Found problems: 85335
Let $ a$, $ b$, $ c$ be positive real numbers such that $ abc=1$.
Prove that
$ \frac{1}{1+b+c}+\frac{1}{1+c+a}+\frac{1}{1+a+b}\leq\frac{1}{2+a}+\frac{1}{2+b}+\frac{1}{2+c}$.
Let $n$ be a positive integer. If the equation $2x+2y+z=n$ has $28$ solutions in positive integers $x, y$ and $z$, then $n$ must be either
$ \textbf{(A)}\ 14\ \text{or}\ 15 \qquad\textbf{(B)}\ 15\ \text{or}\ 16 \qquad\textbf{(C)}\ 16\ \text{or}\ 17 \qquad\textbf{(D)}\ 17\ \text{or}\ 18 \qquad\textbf{(E)}\ 18\ \text{or}\ 19 $
We have a grid of $k^2-k+1$ rows and $k^2-k+1$ columns, where $k=p+1$ and $p$ is prime. For each prime $p$, give a method to put the numbers 0 and 1, one number for each square in the grid, such that on each row there are exactly $k$ 0's, on each column there are exactly $k$ 0's, and there is no rectangle with sides parallel to the sides of the grid with 0s on each four vertices.
At the beginning, three boxes contain $m$, $n$, and $k$ pieces, respectively. Ayşe and Burak are playing a turn-based game with these pieces. At each turn, the player takes at least one piece from one of the boxes. The player who takes the last piece will win the game. Ayşe will be the first player. They are playing the game once for each $(m,n,k)=(1,2012,2014)$, $(2011,2011,2012)$, $(2011,2012,2013)$, $(2011,2012,2014)$, $(2011,2013,2013)$. In how many of them can Ayşe guarantee to win the game?
$ \textbf{(A)}\ 1 \qquad \textbf{(B)}\ 2 \qquad \textbf{(C)}\ 3 \qquad \textbf{(D)}\ 4 \qquad \textbf{(E)}\ 5$
Let $\triangle ABC$ be a right-angled triangle with $\angle BAC = 90^{\circ}$ and let $E$ be the foot of the perpendicular from $A$ to $BC$. Let $Z \ne A$ be a point on the line $AB$ with $AB = BZ$. Let $(c)$ be the circumcircle of the triangle $\triangle AEZ$. Let $D$ be the second point of intersection of $(c)$ with $ZC$ and let $F$ be the antidiametric point of $D$ with respect to $(c)$. Let $P$ be the point of intersection of the lines $FE$ and $CZ$. If the tangent to $(c)$ at $Z$ meets $PA$ at $T$, prove that the points $T$, $E$, $B$, $Z$ are concyclic.
Proposed by [i]Theoklitos Parayiou, Cyprus[/i]
Let $ABC$ be an obtuse triangle with circumcenter $O$ such that $\angle ABC = 15^o$ and $\angle BAC > 90^o$. Suppose that $AO$ meets $BC$ at $D$, and that $OD^2 + OC \cdot DC = OC^2$. Find $\angle C$.
Let's call a number of the form $x^3+y^2$ with natural $x, y$ [i]successful[/i]. Are there infinitely many natural $m$ such that among the numbers from $m + 1$ to $m + 2016^2$ exactly 2017 are successful?
What is the value of
\[ (3x \minus{} 2)(4x \plus{} 1) \minus{} (3x \minus{} 2)4x \plus{} 1\]when $ x \equal{} 4$?
$ \textbf{(A)}\ 0 \qquad \textbf{(B)}\ 1 \qquad \textbf{(C)}\ 10 \qquad \textbf{(D)}\ 11 \qquad \textbf{(E)}\ 12$
Call a set of integers [i]unpredictable[/i] if no four elements in the set form an arithmetic sequence. How many unordered [i]unpredictable[/i] sets of five distinct positive integers $\{a, b, c, d, e\}$ exist such that all elements are strictly less than $12$?
[i]Proposed by Anthony Yang[/i]
In the plane let a finite family of circles be given satisfying the condition: every two circles, either are outside each other, either touch each other from outside and each circle touch at most 6 other circles. Suppose that every circle which does not touch 6 other circles be assigned a real number. Show that there exist at most one assignment to each remaining circle a real number equal to arithmetic mean of 6 numbers assigned to 6 circles which touch it.
Let $ABCD$ be a convex quadrilateral. such that $\angle CAB = \angle CDA$ and $\angle BCA = \angle ACD$. If $M$ be the midpoint of $AB$, prove that $\angle BCM = \angle DBA$.
