[b]p1.[/b] It takes $12$ months for Santa Claus to pack gifts. It would take $20$ months for his apprentice to do the job. If they work together, how long will it take for them to pack the gifts? [b]p2.[/b] All passengers on a bus sit in pairs. Exactly $2/5$ of all men sit with women, exactly $2/3$ of all women sit with men. What part of passengers are men? [b]p3.[/b] There are $100$ colored balls in a box. Every $10$-tuple of balls contains at least two balls of the same color. Show that there are at least $12$ balls of the same color in the box. [b]p4.[/b] There are $81$ wheels in storage marked by their two types, say first and second type. Wheels of the same type weigh equally. Any wheel of the second type is much lighter than a wheel of the first type. It is known that exactly one wheel is marked incorrectly. Show that one can determine which wheel is incorrectly marked with four measurements. [b]p5.[/b] Remove from the figure below the specified number of matches so that there are exactly $5$ squares of equal size left: (a) $8$ matches (b) $4$ matches [img]https://cdn.artofproblemsolving.com/attachments/4/b/0c5a65f2d9b72fbea50df12e328c024a0c7884.png[/img] PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Consider the sytem of equations \[ a_{11}x_1+a_{12}x_2+a_{13}x_3 = 0 \]\[a_{21}x_1+a_{22}x_2+a_{23}x_3 =0\]\[a_{31}x_1+a_{32}x_2+a_{33}x_3 = 0 \] with unknowns $x_1, x_2, x_3$. The coefficients satisfy the conditions: a) $a_{11}, a_{22}, a_{33}$ are positive numbers; b) the remaining coefficients are negative numbers; c) in each equation, the sum ofthe coefficients is positive. Prove that the given system has only the solution $x_1=x_2=x_3=0$.

Find a real number $t$ such that for any set of 120 points $P_1, \ldots P_{120}$ on the boundary of a unit square, there exists a point $Q$ on this boundary with $|P_1Q| + |P_2Q| + \cdots + |P_{120}Q| = t$.

Let $ a_{0},a_{1},\ldots,a_{n}$ be integers, one of which is nonzero, and all of the numbers are not less than $ \minus{} 1$. Prove that if \[ a_{0} \plus{} 2a_{1} \plus{} 2^{2}a_{2} \plus{} \cdots \plus{} 2^{n}a_{n} \equal{} 0,\] then $ a_{0} \plus{} a_{1} \plus{} \cdots \plus{} a_{n} > 0$.

Given any positive integer $n$, let $a_n$ be the real root of equation $x^3+\dfrac{x}{n}=1$. Prove that (1) $a_{n+1}>a_n$; (2) $\sum_{i=1}^{n}\frac{1}{(i+1)^2a_i} <a_n$.

Find all non-constant polynomials $P(x)$ and $Q(x)$ with real coefficients satisfying the equality $P(Q(x))=P(x)Q(x)-P(x)$. [i](I. Voronovich)[/i]

Prove that for all positive real numbers $a$ and $ b$: $$\frac{(a + b)^3}{4} \ge a^2b + ab^2$$

We say that a strictly increasing positive real sequence $a_1,a_2,\cdots $ is an [i]elf sequence[/i] if for any $c>0$ we can find an $N$ such that $a_n<cn$ for $n=N,N+1,\cdots$. Furthermore, we say that $a_n$ is a [i]hat[/i] if $a_{n-i}+a_{n+i}<2a_n$ for $\displaystyle 1\le i\le n-1$. Is it true that every elf sequence has infinitely many hats?

Let $f:\mathbb{R}_{\geq 0} \rightarrow \mathbb{R}_{\geq 0}$ be a continuous function such that $f(0)=0$ and $$f(x)+f(f(x))+f(f(f(x)))=3x$$ for all $x>0$. Show that $f(x)=x$ for all $x>0$.

Investigate the singular points of the differential form $dt = dx/y$ on the compact Riemann surface $y^2/2 + U(x) = E$, where $U$ is a polynomial and $E$ is not a critical value.

Suppose $a_1,a_2, \dots$ is an infinite strictly increasing sequence of positive integers and $p_1, p_2, \dots$ is a sequence of distinct primes such that $p_n \mid a_n$ for all $n \ge 1$. It turned out that $a_n-a_k=p_n-p_k$ for all $n,k \ge 1$. Prove that the sequence $(a_n)_n$ consists only of prime numbers.

Let $n\ge 2$ be a positive integer. There are $n$ real coefficient polynomials $P_1(x),P_2(x),\cdots ,P_n(x)$ which is not all the same, and their leading coefficients are positive. Prove that $$\deg(P_1^n+P_2^n+\cdots +P_n^n-nP_1P_2\cdots P_n)\ge (n-2)\max_{1\le i\le n}(\deg P_i)$$ and find when the equality holds.

Let be three positive integers $ a,b,c $ and a function $ f:\mathbb{N}\longrightarrow\mathbb{N} $ defined as $$ f(n)=\left\{ \begin{matrix} n-a, & n>c\\ f\left( f(n+b) \right) ,& n\le c \end{matrix} \right. . $$ Determine the number of fixed points this function has.

