bx_4^a+\ldots +x_n^ax_1^bx_2^a\] hold for all integers $n>2$ and positive real numbers $x_1,\ldots ,x_n$?

Prove that for each natural number $d$, There is a monic and unique polynomial of degree $d$ like $P$ such that $P(1)$≠$0$ and for each sequence like $a_{1}$,$a_{2}$, $...$ of real numbers that the recurrence relation below is true for them, there is a natural number $k$ such that $0=a_{k}=a_{k+1}= ...$ : $P(n)a_{1} + P(n-1)a_{2} + ... + P(1)a_{n}=0$ $n>1$

The numbers $\frac{50}{1},\frac{50}{2},...\frac{50}{97},\frac{50}{98}$ are written on the board.In each step,two random numbers $a$ and $b$ are chosen and deleted.Then,the number $2ab-a-b+1$ is written instead.What will be the number remained on the board after the last step.

Consider four real numbers $x$, $y$, $a$, and $b$, satisfying $x + y = a + b$ and $x^2 + y^2 = a^2 + b^2$. Prove that $x^n + y^n = a^n + b^n$, for all $n \in \mathbb{N}$.

Kite $ABCD$ has right angles at $B$ and $D$, and $AB<BC$. Points $E\in AB$ and $F\in AD$ satisfy $AE=4$, $EF=7$, and $FA=5$. If $AB=8$ and points $X$ lies outside $ABCD$ while satisfying $XE-XF=1$ and $XE+XF+2XA=23$, then $XA$ may be written as $\tfrac{a-b\sqrt{c}}{d}$ for $a,b,c,d$ positive integers with $\gcd(a^2,b^2,c,d^2)=1$ and $c$ squarefree. Find $a+b+c+d$.

Let $S = \{1, \dots, n\}$. Given a bijection $f : S \to S$ an [i]orbit[/i] of $f$ is a set of the form $\{x, f(x), f(f(x)), \dots \}$ for some $x \in S$. We denote by $c(f)$ the number of distinct orbits of $f$. For example, if $n=3$ and $f(1)=2$, $f(2)=1$, $f(3)=3$, the two orbits are $\{1,2\}$ and $\{3\}$, hence $c(f)=2$. Given $k$ bijections $f_1$, $\ldots$, $f_k$ from $S$ to itself, prove that \[ c(f_1) + \dots + c(f_k) \le n(k-1) + c(f) \] where $f : S \to S$ is the composed function $f_1 \circ \dots \circ f_k$. [i]Proposed by Maria Monks Gillespie[/i]

The product of the positive numbers $a, b, c, d$ and $e$ is equal to $1$. Prove that $$ \frac{a^2}{b^2}+\frac{b^2}{c^2}+\frac{c^2}{d^2}+\frac{d^2}{e^2}+\frac{e^2}{a^2} \geq a+b+c+d+e .$$

Let $ABCD$ be a trapezoid with $AB\parallel CD$ and $\angle D=90^\circ$. Suppose that there is a point $E$ on $CD$ such that $AE=BE$ and that triangles $AED$ and $CEB$ are similar, but not congruent. Given that $\tfrac{CD}{AB}=2014$, find $\tfrac{BC}{AD}$.

Compute the sum of all positive integers whose positive divisors sum to $186.$