Found problems: 85335
There are 12 people such that for every person A and person B there exists a person C that is a friend to both of them. Determine the minimum number of pairs of friends and construct a graph where the edges represent friendships.
Consider a binary matrix $M$(all entries are $0$ or $1$) on $r$ rows and $c$ columns, where every row and every column contain at least one entry equal to $1$. Prove that there exists an entry $M(i,j) = 1$, such that the corresponding row-sum $R(i)$ and column-sum $C(j)$ satisfy $r R(i)\ge c C(j)$.
(Proposed by Gerhard Woeginger, Austria)
An approximate construction of a regular pentagon goes as follows. Inscribe an arbitrary convex pentagon $P_1P_2P_3P_4P_5$ in a circle. Now choose an arror bound $\epsilon > 0$ and apply the following procedure.
(a) Denote $P_0 = P_5$ and $P_6 = P_1$ and construct the midpoint $Q_i$ of the circular arc $P_{i-1}P_{i+1}$ containing $P_i$.
(b) Rename the vertices $Q_1,...,Q_5$ as $P_1,...,P_5$.
(c) Repeat this procedure until the difference between the lengths of the longest and the shortest among the arcs $P_iP_{i+1}$ is less than $\epsilon$.
Prove this procedure must end in a finite time for any choice of $\epsilon$ and the points $P_i$.
Compute \[\int_0^\infty \dfrac{e^{-x}\sin(x)}{x}dx\]
Consider a trihedral angle $Sxyz$ with $\angle xSz = \alpha , \angle xSy = \beta$ and $\angle ySz =\gamma$. Let $A,B,C$ denote the dihedral angles at edges $y, z, x$ respectively.
$(a)$ Prove that $\frac{\sin\alpha}{\sin A}=\frac{\sin\beta}{\sin B}=\frac{\sin\gamma}{\sin C}$
$(b)$ Show that $\alpha + \beta = 180^{\circ}$ if and only if $\angle A + \angle B = 180^{\circ}.$
$(c)$ Assume that $\alpha=\beta =\gamma = 90^{\circ}$. Let $O \in Sz$ be a fixed point such that $SO = a$ and let $M,N$ be variable points on $x, y$ respectively. Prove that $\angle SOM +\angle SON +\angle MON$ is constant and find the locus of the incenter of $OSMN$.
Let $A,B \in \mathbb{R}^{n\times n}$ such that $AB+B+A=0$. Prove that $AB=BA$.
Determine all pairs $(a,b)$ of natural numbers such that the number $$\frac{a^2(b-a)}{b+a}$$ is the square of a prime number.
How many ways are there to fill a $3 \times 3$ grid with the numbers $1$, $2$, $3$, $4$, $5$, $6$, $7$, $8$, and $9$, such that the set of three elements in every row and every column form an arithmetic progression in some order? (Each number must be used exactly once)
What is the average number of pairs of consecutive integers in a randomly selected subset of $5$ distinct integers chosen from the set $\{ 1, 2, 3, …, 30\}$? (For example the set $\{1, 17, 18, 19, 30\}$ has $2$ pairs of consecutive integers.)
$\textbf{(A)}\ \frac{2}{3} \qquad\textbf{(B)}\ \frac{29}{36} \qquad\textbf{(C)}\ \frac{5}{6} \qquad\textbf{(D)}\
\frac{29}{30} \qquad\textbf{(E)}\ 1$
Suppose that $a$ and $b$ are natural numbers such that \[p=\frac{b}{4}\sqrt{\frac{2a-b}{2a+b}}\] is a prime number. What is the maximum possible value of $p$?
Let $F:\mathbb{R}^2\to\mathbb{R}$ and $g:\mathbb{R}\to\mathbb{R}$ be twice continuously differentiable functions with the following properties:
• $F(u,u)=0$ for every $u\in\mathbb{R};$
• for every $x\in\mathbb{R},g(x)>0$ and $x^2g(x)\le 1;$
• for every $(u,v)\in\mathbb{R}^2,$ the vector $\nabla F(u,v)$ is either $\mathbf{0}$ or parallel to the vector $\langle g(u),-g(v)\rangle.$
Prove that there exists a constant $C$ such that for every $n\ge 2$ and any $x_1,\dots,x_{n+1}\in\mathbb{R},$ we have
\[\min_{i\ne j}|F(x_i,x_j)|\le\frac{C}{n}.\]
$k$ is an integer, there exists a triangulation for a regular polygon with $2024$ sides and $2024$ colored dots with $k$ different colors meeting
$(1)$ each color will be used at least once
$(2)$ every small triangle will have at least $2$ dots that will be in the same color.
Try to find the maximum value of$k$
In $\triangle JMT$, $JM=410$, $JT=49$, and $\angle{MJT}>90^\circ$. Let $I$ and $H$ be the incenter and orthocenter of $\triangle JMT$, respectively. The circumcircle of $\triangle JIH$ intersects $\overleftrightarrow{JT}$ at a point $P\neq J$, and $IH=HP$. Find $MT$.