Let $ n > 1$ be a fixed integer. Define functions $ f_0(x) \equal{} 0,$ $ f_1(x) \equal{} 1 \minus{} \cos(x),$ and for $ k > 0,$ \[ f_{k\plus{}1}(x) \equal{} f_k(x) \cdot \cos(x) \minus{} f_{k\minus{}1}(x).\] If $ F(x) \equal{} \sum^n_{r\equal{}1} f_r(x),$ prove that [b](a)[/b] $ 0 < F(x) < 1$ for $ 0 < x < \frac{\pi}{n\plus{}1},$ and [b](b)[/b] $ F(x) > 1$ for $ \frac{\pi}{n\plus{}1} < x < \frac{\pi}{n}.$

Let $f$ be a linear function such that $f(0) = -5$ and $f(f(0)) = -15$. Find the values of $ k \in R$ for which the solutions of the inequality $f(x) \cdot f(k - x) > 0$, lie in an interval of[u][/u] length $2$.

Given a real number $ c > 0$, a sequence $ (x_n)$ of real numbers is defined by $ x_{n \plus{} 1} \equal{} \sqrt {c \minus{} \sqrt {c \plus{} x_n}}$ for $ n \ge 0$. Find all values of $ c$ such that for each initial value $ x_0$ in $ (0, c)$, the sequence $ (x_n)$ is defined for all $ n$ and has a finite limit $ \lim x_n$ when $ n\to \plus{} \infty$.

Let $\mathbb{R}_{>0}$ be the set of all positive real numbers. Find all strictly monotone (increasing or decreasing) functions $f:\mathbb{R}_{>0} \to \mathbb{R}$ such that there exists a two-variable polynomial $P(x, y)$ with real coefficients satisfying $$ f(xy)=P(f(x), f(y)) $$ for all $x, y\in\mathbb{R}_{>0}$.\\ [i]Proposed by Navid Safaei, Iran[/i]

Prove that if \( a \), \( b \), and \( c \) are positive real numbers, then the following inequality holds: \[ \frac{a + 3c}{a + b} + \frac{c + 3a}{b + c} + \frac{4b}{c + a} \geq 6. \]

Real numbers $ a_{1}$, $ a_{2}$, $ \ldots$, $ a_{n}$ are given. For each $ i$, $ (1 \leq i \leq n )$, define \[ d_{i} \equal{} \max \{ a_{j}\mid 1 \leq j \leq i \} \minus{} \min \{ a_{j}\mid i \leq j \leq n \} \] and let $ d \equal{} \max \{d_{i}\mid 1 \leq i \leq n \}$. (a) Prove that, for any real numbers $ x_{1}\leq x_{2}\leq \cdots \leq x_{n}$, \[ \max \{ |x_{i} \minus{} a_{i}| \mid 1 \leq i \leq n \}\geq \frac {d}{2}. \quad \quad (*) \] (b) Show that there are real numbers $ x_{1}\leq x_{2}\leq \cdots \leq x_{n}$ such that the equality holds in (*). [i]Author: Michael Albert, New Zealand[/i]

Determine all pairs of positive integers $(a,b)$, such that the roots of the equations \[x^2-ax+a+b-3=0,\] \[x^2-bx+a+b-3=0,\] are also positive integers.

