If $a \diamondsuit b = \vert a - b \vert \cdot \vert b - a \vert$ then find the value of $1 \diamondsuit (2 \diamondsuit (3 \diamondsuit (4 \diamondsuit 5)))$. [i]Proposed by Muztaba Syed[/i] [hide=Solution] [i]Solution.[/i] $\boxed{9}$ $a\diamondsuit b = (a-b)^2$. This gives us an answer of $\boxed{9}$. [/hide]

Show that there exist infinitely many pairs $(a, b)$ of positive integers with the property that $a+b$ divides $ab+1$, $a-b$ divides $ab-1$, $b>1$ and $a>b\sqrt{3}-1$

A round table has radius $ 4$. Six rectangular place mats are placed on the table. Each place mat has width $ 1$ and length $ x$ as shown. They are positioned so that each mat has two corners on the edge of the table, these two corners being end points of the same side of length $ x$. Further, the mats are positioned so that the inner corners each touch an inner corner of an adjacent mat. What is $ x$? [asy]unitsize(4mm); defaultpen(linewidth(.8)+fontsize(8)); draw(Circle((0,0),4)); path mat=(-2.687,-1.5513)--(-2.687,1.5513)--(-3.687,1.5513)--(-3.687,-1.5513)--cycle; draw(mat); draw(rotate(60)*mat); draw(rotate(120)*mat); draw(rotate(180)*mat); draw(rotate(240)*mat); draw(rotate(300)*mat); label("$x$",(-2.687,0),E); label("$1$",(-3.187,1.5513),S);[/asy]$ \textbf{(A)}\ 2\sqrt {5} \minus{} \sqrt {3} \qquad \textbf{(B)}\ 3 \qquad \textbf{(C)}\ \frac {3\sqrt {7} \minus{} \sqrt {3}}{2} \qquad \textbf{(D)}\ 2\sqrt {3} \qquad \textbf{(E)}\ \frac {5 \plus{} 2\sqrt {3}}{2}$

Let $P(x)$ be a polynomial with rational coefficients such that $P(n)$ is integer for all integers $n$. Moreover: $gcd(P(1), \ldots , P(k), \ldots) = 1$. Prove that every integer $k$ can be represented in infinitely many ways of the form $\pm P(1) \pm P(2) \pm \ldots \pm P(m)$, for some positive integer $m$ and certain choices of $\pm$.

Let $\mathbf M\subseteq\mathbb R^2$ be a set with the following properties: 1) there is a pair $(a,b)\in\mathbf M$ such that $ab(a-b)\neq0,$ 2) if $\left(x_1,y_1\right),\left(x_2,y_2\right)\in\mathbf M$ and $c\in\mathbb R$ then also \[\left(cx_1,cy_1\right),\left(x_1+x_2,y_1+y_2\right),\left(x_1x_2,y_1y_2\right)\in\mathbf M.\] Show that in fact \[\mathbf M=\mathbb R^2.\]

While there do not exist pairwise distinct real numbers $a,b,c$ satisfying $a^2+b^2+c^2 = ab+bc+ca$, there do exist complex numbers with that property. Let $a,b,c$ be complex numbers such that $a^2+b^2+c^2 = ab+bc+ca$ and $|a+b+c| = 21$. Given that $|a-b| = 2\sqrt{3}$, $|a| = 3\sqrt{3}$, compute $|b|^2+|c|^2$. [hide="Clarifications"] [list] [*] The problem should read $|a+b+c| = 21$. An earlier version of the test read $|a+b+c| = 7$; that value is incorrect. [*] $|b|^2+|c|^2$ should be a positive integer, not a fraction; an earlier version of the test read ``... for relatively prime positive integers $m$ and $n$. Find $m+n$.''[/list][/hide] [i]Ray Li[/i]

Two strictly ascending sequences of positive numbers are given. In each sequence, each number starting from the third one is the sum of two preceding ones. It is known that each of the sequences contains at least one number not present in the other sequence. What is the maximum quantity of numbers common for these two sequences? Boris Frenkin

Let be given a tetrahedron whose any two opposite edges are equal. A plane varies so that its intersection with the tetrahedron is a quadrilateral. Find the positions of the plane for which the perimeter of this quadrilateral is minimum, and find the locus of the centroid for those quadrilaterals with the minimum perimeter.

Let $G$ be a graph, not containing $K_4$ as a subgraph and $|V(G)|=3k$ (I interpret this to be the number of vertices is divisible by 3). What is the maximum number of triangles in $G$?

Find the number of ordered pairs of integers $(x, y)$ with $0 \le x, y \le 40$ where $$\frac{x^2-xy^2+1}{41}$$ is an integer.

Consider the acute-angled triangle $ABC$, with orthocentre $H$ and circumcentre $O$. $D$ is the intersection point of lines $AH$ and $BC$ and $E$ lies on $\overline{AH}$ such that $AE=DH$. Suppose $EO$ and $BC$ meet at $F$. Prove that $BD=CF$. [i] (Călin Pop & Vlad Robu) [/i]

In preparation for the family's upcoming vacation, Tony puts together five bags of jelly beans, one bag for each day of the trip, with an equal number of jelly beans in each bag. Tony then pours all the jelly beans out of the five bags and begins making patterns with them. One of the patterns that he makes has one jelly bean in a top row, three jelly beans in the next row, five jelly beans in the row after that, and so on: \[\begin{array}{ccccccccc}&&&&*&&&&\\&&&*&*&*&&&\\&&*&*&*&*&*&&\\&*&*&*&*&*&*&*&\\ *&*&*&*&*&*&*&*&*\\&&&&\vdots&&&&\end{array}\] Continuing in this way, Tony finishes a row with none left over. For instance, if Tony had exactly $25$ jelly beans, he could finish the fifth row above with no jelly beans left over. However, when Tony finishes, there are between $10$ and $20$ rows. Tony then scoops all the jelly beans and puts them all back into the five bags so that each bag once again contains the same number. How many jelly beans are in each bag? (Assume that no marble gets put inside more than one bag.)

