Develop necessary and sufficient conditions that the equation \[ \begin{vmatrix} 0 & a_1 - x & a_2 - x \\ -a_1 - x & 0 & a_3 - x \\ -a_2 - x & -a_3 - x & 0\end{vmatrix} = 0 \qquad (a_i \neq 0) \] shall have a multiple root.

Let $ABCD$ be a rectangle and let $E$ and $F$ be points on $CD$ and $BC$ respectively such that area $(ADE) = 16$, area $(CEF) = 9$ and area $(ABF) = 25$. What is the area of triangle $AEF$ ?

Find all functions $ f\in\mathcal{C}^1 [0,1] $ that satisfy $ f(1)=-1/6 $ and $$ \int_0^1 \left( f'(x) \right)^2 dx\le 2\int_0^1 f(x)dx. $$

Let $ n$ be a positive integer. Find all $ x_1,x_2,\ldots,x_n$ that satisfy the relation: \[ \sqrt{x_1\minus{}1}\plus{}2\cdot \sqrt{x_2\minus{}4}\plus{}3\cdot \sqrt{x_3\minus{}9}\plus{}\cdots\plus{}n\cdot\sqrt{x_n\minus{}n^2}\equal{}\frac{1}{2}(x_1\plus{}x_2\plus{}x_3\plus{}\cdots\plus{}x_n).\]

Let $S$ be a convex region in the euclidean plane containing the origin. Assume that every ray from the origin has at least one point outside $S$. Prove that $S$ is bounded.

The square $DEFG$ is contained in equilateral triangle $ABC$, with $E$ on $\overline{AC}$, $G$ on $\overline{AD}$, and $F$ as the midpoint of $\overline{BC}$. Find $AD$ if $DE = 6$.

Compute the sum of all positive integers $50 \leq n \leq 100$ such that $2n+3 \nmid 2^{n!}-1$.

Let $n$ be a fixed positive integer. Oriol has $n$ cards, each of them with a $0$ written on one side and $1$ on the other. We place these cards in line, some face up and some face down (possibly all on the same side). We begin the following process consisting of $n$ steps: 1) At the first step, Oriol flips the first card 2) At the second step, Oriol flips the first card and second card . . . n) At the last step Oriol flips all the cards Let $s_0, s_1, s_2, \dots, s_n$ be the sum of the numbers seen in the cards at the beggining, after the first step, after the second step, $\dots$ after the last step, respectively. a) Find the greatest integer $k$ such that, no matter the initial card configuration, there exists at least $k$ distinct numbers between $s_0, s_1, \dots, s_n$. b) Find all positive integers $m$ such that, for each initial card configuration, there exists an index $r$ such that $s_r = m$. [i]Proposed by Dorlir Ahmeti[/i]

Two natural numbers have the same decimal digits in different order and have the sum $999\cdots 999$. Is this possible when each of the numbers consists of $1999$ digits?

It turned out for some triangle with sides $a, b$ and $c$, that a circle of radius $r = \frac{a+b+c}{2}$ touches side $c$ and extensions of sides $a$ and $b$. Prove that a circle of radius $ \frac{a+c-b}{2}$ is tangent to $a$ and the extensions of $b$ and $c$.

If $f(x)$ is a real valued function of the real variable $x$, and $f(x)$ is not identically zero, and for all $a$ and $b$ \[f(a+b)+f(a-b)=2f(a)+2f(b),\] then for all $x$ and $y$ $\textbf{(A) }f(0)=1\qquad\textbf{(B) }f(-x)=-f(x)\qquad$ $\textbf{(C) }f(-x)=f(x)\qquad\textbf{(D) }f(x+y)=f(x)+f(y)\qquad$ $\textbf{(E) }\text{there is a positive real number }T\text{ such that }f(x+T)=f(x)$

Prove the following inequality for any integer $ N\geq 4$. \[ \sum_{p\equal{}4}^N \frac{p^2\plus{}2}{(p\minus{}2)^4}<5\]

[color=darkred]Let $ p$ be a prime divisor of $ n\ge 2$. Prove that there exists a set of natural numbers $ A \equal{} \{a_1,a_2,...,a_n\}$ such that product of any two numbers from $ A$ is divisible by the sum of any $ p$ numbers from $ A$.[/color]

Circles $ w_{1}$ and $ w_{2}$ with centres $ O_{1}$ and $ O_{2}$ are externally tangent at point $ D$ and internally tangent to a circle $ w$ at points $ E$ and $ F$ respectively. Line $ t$ is the common tangent of $ w_{1}$ and $ w_{2}$ at $ D$. Let $ AB$ be the diameter of $ w$ perpendicular to $ t$, so that $ A, E, O_{1}$ are on the same side of $ t$. Prove that lines $ AO_{1}$, $ BO_{2}$, $ EF$ and $ t$ are concurrent.

A chess tournament is held with the participation of boys and girls. The girls are twice as many as boys. Each player plays against each other player exactly once. By the end of the tournament, there were no draws and the ratio of girl winnings to boy winnings was $\frac{7}{9}$. How many players took part at the tournament?

