Found problems: 85335
For a finite set $A$ of positive integers, a partition of $A$ into two disjoint nonempty subsets $A_1$ and $A_2$ is $\textit{good}$ if the least common multiple of the elements in $A_1$ is equal to the greatest common divisor of the elements in $A_2$. Determine the minimum value of $n$ such that there exists a set of $n$ positive integers with exactly $2015$ good partitions.
Let $a,b,c$ be pairwise distinct real numbers. Show that
\[
(\frac{2a-b}{a-b})^2+(\frac{2b-c}{b-c})^2+(\frac{2c-a}{c-a})^2 \ge 5.
\]
For positive integers $a$ and $k$, define the sequence $a_1,a_2,\ldots$ by \[a_1=a,\qquad\text{and}\qquad a_{n+1}=a_n+k\cdot\varrho(a_n)\qquad\text{for } n=1,2,\ldots\] where $\varrho(m)$ denotes the product of the decimal digits of $m$ (for example, $\varrho(413)=12$ and $\varrho(308)=0$). Prove that there are positive integers $a$ and $k$ for which the sequence $a_1,a_2,\ldots$ contains exactly $2009$ different numbers.
On the coordinate plane in the first quarter there are $100$ non-intersecting single unit segments parallel to the coordinate axes. These segments aremirrors (on both sides), they reflect the light according to the rule. "The angle of incidence is equal to the angle of reflection." (If you hit the edge of the mirror, the beam of light does not change its direction.) From the point lying in the unit circle with the center at the origin, a ray of light in the direction of the bisector of the first coordinate angle. Prove that, that this initial point can be chosen so that the ray is reflected from the mirrors not more than $150$ times.
Draw a circle that has a given radius $R$ and is tangent to a given line and a given circle. How many solutions does this problem have?
Let $f(n)$ denote the number of permutations $(a_{1}, a_{2}, \ldots ,a_{n})$ of the set $\{1,2,\ldots,n\}$, which satisfy the conditions: $a_{1}=1$ and $|a_{i}-a_{i+1}|\leq2$, for any $i=1,2,\ldots,n-1$. Prove that $f(2006)$ is divisible by 3.
The numbers $1,2,\dots,9$ are randomly placed into the $9$ squares of a $3 \times 3$ grid. Each square gets one number, and each of the numbers is used once. What is the probability that the sum of the numbers in each row and each column is odd?
$\textbf{(A) }1/21\qquad\textbf{(B) }1/14\qquad\textbf{(C) }5/63\qquad\textbf{(D) }2/21\qquad\textbf{(E) } 1/7$
In rectangle $ABCD$, $DC = 2CB$ and points $E$ and $F$ lie on $\overline{AB}$ so that $\overline{ED}$ and $\overline{FD}$ trisect $\angle ADC$ as shown. What is the ratio of the area of $\triangle DEF$ to the area of rectangle $ABCD$?
[asy]
draw((0, 0)--(0, 1)--(2, 1)--(2, 0)--cycle);
draw((0, 0)--(sqrt(3)/3, 1));
draw((0, 0)--(sqrt(3), 1));
label("A", (0, 1), N);
label("B", (2, 1), N);
label("C", (2, 0), S);
label("D", (0, 0), S);
label("E", (sqrt(3)/3, 1), N);
label("F", (sqrt(3), 1), N);
[/asy]
${ \textbf{(A)}\ \ \frac{\sqrt{3}}{6}\qquad\textbf{(B)}\ \frac{\sqrt{6}}{8}\qquad\textbf{(C)}\ \frac{3\sqrt{3}}{16}\qquad\textbf{(D)}}\ \frac{1}{3}\qquad\textbf{(E)}\ \frac{\sqrt{2}}{4}$
For a finite simple graph $G$, we define $G'$ to be the graph on the same vertex set as $G$, where for any two vertices $u \neq v$, the pair $\{u,v\}$ is an edge of $G'$ if and only if $u$ and $v$ have a common neighbor in $G$.
Prove that if $G$ is a finite simple graph which is isomorphic to $(G')'$, then $G$ is also isomorphic to $G'$.
[i]Mehtaab Sawhney and Zack Chroman[/i]
The plane is partitioned into congruent equilateral triangles such that any two of them which are not disjoint have either a common vertex or a common side. Is there a circle containing exactly 1988 points in its interior?
Let $ABC$ be a triangle with circumradius $1$. If the center of the circle passing through $A$, $C$, and the orthocenter of $\triangle ABC$ lies on the circumcircle of $\triangle ABC$, what is $|AC|$?
