Found problems: 85335
Estimate the number of ordered pairs $(a,b)$ of relatively prime positive integers such that $a+b<1412.$ Your score is determined by the function $max\{0, 20 - \lfloor \frac{|A - E|}{500}\rfloor\}$where $A$ is the actual answer, and $E$ is your estimate.
[i]Lightning 5.1[/i]
Consider the paper triangle whose vertices are $(0,0), (34,0),$ and $(16,24).$ The vertices of its midpoint triangle are the midpoints of its sides. A triangular pyramid is formed by folding the triangle along the sides of its midpoint triangle. What is the volume of this pyramid?
Suppose $2$ cars are going into a turn the shape of a half-circle. Car $ 1$ is traveling at $50$ meters per second and is hugging the inside of the turn, which has radius $200$ meters. Car $2$ is trying to pass Car $ 1$ going along the turn, but in order to do this, he has to move to the outside of the turn, which has radius $210$. Suppose that both cars come into the turn side by side, and that they also end the turn being side by side. What was the average speed of Car $2$, in meters per second, throughout the turn?
For any non-empty numerical numbers $A$ and $B$, denote
$$A + B = \{a + b | a \in A, b \in B\} $$
a) Determine the largest natural number not $p$ with the property:
[i] there exists[/i] $A,B \subset N$ [i]such that[/i] $card \, A = card\, B = p$ [i]and [/i] $A+B = \{0, 1, 2,..., 2012\}$
b) Determine the smallest natural number $n$ with the property:
[i] there exists[/i] $A,B \subset N$ [i]such that[/i] $card \, A = card\, B $ [i]and [/i] $A+B =\{0, 1, 2,..., 2012\}$
Let $ a>0$ and $ f: [0,\infty) \to [0,a]$ be a continuous function on $ (0,\infty)$ and having Darboux property on $ [0,\infty)$. Prove that if $ f(0)\equal{}0$ and for all nonnegative $ x$ we have
\[ xf(x) \geq \int^x_0 f(t) dt ,\] then $ f$ admits primitives on $ [0,\infty)$.
Simplify $\frac{1}{\sqrt[3]{81} + \sqrt[3]{72} + \sqrt[3]{64}}$
Let $T=\text{TNFTPP}$, and let $R=T-914$. Let $x$ be the smallest real solution of \[3x^2+Rx+R=90x\sqrt{x+1}.\] Find the value of $\lfloor x\rfloor$.
Alex has one-pound red bricks and two-pound blue bricks, and has 360 total pounds of brick. He observes that it is impossible to rearrange the bricks into piles that all weigh three pounds, but he can put them in piles that each weigh five pounds. Finally, when he tries to put them into piles that all have three bricks, he has one left over. If Alex has $r$ red bricks, find the number of values $r$ could take on.
Given a polynomial $f(x)$ with rational coefficients, of degree $d \ge 2$, we define the sequence of sets $f^0(\mathbb{Q}), f^1(\mathbb{Q}), \ldots$ as $f^0(\mathbb{Q})=\mathbb{Q}$, $f^{n+1}(\mathbb{Q})=f(f^{n}(\mathbb{Q}))$ for $n\ge 0$. (Given a set $S$, we write $f(S)$ for the set $\{f(x)\mid x\in S\})$.
Let $f^{\omega}(\mathbb{Q})=\bigcap_{n=0}^{\infty} f^n(\mathbb{Q})$ be the set of numbers that are in all of the sets $f^n(\mathbb{Q})$, $n\geq 0$. Prove that $f^{\omega}(\mathbb{Q})$ is a finite set.
[i]Dan Schwarz, Romania[/i]
If $3(4x + 5\pi) = P$, then $6(8x + 10\pi) = $
$ \textbf{(A)}\ 2P\qquad\textbf{(B)}\ 4P\qquad\textbf{(C)}\ 6P\qquad\textbf{(D)}\ 8P\qquad\textbf{(E)}\ 18P $
Suppose that $|x+y|+|x-y|=2$. What is the maximum possible value of $x^2-6x+y^2$?
$ \textbf{(A)}\ 5 \qquad
\textbf{(B)}\ 6 \qquad
\textbf{(C)}\ 7 \qquad
\textbf{(D)}\ 8 \qquad
\textbf{(E)}\ 9
$
An equilateral triangle with side $n$ is built with $n^{2}$ [i]plates[/i] - equilateral triangles with side $1$. Each plate has one side black, and the other side white. We name [i]the move[/i] the following operation: we choose a plate $P$, which has common sides with at least two plates, whose visible side is the same color as the visible side of $P$. Then, we turn over plate $P$.