Let $M$ be the least common multiple of all the integers $10$ through $30,$ inclusive. Let $N$ be the least common multiple of $M,$ $32,$ $33,$ $34,$ $35,$ $36,$ $37,$ $38,$ $39,$ and $40.$ What is the value of $\frac{N}{M}?$ $(\textbf{A})\: 1\qquad(\textbf{B}) \: 2\qquad(\textbf{C}) \: 37\qquad(\textbf{D}) \: 74\qquad(\textbf{E}) \: 2886$

There is a $50\times 50$ grid board.. Carlos is going to write a number in each box with the following procedure. He first chooses $100$ distinct numbers that we denote $f_1,f_2,f_3,…,f_{50},c_1,c_2,c_3,…,c_{50}$ among which there are exactly $50$ that they are rational. Then he writes in each box ($i,j)$ the number $f_i \cdot c_j$ (the multiplication of $f_i$ by $c_j$). Determine the maximum number of rational numbers that the squares on the board can contain.

$n(\geq 2)$ is a given integer and $T$ is set of all $n\times n$ matrices whose entries are elements of the set $S=\{1,\cdots,2017\}$. Evaluate the following value. \[\sum_{A\in T} \text{det}(A)\]

[u]Round 1 [/u] [b]p1.[/b] A small pizza costs $\$4$ and has $6$ slices. A large pizza costs $\$9$ and has $14$ slices. If the MMATHS organizers got at least $400$ slices of pizza (having extra is okay) as cheaply as possible, how many large pizzas did they buy? [b]p2.[/b] Rachel flips a fair coin until she gets a tails. What is the probability that she gets an even number of heads before the tails? [b]p3.[/b] Find the unique positive integer $n$ that satisfies $n! \cdot (n + 1)! = (n + 4)!$. [u]Round 2 [/u] [b]p4.[/b] The Portland Malt Shoppe stocks $10$ ice cream flavors and $8$ mix-ins. A milkshake consists of exactly $1$ flavor of ice cream and between $1$ and $3$ mix-ins. (Mix-ins can be repeated, the number of each mix-in matters, and the order of the mix-ins doesn’t matter.) How many different milkshakes can be ordered? [b]p5.[/b] Find the minimum possible value of the expression $(x)^2 + (x + 3)^4 + (x + 4)^4 + (x + 7)^2$, where $x$ is a real number. [b]p6.[/b] Ralph has a cylinder with height $15$ and volume $\frac{960}{\pi}$ . What is the longest distance (staying on the surface) between two points of the cylinder? [u]Round 3 [/u] [b]p7.[/b] If there are exactly $3$ pairs $(x, y)$ satisfying $x^2 + y^2 = 8$ and $x + y = (x - y)^2 + a$, what is the value of $a$? [b]p8.[/b] If $n$ is an integer between $4$ and $1000$, what is the largest possible power of $2$ that $n^4 - 13n^2 + 36$ could be divisible by? (Your answer should be this power of $2$, not just the exponent.) [b]p9.[/b] Find the sum of all positive integers $n \ge 2$ for which the following statement is true: “for any arrangement of $n$ points in three-dimensional space where the points are not all collinear, you can always find one of the points such that the $n - 1$ rays from this point through the other points are all distinct.” [u]Round 4 [/u] [b]p10.[/b] Donald writes the number $12121213131415$ on a piece of paper. How many ways can he rearrange these fourteen digits to make another number where the digit in every place value is different from what was there before? [b]p11.[/b] A question on Joe’s math test asked him to compute $\frac{a}{b} +\frac34$ , where $a$ and $b$ were both integers. Because he didn’t know how to add fractions, he submitted $\frac{a+3}{b+4}$ as his answer. But it turns out that he was right for these particular values of $a$ and $b$! What is the largest possible value that a could have been? [b]p12.[/b] Christopher has a globe with radius $r$ inches. He puts his finger on a point on the equator. He moves his finger $5\pi$ inches North, then $\pi$ inches East, then $5\pi$ inches South, then $2\pi$ inches West. If he ended where he started, what is the largest possible value of $r$? PS. You should use hide for answers. Rounds 5-7 have be posted [url=https://artofproblemsolving.com/community/c4h2789002p24519497]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Find all fuctions $f,g:\mathbb{R}\rightarrow \mathbb{R}$ such that: $f(x-3f(y))=xf(y)-yf(x)+g(x) \forall x,y\in\mathbb{R}$ and $g(1)=-8$

For a natural number $n$, with $v_2(n)$ we denote the largest integer $k\geq0$ such that $2^k|n$. Let us assume that the function $f\colon\mathbb{N}\to\mathbb{N}$ meets the conditions: $(i)$ $f(x)\leq3x$ for all natural numbers $x\in\mathbb{N}$. $(ii)$ $v_2(f(x)+f(y))=v_2(x+y)$ for all natural numbers $x,y\in\mathbb{N}$. Prove that for every natural number $a$ there exists exactly one natural number $x$ such that $f(x)=3a$.

Let $A=\sqrt{7+2\sqrt{10}} - \sqrt{7-2\sqrt{10}}.$ We can express $A$ as $a\sqrt{b},$ where $a,b$ are integers and $b$ is square-free. Compute $a+b.$

In a circle $O$, there are six points, $A,B,C,D,E, F$ in a counterclockwise order. $BD \perp CF$, and $CF,BE,AD$ are concurrent. Let the perpendicular from $B$ to $AC$ be $M$, and the perpendicular from $D$ to $CE$ be $N$. Prove that $AE // MN$.

Let $g(x)$ be a fixed polynomial with real coefficients and define $f(x)$ by $f(x) =x^2 + xg(x^3)$. Show that $f(x)$ is not divisible by $x^2 - x + 1$.

A convex, closed figure lies inside a given circle. The figure is seen from every point of the circumference at a right angle (that is, the two rays drawn from the point and supporting the convex figure are perpendicular). Prove that the center of the circle is a center of symmetry of the figure.