MathDB

2016 Algebra #10

Source:

12/24/2016

Let

a,b

and

c

be real numbers such that \begin{align*} a^2+ab+b^2&=9 \\ b^2+bc+c^2&=52 \\ c^2+ca+a^2&=49. \end{align*} Compute the value of

\dfrac{49b^2+39bc+9c^2}{a^2}

.

2016 Combo #10

Source:

12/30/2016

Kristoff is planning to transport a number of indivisible ice blocks with positive integer weights from the north mountain to Arendelle. He knows that when he reaches Arendelle, Princess Anna and Queen Elsa will name an ordered pair

(p,q)

of nonnegative integers satisfying

p + q \le 2016

. Kristoff must then give Princess Anna \emph{exactly}

p

kilograms of ice. Afterward, he must give Queen Elsa \emph{exactly}

q

kilograms of ice.
What is the minimum number of blocks of ice Kristoff must carry to guarantee that he can always meet Anna and Elsa's demands, regardless of which

p

and

q

are chosen?

2016 Geo #10

Source:

12/30/2016

The incircle of a triangle

ABC

is tangent to

BC

at

D

. Let

H

and

\Gamma

denote the orthocenter and circumcircle of

\triangle ABC

. The

B

-mixtilinear incircle, centered at

O_B

, is tangent to lines

BA

and

BC

and internally tangent to

\Gamma

. The

C

-mixtilinear incircle, centered at

O_C

, is defined similarly. Suppose that

\overline{DH} \perp \overline{O_BO_C}

,

AB = \sqrt3

and

AC = 2

. Find

BC

.

2016 Guts #10

Source:

12/24/2016

Let

ABC

be a triangle with

AB=13

,

BC=14

,

CA=15

. Let

O

be the circumcenter of

ABC

. Find the distance between the circumcenters of triangles

AOB

and

AOC

.

2016 Team #10

Source:

12/30/2016

Let

ABC

be a triangle with incenter

I

whose incircle is tangent to

\overline{BC}

,

\overline{CA}

,

\overline{AB}

at

D

,

E

,

F

. Point

P

lies on

\overline{EF}

such that

\overline{DP} \perp \overline{EF}

. Ray

BP

meets

\overline{AC}

at

Y

and ray

CP

meets

\overline{AB}

at

Z

. Point

Q

is selected on the circumcircle of

\triangle AYZ

so that

\overline{AQ} \perp \overline{BC}

.
Prove that

P

,

I

,

Q

are collinear.

incirclecollinearity

2016 General #10: Quadrilaterals

Source:

11/15/2016

Quadrilateral

ABCD

satisfies

AB = 8, BC = 5, CD = 17, DA = 10

. Let

E

be the intersection of

AC

and

BD

. Suppose

BE : ED = 1 : 2

. Find the area of

ABCD

.

HMMT

2016 Theme #10: Game with points

Source:

11/22/2016

We have

10

points on a line

A_1,A_2\ldots A_{10}

in that order. Initially there are

n

chips on point

A_1

. Now we are allowed to perform two types of moves. Take two chips on

A_i

, remove them and place one chip on

A_{i+1}

, or take two chips on

A_{i+1}

, remove them, and place a chip on

A_{i+2}

and

A_i

. Find the minimum possible value of

n

such that it is possible to get a chip on

A_{10}

through a sequence of moves.

HMMT

10

Problems(7)