$
\textbf{(A)}\ 2
\qquad\textbf{(B)}\ 3
\qquad\textbf{(C)}\ \dfrac 32
\qquad\textbf{(D)}\ \sqrt 2
\qquad\textbf{(E)}\ \sqrt 3
$
Prove that the number of pairs $ \left(\alpha,S\right)$ of a permutation $ \alpha$ of $ \{1,2,\dots,n\}$ and a subset $ S$ of $ \{1,2,\dots,n\}$ such that
\[ \forall x\in S: \alpha(x)\not\in S\]
is equal to $ n!F_{n \plus{} 1}$ in which $ F_n$ is the Fibonacci sequence such that $ F_1 \equal{} F_2 \equal{} 1$
Let be an element $ e $ from a group $ (G,\cdot ) , $ and let be the operation $ *:G^2\longrightarrow G $ defined as $ x*y=x\cdot a\cdot y . $ Prove that $ (G,*) $ is a group and is isomorphic with $ (G,\cdot ) . $
Given triangle $ABC, BC$ is extended beyond $B$ to the point $D$ such that $BD = BA$. The bisectors of the exterior angles at vertices $B$ and $C$ intersect at the point $M$. Prove that quadrilateral $ADMC$ is cyclic. (4)
Suppose Pat and Rick are playing a game in which they take turns writing numbers from $\{1, 2, \dots, 97\}$ on a blackboard. In each round, Pat writes a number, then Rick writes a number; Rick wins if the sum of all the numbers written on the blackboard after $n$ rounds is divisible by 100. Find the minimum positive value of $n$ for which Rick has a winning strategy.
For $1\leq n\leq 2016$, how many integers $n$ satisfying the condition: the reminder divided by $20$ is smaller than the one divided by $16$.
Let $ G$ be a simple graph with vertex set $ V\equal{}\{0,1,2,3,\cdots ,n\plus{}1\}$ .$ j$and$ j\plus{}1$ are connected by an edge for $ 0\le j\le n$. Let $ A$ be a subset of $ V$ and $ G(A)$ be the induced subgraph associated with $ A$. Let $ O(G(A))$ be number of components of $ G(A)$ having an odd number of vertices.
Let
$ T(p,r)\equal{}\{A\subset V \mid 0.n\plus{}1 \notin A,|A|\equal{}p,O(G(A))\equal{}2r\}$ for $ r\le p \le 2r$.
Prove That $ |T(p,r)|\equal{}{n\minus{}r \choose{p\minus{}r}}{n\minus{}p\plus{}1 \choose{2r\minus{}p}}$.
Let $m$ be a positive integer less than $2015$. Suppose that the remainder when $2015$ is divided by $m$ is $n$. Compute the largest possible value of $n$.
[i] Proposed by Michael Ren [/i]
[b]p1.[/b] Martin’s car insurance costed $\$6000$ before he switched to Geico, when he saved $15\%$ on car insurance. When Mayhem switched to Allstate, he, a safe driver, saved $40\%$ on car insurance. If Mayhem and Martin are now paying the same amount for car insurance, how much was Mayhem paying before he switched to Allstate?
[b]p2.[/b] The $7$-digit number $N$ can be written as $\underline{A} \,\, \underline{2} \,\,\underline{0} \,\,\underline{B} \,\,\underline{2} \,\, \underline{1} \,\,\underline{5}$. How many values of $N$ are divisible by $9$?
[b]p3.[/b] The solutions to the equation $x^2-18x-115 = 0$ can be represented as $a$ and $b$. What is $a^2+2ab+b^2$?
[b]p4.[/b] The exterior angles of a regular polygon measure to $4$ degrees. What is a third of the number of sides of this polygon?
[b]p5.[/b] Charlie Brown is having a thanksgiving party.
$\bullet$ He wants one turkey, with three different sizes to choose from.
$\bullet$ He wants to have two or three vegetable dishes, when he can pick from Mashed Potatoes, Saut´eed Brussels Sprouts, Roasted Butternut Squash, Buttery Green Beans, and Sweet Yams;
$\bullet$ He wants two desserts out of Pumpkin Pie, Apple Pie, Carrot Cake, and Cheesecake.
How many different combinations of menus are there?
[b]p6.[/b] In the diagram below, $\overline{AD} \cong \overline{CD}$ and $\vartriangle DAB$ is a right triangle with $\angle DAB = 90^o$. Given that the radius of the circle is $6$ and $m \angle ADC = 30^o$, if the length of minor arc $AB$ is written as $a\pi$, what is $a$?
