Given $n$ points in space such that the triangle formed from any three of the points has an angle greater than $120$ degrees. Prove that the points can be labeled $1, 2, 3, \ldots, n$ so that the angle defined by $i$, $i+1$, $i+2$ is greater than $120$ degrees for $i = 1, 2, \ldots, n-2$.
Given the equation $x^{3} + a x^{2} + b x + c = 0$, the first player gives one of $a$, $b$, $c$ an integral value. Then the second player gives one of the remaining coefficients an integral value, and finally the first player gives the remaining coefficient an integral value. The first player's objective is to ensure that the equation has three integral roots (not necessarily distinct). The second player's objective is to prevent this. Who wins?
What is the smallest positive integer $a$ such that we can find integers $b$ and $c$ so that $a x^2 + b x + c$ has two distinct positive roots less than $1$?
Every city in a certain state is directly connected by air with at most three other cities in the state, but one can get from any city to any other city with at most one change of plane. What is the maximum possible number of cities?
Given positive numbers $a$, $b$, $c$, $d$ prove that at least one of the inequalities does not hold:
$a + b \lt c + d$ ;
$(a + b)(c + d) \lt ab + cd$ ;
$(a + b)cd \lt ab(c + d)$ .
$n$ is an integer. Prove that the sum of all fractions $1/rs$, where $r$ and $s$ are relatively prime integers satisfying $0 \lt r \lt s \leq n$, $r + s \gt n$, is $1/2$.
A finite sequence of 0s and 1s has the following properties: (1) for any $i \lt j$, the sequences of length 5 beginning at position $i$ and position $j$ are different; (2) if you add an additional digit at either the start or end of the sequence, then (1) no longer holds. Prove that the first 4 digits of the sequence are the same as the last 4 digits.
20 teams compete in a competition. What is the smallest number of games that must be played to ensure that given any three teams at least two play each other?
A wolf is in the center of a square field and there is a dog at each corner. The wolf can run anywhere in the field, but the dogs can only run along the sides. The dogs' speed is $\frac{3}{2}$ times the wolf's speed. The wolf can kill a single dog, but two dogs together can kill the wolf. Prove that the dogs can prevent the wolf escaping.
A regular $n$-gon is inscribed in a circle radius $R$. The distance from the center of the circle to the center of a side is $h_{n}$. Prove that $(n + 1) h_{n + 1} - n h_{n} \gt R$.
In the quadrilateral $ABCD$, $BC$ is parallel to $AD$. The point $E$ lies on the segment $AD$ and the perimeters of $ABE$, $BCE$ and $CDE$ are equal. Prove that $BC = AD / 2$.