Found problems: 85335
In a group of $m$ girls and $n$ boys, any two persons either know each other or do not know each other. For any two boys and any two girls, there are at least one boy and one girl among them,who do not know each other. Prove that the number of unordered pairs of (boy, girl) who know each other does not exceed $m+\frac{n(n-1)}{2}$.
Let $A$ be the number of arrangements of the letters in JOHNS HOPKINS such that no two Os are adjacent, no two Hs are adjacent, no two Ns are adjacent, and no two Ss are adjacent. Find $\frac{A}{8!}$.
Evaluate $ \int_0^{n\pi} \sqrt{1\minus{}\sin t}\ dt\ (n\equal{}1,\ 2,\ \cdots).$
Find the range value of $a$, satisfyin that $\forall x\in\mathbb{R},\theta\in\left[0,\frac{\pi}{2}\right]$,
$$(x+3+2\sin\theta\cos\theta)^2+(x+a\sin\theta+a\cos\theta)^2\geq\frac{1}{8}.$$
Let S be a non-empty set with an associative operation that is left and right cancellative (xy=xz implies y=z, and yx = zx implies y = z). Assume that for every a in S the set {a^n : n = 0,1,2...} is finite. Must S be a group?
I haven't had much group theory at this point...
Prove that if $m=5^n+3^n+1$ is a prime, then $12$ divides $n$.
Let $AB$ be a segment of unit length and let $C, D$ be variable points of this segment. Find the maximum value of the product of the lengths of the six distinct segments with endpoints in the set $\{A,B,C,D\}.$
How many numbers of $ n $ digits formed only with $ 1,9,8 $ and $ 6 $ divide themselves by $ 3 $ ?
Let $a$ be a positive real number, $n$ a positive integer, and define the [i]power tower[/i] $a\uparrow n$ recursively with $a\uparrow 1=a$, and $a\uparrow(i+1)=a^{a\uparrow i}$ for $i=1,2,3,\ldots$. For example, we have $4\uparrow 3=4^{(4^4)}=4^{256}$, a number which has $155$ digits. For each positive integer $k$, let $x_k$ denote the unique positive real number solution of the equation $x\uparrow k=10\uparrow (k+1)$. Which is larger: $x_{42}$ or $x_{43}$?
Prove that for all positive integers $n$ and all positive reals $a,b,c$ the following inequality holds: \[\frac{a^{n+1}}{a^{n}+a^{n-1}b+\ldots+b^{n}}+\frac{b^{n+1}}{b^{n}+b^{n-1}c+\ldots+c^{n}}+\frac{c^{n+1}}{c^{n}+c^{n-1}a+\ldots+a^{n}}\\ \ge \frac{a+b+c}{n+1}\]
Prove that
$$\sqrt{x-1}+\sqrt{2x+9}+\sqrt{19-3x}<9$$
for all real $x$ for which the left-hand side is well defined.
$O$ is a point in triangle $ABC$. We draw perpendicular from $O$ to $BC,AC,AB$ which intersect $BC,AC,AB$ at $A_{1},B_{1},C_{1}$. Prove that $O$ is circumcenter of triangle $ABC$ iff perimeter of $ABC$ is not less than perimeter of triangles $AB_{1}C_{1},BC_{1}A_{1},CB_{1}A_{1}$.
A vintage tram departs a stop with a certain number of boys and girls on board. At the first stop, a third of the girls get out and their places are taken by boys. At the next stop, a third of the boys get out and their places are taken by girls. There are now two more girls than boys and as many boys as there originally were girls. How many boys and girls were there on board at the start?
A $9\times 9$ table is filled with zeroes.In every step we can either take a row add $1$ to every cell and shift it one unit to right or take a column reduce every cell by $1$ and shift it one cell down. Can the table with the top right $-1$ and bottom left $+1$ and all other cells zero be reached?