[hide=R stands for Ramanujan , P stands for Pascal]they had two problem sets under those two names[/hide] [b]R1.[/b] What is $11^2 - 9^2$? [b]R2.[/b] Write $\frac{9}{15}$ as a decimal. [b]R3.[/b] A $90^o$ sector of a circle is shaded, as shown below. What percent of the circle is shaded? [b]R4.[/b] A fair coin is flipped twice. What is the probability that the results of the two flips are different? [b]R5.[/b] Wayne Dodson has $55$ pounds of tungsten. If each ounce of tungsten is worth $75$ cents, and there are $16$ ounces in a pound, how much money, in dollars, is Wayne Dodson’s tungsten worth? [b]R6.[/b] Tenley Towne has a collection of $28$ sticks. With these $28$ sticks he can build a tower that has $1$ stick in the top row, $2$ in the next row, and so on. Let $n$ be the largest number of rows that Tenley Towne’s tower can have. What is n? [b]R7.[/b] What is the sum of the four smallest primes? [b]R8 / P1.[/b] Let $ABC$ be an isosceles triangle such that $\angle B = 42^o$. What is the sum of all possible degree measures of angle $A$? [b]R9.[/b] Consider a line passing through $(0, 0)$ and $(4, 8)$. This line passes through the point $(2, a)$. What is the value of $a$? [b]R10 / P2.[/b] Brian and Stan are playing a game. In this game, Brian rolls a fair six-sided die, while Stan rolls a fair four-sided die. Neither person shows the other what number they rolled. Brian tells Stan, “The number I rolled is guaranteed to be higher than the number you rolled.” Stan now has to guess Brian’s number. If Stan plays optimally, what is the probability that Stan correctly guesses the number that Brian rolled? [b]R11.[/b] Guang chooses $4$ distinct integers between $0$ and $9$, inclusive. How many ways can he choose the integers such that every pair of chosen integers sums up to an even number? [b]R12 / P4.[/b] David is trying to write a problem for MBMT. He assigns degree measures to every interior angle in a convex $n$-gon, and it so happens that every angle he assigned is less than $144$ degrees. He tells Pratik the value of $n$ and the degree measures in the $n$-gon, and to David’s dismay, Pratik claims that such an $n$-gon does not exist. What is the smallest value of $n \ge 3$ such that Pratik’s claim is necessarily true? [b]R13 / P3.[/b] Consider a triangle $ABC$ with side lengths of $5$, $5$, and $2\sqrt5$. There exists a triangle with side lengths of $5, 5$, and $x$ ($x \ne 2\sqrt5$) which has the same area as $ABC$. What is the value of $x$? [b]R14 / P5.[/b] A mother has $11$ identical apples and $9$ identical bananas to distribute among her $3$ kids. In how many ways can the fruits be allocated so that each child gets at least one apple and one banana? [b]R15 / P7.[/b] Find the sum of the five smallest positive integers that cannot be represented as the sum of two not necessarily distinct primes. [b]P6.[/b] Srinivasa Ramanujan has the polynomial $P(x) = x^5 - 3x^4 - 5x^3 + 15x^2 + 4x - 12$. His friend Hardy tells him that $3$ is one of the roots of $P(x)$. What is the sum of the other roots of $P(x)$? [b]P8.[/b] $ABC$ is an equilateral triangle with side length $10$. Let $P$ be a point which lies on ray $\overrightarrow{BC}$ such that $PB = 20$. Compute the ratio $\frac{PA}{PC}$. [b]P9.[/b] Let $ABC$ be a triangle such that $AB = 10$, $BC = 14$, and $AC = 6$. The median $CD$ and angle bisector $CE$ are both drawn to side $AB$. What is the ratio of the area of triangle $CDE$ to the area of triangle $ABC$? [b]P10.[/b] Find all integer values of $x$ between $0$ and $2017$ inclusive, which satisfy $$2016x^{2017} + 990x^{2016} + 2x + 17 \equiv 0 \,\,\, (mod \,\,\, 2017).$$ [b]P11.[/b] Let $x^2 + ax + b$ be a quadratic polynomial with positive integer roots such that $a^2 - 2b = 97$. Compute $a + b$. [b]P12.[/b] Let $S$ be the set $\{2, 3, ... , 14\}$. We assign a distinct number from $S$ to each side of a six-sided die. We say a numbering is predictable if prime numbers are always opposite prime numbers and composite numbers are always opposite composite numbers. How many predictable numberings are there? (Rotations of a die are not distinct) [b]P13.[/b] In triangle $ABC$, $AB = 10$, $BC = 21$, and $AC = 17$. $D$ is the foot of the altitude from $A$ to $BC$, $E$ is the foot of the altitude from $D$ to $AB$, and $F$ is the foot of the altitude from $D$ to $AC$. Find the area of the smallest circle that contains the quadrilateral $AEDF$. [b]P14.[/b] What is the greatest distance between any two points on the graph of $3x^2 + 4y^2 + z^2 - 12x + 8y + 6z = -11$? [b]P15.[/b] For a positive integer $n$, $\tau (n)$ is defined to be the number of positive divisors of $n$. Given this information, find the largest positive integer $n$ less than $1000$ such that $$\sum_{d|n} \tau (d) = 108.$$ In other words, we take the sum of $\tau (d)$ for every positive divisor $d$ of $n$, which has to be $108$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

The values of $ a$ in the equation: $ \log_{10}(a^2 \minus{} 15a) \equal{} 2$ are: $ \textbf{(A)}\ \frac {15\pm\sqrt {233}}{2} \qquad\textbf{(B)}\ 20, \minus{} 5 \qquad\textbf{(C)}\ \frac {15 \pm \sqrt {305}}{2}$ $ \textbf{(D)}\ \pm20 \qquad\textbf{(E)}\ \text{none of these}$

Let $n$ be an integer greater than $1$. Define \[x_1 = n, y_1 = 1, x_{i+1} =\left[ \frac{x_i+y_i}{2}\right] , y_{i+1} = \left[ \frac{n}{x_{i+1}}\right], \qquad \text{for }i = 1, 2, \ldots\ ,\] where $[z]$ denotes the largest integer less than or equal to $z$. Prove that \[ \min \{x_1, x_2, \ldots, x_n \} =[ \sqrt n ]\]

What is the remainder when $x^{51}+51$ is divided by $x+1$? $ \textbf{(A)}\ 0 \qquad\textbf{(B)}\ 1 \qquad\textbf{(C)}\ 49 \qquad\textbf{(D)}\ 50 \qquad\textbf{(E)}\ 51 $

Let $\left ( a_{n} \right )_{n \geq 1}$ be an increasing sequence of positive integers such that $a_1=1$, and for all positive integers $n$, $a_{n+1}\leq 2n$. Prove that for every positive $n$; there exists positive integers $p$ and $q$ such that $n=a_{p}-a_{q}$.