Found problems: 85335
2 curves $ y \equal{} x^3 \minus{} x$ and $ y \equal{} x^2 \minus{} a$ pass through the point $ P$ and have a common tangent line at $ P$. Find the area of the region bounded by these curves.
Two circles $\omega_1$ and $\omega_2$ intersecting at points $X{}$ and $Y{}$ are inside the circle $\Omega$ and touch it at points $A{}$ and $B{}$, respectively; the segments $AB$ and $XY$ intersect. The line $AB$ intersects the circles $\omega_1$ and $\omega_2$ again at points $C{}$ and $D{}$, respectively. The circle inscribed in the curved triangle $CDX$ touches the side $CD$ at the point $Z{}$. Prove that $XZ$ is a bisector of $\angle AXB{}$.
A set $C$ of positive integers is called good if for every integer $k$ there exist distinct $a, b \in C$ such that the numbers $a+k$ and $b+k$ are not relatively prime. Prove that if the sum of the elements of a good set $C$ equals $2003$, then there exists $c \in C$ such that the set $C-\{c\}$ is good.
Three circles are pair-wise tangent to each other. Prove that the circle passing through the three tangent points is perpendicular to each of the initial three circles.
Denote $\langle x\rangle$ the distance of the real number x from the nearest integer. Let f be a linear, 1 periodic, continuous real function. Prove that there exist natural n and real numbers $a_1 , ..., a_n , b_1 , ..., b_n , c_1 , ..., c_n$ such that $$f(x) = \sum_{i = 1}^n c_i \langle a_ix + b_i \rangle$$ for every x iff there is a k such that $$\sum_{j = 1}^{2^k} f \left(x+{j\over2^k}\right)$$ is constant.
$ABCD$ is a regular tetrahedron of side length $4.$ Four congruent spheres are inside $ABCD$ such that each sphere is tangent to exactly three of the faces, the spheres have distinct centers, and the four spheres are concurrent at one point. Let $v$ be the volume of one of the spheres. If $v^2$ can be written as $\tfrac{a}{b}\pi^2,$ where $a$ and $b$ are relatively prime positive integers, find $a+b.$
Let $a$ be a positive integer which is not a perfect square, and consider the equation \[k = \frac{x^2-a}{x^2-y^2}.\] Let $A$ be the set of positive integers $k$ for which the equation admits a solution in $\mathbb Z^2$ with $x>\sqrt{a}$, and let $B$ be the set of positive integers for which the equation admits a solution in $\mathbb Z^2$ with $0\leq x<\sqrt{a}$. Show that $A=B$.
Suppose $a$ and $b$ are digits which are not both nine and not both zero. If the repeating decimal $.\overline{AB}$ is expressed as a fraction in lowest terms, how many possible denominators are there?
$\text{(A) }2\qquad\text{(B) }3\qquad\text{(C) }4\qquad\text{(D) }5\qquad\text{(E) }6$
Let $A, B, C$ be points on a unit circle such that $|AB|=|BC|$ and $m(\widehat{ABC})=72^\circ$. Let $D$ be a point such that $\triangle BCD$ is equilateral. If $AD$ meets the circle at $D$, what is $|DE|$?
$
\textbf{(A)}\ \dfrac 12
\qquad\textbf{(B)}\ \dfrac {\sqrt 3}2
\qquad\textbf{(C)}\ \dfrac {\sqrt 2}2
\qquad\textbf{(D)}\ \sqrt 3 -1
\qquad\textbf{(E)}\ \text{None of the above}
$
There are 1$2$ people labeled $1, ..., 12$ working together on $12$ missions, with people $1, ... , i $working on the $i$th mission. There is exactly one spy among them. If the spy is not working on a mission, it will be a huge success, but if the spy is working on the mission, it will fail with probability $1/2$. Given that the first $11$ missions succeed, and the $12$th mission fails, what is the probability that person $12$ is the spy?
