Found problems: 85335
Assume that $x, y \ge 0$. Show that $x^2 + y^2 + 1 \le \sqrt{(x^3 + y + 1)(y^3 + x + 1)}$.
A [i]complex set[/i], along with its [i]complexity[/i], is defined recursively as the following:
[list]
[*]The set $\mathbb{C}$ of complex numbers is a complex set with complexity $1$.
[*]Given two complex sets $C_1, C_2$ with complexity $c_1, c_2$ respectively, the set of all functions $f:C_1\rightarrow C_2$ is a complex set denoted $[C_1, C_2]$ with complexity $c_1 + c_2$.
[/list]
A [i]complex expression[/i], along with its [i]evaluation[/i] and its [i]complexity[/i], is defined recursively as the following:
[list]
[*]A single complex set $C$ with complexity $c$ is a complex expression with complexity $c$ that evaluates to itself.
[*]Given two complex expressions $E_1, E_2$ with complexity $e_1, e_2$ that evaluate to $C_1$ and $C_2$ respectively, if $C_1 = [C_2, C]$ for some complex set $C$, then $(E_1, E_2)$ is a complex expression with complexity $e_1+e_2$ that evaluates to $C$.
[/list]
For a positive integer $n$, let $a_n$ be the number of complex expressions with complexity $n$ that evaluate to $\mathbb{C}$. Let $x$ be a positive real number. Suppose that \[a_1+a_2x+a_3x^2+\dots = \dfrac{7}{4}.\] Then $x=\frac{k\sqrt{m}}{n}$, where $k$,$m$, and $n$ are positive integers such that $m$ is not divisible by the square of any integer greater than $1$, and $k$ and $n$ are relatively prime. Compute $100k+10m+n$.
[i]Proposed by Luke Robitaille and Yannick Yao[/i]
Bill bought 13 notebooks, 26 pens, and 19 markers for 25 dollars. Paula bought 27 notebooks, 18 pens, and 31 markers for 31 dollars. How many dollars would it cost Greg to buy 24 notebooks, 120 pens, and 52 markers?
Find all positive integers $ n$ such that there exists a unique integer $ a$ such that $ 0\leq a < n!$ with the following property:
\[ n!\mid a^n \plus{} 1
\]
[i]Proposed by Carlos Caicedo, Colombia[/i]
In the diagram $ABCDEFG$ is a regular heptagon (a $7$ sided polygon). Shown is the star $AEBFCGD$. The degree measure of the obtuse angle formed by $AE$ and $CG$ is $\dfrac{m}{n}$ where m and n are relatively prime positive integers. Find $m + n$.
[asy]
size(150);
defaultpen(linewidth(1));
string lab[]={"A","B","C","D","E","F","G"};
real r = 360/7;
pair A=dir(90-r),B=dir(90),C=dir(90+r),D=dir(90+2*r),E=dir(90+3*r),F=dir(90+4*r),G=dir(90+5*r);
draw(A--E--B--F--C--G--D--cycle);
for(int k = -1;k <= 5;++k) {
label("$"+lab[k+1]+"$",dir(90+k*r),dir(90+k*r));
}
[/asy]
Let the sequence $\{a_n\}_{n\ge 1}$ be defined by
$$a_n=\tan(n\theta)$$
where $\tan\theta =2$. Show that for all $n$, $a_n$ is a rational number which can be written with an odd denominator.
Let $P,Q,R$ be the polynomials with real or complex coefficients such that at least one of them is not constant. If $P^n+Q^n+R^n = 0$, prove that $n < 3.$
For any quadrilateral with the side lengths $a,$ $b,$ $c,$ $d$ and the area $S,$ prove the inequality$S\leq \frac{a+c}{2}\cdot \frac{b+d}{2}.$
A group of $16$ people will be partitioned into $4$ indistinguishable $4$-person committees. Each committee will have one chairperson and one secretary. The number of different ways to make these assignments can be written as $3^r M,$ where $r$ and $M$ are positive integers and $M$ is not divisible by $3.$ What is $r?$
$\textbf{(A) }5 \qquad\textbf{(B) }6\qquad\textbf{(C) }7\qquad\textbf{(D) }8\qquad\textbf{(E) }9$
At the start of a game there are three boxes with $2008, 2009$ and $2010$ game pieces
Anja and Bernd play in turns according to the following rule:
[i]When it is your turn, select two boxes, empty them and then distribute the pieces from the third box to the three boxes, such that no box may remain empty.If you can no longer complete a turn, you have lost. [/i]
Who has a winning strategy when Anja starts?
Determine all pairs $P(x),Q(x)$ of complex polynomials with leading coefficient $1$ such that $P(x)$ divides $Q(x)^2+1$ and $Q(x)$ divides $P(x)^2+1$.