For any $n\geq 2$ decide whether there exists an innitial configuration of plates permitting for an infinite sequence of moves.
Let $G$ be the complete graph on $2015$ vertices. Each edge of $G$ is dyed red, blue or white. For a subset $V$ of vertices of $G$, and a pair of vertices $(u,v)$, define \[ L(u,v) = \{ u,v \} \cup \{ w | w \in V \ni \triangle{uvw} \text{ has exactly 2 red sides} \}\]Prove that, for any choice of $V$, there exist at least $120$ distinct values of $L(u,v)$.
Find all real numbers $r$, such that the inequality
\[r(ab+bc+ca)+(3-r)\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right)\geq 9\]
holds for any real $a,b,c>0$.
Let $f(n)=n+\lfloor \sqrt{n}\rfloor$. Prove that, for every positive integer $m$, the sequence \[m, f(m), f(f(m)), f(f(f(m))), \cdots\] contains at least one square of an integer.
Given two odd natural numbers $ a,b$ prove that for each $ n\in\mathbb{N}$ there exists $ m\in\mathbb{N}$ such that either $ a^mb^2-1$ or $ b^ma^2-1$ is multiple of $ 2^n.$
Find all natural numbers $a$ and $b$ such that \[a|b^2, \quad b|a^2 \mbox{ and } a+1|b^2+1.\]
Let $a$ and $b$ be real numbers such that $a + b \ne 0$. Solve the equation
$$\frac{1}{(x + a)^2 - b^2} +\frac{1}{(x +b)^2 - a^2}=\frac{1}{x^2 -(a + b)^2}+\frac{1}{x^2-(a -b)^2}$$
The positive integers $a_1,a_2, \dots, a_n$ are aligned clockwise in a circular line with $n \geq 5$. Let $a_0=a_n$ and $a_{n+1}=a_1$. For each $i \in \{1,2,\dots,n \}$ the quotient \[ q_i=\frac{a_{i-1}+a_{i+1}}{a_i} \] is an integer. Prove \[ 2n \leq q_1+q_2+\dots+q_n < 3n. \]
Let $ABC$ be a triangle with $\angle B > \angle C$. Let $P$ and $Q$ be two different points on line $AC$ such that $\angle PBA = \angle QBA = \angle ACB $ and $A$ is located between $P$ and $C$. Suppose that there exists an interior point $D$ of segment $BQ$ for which $PD=PB$. Let the ray $AD$ intersect the circle $ABC$ at $R \neq A$. Prove that $QB = QR$.
Let $a_n, b_n$ be sequences of positive reals such that,$$a_{n+1}= a_n + \frac{1}{2b_n}$$ $$b_{n+1}= b_n + \frac{1}{2a_n}$$ for all $n\in\mathbb N$.
Prove that, $\text{max}\left(a_{2018}, b_{2018}\right) >44$.
Find the smallest constant $C > 0$ for which the following statement holds: among any five positive real numbers $a_1,a_2,a_3,a_4,a_5$ (not necessarily distinct), one can always choose distinct subscripts $i,j,k,l$ such that
\[ \left| \frac{a_i}{a_j} - \frac {a_k}{a_l} \right| \le C. \]
There are three stacks of tokens on the table: the first contains $a,$ the second contains $b,$ and the third contains $c$ where $a \ge b \ge c.$ Players $A$ and $B$ take turns playing a game of swapping tokens. $A$ starts first. On each turn, the player who gets his turn chooses two stacks, then takes at least one token from the stack with the lowest number of tokens and places them on the stack with the highest number of tokens. If the number of tokens in the two piles he/she chooses is equal, then he/she takes at least one token from any of them and puts it in the other. If only one pile remains after a player's move, that player is considered a winner. At what values of $a, b, c$ who has the winning strategy ($A$ or $B$)?
Let $\mathbb{Z}$ be the set of integers. Find all functions $f: \mathbb{Z}\rightarrow \mathbb{Z}$ such that, for any two integers $m, n$, $$f(m^2)+f(mf(n))=f(m+n)f(m).$$
[i]Proposed by Victor Domínguez and Pablo Valeriano[/i]
Let $N$ be the number of sequences of positive integers greater than $ 1$ where the product of all of the terms of the sequence is $12^{64}$. If $N$ can be expressed as $a(2^b)$ ), where $a$ is an odd positive integer, determine $b$.