Find the smallest integer $k > 1$ for which $n^k-n$ is a multiple of $2010$ for every integer positive $n$.
Let $ABC$ be an acute non-isosceles triangle with circumcircle $\omega$, circumcenter $O$ and orthocenter $H$. We draw a line perpendicular to $AH$ through $O$ and a line perpendicular to $AO$ through $H$. Prove that the points of intersection of these lines with sides $AB$ and $AC$ lie on a circle, which is tangent to $\omega$.
[i]Proposed by A. Kuznetsov[/i]
7. There are 100 chess bishops on white squares of a $100 \times 100$ chess board. Some of them are white and some of them are black. They can move in any order and capture the bishops of opposing color. What number of moves is sufficient for sure to retain only one bishop on the chess board?
[b]p1.[/b] Compute $21 \cdot 21 - 20 \cdot 20$.
[b]p2.[/b] A square has side length $2$. If the square is scaled by a factor of $n$, the perimeter of the new square is equal to the area of the original square. Find $10n$.
[b]p3.[/b] Kevin has $2$ red marbles and $2$ blue marbles in a box. He randomly grabs two marbles. The probability that they are the same color can be expressed as $\frac{a}{b}$ for relatively prime integers $a$ and $b$. Find $a +b$.
[b]p4.[/b] In a classroom, if the teacher splits the students into groups of $3$ or $4$, there is one student left out. If the students formgroups of $5$, every student is in a group. What is the fewest possible number of students in this classroom?
[b]p5.[/b] Find the sum of all positive integer values of $x$ such that $\lfloor \sqrt{x!} \rfloor = x$.
[b]p6.[/b] Find the number of positive integer factors of $2021^{(2^0+2^1)} \cdot 1202^{(1^2+0^2)}$.
[b]p7.[/b] Let $n$ be the number of days over a $13$ year span. Find the difference between the greatest and least possible values of $n$. Note: All years divisible by $4$ are leap years unless they are divisible by 100 but not $400$. For example, $2000$ and $2004$ are leap years, but $1900$ is not.
[b]p8.[/b] In isosceles $\vartriangle ABC$, $AB = AC$, and $\angle ABC = 72^o$. The bisector of $\angle ABC$ intersects $AC$ at $D$. Given that $BC = 30$, find $AD$.
[b]p9.[/b] For an arbitrary positive value of $x$, let $h$ be the area of a regular hexagon with side length $x$ and let $s$ be the area of a square with side length $x$. Find the value of $\left \lfloor \frac{10h}{s} \right \rfloor$.
[b]p10.[/b] There is a half-full tub of water with a base of $4$ inches by $5$ inches and a height of $8$ inches. When an infinitely long stick with base $1$ inch by $1$ inch is inserted vertically into the bottom of the tub, the number of inches the water level rises by can be written as $\frac{a}{b}$ where $a$ and $b$ are relatively prime positive integers. Find $a +b$.
[b]p11.[/b] Find the sum of all $4$-digit numbers with digits that are a permutation of the digits in $2021$. Note that positive integers cannot have first digit $0$.
[b]p12.[/b] A $10$-digit base $8$ integer is chosen at random. The probability that it has $30$ digits when written in base $2$ can be expressed as $\frac{a}{b}$, where $a$ and $b$ are relatively prime positive integers. Find $a +b$.
[b]p13.[/b] Call a natural number sus if it can be expressed as $k^2 +k +1$ for some positive integer $k$. Find the sum of all sus integers less than $2021$.
[b]p14.[/b] In isosceles triangle $ABC$, $D$ is the intersection of $AB$ and the perpendicular to $BC$ through $C$. Given that $CD = 5$ and $AB = BC = 1$, find $\sec^2 \angle ABC$.
[b]p15.[/b] Every so often, the minute and hour hands of a clock point in the same direction. The second time this happens after 1:00 is a b minutes later, where a and b are relatively prime positive integers. Find a +b.
[b]p16.[/b] The $999$-digit number $N = 123123...123$ is composed of $333$ iterations of the number $123$. Find the least nonnegative integerm such that $N +m$ is a multiple of $101$.
[b]p17.[/b] The sum of the reciprocals of the divisors of $2520$ can be written as $\frac{a}{b}$, where $a$ and $b$ are relatively prime positive integers. Find $a +b$.
[b]p18.[/b] Duncan, Paul, and $6$ Atreides guards are boarding three helicopters. Duncan, Paul, and the guards enter the helicopters at random, with the condition that Duncan and Paul do not enter the same helicopter. Note that not all helicoptersmust be occupied. The probability that Paul has more guards with him in his helicopter than Duncan does can be written as $\frac{a}{b}$ where $a$ and $b$ are relatively prime positive integers. Find $a +b$.
[b]p19.[/b] Let the minimum possible distance from the origin to the parabola $y = x^2 -2021$ be $d$. The value of d2 can be expressed as $\frac{a}{b}$ where $a$ and $b$ are relatively prime positive integers. Find $a +b$.
