$P, A, B, C,$ and $D$ are five distinct points in space such that $\angle APB = \angle BPC = \angle CPD = \angle DPA = \theta$, where $\theta$ is a given acute angle. Determine the greatest and least values of $\angle APC + \angle BPD$.
A difficult mathematical competition consisted of a Part I and a Part II with a combined total of $28$ problems. Each contestant solved $7$ problems altogether. For each pair of problems, there were exactly two contestants who solved both of them. Prove that there was a contestant who, in Part I, solved either no problems or at least four problems.
$P(x)$ is a polynomial of degree $3n$ such that
\begin{eqnarray*}
P(0) = P(3) = \cdots &=& P(3n) = 2, \\
P(1) = P(4) = \cdots &=& P(3n-2) = 1, \\
P(2) = P(5) = \cdots &=& P(3n-1) = 0, \quad\text{ and }\\
&& P(3n+1) = 730.\end{eqnarray*}
Determine $n$.
Determine whether or not there are any positive integral solutions of the simultaneous equations
\[x_1^2+x_2^2+\cdots+x_{1985}^2=y^3\\\\
x_1^3+x_2^3+\cdots+x_{1985}^3=z^2\]
with distinct integers $x_1,x_2,\cdots,x_{1985}$.
Determine each real root of
$x^4-(2\cdot10^{10}-1)x^3-x+10^{20}+10^{10}-1=0$
correct to four decimal places.
Let $A,B,C,D$ denote four points in space such that at most one of the distances $AB,AC,AD,BC,BD,CD$ is greater than $1$. Determine the maximum value of the sum of the six distances.
Let $a_1,a_2,a_3,\cdots$ be a non-decreasing sequence of positive integers. For $m\ge1$, define $b_m=\min\{n: a_n \ge m\}$, that is, $b_m$ is the minimum value of $n$ such that $a_n\ge m$. If $a_{19}=85$, determine the maximum value of
$a_1+a_2+\cdots+a_{19}+b_1+b_2+\cdots+b_{85}$.
$0\le a_1\le a_2\le a_3\le \cdots$ is an unbounded sequence of integers. Let $b_n = m$ if $a_m$ is the first member of the sequence to equal or exceed $n$. Given that $a_{19}=85$, what is the maximum possible value of $a_1+a_2+\cdots a_{19}+b_1+b_2+\cdots b_{85}$
$(\text{a})$ Do there exist 14 consecutive positive integers each of which is divisible by one or more primes $p$ from the interval $2\le p \le 11$?
$(\text{b})$ Do there exist 21 consecutive positive integers each of which is divisible by one or more primes $p$ from the interval $2\le p \le 13$?
During a certain lecture, each of five mathematicians fell asleep exactly twice. For each pair of mathematicians, there was some moment when both were asleep simultaneously. Prove that, at some moment, three of them were sleeping simultaneously.
What is the smallest integer $n$, greater than one, for which the root-mean-square of the first $n$ positive integers is an integer?
$\mathbf{Note.}$ The root-mean-square of $n$ numbers $a_1, a_2, \cdots, a_n$ is defined to be
\[\left[\frac{a_1^2 + a_2^2 + \cdots + a_n^2}n\right]^{1/2}\]
Two distinct circles $K_1$ and $K_2$ are drawn in the plane. They intersect at points $A$ and $B$, where $AB$ is the diameter of $K_1$. A point $P$ on $K_2$ and inside $K_1$ is also given.
Using only a "T-square" (i.e. an instrument which can produce a straight line joining two points and the perpendicular to a line through a point on or off the line), find a construction for two points $C$ and $D$ on $K_1$ such that $CD$ is perpendicular to $AB$ and $\angle CPD$ is a right angle.
By a partition $\pi$ of an integer $n\ge 1$, we mean here a representation of $n$ as a sum of one or more positive integers where the summands must be put in nondecreasing order. (E.g., if $n=4$, then the partitions $\pi$ are $1+1+1+1$, $1+1+2$, $1+3, 2+2$, and $4$).