[i]Proposed by Rodrigo Angelo, Princeton University and Matheus Secco, PUC, Rio de Janeiro[/i]
If $d\in \mathbb{Q}$, is there always an $\omega \in \mathbb{C}$ such that $\omega ^n=1$ for some $n\in \mathbb{N}$ and $\mathbb{Q}(\sqrt{d})\subseteq \mathbb{Q}(\omega)$?
Players $A$ and $B$ play the following game. Each of them throws a dice, and if the outcomes are $x$ and $y$ respectively, a list of all two digit numbers $10a + b$ with $a,b\in \{1,..,6\}$ and $10a + b \le 10x + y$ is created. Then the players alternately reduce the list by replacing a pair of numbers in the list by their absolute difference, until only one number remains. If the remaining number is of the same parity as the outcome of $A$’s throw, then $A$ is proclaimed the winner. What is the probability that $A$ wins the game?
There is an interval $[a, b]$ that is the solution to the inequality \[|3x-80|\le|2x-105|\] Find $a + b$.
Let $a, b, c$ be the side lengths of a scalene triangle and let $O_a, O_b$ and $O_c$ be three concentric circles with radii $a, b$ and $c$ respectively.
(a) How many equilateral triangles with different areas can be constructed such that the lines containing the sides are tangent to the circles?
(b) Find the possible areas of such triangles.
Source: 1976 Euclid Part A Problem 5
-----
If $\log_8 m+\log_8 \frac{1}{6}=\frac{2}{3}$, then $m$ equals
$\textbf{(A) } \frac{1}{2} \qquad \textbf{(B) } \frac{2}{3} \qquad \textbf{(C) } \frac{23}{6} \qquad \textbf{(D) } 4 \qquad \textbf{(E) } 24$
$f(n)$ is the sum of all integers less than $n$ and relatively prime to $n$. Find all integers $n$ such that there exist integers $k$ and $\ell$ such that $f(n^k) = n^{\ell}$.
Let $ABC$ be a triangle with $AB = 13$, $BC = 14$, and $CA = 15$. Let $\Gamma$ be the circumcircle of $ABC$, let $O$ be its circumcenter, and let $M$ be the midpoint of minor arc $BC$. Circle $\omega_1$ is internally tangent to $\Gamma$ at $A$, and circle $\omega_2$, centered at $M$, is externally tangent to $\omega_1$ at a point $T$. Ray $AT$ meets segment $BC$ at point $S$, such that $BS - CS = \dfrac4{15}$. Find the radius of $\omega_2$
It is easy to prove that in a right triangle the sum of the radii of the incircle and three excircles is equal to the perimeter. Prove that the opposite statement is also true.
[i]Proposed by I. Weinstein[/i]
Consider the $ \mathbb {Z} / (10) $ additive group automorphism group of integers module $10$, that is,
$ A = \left \{\phi: \mathbb {Z} / (10) \to \mathbb {Z} / (10) | \phi-automorphism \right \}$
On a stormy night ten guests came to dinner party and left their shoes outside the room in order to keep the carpet clean. After the dinner there was a blackout, and the gusts leaving one by one, put on at random, any pair of shoes big enough for their feet. (Each pair of shoes stays together). Any guest who could not find a pair big enough spent the night there. What is the largest number of guests who might have had to spend the night there?
Let $\omega$ be the circumscribed circle of triangle $ABC$, and let $\omega'$ 'be the circle tangent to the side $BC$ and the extensions of the sides $AB$ and $AC$. The common tangents to the circles $\omega$ and $\omega'$ intersect the line $BC$ at points $D$ and $E$. Prove that $\angle BAD = \angle CAE$.
Let $ABCD$ be a cyclic quadrilateral satysfing $AB=AD$ and $AB+BC=CD$. Determine $\measuredangle CDA$.
Given an isosceles triangle $ABC$ with $AB = AC$. Let $O$ be the circumcenter of $ABC$, $D$ the midpoint of $AB$ and $E$ the centroid of $ACD$. Prove that $CD \perp EO$.
Consider the real number axis (i. e. the $x$-axis of a Cartesian coordinate system). We mark the points $1$, $2$, ..., $2n$ on this axis. A flea starts at the point $1$. Now it jumps along the real number axis; it can jump only from a marked point to another marked point, and it doesn't visit any point twice. After the ($2n-1$)-th jump, it arrives at a point from where it cannot jump any more after this rule, since all other points are already visited. Hence, with its $2n$-th jump, the flea breaks this rule and gets back to the point $1$. Assume that the sum of the (non-directed) lengths of the first $2n-1$ jumps of the flea was $n\left(2n-1\right)$. Show that the length of the last ($2n$-th) jump is $n$.