[b]p20.[/b] In quadrilateral $ABCD$ with interior point $E$ and area $49 \sqrt3$, $\frac{BE}{CE}= 2 \sqrt3$, $\angle ABC = \angle BCD = 90^o$, and $\vartriangle ABC \sim \vartriangle BCD \sim \vartriangle BEC$. The length of $AD$ can be expressed aspn where $n$ is a positive integer. Find $n$.
[b]p21.[/b] Find the value of
$$\sum^{\infty}_{i=1}\left( \frac{i^2}{2^{i-1}}+\frac{i^2}{2^{i}}+\frac{i^2}{2^{i+1}}\right)=\left( \frac{1^2}{2^{0}}+\frac{1^2}{2^{1}}+\frac{1^2}{2^{2}}\right)+\left( \frac{2^2}{2^{1}}+\frac{2^2}{2^{2}}+\frac{2^2}{2^{3}}\right)+\left( \frac{3^2}{2^{2}}+\frac{2^2}{2^{3}}+\frac{2^2}{2^{4}}\right)+...$$
[b]p22.[/b] Five not necessarily distinct digits are randomly chosen in some order. Let the probability that they form a nondecreasing sequence be $\frac{a}{b}$ , where $a$ and $b$ are relatively prime positive integers. Find the remainder when $a +b$ is divided by$ 1000$.
[b]p23.[/b] Real numbers $a$, $b$, $c$, and d satisfy $$ac -bd = 33$$
$$ad +bc = 56.$$ Given that $a^2 +b^2 = 5$, find the sum of all possible values of $c^2 +d^2$.
[b]p24.[/b] Jeff has a fair tetrahedral die with sides labeled $0$, $1$, $2$, and $3$. He continuously rolls the die and record the numbers rolled in that order. For example, if he rolls a $1$, then rolls a $2$, and then rolls a $3$, he writes down $123$. He keeps rolling the die until he writes the substring $2021$. What is the expected number of times he rolls the die?
[b]p25.[/b] In triangle $ABC$, $BC = 2\sqrt3$, and $AB = AC = 4\sqrt3$. Circle $\omega$ with center $O$ is tangent to segment $AB$ at $T$ , and $\omega$ is also tangent to ray $CB$ past $B$ at another point. Points $O, T$ , and $C$ are collinear. Let $r$ be the radius of $\omega$. Given that $r^2 = \frac{a}{b}$ where $a$ and $b$ are relatively prime positive integers, find $a +b$.
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
There are some papers of the size $5\times 5$ with two sides which are divided into unit squares for both sides. One uses $n$ colors to paint each cell on the paper, one cell by one color, such that two cells on the same positions for two sides are painted by the same color. Two painted papers are consider as the same if the color of two corresponding cells are the same. Prove that there are no more than
$$\frac{1}{8}\left( {{n}^{25}}+4{{n}^{15}}+{{n}^{13}}+2{{n}^{7}} \right)$$ pairwise distinct papers that painted by this way.
A rock travelled through an n x n board, stepping at each turn to the cell neighbouring the previous one by a side, so that each cell was visited once. Bob has put the integer numbers from 1 to n^2 into the cells, corresponding to the order in which the rook has passed them. Let M be the greatest difference of the numbers in neighbouring by side cells. What is the minimal possible value of M?
A set $P$ consists of $2005$ distinct prime numbers. Let $A$ be the set of all possible products of $1002$ elements of $P$ , and $B$ be the set of all products of $1003$ elements of $P$ . Find a one-to-one correspondance $f$ from $A$ to $B$ with the property that $a$ divides $f (a)$ for all $a \in A.$
Given any $60$ points on a circle of radius $1$, prove that there is a point on the circle the sum of whose distances to these $60$ points is at most $80$.
Prove that for any positive even integer $n$ larger than $38$, $n$ can be written as $a\times b+c\times d$ where $a, b, c, d$ are odd integers larger than $1$.
A right-angled triangle has perimeter $60$ and the altitude of the hypotenuse has a length $12$. Determine the lengths of the sides.
Bill plays a game in which he rolls two fair standard six-sided dice with sides labeled one through six. He wins if the number on one of the dice is three times the number on the other die. If Bill plays this game three times, compute the probability that he wins at least once.
Wesyu is a farmer, and she's building a cao (a relative of the cow) pasture. Shw starts with a triangle $A_0A_1A_2$ where angle $A_0$ is $90^\circ$, angle $A_1$ is $60^\circ$, and $A_0A_1$ is $1$. She then extends the pasture. FIrst, she extends $A_2A_0$ to $A_3$ such that $A_3A_0=\dfrac12A_2A_0$ and the new pasture is triangle $A_1A_2A_3$. Next, she extends $A_3A_1$ to $A_4$ such that $A_4A_1=\dfrac16A_3A_1$. She continues, each time extending $A_nA_{n-2}$ to $A_{n+1}$ such that $A_{n+1}A_{n-2}=\dfrac1{2^n-2}A_nA_{n-2}$. What is the smallest $K$ such that her pasture never exceeds an area of $K$?