Found problems: 85335
Show that $\frac{(n - m)!}{m!} \le \left(\frac{n}{2} + \frac{1}{2}\right)^{n-2m}$ for positive integers $m, n$ with $2m \le n$.
The number of teeth in three meshed gears $A$, $B$, and $C$ are $x$, $y$, and $z$, respectively. (The teeth on all gears are the same size and regularly spaced.) The angular speeds, in revolutions per minutes of $A$, $B$, and $C$ are in the proportion
$\text{(A)} \ x: y: z ~~\text{(B)} \ z: y: x ~~ \text{(C)} \ y: z: x~~ \text{(D)} \ yz: xz: xy ~~ \text{(E)} \ xz: yx: zy$
Let $\mathcal{C}$ be a circle with center $O$, and let $A$ be a point outside the circle. Let the two tangents from the point $A$ to the circle $\mathcal{C}$ meet this circle at the points $S$ and $T$, respectively. Given a point $M$ on the circle $\mathcal{C}$ which is different from the points $S$ and $T$, let the line $MA$ meet the perpendicular from the point $S$ to the line $MO$ at $P$.
Prove that the reflection of the point $S$ in the point $P$ lies on the line $MT$.
A contest has six problems worth seven points each. On any given problem, a contestant can score either $0$, $1$, or $7$ points. How many possible total scores can a contestant achieve over all six problems?
A new meme is circling around social media known as the [i]DaDerek Convertible[/i]. The license plate number of the [i]DaDerek Convertible[/i] is such that the product of its nonzero digits times $5$ is equal to itself. Given that its license plate number has less than or equal to $3$ digits and that it has at least one nonzero digit, find the [i]DaDerek Convertible[/i]'s license plate number.
Let $ ABCD$ be an isosceles trapezoid with $ AD \parallel BC$. Its diagonals $ AC$ and $ BD$ intersect at point $ M$. Points $ X$ and $ Y$ on the segment $ AB$ are such that $ AX \equal{} AM$, $ BY \equal{} BM$. Let $ Z$ be the midpoint of $ XY$ and $ N$ is the point of intersection of the segments $ XD$ and $ YC$. Prove that the line $ ZN$ is parallel to the bases of the trapezoid.
[i]Author: A. Akopyan, A. Myakishev[/i]
Consider two circles, tangent at $T$, both inscribed in a rectangle of height $2$ and width $4$. A point $E$ moves counterclockwise around the circle on the left, and a point $D$ moves clockwise around the circle on the right. $E$ and $D$ start moving at the same time; $E$ starts at $T$, and $D$ starts at $A$, where $A$ is the point where the circle on the right intersects the top side of the rectangle. Both points move with the same speed. Find the locus of the midpoints of the segments joining $E$ and $D$.
[b](a)[/b] Consider the set of all triangles $ABC$ which are inscribed in a circle with radius $R.$ When is $AB^2+BC^2+CA^2$ maximum? Find this maximum.
[b](b)[/b] Consider the set of all tetragonals $ABCD$ which are inscribed in a sphere with radius $R.$ When is the sum of squares of the six edges of $ABCD$ maximum? Find this maximum, and in this case prove that all of the edges are equal.
$n$ people (with names $1,2,\dots,n$) are around a table. Some of them are friends. At each step 2 friend can change their place. Find a necessary and sufficient condition for friendship relation between them that with these steps we can always reach to all of posiible permutations.
The positive numbers $a, b, c,d,e$ are such that the following identity hold for all real number $x$:
$(x + a)(x + b)(x + c) = x^3 + 3dx^2 + 3x + e^3$. Find the smallest value of $d$.
Let $ABC$ be an equilateral triangle. $N$ is a point on the side $AC$ such that $\vec{AC} = 7\vec{AN}$, $M$ is a point on the side $AB$ such that $MN$ is parallel to $BC$ and $P$ is a point on the side $BC$ such that $MP$ is parallel to $AC$. Find the ratio of areas $\frac{ (MNP)}{(ABC)}$
Can the numbers from \( 1 \) to \( 2025 \) be arranged in a circle such that the difference between any two adjacent numbers has the form \( 2^k \) for some non-negative integer \( k \)? For different adjacent pairs of numbers, the values of \( k \) may be different.