Find all polynomials \(P(x)\) with real coefficients such that for all nonzero real numbers \(x\), \[P(x)+P\left(\frac1x\right) =\frac{P\left(x+\frac1x\right) +P\left(x-\frac1x\right)}2.\]
[i]Proposed by Holden Mui[/i]
Find all sequence of consecutive positive numbers which the sum of them is equal with $2019$.
Let two positive integers $x, y$ satisfy the condition $44 /( x^2 + y^2)$.
Determine the smallest value of $T = x^3 + y^3$.
Fridolin the frog jumps on the number line: He starts at $0$, then jumps in some order on each of the numbers $1,2,\dots,9$ exactly once and finally returns with his last jump to $0$. Can the total distance he travelled with these $10$ jumps be a) $20$, b) $25$?
Find the smallest natural number $n$ for which the number $n^2-n+11$ has exactly four prime factors (not necessarily distinct).
A cloakroom in a cinema works with some breaks. The total time cloakroom worked today is 8 hours. The schedule of the cloakroom is such that it is possible to show any film of duration at most 12 hours such that the cloakroom will be open at least one hour before and after the film (the films are shown without breaks).
Find the minimal possible amount of breaks in the schedule of cloakroom.
[i]A. Voidelevich[/i]
A merchant buys a number of oranges at $ 3$ for $ 10$ cents and an equal number at $ 5$ for $ 20$ cents. To "break even" he must sell all at:
$ \textbf{(A)}\ \text{8 for 30 cents} \qquad
\textbf{(B)}\ \text{3 for 11 cents} \qquad
\textbf{(C)}\ \text{5 for 18 cents} \\
\textbf{(D)}\ \text{11 for 40 cents} \qquad
\textbf{(E)}\ \text{13 for 50 cents}$
Let $ACBD$ be a asquare and $K,L,M,N$ be points of $AB,BC,CD,DA$ respectively. If $O$ is the center of the square , prove that the expression $$ \overrightarrow{OK}\cdot \overrightarrow{OL}+\overrightarrow{OL}\cdot\overrightarrow{OM}+\overrightarrow{OM}\cdot\overrightarrow{ON}+\overrightarrow{ON}\cdot\overrightarrow{OK}$$
is independent of positions of $K,L,M,N$, (i.e. is constant )
In a quadrilateral $ABCD$ we have $AB||CD$ and $AB=2CD$. A line $\ell$ is perpendicular to $CD$ and contains the point $C$. The circle with centre $D$ and radius $DA$ intersects the line $\ell$ at points $P$ and $Q$. Prove that $AP\perp BQ$.
The positive integers $x_1, x_2, ... , x_7$ satisfy $x_6 = 144$, $x_{n+3} = x_{n+2}(x_{n+1}+x_n)$ for $n = 1, 2, 3, 4$. Find $x_7$.
[hide=B stands for Bernoulli, G stands for Germain]they had two problem sets under those two names[/hide]
[u]Set 1[/u]
[b]B1 / G1[/b] Find $20^3 + 2^2 + 3^1$.
[b]B2[/b] A piece of string of length $10$ is cut $4$ times into strings of equal length. What is the length of each small piece of string?
[b]B3 / G2[/b] What is the smallest perfect square that is also a perfect cube?
[b]B4[/b] What is the probability a $5$-sided die with sides labeled from $1$ through $5$ rolls an odd number?
[b]B5 / G3[/b] Hanfei spent $14$ dollars on chicken nuggets at McDonalds. $4$ nuggets cost $3$ dollars, $6$ nuggets cost $4$ dollars, and $12$ nuggets cost $9$ dollars. How many chicken nuggets did Hanfei buy?
[u]Set 2[/u]
[b]B6[/b] What is the probability a randomly chosen positive integer less than or equal to $15$ is prime?
[b]B7[/b] Andrew flips a fair coin with sides labeled 0 and 1 and also rolls a fair die with sides labeled $1$ through $6$. What is the probability that the sum is greater than $5$?
[b]B8 / G4[/b] What is the radius of a circle with area $4$?
