[b]p1.[/b] Compute $2017 + 7201 + 1720 + 172$. [b]p2. [/b]A number is called [i]downhill [/i]if its digits are distinct and in descending order. (For example, $653$ and $8762$ are downhill numbers, but $97721$ is not.) What is the smallest downhill number greater than 86432? [b]p3.[/b] Each vertex of a unit cube is sliced off by a planar cut passing through the midpoints of the three edges containing that vertex. What is the ratio of the number of edges to the number of faces of the resulting solid? [b]p4.[/b] In a square with side length $5$, the four points that divide each side into five equal segments are marked. Including the vertices, there are $20$ marked points in total on the boundary of the square. A pair of distinct points $A$ and $B$ are chosen randomly among the $20$ points. Compute the probability that $AB = 5$. [b]p5.[/b] A positive two-digit integer is one less than five times the sum of its digits. Find the sum of all possible such integers. [b]p6.[/b] Let $$f(x) = 5^{4^{3^{2^{x}}}}.$$ Determine the greatest possible value of $L$ such that $f(x) > L$ for all real numbers $x$. [b]p7.[/b] If $\overline{AAAA}+\overline{BB} = \overline{ABCD}$ for some distinct base-$10$ digits $A, B, C, D$ that are consecutive in some order, determine the value of $ABCD$. (The notation $\overline{ABCD}$ refers to the four-digit integer with thousands digit $A$, hundreds digit $B$, tens digit $C$, and units digit $D$.) [b]p8.[/b] A regular tetrahedron and a cube share an inscribed sphere. What is the ratio of the volume of the tetrahedron to the volume of the cube? [b]p9.[/b] Define $\lfloor x \rfloor$ as the greatest integer less than or equal to x, and ${x} = x - \lfloor x \rfloor$ as the fractional part of $x$. If $\lfloor x^2 \rfloor =2 \lfloor x \rfloor$ and $\{x^2\} =\frac12 \{x\}$, determine all possible values of $x$. [b]p10.[/b] Find the largest integer $N > 1$ such that it is impossible to divide an equilateral triangle of side length $ 1$ into $N$ smaller equilateral triangles (of possibly different sizes). [b]p11.[/b] Let $f$ and $g$ be two quadratic polynomials. Suppose that $f$ has zeroes $2$ and $7$, $g$ has zeroes $1$ and $ 8$, and $f - g$ has zeroes $4$ and $5$. What is the product of the zeroes of the polynomial $f + g$? [b]p12.[/b] In square $PQRS$, points $A, B, C, D, E$, and $F$ are chosen on segments $PQ$, $QR$, $PR$, $RS$, $SP$, and $PR$, respectively, such that $ABCDEF$ is a regular hexagon. Find the ratio of the area of $ABCDEF$ to the area of $PQRS$. [b]p13.[/b] For positive integers $m$ and $n$, define $f(m, n)$ to be the number of ways to distribute $m$ identical candies to $n$ distinct children so that the number of candies that any two children receive differ by at most $1$. Find the number of positive integers n satisfying the equation $f(2017, n) = f(7102, n)$. [b]p14.[/b] Suppose that real numbers $x$ and $y$ satisfy the equation $$x^4 + 2x^2y^2 + y^4 - 2x^2 + 32xy - 2y^2 + 49 = 0.$$ Find the maximum possible value of $\frac{y}{x}$. [b]p15.[/b] A point $P$ lies inside equilateral triangle $ABC$. Let $A'$, $B'$, $C'$ be the feet of the perpendiculars from $P$ to $BC, AC, AB$, respectively. Suppose that $PA = 13$, $PB = 14$, and $PC = 15$. Find the area of $A'B'C'$. PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Find the real numbers $a$, $b$, $c$ and $d$ that satisfy the following equations: $$\left \{\begin{matrix} a\cdot b+c+d & = & 6, \\ b\cdot c+d+a & = & 2, \\ c\cdot d+a+b & = & 5, \\ d\cdot a+b+c & = & 3. \end{matrix}\right .$$

Compute the number of nonempty subsets $S$ of $\{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\}$ such that $\frac{\max \,\, S + \min \,\,S}{2}$ is an element of $S$.

Prove that for any positive integer $ n$, there exists only $ n$ degree polynomial $ f(x),$ satisfying $ f(0) \equal{} 1$ and $ (x \plus{} 1)[f(x)]^2 \minus{} 1$ is an odd function.

Let $n$ be an integer greater than $3$. Prove that all the roots of the polynomial $P(x) = x^n - 5x^{n-1} + 12x^{n-2}- 15x^{n-3} + a_{n-4}x^{n-4} +...+ a_0$ cannot be both real and positive.