[i]Proposed by Anton Trygub[/i]
Determine the volume of the body obtained by cutting the ball of radius $R$ by the trihedron with vertex in the center of that ball, it its dihedral angles are $\alpha, \beta, \gamma.$
[b]a)[/b] Let $ AA',BB',CC' $ be the altitudes of a triangle $ ABC. $ Prove that
$$ \frac{BC}{AA'}\cdot \overrightarrow{AA'} +\frac{AC}{BB'}\cdot \overrightarrow{BB'} +\frac{AB}{CC'}\cdot \overrightarrow{CC'} =0. $$
[b]b)[/b] The sum of the vectors that are perpendicular to the sides of a convex polygon and have equal lengths as those sides, respectively, is $ 0. $
Does there exist a positive odd integer $n$ so that there are primes $p_1$, $p_2$ dividing $2^n-1$ with $p_1-p_2=2$?
Given positive real numbers $x_1, x_2, \ldots , x_n$ find the polygon $A_0A_1\ldots A_n$ with $A_iA_{i+1} = x_{i+1}$ and which has greatest area.
The rows and columns of a $2^n \times 2^n$ table are numbered from $0$ to $2^{n}-1.$ The cells of the table have been coloured with the following property being satisfied: for each $0 \leq i,j \leq 2^n - 1,$ the $j$-th cell in the $i$-th row and the $(i+j)$-th cell in the $j$-th row have the same colour. (The indices of the cells in a row are considered modulo $2^n$.) Prove that the maximal possible number of colours is $2^n$.
[i]Proposed by Hossein Dabirian, Sepehr Ghazi-nezami, Iran[/i]
For positive integers $n$, let $c_n$ be the smallest positive integer for which $n^{c_n}-1$ is divisible by $210$, if such a positive integer exists, and $c_n = 0$ otherwise. What is $c_1 + c_2 + \dots + c_{210}$?
In a triangle $ ABC$, where $ a \equal{} BC$, $ b \equal{} CA$ and $ c \equal{} AB$, it is known that: $ a \plus{} b \minus{} c \equal{} 2$ and $ 2ab \minus{} c^2 \equal{} 4$. Prove that $ ABC$ is an equilateral triangle.
A chess king was placed on a square of an \(8 \times 8\) board and made $64$ moves so that it visited all squares and returned to the starting square. At every moment, the distance from the center of the square the king was on to the center of the board was calculated. A move is called $\emph{pleasant}$ if this distance becomes smaller after the move. Find the maximum possible number of pleasant moves. (The chess king moves to a square adjacent either by side or by corner.)
Let $ABC$ be a triangle such that $\angle ACB=2\angle ABC$. Let $D$ be the point on the side $BC$ such that $CD=2BD$. The segment $AD$ is extended to $E$ so that $AD=DE$. Prove that \[ \angle ECB+180^{\circ }=2\angle EBC. \]
Find all ordered pairs of integers $(x,y)$ such that $5^x = 1 + 4y + y^4$.
Call an ordered triple $(a, b, c)$ of integers feral if $b -a, c - a$ and $c - b$ are all prime.
Find the number of feral triples where $1 \le a < b < c \le 20$.
Does there exist a triangle $ ABC$ satisfying the following two conditions:
(a) ${ \sin^2A + \sin^2B + \sin^2C = \cot A + \cot B + \cot C}$
(b) $ S\ge a^2 - (b - c)^2$ where $ S$ is the area of the triangle $ ABC$.
Among the divisors of a natural number $n$, we have numbers such that when they are devided by $2013$, give us remainders $1001, 1002, \ldots, 2012$. Prove that among the divisors of the number $n^2$, there exist numbers such that when they are divided by $2013$, give us reminders $1, 2, 3, \ldots, 2012$.