For any partition $\pi$, define $A(\pi)$ to be the number of $1$'s which appear in $\pi$, and define $B(\pi)$ to be the number of distinct integers which appear in $\pi$. (E.g., if $n=13$ and $\pi$ is the partition $1+1+2+2+2+5$, then $A(\pi)=2$ and $B(\pi) = 3$).
Prove that, for any fixed $n$, the sum of $A(\pi)$ over all partitions of $\pi$ of $n$ is equal to the sum of $B(\pi)$ over all partitions of $\pi$ of $n$.
Find all solutions to $(m^2+n)(m + n^2)= (m - n)^3$, where m and n are non-zero integers.
The feet of the angle bisectors of $\Delta ABC$ form a right-angled triangle. If the right-angle is at $X$, where $AX$ is the bisector of $\angle A$, find all possible values for $\angle A$.
$X$ is the smallest set of polynomials $p(x)$ such that:
1. $p(x) = x$ belongs to $X$.
2. If $r(x)$ belongs to $X$, then $x\cdot r(x)$ and $(x + (1 - x) \cdot r(x) )$ both belong to $X$.
Show that if $r(x)$ and $s(x)$ are distinct elements of $X$, then $r(x) \neq s(x)$ for any $0 < x < 1$.
M is the midpoint of XY. The points P and Q lie on a line through Y on opposite sides of Y, such that $|XQ| = 2|MP|$ and $\frac{|XY|}2 < |MP| < \frac{3|XY|}2$. For what value of $\frac{|PY|}{|QY|}$ is $|PQ|$ a minimum?
$a_1, a_2, \cdots, a_n$ is a sequence of 0's and 1's. T is the number of triples $(a_i, a_j, a_k)$ with $ii$ with $a_j\neq a_i$. Show that $T=\sum_{i=1}^n f(i)\cdot\left(\frac{f(i)-1}2\right)$. If n is odd, what is the smallest value of T?
The repeating decimal $ 0.ab \cdots k \overline {pq \cdots u} = \dfrac {m}{n} $, where $m$ and $n$ are relatively prime integers, and there is at least one decimal before the repeating part. Show that $n$ is divisible by $2$ or $5$ (or both). (For example, $ 0.011 \overline {36} = 0.01136363636 \cdots = \dfrac {1}{88} $, and $88$ is divisible by $2$.)
The cubic polynomial $ x^3 + ax^2 + bx + c $ has real coefficients and three real roots $ r \ge s \ge t $. Show that $ k = a^2 - 3b \ge 0 $ and that $ \sqrt {k} \le r - t $.
Let $X$ be the set $ \{ 1, 2, \cdots, 20 \} $ and let $P$ be the set of all 9-element subsets of $X$. Show that for any map $ f: P \mapsto X $ we can find a 10-element subset $Y$ of $X$, such that $ f(Y - \{ k \}) \ne k $ for any $k$ in $Y$.
$ \triangle ABC $ is a triangle with incenter $I$. Show that the circumcenters of $ \triangle IAB, \triangle IBC, \triangle ICA $ lie on a circle whose center is the circumcenter of $ \triangle ABC $.
A polynomial product of the form \[(1-z)^{b_1}(1-z^2)^{b_2}(1-z^3)^{b_3}(1-z^4)^{b_4}(1-z^5)^{b_5}\cdots(1-z^{32})^{b_{32}},\]where the $b_k$ are positive integers, has the surprising property that if we multiply it out and discard all terms involving $z$ to a power larger than $32$, what is left is just $1-2z$. Determine, with proof, $b_{32}$.
For each positive integer $n$, let
\begin{eqnarray*} S_n &=& 1 + \frac 12 + \frac 13 + \cdots + \frac 1n, \\ T_n &=& S_1 + S_2 + S_3 + \cdots + S_n, \\ U_n &=& \frac{T_1}{2} + \frac{T_2}{3} + \frac{T_3}{4} + \cdots + \frac{T_n}{n+1}. \end{eqnarray*} Find, with proof, integers $0 < a, b,c, d < 1000000$ such that $T_{1988} = a S_{1989} - b$ and $U_{1988} = c S_{1989} - d$.
The 20 members of a local tennis club have scheduled exactly 14 two-person games among themselves, with each member playing in at least one game. Prove that within this schedule there must be a set of 6 games with 12 distinct players.