[u]Round 1[/u] [b]p1.[/b] The fractal T-shirt for this year's Duke Math Meet is so complicated that the printer broke trying to print it. Thus, we devised a method for manually assembling each shirt - starting with the full-size 'base' shirt, we paste a smaller shirt on top of it. And then we paste an even smaller shirt on top of that one. And so on, infinitely many times. (As you can imagine, it took a while to make all the shirts.) The completed T-shirt consists of the original 'base' shirt along with all of the shirts we pasted onto it. Now suppose the base shirt requires $2011$ $cm^2$ of fabric to make, and that each pasted-on shirt requires $4/5$ as much fabric as the previous one did. How many $cm^2$ of fabric in total are required to make one complete shirt? [b]p2.[/b] A dog is allowed to roam a yard while attached to a $60$-meter leash. The leash is anchored to a $40$-meter by $20$-meter rectangular house at the midpoint of one of the long sides of the house. What is the total area of the yard that the dog can roam? [b]p3.[/b] $10$ birds are chirping on a telephone wire. Bird $1$ chirps once per second, bird $2$ chirps once every $2$ seconds, and so on through bird $10$, which chirps every $10$ seconds. At time $t = 0$, each bird chirps. Define $f(t)$ to be the number of birds that chirp during the $t^{th}$ second. What is the smallest $t > 0$ such that $f(t)$ and $f(t + 1)$ are both at least $4$? [u]Round 2[/u] [b]p4.[/b] The answer to this problem is $3$ times the answer to problem 5 minus $4$ times the answer to problem 6 plus $1$. [b]p5.[/b] The answer to this problem is the answer to problem 4 minus $4$ times the answer to problem 6 minus $1$. [b]p6.[/b] The answer to this problem is the answer to problem 4 minus $2$ times the answer to problem 5. [u]Round 3[/u] [b]p7.[/b] Vivek and Daniel are playing a game. The game ends when one person wins $5$ rounds. The probability that either wins the first round is $1/2$. In each subsequent round the players have a probability of winning equal to the fraction of games that the player has lost. What is the probability that Vivek wins in six rounds? [b]p8.[/b] What is the coefficient of $x^8y^7$ in $(1 + x^2 - 3xy + y^2)^{17}$? [b]p9.[/b] Let $U(k)$ be the set of complex numbers $z$ such that $z^k = 1$. How many distinct elements are in the union of $U(1),U(2),...,U(10)$? [u]Round 4[/u] [b]p10.[/b] Evaluate $29 {30 \choose 0}+28{30 \choose 1}+27{30 \choose 2}+...+0{30 \choose 29}-{30\choose 30}$. You may leave your answer in exponential format. [b]p11.[/b] What is the number of strings consisting of $2a$s, $3b$s and $4c$s such that $a$ is not immediately followed by $b$, $b$ is not immediately followed by $c$ and $c$ is not immediately followed by $a$? [b]p12.[/b] Compute $\left(\sqrt3 + \tan (1^o)\right)\left(\sqrt3 + \tan (2^o)\right)...\left(\sqrt3 + \tan (29^o)\right)$. [u]Round 5[/u] [b]p13.[/b] Three massless legs are randomly nailed to the perimeter of a massive circular wooden table with uniform density. What is the probability that the table will not fall over when it is set on its legs? [b]p14.[/b] Compute $$\sum^{2011}_{n=1}\frac{n + 4}{n(n + 1)(n + 2)(n + 3)}$$ [b]p15.[/b] Find a polynomial in two variables with integer coefficients whose range is the positive real numbers. PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

How many real numbers are roots of the polynomial \[x^9 - 37x^8 - 2x^7 + 74x^6 + x^4 - 37x^3 - 2x^2 + 74x?\]

Let the roots of $ax^2+bx+c=0$ be $r$ and $s$. The equation with roots $ar+b$ and $as+b$ is: $ \textbf{(A)}\ x^2-bx-ac=0$ $\qquad\textbf{(B)}\ x^2-bx+ac=0$ $\qquad\textbf{(C)}\ x^2+3bx+ca+2b^2=0$ ${\qquad\textbf{(D)}\ x^2+3bx-ca+2b^2=0 }$ ${\qquad\textbf{(E)}\ x^2+bx(2-a)+a^2c+b^2(a+1)=0} $

How many pairs of positive integers $(a, b)$ satisfy the equation $log_a 16 = b$?

Consider a matrix of size $n\times n$ whose entries are real numbers of absolute value not exceeding $1$. The sum of all entries of the matrix is $0$. Let $n$ be an even positive integer. Determine the least number $C$ such that every such matrix necessarily has a row or a column with the sum of its entries not exceeding $C$ in absolute value. [i]Proposed by Marcin Kuczma, Poland[/i]

Let $f: \mathbb{R}\rightarrow \mathbb{R}$ be a function satisfying $f(x)f(y)=f(x-y)$. Find all possible values of $f(2017)$.

