This week, Coach Newton is drilling the team on conic sections — curves that arise by slicing a cone with a plane in different ways.

### Possible Parabolas

Three fixed points in the plane are specified by their coordinates: *A* = (1,0), *B* = (0, 4), and *C* with *y*-coordinate equal to 0 and *x*-coordinate equal to the answer to last week’s **Hat Trick** problem divided by 1,440. Given a parabola in a plane, a point *X* is called *interior* to the parabola if any ray originating at *X* crosses the parabola at most once.

*What is the area of the region of the plane consisting of the points X which are interior to *all* parabolas through A, B, and C?*

### Circle Crossing

Let *A, B,* and *C* be the three vertices of a triangle with side lengths 4, 5, and 6. Simone chooses a point *D* on the side (*BC*) opposite *A*, *E* on the side opposite *B*, and *F* on the side opposite *C*. None of *D, E,* or *F* coincide with *A, B,* or *C*. Simone then draws three circles, *P* through *E, F,* and *A*, *Q* through *D, F,* and *B*, and *R* through *D, E,* and *C*. She notices that each pair of these circles intersects in two points..

*What is the minimum distance between an intersection point of one pair of these circles and an intersection point of another pair of these circles?*

## Solution to week 35

**Heart Trick**. When Dariel’s teammate has finished turning over any 13 cards, some number *N* of the hearts were flipped, so there are 13 – *N* hearts face up, and the other cards that were flipped were spades, so 13 – *N* spades were flipped, so there are *N* spades face up. When Dariel flips over all of the hearts, there are then *N* hearts face up, so the number of spades face up minus the number of hearts face up is **0**.

Coach Newton suggests:

This one, you can try at home! After someone has flipped over their thirteen cards at random, say that even though you’re blindfolded, you’re going to flip over some cards to bring the hearts and spades back into balance, with an equal number of each face up, and then make good on your claim. Do the whole thing blindfolded, and start with the cards face down instead of face up, and the trick will seem even more mysterious when you succeed in balancing the hearts and the spades.

## Recent Weeks

**Week 35**: Hat Trick & Heart Trick, solutions to Test Test, Simultaneous Remainders, Different Dice, & No-Dispute Knights

**Week 34**: Different Dice & No-Dispute Knights, solution to Subtle Sequences

**Week 33**: Simultaneous Remainders & Subtle Sequences, solution to Late Show

**Week 32**: Test Test & Late Show, solutions to Talking Midpoints, Split Vote, Dream Sequence, & Alien Arithmetic

**Week 31**: Alien Arithmetic & Dream Sequence, solution to Treasure Trek

**Week 30**: Split Vote & Treasure Trek, solution to Patriotic Packing

Links to all of the puzzles and solutions are on the Complete Varsity Math page.

**Come back next week** for answers and more puzzles.

[asciimathsf]