For each positive integer $n$, let $S_n = \sum_{k=1}^nk^3$, and let $d(n)$ be the number of positive divisors of $n$. For how many positive integers $m$, where $m\leq 25$, is there a solution $n$ to the equation $d(S_n) = m$?
Let
(i) $a_1,a_2,a_3$ is an arithmetic progression and $a_1+a_2+a_3=18$
(ii) $b_1,b_2,b_3$ is a geometric progression and $b_1b_2b_3=64$
If $a_1+b_1,a_2+b_2,a_3+b_3$ are all positive integers and it is a ageometric progression, then find the maximum value of $a_3$.
Prove that all positive real $x, y, z$ satisfy the inequality $x^y + y^z + z^x > 1$.
(D. Bazylev)
Find the smallest positive integer $n$ with the following property: for every sequence of positive integers $a_1,a_2,\ldots , a_n$ with $a_1+a_2+\ldots +a_n=2013$, there exist some (possibly one) consecutive term(s) in the sequence that add up to $70$.
Prove by the method of induction that:
[b]a)[/b] $ a!b! $ divides $ (a+b)! , $ for any natural numbers $ a,b. $
[b]b)[/b] $ p $ divides $ (-1)^{k+1} +\binom{p-1}{k} , $ for any odd primes $ p $ and $ k\in\{ 0,1,\ldots ,p-1\} . $
Non-zero numbers $a, b, c$ are such that $ax^2+bx+c > cx$ for any $x$.
Prove that $cx^2-bx + a > cx-b$ for any $x$.
The function $f : \mathbb{R}\to\mathbb{R}$ satisfies $f(x^2)f^{\prime\prime}(x)=f^\prime (x)f^\prime (x^2)$ for all real $x$. Given that $f(1)=1$ and $f^{\prime\prime\prime}(1)=8$, determine $f^\prime (1)+f^{\prime\prime}(1)$.
Let $ABC$ an acute triangle and $D$ and $E$ the feet of heights by $A$ and $B$, respectively, and let $M$ be the midpoint of $AC$. The circle that passes through $D$ and $B$ and is tangent to $BE$ in $B$ intersects the line $BM$ in $F, F\neq B$. Show that $FM$ is the angle bisector of $\angle AFD$.
Let $ABC$ be an obtuse triangle, with $AB$ being the largest side.
Draw the angle bisector of $\measuredangle BAC$. Then, draw the perpendiculars to this angle bisector from vertices $B$ and $C$, and call their feet $P$ and $Q$, respectively.
$D$ is the point in the line $BC$ such that $AD \perp AP$.
Prove that the lines $AD$, $BQ$ and $PC$ are concurrent.
Let $(a_n )_{n\ge 1}$ be a sequence of integers that satisfies
\[a_n = a_{n-1} -\text{min}(a_{n-2} , a_{n-3} )\]
for all $n \ge 4$. Prove that for every positive integer $k$, there is an $n$ such that $a_n$ is divisible by $3^k$ .
A set $G$ with elements $u,v,w...$ is a Group if the following conditions are fulfilled:
$(\text{i})$ There is a binary operation $\circ$ defined on $G$ such that $\forall \{u,v\}\in G$ there is a $w\in G$ with $u\circ v = w$.
$(\text{ii})$ This operation is associative; i.e. $(u\circ v)\circ w = u\circ (v\circ w)$ $\forall\{u,v,w\}\in G$.
$(\text{iii})$ $\forall \{u,v\}\in G$, there exists an element $x\in G$ such that $u\circ x = v$, and an element $y\in G$ such that $y\circ u = v$.
Let $K$ be a set of all real numbers greater than $1$. On $K$ is defined an operation by $ a\circ b = ab-\sqrt{(a^2-1)(b^2-1)}$. Prove that $K$ is a Group.
Consider a set of \( n \geq 3 \) points in the plane where no three are collinear. Prove that the points can be labeled as \( P_1, P_2, \dots, P_n \) so that the angles \( \angle P_i P_{i+1} P_{i+2} \) are less than \( 90^\circ \) for all \( i \).
[i]20 problems for 20 minutes.[/i]
[b]p1.[/b] Euclid eats $\frac17$ of a pie in $7$ seconds. Euler eats $\frac15$ of an identical pie in $10$ seconds. Who eats faster?
[b]p2.[/b] Given that $\pi = 3.1415926...$ , compute the circumference of a circle of radius 1. Express your answer as a decimal rounded to the nearest hundred thousandth (i.e. $1.234562$ and $1.234567$ would be rounded to $1.23456$ and $1.23457$, respectively).
