Prove that for every positive integer $n \geq 2$ the following inequality holds: $e^{n-1}n!<n^{n+\frac{1}{2}}$

Five people are standing in a straight line, and the distance between any two people is a unique positive integer number of units. Find the least possible distance between the leftmost and rightmost people in the line in units.

In an $m\times n$ table there is a policeman in cell $(1,1)$, and there is a thief in cell $(i,j)$. A move is going from a cell to a neighbor (each cell has at most four neighbors). Thief makes the first move, then the policeman moves and ... For which $(i,j)$ the policeman can catch the thief?

Let $ABCD$ be a convex quadrilateral with non-parallel sides $BC$ and $AD$. Assume that there is a point $E$ on the side $BC$ such that the quadrilaterals $ABED$ and $AECD$ are circumscribed. Prove that there is a point $F$ on the side $AD$ such that the quadrilaterals $ABCF$ and $BCDF$ are circumscribed if and only if $AB$ is parallel to $CD$.

Find the largest real number $x$ such that \[\left(\dfrac{x}{x-1}\right)^2+\left(\dfrac{x}{x+1}\right)^2=\dfrac{325}{144}.\]

For a given positive integer $n$, find the greatest positive integer $k$, such that there exist three sets of $k$ non-negative distinct integers, $A=\{x_1,x_2,\cdots,x_k\}, B=\{y_1,y_2,\cdots,y_k\}$ and $C=\{z_1,z_2,\cdots,z_k\}$ with $ x_j\plus{}y_j\plus{}z_j\equal{}n$ for any $ 1\leq j\leq k$. [size=85][color=#0000FF][Moderator edit: LaTeXified][/color][/size]

Find the sum of the first 5 positive integers $n$ such that $n^2 - 1$ is the product of 3 distinct primes.

For real number $ a$ with $ |a|>1$, evaluate $ \int_0^{2\pi} \frac{d\theta}{(a\plus{}\cos \theta)^2}$.

Consider triangle $ABC$ with $\angle A=2\angle B$. The angle bisectors from $A$ and $C$ intersect at $D$, and the angle bisector from $C$ intersects $\overline{AB}$ at $E$. If $\dfrac{DE}{DC}=\dfrac13$, compute $\dfrac{AB}{AC}$.

A regular polygon of $n$ sides is inscribed in a circle of radius $R$. The area of the polygon is $3R^2$. Then $n$ equals: ${{ \textbf{(A)}\ 8\qquad\textbf{(B)}\ 10\qquad\textbf{(C)}\ 12\qquad\textbf{(D)}\ 15}\qquad\textbf{(E)}\ 18} $

The kid cut out of grid paper with the side of the cell $1$ rectangle along the grid lines and calculated its area and perimeter. Carlson snatched his scissors and cut out of this rectangle along the lines of the grid a square adjacent to the boundary of the rectangle. - My rectangle ... - kid sobbed. - There is something strange about this figure! - Nonsense, do not mention it - Carlson said - waving his hand carelessly. - Here you see, in this figure the perimeter is the same as the area of ​​the rectangle was, and the area is the same as was the perimeter! What size square did Carlson cut out?

Find all functions $f:\mathbb{Q}^+\to\mathbb{Q}^+$ such that \[xy(f(x)-f(y))|x-f(f(y))\] holds for all positive rationals $x$, $y$ (we define that $a|b$ if and only if exist $n \in \mathbb{Z}$ such that $b=an$) [i]Proposed by supercarry & windleaf1A[/i]

A rectangular piece of paper has the side lengths $12$ and $15$. A corner is bent about as shown in the figure. Determine the area of the gray triangle. [img]https://1.bp.blogspot.com/-HCfqWF0p_eA/XzcIhnHS1rI/AAAAAAAAMYg/KfY14frGPXUvF-H6ZVpV4RymlhD_kMs-ACLcBGAsYHQ/s0/1993%2BMohr%2Bp2.png[/img]

Five real numbers of absolute values not greater than $1$ and having the sum equal to $1$ are written on the circumference of a circle. Prove that one can choose three consecutively disposed numbers $a, b, c$, such that all the sums $a + b,b + c$ and $a + b + c$ are nonnegative.

