MathNet 1962 — Part 01

Given a number with 1998 digits which is divisible by 9. Let $x$ be the sum of its digits, let $y$ be the sum of the digits of $x$, and $z$ the sum of the digits of $y$. Find $z$.

Given a fixed circle $C$ and a line $L$ through the center $O$ of $C$. Take a variable point $P$ on $L$ and let $K$ be the circle center $P$ through $O$. Let $T$ be the point where a common tangent to $C$ and $K$ meets $K$. What is the locus of $T$?

Given four positive real numbers $a$, $b$, $c$, $d$ such that $abcd = 1$, prove that
$$
a^2 + b^2 + c^2 + d^2 + ab + ac + ad + bc + bd + cd \geq 10.
$$

The triangle $ABC$ satisfies $0 \leq AB \leq 1 \leq BC \leq 2 \leq CA \leq 3$. What is the maximum area it can have?

$AB = BC$ and $M$ is the midpoint of $AC$. $H$ is chosen on $BC$ so that $MH$ is perpendicular to $BC$. $P$ is the midpoint of $MH$. Prove that $AH$ is perpendicular to $BP$.

Given $a_0$, $a_1$, ..., $a_n$, satisfying $a_0 = a_n = 0$, and $a_{k - 1} - 2a_k + a_{k + 1} \geq 0$ for $k = 1, 2, ..., n-1$. Prove that all the numbers are negative or zero.

Given unequal integers $x$, $y$, $z$ prove that $\left(x - y\right)^{5} + \left(y - z\right)^{5} + \left(z - x\right)^{5}$ is divisible by $5(x - y)(y - z)(z - x)$.

Given the lengths $AB$ and $BC$ and the fact that the medians to those two sides are perpendicular, construct the triangle $ABC$.

Prove that there are no integers $a$, $b$, $c$, $d$ such that the polynomial $ax^3 + bx^2 + cx + d$ equals $1$ at $x = 19$ and $2$ at $x = 62$.

Given integers $a_0$, $a_1$, ..., $a_{100}$, satisfying $a_1 \gt a_0$, $a_1 \gt 0$, and $a_{r + 2} = 3 a_{r + 1} - 2 a_r$ for $r = 0$, $1$, ..., $98$. Prove $a_{100} \gt 2^{99}$.

$ABCD$ is any convex quadrilateral. Construct a new quadrilateral as follows. Take $A'$ so that $A$ is the midpoint of $DA'$; similarly, $B'$ so that $B$ is the midpoint of $AB'$; $C'$ so that $C$ is the midpoint of $BC'$; and $D'$ so that $D$ is the midpoint of $CD'$. Show that the area of $A'B'C'D'$ is five times the area of $ABCD$.

Given a fixed regular pentagon $ABCDE$ with side $1$. Let $M$ be an arbitrary point inside or on it. Let the distance from $M$ to the closest vertex be $\mathbf{r}_1$, to the next closest be $\mathbf{r}_2$ and so on, so that the distances from $M$ to the five vertices satisfy $\mathbf{r}_1 \leq \mathbf{r}_2 \leq \mathbf{r}_3 \leq \mathbf{r}_4 \leq \mathbf{r}_5$. Find


a) the locus of $M$ which gives $\mathbf{r}_3$ the minimum possible value,


b) the locus of $M$ which gives $\mathbf{r}_3$ the maximum possible value.

Given an $n \times n$ array of numbers. $n$ is odd and each number in the array is $1$ or $-1$. Prove that the number of rows and columns containing an odd number of $-1$s cannot total $n$.