Found problems: 85335
One evening a theater sold 300 tickets for a concert. Each ticket sold for \$40, and all tickets were purchased using \$5, \$10, and \$20 bills. At the end of the evening the theater had received twice as many \$10 bills as \$20 bills, and 20 more \$5 bills than \$10 bills. How many bills did the theater receive altogether?
Find the number of ordered pairs $(P(x),Q(x))$ of polynomials with integer coefficients such that
\[
P(x)^2+Q(x)^2=\left(x^{4096}-1\right)^2.
\]
[i]Proposed by Michael Kural[/i]
The roots of $ x(x^2\plus{}8x\plus{}16)(4\minus{}x)\equal{}0$ are:
$ \textbf{(A)}\ 0 \qquad\textbf{(B)}\ 0,4 \qquad\textbf{(C)}\ 0,4,\minus{}4 \qquad\textbf{(D)}\ 0,4,\minus{}4,\minus{}4 \qquad\textbf{(E)}\ \text{none of these}$
Let $a_1, a_2, a_3, ...$ be a sequence of positive real numbers. Prove that for any positive integer $n$ the inequality holds $\sum_{i=1}^n b_i^2 \le 4 \sum_{i=1}^n a_i^2$ where $b_i$ is the arithmetic mean of the numbers $a_1, a_2, ..., a_n$
A ticket to a school play costs $ x$ dollars, where $ x$ is a whole number. A group of 9th graders buys tickets costing a total of $ \$48$, and a group of 10th graders buys tickets costing a total of $ \$64$. How many values of $ x$ are possible?
$ \textbf{(A)}\ 1\qquad\textbf{(B)}\ 2\qquad\textbf{(C)}\ 3\qquad\textbf{(D)}\ 4\qquad\textbf{(E)}\ 5$
If $a$ and $b$ satisfy the equations $a +\frac1b=4$ and $\frac1a+b=\frac{16}{15}$, determine the product of all possible values of $ab$.
Find the sum of the absolute values of the roots of $x^4 - 4x^3 - 4x^2 + 16x - 8 = 0$.
$\forall k\in N \,\,\, \exists n(k) \in N, a(k):0<a(k)<1 [(1+\sqrt2)^{2k+1}=n(k)+a(k)]$
Prove: $(n(k) + a(k))a(k) = 1$, for all $k \in N$, and calculate $\lim_{k \to \infty }a(k)$
Quadrilateral $ABCD$ is neither a kite nor a rectangle. It is known that its sidelengths are integers, $AB = 6$, $BC = 7$, and $\angle B = \angle D = 90^o$. Find the area of$ ABCD$.
In a convex quadrilateral $ABCD$, let $M$, $N$, $P$, and $Q$ be the midpoints of $AB$, $BC$, $CD$, and $DA$ respectively. If $MP$ and $NQ$ divide $ABCD$ in four quadrilaterals with the same area, prove that $ABCD$ is a parallelogram.
The altitueds $A {{A} _ {1}} $, $B {{B} _ {1}}$ and $C {C} _ 1$ are drawn in the acute triangle $ABC$. . The perpendicular $AK$ is drawn from the vertex $A$ on the line ${{A} _ {1}} {{B} _ {1}}$, and the perpendicular $BL$ is drawn from the vertex $B$ on the line ${{C} _ {1}} {{B} _ {1}}$. Prove that ${{A} _ {1}} K = {{B} _ {1}} L$.
(Maria Rozhkova)
Is it true that if a subgroup $G \leq \text{Sym}(\mathbb{N})$ is $n$-transitive for every positive integer $n$, then every group automorphism of $G$ extends to a group automorphism of $\text{Sym}(\mathbb{N})$?
Let $a,b,c$ be positive real numbers such that $$\frac{a}{1+a}+\frac{b}{1+b}+\frac{c}{1+c}=1.$$ Prove that $abc \le \frac{1}{8}$.
Solve for $x \in [0,2\pi[$: \[\sin x < \cos x < \tan x < \cot x\]
In trapezoid $ ABCD$, $ \overline{AB}$ and $ \overline{CD}$ are perpendicular to $ \overline{AD}$, with $ AB\plus{}CD\equal{}BC$, $ AB<CD$, and $ AD\equal{}7$. What is $ AB\cdot CD$?
$ \textbf{(A)}\ 12 \qquad
\textbf{(B)}\ 12.25 \qquad
\textbf{(C)}\ 12.5 \qquad
\textbf{(D)}\ 12.75 \qquad
\textbf{(E)}\ 13$
Find minimum of $x+y+z$ where $x$, $y$ and $z$ are real numbers such that $x \geq 4$, $y \geq 5$, $z \geq 6$ and $x^2+y^2+z^2 \geq 90$
Let $H_1, H_2,\ldots, H_m$ be non-empty subsets of the positive integers, and let $S$ denote their union. Prove that
\[\sum_{i=1}^m \sum_{(a,b)\in H_i^2}\gcd(a,b)\ge\frac1m \sum_{(a,b)\in S^2}\gcd(a,b).\]
[i]Proposed by Dávid Matolcsi, Berkeley[/i]
The triangle $ABC$ is given. Consider the point $C'{}$ on the side $AB$ such that the segment $CC'$ divides the triangle into two triangles with equal radii of inscribed circles. Denote by $t_c$ the length of the segment $CC'$. Similarly, we define $t_a$ and $t_b$. Express the area of triangle $ABC$ in terms of $t_a,t_b$ and $t_c$.
[i]Proposed by K. Mosevich[/i]
Two given circles intersect in two points $ P$ and $ Q$. Show how to construct a segment $ AB$ passing through $ P$ and terminating on the circles such that $ AP \cdot PB$ is a maximum.
In a plane a set of $n\geq 3$ points is given. Each pair of points is connected by a segment. Let $d$ be the length of the longest of these segments. We define a diameter of the set to be any connecting segment of length $d$. Prove that the number of diameters of the given set is at most $n$.
Four rectangular paper strips of length $10$ and width $1$ are put flat on a table and overlap perpendicularly as shown. How much area of the table is covered?
[asy]
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draw((1,4)--(3,4)--(3,2)--(1,2)--(4,2)--(3,2)--(3,6)--(4,6)--(4,5)--(5,5)--(5,4)--(4,4));[/asy]
$ \textbf{(A)}\ 36 \qquad \textbf{(B)}\ 40 \qquad \textbf{(C)}\ 44 \qquad \textbf{(D)}\ 98 \qquad \textbf{(E)}\ 100 $
What is the largest possible area of a triangle with largest side length $39$ and inradius $10$?
Let $ABC$ be a triangle and $A', B', C'$ the points in which its incircle touches the sides $BC, CA, AB$, respectively. We denote by $I$ the incenter and by $P$ its projection onto $AA' $. Let $M$ be the midpoint of the line segment $[A'B']$ and $N$ be the intersection point of the lines $MP$ and $AC$. Prove that $A'N $is parallel to $B'C'$
For how many positive integers $n$, $1\leq n\leq 2008$, can the set \[\{1,2,3,\ldots,4n\}\] be divided into $n$ disjoint $4$-element subsets such that every one of the $n$ subsets contains the element which is the arithmetic mean of all the elements in that subset?
Find all positive integers $n$ for which $$3x^n + n(x + 2) - 3 \geq nx^2$$ holds for all real numbers $x.$