[b]B9[/b] What is the maximum number of equilateral triangles on a piece of paper that can share the same corner?
[b]B10 / G5[/b] Bob likes to make pizzas. Bab also likes to make pizzas. Bob can make a pizza in $20$ minutes. Bab can make a pizza in $30$ minutes. If Bob and Bab want to make $50$ pizzas in total, how many hours would that take them?
[u]Set 3[/u]
[b]B11 / G6[/b] Find the area of an equilateral rectangle with perimeter $20$.
[b]B12 / G7[/b] What is the minimum possible number of divisors that the sum of two prime numbers greater than $2$ can have?
[b]B13 / G8[/b] Kwu and Kz play rock-paper-scissors-dynamite, a variant of the classic rock-paperscissors in which dynamite beats rock and paper but loses to scissors. The standard rock-paper-scissors rules apply, where rock beats scissors, paper beats rock, and scissors beats paper. If they throw out the same option, they keep playing until one of them wins. If Kz randomly throws out one of the four options with equal probability, while Kwu only throws out dynamite, what is the probability Kwu wins?
[b]B14 / G9[/b] Aven has $4$ distinct baguettes in a bag. He picks three of the bagged baguettes at random and lays them on a table in random order. How many possible orderings of three baguettes are there on the table?
[b]B15 / G10[/b] Find the largest $7$-digit palindrome that is divisible by $11$.
PS. You should use hide for answers. Rest problems have been posted [url=https://artofproblemsolving.com/community/c3h3132170p28376644]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
[asy]
size(120);
path c = Circle((0, 0), 1);
pair A = dir(20), B = dir(130), C = dir(240), D = dir(330);
draw(c);
pair F = 3(A-B) + B;
pair G = 3(D-C) + C;
pair E = intersectionpoints(B--F, C--G)[0];
draw(B--E--C--A);
label("$A$", A, NE);
label("$B$", B, NW);
label("$C$", C, SW);
label("$D$", D, SE);
label("$E$", E, E);
//Credit to MSTang for the diagram[/asy]
In the adjoining figure $\measuredangle E=40^\circ$ and arc $AB$, arc $BC$, and arc $CD$ all have equal length. Find the measure of $\measuredangle ACD$.
$\textbf{(A) }10^\circ\qquad\textbf{(B) }15^\circ\qquad\textbf{(C) }20^\circ\qquad\textbf{(D) }\left(\frac{45}{2}\right)^\circ\qquad \textbf{(E) }30^\circ$
Let $ ABCD$ be a trapezoid with $ BC\parallel AD$, and let $ O$ be the point of intersection of its diagonals $ AC$ and $ BD$. Prove that $ \left\vert ABCD\right\vert \equal{}\left( \sqrt{\left\vert BOC\right\vert }\plus{}\sqrt{\left\vert DOA\right\vert }\right) ^{2}$.
[hide="Source of the problem"][i]Source of the problem:[/i] exercise 8 in: V. Alekseev, V. Galkin, V. Panferov, V. Tarasov, [i]Zadachi o trapezijah[/i], Kvant 6/2000, pages 37-4.[/hide]
Find all triples $ (x,y,z)$ of real numbers that satisfy the system of equations
\[ \begin{cases}x^3 \equal{} 3x\minus{}12y\plus{}50, \\ y^3 \equal{} 12y\plus{}3z\minus{}2, \\ z^3 \equal{} 27z \plus{} 27x. \end{cases}\]
[i]Razvan Gelca.[/i]
Riders in a carnival ride stand with their backs against the wall of a circular room of diameter $\text{8.0 m}$. The room is spinning horizontally about an axis through its center at a rate of $\text{45 rev/min}$ when the floor drops so that it no longer provides any support for the riders. What is the minimum coefficient of static friction between the wall and the rider required so that the rider does not slide down the wall?
(a) $\text{0.0012}$
(b) $\text{0.056}$
(c) $\text{0.11}$
(d) $\text{0.53}$
(e) $\text{8.9}$