There are 47 students in a classroom with seats arranged in 6 rows $ \times$ 8 columns, and the seat in the $ i$-th row and $ j$-th column is denoted by $ (i,j).$ Now, an adjustment is made for students’ seats in the new school term. For a student with the original seat $ (i,j),$ if his/her new seat is $ (m,n),$ we say that the student is moved by $ [a, b] \equal{} [i \minus{} m, j \minus{} n]$ and define the position value of the student as $ a\plus{}b.$ Let $ S$ denote the sum of the position values of all the students. Determine the difference between the greatest and smallest possible values of $ S.$

Does there exist a function $f: \mathbb R \to \mathbb R $ satisfying the following conditions: (i) for each real $y$ there is a real $x$ such that $f(x)=y$ , and (ii) $f(f(x)) = (x - 1)f(x) + 2$ for all real $x$ ? [i]Proposed by Igor I. Voronovich, Belarus[/i]

Show that there is a positive integer n such that the first 1992 digits of $n^{1992}$ are 1.

Solve for $x$ if $\sqrt{x + 1}+ \sqrt{x} = 5.$ [i]Proposed by Eric Oh[/i]

How many real numbers $x$ are solutions to the equation $|x - 2| - 4 =\frac{1}{|x - 3|}$ ?

Let $a$ and $b$ be positive integers. Prove that the numbers $an^2+b$ and $a(n+1)^2+b$ are both perfect squares only for finitely many integers $n$.

For positive integers $a$ and $b$, an $(a,b)$-shuffle of a deck of $a+b$ cards is any shuffle that preserves the relative order of the top $a$ cards and the relative order of the bottom $b$ cards. Let $n$, $k$, $a_1$, $a_2$, $\dots$, $a_k$, $b_1$, $b_2$, $\dots$, $b_k$ be fixed positive integers such that $a_i+b_i=n$ for all $1\leq i\leq k$. Big Bird has a deck of $n$ cards and will perform an $(a_i,b_i)$-shuffle for each $1\leq i\leq k$, in ascending order of $i$. Suppose that Big Bird can reverse the order of the deck. Prove that Big Bird can also achieve any of the $n!$ permutations of the cards. [i]Linus Tang[/i]

The USAMO is a $6$ question test. For each question, you submit a positive integer number $p$ of pages on which your solution is written. On the $i$th page of this question, you write the fraction $i/p$ to denote that this is the $i$th page out of $p$ for this question. When you turned in your submissions for the $2017$ USAMO, the bored proctor computed the sum of the fractions for all of the pages which you turned in. Surprisingly, this number turned out to be $2017$. How many pages did you turn in? [i]Proposed by Tristan Shin[/i]

Given a regular pentagon, its five diagonals are drawn. How many triangles do appear in the figure? Classify the set of triangles in classes of equal triangles.

At the Berkeley Math Tournament, teams are composed of $6$ students, each of whom pick two distinct subject tests out of $5$ choices. How many different distributions across subjects are possible for a team?

Let $A_1A_2A_3A_4$ be a tetrahedron. For any permutation $(i, j,k,h)$ of $1,2,3,4$ denote: - $P_i$ – the orthogonal projection of $A_i$ on $A_jA_kA_h$; - $B_{ij}$ – the midpoint of the edge $A_iAj$, - $C_{ij}$ – the midpoint of segment $P_iP_j$ - $\beta_{ij}$– the plane $B_{ij}P_hP_k$ - $\delta_{ij}$ – the plane $B_{ij}P_iP_j$ - $\alpha_{ij}$ – the plane through $C_{ij}$ orthogonal to $A_kA_h$ - $\gamma_{ij}$ – the plane through $C_{ij}$ orthogonal to $A_iA_j$. Prove that if the points $P_i$ are not in a plane, then the following sets of planes are concurrent: (a) $\alpha_{ij}$, (b) $\beta_{ij}$, (c) $\gamma_{ij}$, (d) $\delta_{ij}$.

Given a triangle $ABC$ and three circles $x$, $y$ and $z$ such that $A \in y \cap z$, $B \in z \cap x$ and $C \in x \cap y$. The circle $x$ intersects the line $AC$ at the points $X_b$ and $C$, and intersects the line $AB$ at the points $X_c$ and $B$. The circle $y$ intersects the line $BA$ at the points $Y_c$ and $A$, and intersects the line $BC$ at the points $Y_a$ and $C$. The circle $z$ intersects the line $CB$ at the points $Z_a$ and $B$, and intersects the line $CA$ at the points $Z_b$ and $A$. (Yes, these definitions have the symmetries you would expect.) Prove that the perpendicular bisectors of the segments $Y_a Z_a$, $Z_b X_b$ and $X_c Y_c$ concur.

Let $ABC$ be an acute-angled triangle with circumradius $R$ less than the sides of $ABC$, let $H$ and $O$ be the orthocenter and circuncenter of $ABC$, respectively. The angle bisectors of $\angle ABH$ and $\angle ACH$ intersects in the point $A_1$, analogously define $B_1$ and $C_1$. If $E$ is the midpoint of $HO$, prove that $EA_1+EB_1+EC_1=p-\frac{3R}{2}$ where $p$ is the semiperimeter of $ABC$