San Diego Power Contest 2018-21 (SDPC) 13p

geometry problems from San Diego Power Contest , a Correspondence olympiad for members of San Diego Math Circle twice a year, with aops links in the names

aops contest collection: here

2018 - 2021

MS / HS stand for Middle  /High  School

2017-18 SDPC p3  (MS - HS)

Let $n > 2$ be a fixed positive integer. For a set $S$ of $n$ points in the plane, let $P(S)$ be the set of perpendicular bisectors of pairs of distinct points in $S$. Call set $S$ complete if no two (distinct) pairs of points share the same perpendicular bisector, and every pair of lines in $P(S)$ intersects. Let $f(S)$ be the number of distinct intersection points of pairs of lines in $P(S)$.

(a) Find all complete sets $S$ such that $f(S) = 1$.

(b) Let $S$ be a complete set with $n$ points. Show that if $f(S)>1$, then $f(S) \geq n$.

2017-18 SDPC p6 (HS)

Let $ABC$ be an acute triangle with circumcenter $O$. Let the parallel to $BC$ through $A$ intersect line $BO$ at $B_A$ and $CO$ at $C_A$. Lines $B_AC$ and $BC_A$ intersect at $A'$. Define $B'$ and $C'$ similarly.

(a) Prove that the the perpendicular from $A'$ to $BC$, the perpendicular from $B'$ to $AC$, and $C'$ to $AB$ are concurrent.

(b) Prove that likes $AA'$, $BB'$, and $CC'$ are concurrent.

2018-19 Fall SDPC p1 (MS)

An isosceles triangle $T$ has the following property: it is possible to draw a line through one of the three vertices of $T$ that splits it into two smaller isosceles triangles $R$ and $S$, neither of which are similar to $T$. Find all possible values of the vertex (apex) angle of $T$.

2018-19 Fall SDPC p7 (MS-HS)

The incircle of $\triangle{ABC}$ touches $BC$, $CA$, $AB$ at $D$, $E$, $F$, respectively. Point $P$ is chosen on $EF$ such that $AP$ is parallel to $BC$, and $AD$ intersects the incircle of $\triangle{ABC}$ again at $G$. Show that $\angle AGP = 90^{\circ}$.

2018-19 Winter SDPC p7 (HS)

In triangle $ABC$, let $D$ be on side $BC$. The line through $D$ parallel to $AB,AC$ meet $AC,AB$ at $E,F$, respectively.

(a) Show that if $D$ varies on line $BC$, the circumcircle of $AEF$ passes through a fixed point $T$.

(b) Show that if $D$ lies on line $AT$, then the circumcircle of $AEF$ is tangent to the circumcircle of $BTC$.

2019-20 Fall SDPC p4 (MS-HS)

Let $\triangle{ABC}$ be an acute, scalene triangle with orthocenter $H$, and let $AH$ meet the circumcircle of $\triangle{ABC}$ at a point $D \neq A$. Points $E$ and $F$ are chosen on $AC$ and $AB$ such that $DE \perp AC$ and $DF \perp AB$. Show that $BE$, $CF$, and the line through $H$ parallel to $EF$ concur.

2019-20 Fall SDPC p6 (MS-HS)

Let $ABCD$ be an isosceles trapezoid inscribed in circle $\omega$, such that $AD \| BC$. Point $E$ is chosen on the arc $BC$ of $\omega$ not containing $A$. Let $BC$ and $DE$ intersect at $F$. Show that if $E$ is chosen such that $EB = EC$, the area of $AEF$ is maximized.

2019-20 Winter SDPC p4 (MS-HS)

Farmer John ties his goat to a number of ropes of varying lengths in the Euclidean plane. If he ties the goat to $k$ ropes centered at $Q_1$, $Q_2$, ... $Q_k$ with lengths $\ell_1$, $\ell_2$, ... $\ell_k$ (respectively), the goat can reach any point $R$ such that $\ell_j \geq RQ_j$ for all $j \in \{1,2,3, \ldots k\}$.

Suppose that Farmer John has planted grass at a finite set of points $P_1$, $P_2$, ... $P_n$, and sets the ropes such that the goat can reach all of these points. What is, in terms of the points, the largest possible lower bound on the area of the region that the goat can reach?

2019-20 Winter SDPC p8 (HS)

Let $ABC$ be a triangle with circumcircle $\Gamma$. If the internal angle bisector of $\angle A$ meets $BC$ and $\Gamma$ at $D$ and $E$ respectively. Let $O_1$ be the center of the circle through $A$ and $D$ tangent to $BC$, let the external angle bisector of $\angle A$ meet $\Gamma$ at $F$, and let $FO_1$ meet $\Gamma$ at some point $P \neq F$. Show that the circumcircle of $DEP$ is tangent to $BC$.

2020-21 Fall SDPC p4 (MS-HS)

Let $ABC$ be an acute scalene triangle, let $D$ be a point on the $A$-altitude, and let the circle with diameter $AD$ meet $AC$, $AB$, and the circumcircle of $ABC$ at $E$, $F$, $G$, respectively. Let $O$ be the circumcenter of $ABC$, let $AO$ meet $EF$ at $T$, and suppose the circumcircles of $ABC$ and $GTO$ meet at $X \neq G$. Then, prove that $AX$, $DG$, and $EF$ concur.

2020-21 Fall SDPC p5 (MS)

Let $ABC$ be a triangle with area $1$. Let $D$ be a point on segment $BC$. Let points $E$ and $F$ on $AC$ and $AB$, respectively, satisfy $DE || AB$ and $DF || AC$. Compute, with proof, the area of the quadrilateral with vertices at $E$, $F$, the midpoint of $BD$, and the midpoint of $CD$.

2020-21 Winter SDPC p3 

Let $ABCD$ be a quadrilateral, let $P$ be the intersection of $AB$ and $CD$, and let $O$ be the intersection of the perpendicular bisectors of $AB$ and $CD$. Suppose that $O$ does not lie on line $AB$ and $O$ does not lie on line $CD$. Let $B'$ and $D'$ be the reflections of $B$ and $D$ across $OP$. Show that if $AB'$ and $CD'$ meet on $OP$, then $ABCD$ is cyclic.

2020-21 Winter SDPC p6

Let $ABC$ be an acute, scalene triangle, and let $P$ be an arbitrary point in its interior. Let $\mathcal{P}_A$ be the parabola with focus $P$ and directrix $BC$, and define $\mathcal{P}_B$ and $\mathcal{P}_C$ similarly.

(a) Show that if $Q$ is an intersection point of $\mathcal{P}_B$ and $\mathcal{P}_C$, then $P$ and $Q$ are on the same side of $AB$, and $P$ and $Q$ are on the same side of $AC$.

(b) You are given that $\mathcal{P}_B$ and $\mathcal{P}_C$ intersect at exactly two points. Let $\ell_A$ be the line between these points, and define $\ell_B$ and $\ell_C$ similarly. Show that $\ell_A$, $\ell_B$, and $\ell_C$ concur.

Note: A parabola with focus point $X$ and directrix line $\ell$ is the set of all points $Z$ that are the same distance from $X$ and $\ell$.