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## PROBLEMS FOR DECEMBER

Edward J. Barbeau
Department of Mathematics
University of Toronto
Toronto, ON M5S 3G3

Notes. An isosceles tetrahedron is one for which the three pairs of oppposite edges are equal. For integers $a$, $b$ and $n$, $a\equiv b$, modulo $n$, iff $a-b$ is a multiple of $n$.

192.
Let $\mathrm{ABC}$ be a triangle, $D$ be the midpoint of $\mathrm{AB}$ and $E$ a point on the side $\mathrm{AC}$ for which $\mathrm{AE}=2\mathrm{EC}$. Prove that $\mathrm{BE}$ bisects the segment $\mathrm{CD}$.

193.
Determine the volume of an isosceles tetrahedron for which the pairs of opposite edges have lengths $a$, $b$, $c$. Check your answer independently for a regular tetrahedron.

194.
Let $\mathrm{ABC}$ be a triangle with incentre $I$. Let $M$ be the midpoint of $\mathrm{BC}$, $U$ be the intersection of $\mathrm{AI}$ produced with $\mathrm{BC}$, $D$ be the foot of the perpendicular from $I$ to $\mathrm{BC}$ and $P$ be the foot of the perpendicular from $A$ to $\mathrm{BC}$. Prove that

$‖\mathrm{PD}‖‖\mathrm{DM}‖=‖\mathrm{DU}‖‖\mathrm{PM}‖ .$

195.
Let $\mathrm{ABCD}$ be a convex quadrilateral and let the midpoints of $\mathrm{AC}$ and $\mathrm{BD}$ be $P$ and $Q$ respectively, Prove that

$‖\mathrm{AB}‖{}^{2}+‖\mathrm{BC}‖{}^{2}+‖\mathrm{CD}‖{}^{2}+‖\mathrm{DA}‖{}^{2}=‖\mathrm{AC}‖{}^{2}+‖\mathrm{BD}‖{}^{2}+4‖\mathrm{PQ}‖{}^{2} .$

196.
Determine five values of $p$ for which the polynomial ${x}^{2}+2002x-1002p$ has integer roots.

197.
Determine all integers $x$ and $y$ that satisfy the equation ${x}^{3}+9\mathrm{xy}+127={y}^{3}$.

198.
Let $p$ be a prime number and let $f\left(x\right)$ be a polynomial of degree $d$ with integer coefficients such that $f\left(0\right)=0$ and $f\left(1\right)=1$ and that, for every positive integer $n$, $f\left(n\right)\equiv 0$ or $f\left(n\right)\equiv 1$, modulo $p$. Prove that $d\ge p-1$. Give an example of such a polynomial.

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