Given a polynomial $p(x) =Ax^3+x^2-A$ with $A \neq 0$. Show that for every different real number $a,b,c$, at least one of $ap(b)$, $bp(a)$, and $cp(a)$ not equal to 1.

Determine all real numbers $x$ which satisfy \[ x = \sqrt{a - \sqrt{a+x}} \] where $a > 0$ is a parameter.

Suppose that $x$ and $y$ are nonzero real numbers such that \[\frac{3x+y}{x-3y}= -2.\] What is the value of \[\frac{x+3y}{3x-y}?\] $\textbf{(A) } {-3} \qquad \textbf{(B) } {-1} \qquad \textbf{(C) } 1 \qquad \textbf{(D) }2 \qquad \textbf{(E) } 3$

The number obtained from the last two nonzero digits of $ 90!$ is equal to $ n$. What is $ n$? $ \textbf{(A)}\ 12 \qquad \textbf{(B)}\ 32 \qquad \textbf{(C)}\ 48 \qquad \textbf{(D)}\ 52 \qquad \textbf{(E)}\ 68$

Find all functions $f:\mathbb{R}^{+}\rightarrow \mathbb{R}^{+}$ such that $x,y\in \mathbb{R}^{+},$ \[ f\left(\frac{y}{f(x+1)}\right)+f\left(\frac{x+1}{xf(y)}\right)=f(y) \]

Suppose in a given collection of $2016$ integer, the sum of any $1008$ integers is positive. Show that sum of all $2016$ integers is positive.

[b]a)[/b] Solve the system $$ \left\{\begin{matrix} 3^y-4^x=11\\ \log_4{x} +\log_3 y =3/2\end{matrix}\right. $$ [b]b)[/b] Solve the equation $ \quad 9^{\log_5 (x-2)} -5^{\log_9 (x+2)} = 4. $

Let $d \geq 1$ be an integer that is not the square of an integer. Prove that for every integer $n \geq 1,$ \[(n \sqrt d +1) \cdot | \sin(n \pi \sqrt d )| \geq 1\]

We consider the equation $x^2 + (a + b - 1)x + ab - a - b = 0$, where $a$ and $b$ are positive integers with $a \leq b$. a) Show that the equation has $2$ distinct real solutions. b) Prove that if one of the solutions is an integer, then both solutions are non-positive integers and $b < 2a.$

Let us define a $\textit{triangle equivalence}$ a group of numbers that can be arranged as shown $a+b=c$ $d+e+f=g+h$ $i+j+k+l=m+n+o$ and so on... Where at the $j$-th row, the left hand side has $j+1$ terms and the right hand side has $j$ terms. Now, we are given the first $N^2$ positive integers, where $N$ is a positive integer. Suppose we eliminate any one number that has the same parity with $N$. Prove that the remaining $N^2-1$ numbers can be formed into a $\textit{triangle equivalence}$. For example, if $10$ is eliminated from the first $16$ numbers, the remaining numbers can be arranged into a $\textit{triangle equivalence}$ as shown. $1+3=4$ $2+5+8=6+9$ $7+11+12+14=13+15+16$

Show that if $2023$ real numbers $x_1,x_2,\dots,x_{2023}$ satisfy $x_1\geq x_2\geq\dots\geq x_{2023}\geq0,$ then $$x_1^2+3x_2^2+5x_3^2+\cdots+(2\cdot2023-1)\cdot x^2_{2023}\leq(x_1+x_2+\cdots+x_{2023})^2.$$ When does the equality take place?

Let $a, b_1, b_2, \dots, b_n, c_1, c_2, \dots, c_n$ be real numbers such that \[x^{2n} + ax^{2n - 1} + ax^{2n - 2} + \dots + ax + 1 = \prod_{i = 1}^{n}{(x^2 + b_ix + c_i)}\] Prove that $c_1 = c_2 = \dots = c_n = 1$. As a consequence, all complex zeroes of this polynomial must lie on the unit circle.

Let be two nonzero real numbers $ a,b, $ and a function $ f:\mathbb{R}\longrightarrow [0,\infty ) $ satisfying the functional equation $$ f(x+a+b)+f(x)=f(x+a)+f(x+b) . $$ [b]1)[/b] Prove that $ f $ is periodic if $ a/b $ is rational. [b]2)[/b] If $ a/b $ is not rational, could $ f $ be nonperiodic?

Prove that $x^{12} - x^9 + x^4 - x + 1 > 0$ for all $x$.