[img]https://cdn.artofproblemsolving.com/attachments/d/9/ea57032a30c16f4402886af086064261d6828b.png[/img]
[b]p7.[/b] This Halloween, Owen and his two friends dressed up as guards from Squid Game. They needed to make three masks, which were black circles with a white equilateral triangle, circle, or square inscribed in their upper halves. Resourcefully, they used black paper circles with a radius of $5$ inches and white tape to create these masks. Ignoring the width of the tape, how much tape did they use? If the length can be expressed $a\sqrt{b}+c\sqrt{d}+ \frac{e}{f} \pi$ such that $b$ and $d$ are not divisible by the square of any prime, and $e$ and $f$ are relatively prime, find $a + b + c + d + e + f$.
[img]https://cdn.artofproblemsolving.com/attachments/0/c/bafe3f9939bd5767ba5cf77a51031dd32bbbec.png[/img]
[b]p8.[/b] Given $LCM (10^8, 8^{10}, n) = 20^{15}$, where $n$ is a positive integer, find the total number of possible values of $n$.
[b]p9.[/b] If one can represent the infinite progression $\frac{1}{11} + \frac{2}{13} + \frac{3}{121} + \frac{4}{169} + \frac{5}{1331} + \frac{6}{2197}+ ...$ as $\frac{a}{b}$ , where $a$ and $b$ are relatively prime positive integers, what is $a$?
[b]p10.[/b] Consider a tiled $3\times 3$ square without a center tile. How many ways are there to color the squares such that no two colored squares are adjacent (vertically or horizontally)? Consider rotations of an configuration to be the same, and consider the no-color configuration to be a coloring.
[b]p11.[/b] Let $ABC$ be a triangle with $AB = 4$ and $AC = 7$. Let $AD$ be an angle bisector of triangle $ABC$. Point $M$ is on $AC$ such that $AD$ intersects $BM$ at point $P$, and $AP : PD = 3 : 1$. If the ratio $AM : MC$ can be expressed as $\frac{a}{b}$ such that $a$, $b$ are relatively prime positive integers, find $a + b$.
[b]p12.[/b] For a positive integer $n$, define $f(n)$ as the number of positive integers less than or equal to $n$ that are coprime with $n$. For example, $f(9) = 6$ because $9$ does not have any common divisors with $1$, $2$, $4$, $5$, $7$, or $8$. Calculate: $$\sum^{100}_{i=2} \left( 29^{f(i)}\,\,\, mod \,\,i \right).$$
[b]p13.[/b] Let $ABC$ be an equilateral triangle. Let $P$ be a randomly selected point in the incircle of $ABC$. Find $a+b+c+d$ if the probability that $\angle BPC$ is acute can be expressed as $\frac{a\sqrt{b} -c\pi}{d\pi }$ for positive integers $a$, $b$, $c$, $d$ where $gcd(a, c, d) = 1$ and $b$ is not divisible by the square of any prime.
[b]p14.[/b] When the following expression is simplified by expanding then combining like terms, how many terms are in the resulting expression? $$(a + b + c + d)^{100} + (a + b - c - d)^{100}$$
[b]p15.[/b] Jerry has a rectangular box with integral side lengths. If $3$ units are added to each side of the box, the volume of the box is tripled. What is the largest possible volume of this box?
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
Four positive integers $x,y,z$ and $t$ satisfy the relations
\[ xy - zt = x + y = z + t. \]
Is it possible that both $xy$ and $zt$ are perfect squares?
In the convex pentagon $ABCDE$: $\angle A = \angle C = 90^o$, $AB = AE, BC = CD, AC = 1$. Find the area of the pentagon.
Consider a finite collection of 3-element sets $A_i$, no two of which share more than one element, whose union has cardinality 2017. Show that the elements of this union can be coloured with two colors, blue and red, so that at least 64 elements are blue and each $A_i$ has at least one red element.
Let $ABCD$ be a cyclic quadrilateral with $AB = 5$, $BC = 10$, $CD = 11$, and $DA = 14$. The value of $AC + BD$ can be written as $\tfrac{n}{\sqrt{pq}}$, where $n$ is a positive integer and $p$ and $q$ are distinct primes. Find $n + p + q$.
Show that the inequality \[\sum_{i=1}^n \sum_{j=1}^n \sqrt{|x_i-x_j|}\leqslant \sum_{i=1}^n \sum_{j=1}^n \sqrt{|x_i+x_j|}\]holds for all real numbers $x_1,\ldots x_n.$
Let $O$ be the center of the circle of the triangle $ABC$ with $AB \ne AC$. Furthermore, let $S$ and $T$ be points on the rays $AB$ and $AC$, such that $\angle ASO = \angle ACO$ and $\angle ATO = \angle ABO$. Show that $ST$ bisects the segment $BC$.