Found problems: 85335
Find the smallest positive integer $n$ such that the decimal representation of $n!(n+1)!(2n+1)! - 1$ has its last $30$ digits all equal to $9$.
Let be distinct points on a plane, four of which form a quadrangle, and three of which are in the interior or boundary of this quadrangle. Show that the diagonals of this quadrangle are longer than the double of the minimum of the distances between any two of these seven points.
[i]Paul Erdős[/i]
[hide=Side note]If the quadrangle is convex, the constant from the inequality can be improved from $ 2 $ to $ \sqrt{\frac{3\pi}{2}}. $[/hide]
Given a quadrilateral $ABCD$, around which you can circumscribe a circle. The perpendicular bisectors of sides $AD$ and $CD$ intersect at point $Q$ and intersect sides $BC$ and $AB$ at points $P$ and $K$ resepctively. It turned out that the points $K, B, P, Q$ lie on the same circle. Prove that the points $A, Q, C$ lie on one line.
(Olena Artemchuk)
Members of the Rockham Soccer League buy socks and T-shirts. Socks cost $ \$4$ per pair and each T-shirt costs $ \$5$ more than a pair of socks. Each member needs one pair of socks and a shirt for home games and another pair of socks and a shirt for away games. If the total cost is $ \$2366$, how many members are in the League?
$ \textbf{(A)}\ 77 \qquad
\textbf{(B)}\ 91 \qquad
\textbf{(C)}\ 143 \qquad
\textbf{(D)}\ 182 \qquad
\textbf{(E)}\ 286$
Let $a_n = 6^n + 8^n$. Determine the remainder on dividing $a_{83}$ by 49.
Find the area of the shaded region.
[i]Lightning 1.4[/i]
Solve in non-negative integers the equation
$125.2^n-3^m=271$
Determine all real numbers $a>0$ for which there exists a nonnegative continuous function $f(x)$ defined on $[0,a]$ with the property that the region
$R=\{(x,y): 0\le x\le a, 0\le y\le f(x)\}$
has perimeter $k$ units and area $k$ square units for some real number $k$.
Let $ n$ and $ k$ be positive integers. Please, find an explicit formula for
\[ \sum y_1y_2 \dots y_k,\]
where the summation runs through all $ k\minus{}$tuples positive integers $ (y_1,y_2,\dots,y_k)$ satisfying $ y_1\plus{}y_2\plus{}\dots\plus{}y_k\equal{}n$.
Andy and Bethany have a rectangular array of numbers with $ 40$ rows and $ 75$ columns. Andy adds the numbers in each row. The average of his $ 40$ sums is $ A$. Bethany adds the numbers in each column. The average of her $ 75$ sums is $ B$. What is the value of $ \frac{A}{B}$?
$ \textbf{(A)}\ \frac{64}{225} \qquad
\textbf{(B)}\ \frac{8}{15} \qquad
\textbf{(C)}\ 1 \qquad
\textbf{(D)}\ \frac{15}{8} \qquad
\textbf{(E)}\ \frac{225}{64}$
A pair of integers is special if it is of the form $(n, n-1)$ or $(n-1, n)$ for some positive integer $n$. Let $n$ and $m$ be positive integers such that pair $(n, m)$ is not special. Show $(n, m)$ can be expressed as a sum of two or more different special pairs if and only if $n$ and $m$ satisfy the inequality $ n+m\geq (n-m)^2 $.
Note: The sum of two pairs is defined as $ (a, b)+(c, d) = (a+c, b+d) $.
In quadrilateral $ABCD$, let $AB = 7$, $BC = 11$, $CD = 3$, $DA = 9$, $\angle BAD = \angle BCD = 90^o$, and diagonals $\overline{AC}$ and $\overline{BD}$ intersect at $E$. The ratio $\frac{BE}{DE} = \frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Find $m + n$.
An alphabet has $n$ letters. A word is called [i]differentiated [/i] if it has the following property fulfilled: No letter occurs more than once between two identical letters. For example with the alphabet $\{a, b, c, d\}$ the word [i]abbdacbdd [/i] is not, the word [i]bbacbadcdd [/i] is differentiated.