[b]p3.[/b] Alice bikes to Wonderland, which is $6$ miles from her house. Her bicycle has two wheels, and she also keeps a spare tire with her. If each of the three tires must be used for the same number of miles, for how many miles will each tire be used?
[b]p4.[/b] Simplify $\frac{2010 \cdot 2010}{2011}$ to a mixed number. (For example, $2\frac12$ is a mixed number while $\frac52$ and $2.5$ are not.)
[b]p5.[/b] There are currently $175$ problems submitted for $EMC^2$. Chris has submitted $51$ of them. If nobody else submits any more problems, how many more problems must Chris submit so that he has submitted $\frac13$ of the problems?
[b]p6.[/b] As shown in the diagram below, points $D$ and $L$ are located on segment $AK$, with $D$ between $A$ and $L$, such that $\frac{AD}{DK}=\frac{1}{3}$ and $\frac{DL}{LK}=\frac{5}{9}$. What is $\frac{DL}{AK}$?
[img]https://cdn.artofproblemsolving.com/attachments/9/a/3f92bd33ffbe52a735158f7ebca79c4c360d30.png[/img]
[b]p7.[/b] Find the number of possible ways to order the letters $G, G, e, e, e$ such that two neighboring letters are never $G$ and $e$ in that order.
[b]p8.[/b] Find the number of odd composite integers between $0$ and $50$.
[b]p9.[/b] Bob tries to remember his $2$-digit extension number. He knows that the number is divisible by $5$ and that the first digit is odd. How many possibilities are there for this number?
[b]p10.[/b] Al walks $1$ mile due north, then $2$ miles due east, then $3$ miles due south, and then $4$ miles due west. How far, in miles, is he from his starting position? (Assume that the Earth is flat.)
[b]p11.[/b] When n is a positive integer, $n!$ denotes the product of the first $n$ positive integers; that is, $n! = 1 \cdot 2 \cdot 3 \cdot ... \cdot n$. Given that $7! = 5040$, compute $8! + 9! + 10!$.
[b]p12.[/b] Sam's phone company charges him a per-minute charge as well as a connection fee (which is the same for every call) every time he makes a phone call. If Sam was charged $\$4.88$ for an $11$-minute call and $\$6.00$ for a $19$-minute call, how much would he be charged for a $15$-minute call?
[b]p13.[/b] For a positive integer $n$, let $s_n$ be the sum of the n smallest primes. Find the least $n$ such that $s_n$ is a perfect square (the square of an integer).
[b]p14.[/b] Find the remainder when $2011^{2011}$ is divided by $7$.
[b]p15.[/b] Let $a, b, c$, and $d$ be $4$ positive integers, each of which is less than $10$, and let $e$ be their least common multiple. Find the maximum possible value of $e$.
[b]p16.[/b] Evaluate $100 - 1 + 99 - 2 + 98 - 3 + ... + 52 - 49 + 51 - 50$.
[b]p17.[/b] There are $30$ basketball teams in the Phillips Exeter Dorm Basketball League. In how ways can $4$ teams be chosen for a tournament if the two teams Soule Internationals and Abbot United cannot be chosen at the same time?
[b]p18.[/b] The numbers $1, 2, 3, 4, 5, 6$ are randomly written around a circle. What is the probability that there are four neighboring numbers such that the sum of the middle two numbers is less than the sum of the other two?
[b]p19.[/b] What is the largest positive $2$-digit factor of $3^{2^{2011}} - 2^{2^{2011}}$?
[b]p20.[/b] Rhombus $ABCD$ has vertices $A = (-12,-4)$, $B = (6, b)$, $C = (c,-4)$ and $D = (d,-28)$, where $b$, $c$, and $d$ are integers. Find a constant $m$ such that the line y = $mx$ divides the rhombus into two regions of equal area.
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
a) Let $x_1,x_2,x_3,y_1,y_2,y_3\in \mathbb{R}$ and $a_{ij}=\sin(x_i-y_j),\ i,j=\overline{1,3}$ and $A=(a_{ij})\in \mathcal{M}_3$ Prove that $\det A=0$.
b) Let $z_1,z_2,\ldots,z_{2n}\in \mathbb{C}^*,\ n\ge 3$ such that $|z_1|=|z_2|=\ldots=|z_{n+3}|$ and $\arg z_1\ge \arg z_2\ge \ldots\ge \arg(z_{n+3})$. If $b_{ij}=|z_i-z_{j+n}|,\ i,j=\overline{1,n}$ and $B=(b_{ij})\in \mathcal{M}_n$, prove that $\det B=0$.