Let $(a_i,b_i)$ be pairwise distinct pairs of positive integers for $1\le i\le n$. Prove that
\[(a_1+a_2+\ldots+a_n)(b_1+b_2+\ldots+b_n)>\frac29 n^3,\]
and show that the statement is sharp, i.e. for an arbitrary $c>\frac29$ it is possible that
\[(a_1+a_2+\ldots+a_n)(b_1+b_2+\ldots+b_n)<cn^3.\]
[i]Submitted by Péter Pál Pach, Budapest, based on an OKTV problem[/i]
Let $ P$ be a polyhedron where every face is a regular polygon, and every edge has length $ 1$. Each vertex of $ P$ is incident to two regular hexagons and one square. Choose a vertex $ V$ of the polyhedron. Find the volume of the set of all points contained in $ P$ that are closer to $ V$ than to any other vertex.
Calculate the product $\tfrac13 \times \tfrac24 \times \tfrac35 \times \cdots \times \tfrac{18}{20} \times \tfrac{19}{21}$.
$\textbf{(A) }\dfrac{1}{210}\qquad\textbf{(B) }\dfrac{1}{190}\qquad\textbf{(C) }\dfrac{1}{21}\qquad\textbf{(D) }\dfrac{1}{20}\qquad\textbf{(E) }\dfrac{1}{10}$
Let $P$ be a point chosen on the interior of side $\overline{BC}$ of triangle $\triangle ABC$ with side lengths $\overline{AB} = 10, \overline{BC} = 10, \overline{AC} = 12$. If $X$ and $Y$ are the feet of the perpendiculars from $P$ to the sides $AB$ and $AC$, then the minimum possible value of $PX^2 + PY^2$ can be expressed as $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Find $m+n$.
[i]Proposed by Andrew Wen[/i]
A white bug sits in one corner square of a $1000$ × $n$ chessboard, where $n$ is an odd positive integer and $n > 2020$. In the two nearest corner squares there are two black chess bishops. On each move, the bug either steps into a square adjacent by side or moves as a chess knight. The bug wishes to reach the opposite corner square by never visiting a square occupied or attacked by a bishop, and visiting every other square exactly once. Show that the number of ways for the bug to attain its goal does not depend on $n$.
In an attempt to copy down from the board a sequence of six positive integers in arithmetic progression, a student wrote down the five numbers, \[ 113,137,149,155,173, \] accidentally omitting one. He later discovered that he also miscopied one of them. Can you help him and recover the original sequence?
Let us define a sequence $\{a_n\}_{n\ge 1}$. Define as follows:
\[ a_1=2\text{ and }a_{n+1}=2^{a_n}\text{ for }n\ge 1 \]
Show this :
\[ a_{n}\equiv a_{n-1}\pmod n \]
In $\triangle ABC$, $H$ is the orthocenter, and $AD,BE$ are arbitrary cevians. Let $\omega_1, \omega_2$ denote the circles with diameters $AD$ and $BE$, respectively. $HD,HE$ meet $\omega_1,\omega_2$ again at $F,G$. $DE$ meets $\omega_1,\omega_2$ again at $P_1,P_2$ respectively. $FG$ meets $\omega_1,\omega_2$ again $Q_1,Q_2$ respectively. $P_1H,Q_1H$ meet $\omega_1$ at $R_1,S_1$ respectively. $P_2H,Q_2H$ meet $\omega_2$ at $R_2,S_2$ respectively. Let $P_1Q_1\cap P_2Q_2 = X$, and $R_1S_1\cap R_2S_2=Y$. Prove that $X,Y,H$ are collinear.
[i]Ray Li.[/i]
The centre of a circle is the origin of $N$ vectors whose ends divide the circle in $N$ equal arcs . Some of the vectors are blue and some are red . We calculate the sum of the angles formed between each pair consisting of a red vector and a blue vector (the angle being measured anticlockwise from red to blue) and divide this sum by the total number of such angles . Prove that the "mean angle" thus obtained is $180^o$.
(V. P roizvolov)
Let $\vartriangle ABC$ be a triangle with an obtuse angle $\angle ACB$. The incircle of $\vartriangle ABC$ centered at $I$ is tangent to the sides $AB, BC, CA$ at $D, E, F$ respectively. Lines $AI$ and $BI$ intersect $EF$ at $M$ and $N$ respectively. Let $G$ be the midpoint of $AB$. Show that $M, N, G, D$ lie on a circle.
Determine all polynomials $P(x)$ with real coefficients such that $(x+1)P(x-1)-(x-1)P(x)$ is a constant polynomial.