(a) Each differentiated word has a maximum of $3n$ letters.
(b) How many differentiated words with exactly $3n$ letters are ther
Let $n$ be a positive integer and let $\alpha_n $ be the number of $1$'s within binary representation of $n$.
Show that for all positive integers $r$,
\[2^{2n-\alpha_n}\phantom{-1} \bigg|^{\phantom{0}}_{\phantom{-1}} \sum_{k=-n}^{n} \binom{2n}{n+k} k^{2r}.\]
Find the sum of all the possible values of xy such that x and y are positive integers satisfying $(x^2 + 1)(y^2 + 1) + 2(x -y)(1 - xy) = 4(1 + xy) + 140$.
There are $ 2000 $ people, and some of them have called each other. Two people can call each other at most $1$ time. For any two groups of three people $ A$ and $ B $ which $ A \cap B = \emptyset $, there exist one person from each of $A$ and $B$ that haven't called each other. Prove that the number of two people called each other is less than $ 201000 $.
$P$ is a point on hyperbola $\frac{x^2}{16}-\frac{y^2}{9}=1$, if the distance from $P$ to right directrix is the arithmetic mean of the distance from $P$ to two focal points, then the $x$-axis of $P$ is________.
What is the probability of having $2$ adjacent white balls or $2$ adjacent blue balls in a random arrangement of $3$ red, $2$ white and $2$ blue balls?
$
\textbf{(A)}\ \dfrac{2}{5}
\qquad\textbf{(B)}\ \dfrac{3}{7}
\qquad\textbf{(C)}\ \dfrac{16}{35}
\qquad\textbf{(D)}\ \dfrac{10}{21}
\qquad\textbf{(E)}\ \dfrac{5}{14}
$
For all $a,b,c\in \bb{R}^+ $ such that $a+b+c=1$ and $ ( \frac{1}{(a+b)^2}+\frac{1}{(b+c)^2}+\frac{1}{(c+a)^2} )(a-bc)(b-ac)(c-ab)\le M \cdot abc$. Find min $M$
On a circumference at some points sit $12$ grasshoppers. The points divide the circumference into $12$ arcs. By a signal each grasshopper jumps from its point to the midpoint of its arc (in clockwise direction). In such way new arcs are created. The process repeats for a number of times. Can it happen that at least one of the grasshoppers returns to its initial point after
a) $12$ jumps? (4)
a) $13$ jumps? (3)
Let $\mathbb{R}^+$ denote the set of positive real numbers. Determine all functions $f: \mathbb{R}^+ \to \mathbb{R}^+$ such that for all positive real numbers $x$ and $y$ : \[f(x)f(y+f(x))=f(1+xy)\]
[i]Proposed by Otgonbayar Uuye. [/i]
Prove that if $x$, $y$, and $z$ are non-negative numbers and $x^2+y^2+z^2=1$, then the following inequality is true:
$\frac{x}{1-x^2}+\frac{y}{1-y^2}+\frac{z}{1-z^2 }\geq \frac{3\sqrt{3}}{2}$
Find the lowest odd positive integer with an odd number of divisors and is divisible by $d^2$ and $a+b+c+d+e+f$, where $a, b, c, d, e, f$ are consecutive prime numbers.
Let $ABC$ be a triangle and let $AD,BE,CF$ be its altitudes . $FA_{1},DB_{1},EC_{1}$ are perpendicular segments to $BC,AC,AB$ respectively.
Prove that : $ABC$~$A_{1}B_{1}C_{1}$
The city of Atlantis is built on an island represented by $[ -1, 1]$, with skyline initially given by $f(x) = 1 - |x| $. The sea level is currently $y=0$, but due to global warming, it is rising at a rate of $0.01$ a year. For any position $-1 < x < 1$, while the building at $x$ is not completely submerged, then it is instantaneously being built upward at a rate of $r$ per year, where $r$ is the distance (along the $x$-axis) from this building to the nearest completely submerged building.
How long will it be until Atlantis becomes completely submerged?
[i]Proposed by Ethan Tan[/i]