Acute triangle $ABC$ has circumcircle $\Gamma$. Let $M$ be the midpoint of $BC.$ Points $P$ and $Q$ lie on $\Gamma$ so that $\angle APM = 90^{\circ}$ and $Q \neq A$ lies on line $AM.$ Segments $PQ$ and $BC$ intersect at $S$. Suppose that $BS = 1, CS = 3, PQ = 8\sqrt{\tfrac{7}{37}},$ and the radius of $\Gamma$ is $r$. If the sum of all possible values of $r^2$ can be expressed as $\tfrac ab$ for relatively prime positive integers $a$ and $b,$ compute $100a + b$.

In trapezoid $ABCD$, diagonal $AC$ is the bisector of angle $A$. Point $K$ is the midpoint of diagonal $AC$. It is known that $DC = DK$. Find the ratio of the bases $AD: BC$.

Points $D$ and $E$ are chosen on the sides $AB$ and $AC$ of the triangle $ABC$ in such a way that if $F$ is the intersection point of $BE$ and $CD$, then $AE + EF = AD + DF$. Prove that $AC + CF = AB + BF.$

Let $O$ be a fixed point in the plane. We have $2024$ red points, $2024$ yellow points and $2024$ green points in the plane, where there isn't any three colinear points and all points are distinct from $O$. It is known that for any two colors, the convex hull of the points that are from any of those two colors contains $O$ (it maybe contain it in it's interior or in a side of it). We say that a red point, a yellow point and a green point make a [i]bolivian[/i] triangle if said triangle contains $O$ in its interior or in one of its sides. Determine the greatest positive integer $k$ such that, no matter how such points are located, there is always at least $k$ [i]bolivian[/i] triangles.

Decide whether there exists a set $M$ of positive integers satisfying the following conditions: (i) For any natural number $m>1$ there exist $a, b \in M$ such that $a+b = m.$ (ii) If $a, b, c, d \in M$, $a, b, c, d > 10$ and $a + b = c + d$, then $a = c$ or $a = d.$

Twelve balls are numbered by the numbers $1,2,3,\cdots,12$. Each ball is colored either red or green, so that the following two conditions are satisfied: (i) If two balls marked by different numbers $a$ and $b$ are colored red and $a+b<13$, then the ball marked by the number $a+b$ is colored red, too. (ii) If two balls marked by different numbers $a$ and $b$ are colored green and $a+b<13$, then the ball marked by the number $a+b$ is also colored green. How many ways are there of coloring the balls? Please remember to hide your solution. (by using the hide tags of course.. I don't literally mean that you should hide it :ninja: )

Find the derivative of the solution of the equation $\ddot{x} = \dot{x}^2 + x^3$ with initial condition $x(0) = 0$, $\dot{x}(0) = A$ with respect to $A$ for $A = 0$.

How many multiples of $20$ are also divisors of $17!$?

Let $a_1, a_2, \ldots, a_n$ ($n\ge 3$) be positive integers such that $gcd(a_1, a_2, \ldots, a_n)=1$ and for each $i\in \lbrace 1,2,\ldots, n \rbrace$ we have $a_i|a_1+a_2+\ldots+a_n$. Prove that $a_1a_2\ldots a_n | (a_1+a_2+\ldots+a_n)^{n-2}$.

Let $a_1,a_2,\ldots,a_{2n}$ be positive real numbers such that $a_ja_{n+j}=1$ for the values $j=1,2,\ldots,n$. [list] a. Prove that either the average of the numbers $a_1,a_2,\ldots,a_n$ is at least 1 or the average of the numbers $a_{n+1},a_{n+2},\ldots,a_{2n}$ is at least 1. b. Assuming that $n\ge2$, prove that there exist two distinct numbers $j,k$ in the set $\{1,2,\ldots,2n\}$ such that \[|a_j-a_k|<\frac{1}{n-1}.\] [/list]

Let $F$ be the midpoint of circle arc $AB$, and let $M$ be a point on the arc such that $AM <MB$. The perpendicular drawn from point $F$ on $AM$ intersects $AM$ at point $T$. Show that $T$ bisects the broken line $AMB$, that is $AT =TM+MB$. KöMaL Gy. 2404. (March 1987), Archimedes of Syracuse