MathDB

1

Math Prize 2014 Problem 20

z

such that

\left| z \right| < 30

satisfy the equation

e^z = \frac{z - 1}{z + 1} \, ?

19

1

Math Prize 2014 Problem 19

Let

n

be a positive integer. Let

(a, b, c)

be a random ordered triple of nonnegative integers such that

a + b + c = n

, chosen uniformly at random from among all such triples. Let

M_n

be the expected value (average value) of the largest of

a

,

b

, and

c

. As

n

approaches infinity, what value does

\frac{M_n}{n}

approach?

18

1

Math Prize 2014 Problem 18

For how many integers

k

such that

0 \le k \le 2014

is it true that the binomial coefficient

\binom{2014}{k}

is a multiple of 4?

17

1

Math Prize 2014 Problem 17

Let

ABC

be a triangle. Points

D

,

E

, and

F

are respectively on the sides

\overline{BC}

,

\overline{CA}

, and

\overline{AB}

of

\triangle ABC

. Suppose that

\frac{AE}{AC} = \frac{CD}{CB} = \frac{BF}{BA} = x

for some

x

with

\frac{1}{2} < x < 1

. Segments

\overline{AD}

,

\overline{BE}

, and

\overline{CF}

cut the triangle into 7 nonoverlapping regions: 4 triangles and 3 quadrilaterals. The total area of the 4 triangles equals the total area of the 3 quadrilaterals. Compute the value of

x

.

16

1

Math Prize 2014 Problem 16

If

\sin x + \sin y = \frac{96}{65}

and

\cos x + \cos y = \frac{72}{65}

, then what is the value of

\tan x + \tan y

?

15

1

Math Prize 2014 Problem 15

There are two math exams called A and B. 2014 students took the A exam and/or the B exam. Each student took one or both exams, so the total number of exam papers was between 2014 and 4028, inclusive. The score for each exam is an integer from 0 through 40. The average score of all the exam papers was 20. The grade for a student is the best score from one or both exams that she took. The average grade of all 2014 students was 14. Let

G

be the greatest possible number of students who took both exams. Let

L

be the least possible number of students who took both exams. Compute

G - L

.

14

1

Math Prize 2014 Problem 14

A triangle has area 114 and sides of integer length. What is the perimeter of the triangle?

13

1

Math Prize 2014 Problem 13

Deepali has a bag containing 10 red marbles and 10 blue marbles (and nothing else). She removes a random marble from the bag. She keeps doing so until all of the marbles remaining in the bag have the same color. Compute the probability that Deepali ends with exactly 3 marbles remaining in the bag.

12

1

Math Prize 2014 Problem 12

Let

B

be a

1 \times 2 \times 4

box (rectangular parallelepiped). Let

R

be the set of points that are within distance 3 of some point in

B

. (Note that

R

contains

B

.) What is the volume of

R

?

11

1

Math Prize 2014 Problem 11

Let

R

be the set of points

(x, y)

such that

\lfloor x^2 \rfloor = \lfloor y \rfloor

and

\lfloor y^2 \rfloor = \lfloor x \rfloor

. Compute the area of region

R

. Recall that

\lfloor z \rfloor

is the greatest integer that is less than or equal to

z

.

10

1

Math Prize 2014 Problem 10

An ant is on one face of a cube. At every step, the ant walks to one of its four neighboring faces with equal probability. What is the expected (average) number of steps for it to reach the face opposite its starting face?

9

1

Math Prize 2014 Problem 9

Let

abc

be a three-digit prime number whose digits satisfy

a < b < c

. The difference between every two of the digits is a prime number too. What is the sum of all the possible values of the three-digit number

abc

?

8

1

Math Prize 2014 Problem 8

A triangle has sides of length

\sqrt{13}

,

\sqrt{17}

, and

2 \sqrt{5}

. Compute the area of the triangle.

7

1

Math Prize 2014 Problem 7

If

x

is a real number and

k

is a nonnegative integer, recall that the binomial coefficient

\binom{x}{k}

is defined by the formula

\binom{x}{k} = \frac{x(x - 1)(x - 2) \dots (x - k + 1)}{k!} \, .

Compute the value of

\frac{\binom{1/2}{2014} \cdot 4^{2014}}{\binom{4028}{2014}} \, .

6

1

Math Prize 2014 Problem 6

There are

N

students in a class. Each possible nonempty group of students selected a positive integer. All of these integers are distinct and add up to 2014. Compute the greatest possible value of

N

.

5

1

Math Prize 2014 Problem 5

Say that an integer

n \ge 2

is delicious if there exist

n

positive integers adding up to 2014 that have distinct remainders when divided by

n

. What is the smallest delicious integer?

4

1

Math Prize 2014 Problem 4

Say that an integer

A

is yummy if there exist several consecutive integers (including

A

) that add up to 2014. What is the smallest yummy integer?

3

1

Math Prize 2014 Problem 3

Four different positive integers less than 10 are chosen randomly. What is the probability that their sum is odd?

2

1

Math Prize 2014 Problem 2

Let

x_1

,

x_2

, …,

x_{10}

be 10 numbers. Suppose that

x_i + 2 x_{i + 1} = 1

for each

i

from 1 through 9. What is the value of

x_1 + 512 x_{10}

?

1

Math Prize 2014 Problem 1

The four congruent circles below touch one another and each has radius 1. [asy] unitsize(30); fill(box((-1,-1), (1, 1)), gray); filldraw(circle((1, 1), 1), white); filldraw(circle((1, -1), 1), white); filldraw(circle((-1, 1), 1), white); filldraw(circle((-1, -1), 1), white); [/asy] What is the area of the shaded region?

2014 Math Prize For Girls Problems

Subcontests

Math Prize 2014 Problem 20

Math Prize 2014 Problem 19

Math Prize 2014 Problem 18

Math Prize 2014 Problem 17

Math Prize 2014 Problem 16

Math Prize 2014 Problem 15

Math Prize 2014 Problem 14

Math Prize 2014 Problem 13

Math Prize 2014 Problem 12

Math Prize 2014 Problem 11

Math Prize 2014 Problem 10

Math Prize 2014 Problem 9

Math Prize 2014 Problem 8

Math Prize 2014 Problem 7

Math Prize 2014 Problem 6

Math Prize 2014 Problem 5

Math Prize 2014 Problem 4

Math Prize 2014 Problem 3

Math Prize 2014 Problem 2

Math Prize 2014 Problem 1