MathDB

p1. Hannah is training for a marathon. This week, she ran

50

miles. In each of the next

8

weeks, she plans on running

5

miles more than in the previous week. How many total miles will she run over the course of her 9 weeks of training?
p2. An ant is standing at the bottom left corner of a

3

3

grid. How many ways can it get to the top right corner if it can only move up, right, and left, and it is not allowed to cross the same edge twice? https://cdn.artofproblemsolving.com/attachments/8/b/a28d64f3c14388cda81a603c0073ca60f91226.png
p3. If

56

35\%

of x, what is

55\%

x

?
p4. Diana covers a large square of area

36

with six non-overlapping smaller squares (which can have different sizes). What is the area of the largest of these six smaller squares?
p5. Find the largest value of

x

satisfying

|x^2 + 2x - 15| = |x^2 + 6x - 9|

.
p6. In the diagram below, all seven of the small rectangles are congruent. If the perimeter of the large rectangle is

65

, what is its area? https://cdn.artofproblemsolving.com/attachments/6/1/ccdac7ac6196f43ccfe91c3f117ce2439b4919.png
p7. Find the value of

x

that satisfies

\frac{x-5}{107}+\frac{x - 7}{105}+\frac{x - 9}{103}+\frac{x - 11}{101}= \frac{x - 104}{4}+\frac{x - 108}{2}

p8. Let

\vartriangle ABC

be a right triangle with hypotenuse

\overline{AC}

. Construct three squares: one with

\overline{AB}

as a side, one with

\overline{AC}

as a side, and one with

\overline{BC}

as a side. Inscribe a circle in each of the three squares. The area of the circle that is tangent to

\overline{AB}

18

, and the area of the circle that is tangent to

\overline{BC}

24

. What is the area of the circle that is tangent to

\overline{AC}

?
p9. Emma checks her email at least once every day but no more than

10

times in any

3

consecutive days. If she checked her email

25

times over the course of last week (

7

consecutive days), what is the largest number of times she could have checked it on the second day of last week?
p10.

12

balls labeled with the integers

1

through

12

, are placed in a box. Alexandra randomly takes out

3

of them, sets aside the largest, and repeats this procedure (without replacement) until there are no balls left in the box. What is the probability that the

4

balls set aside are labeled

9

10

11

, and

12

?
p11. Let

x

be the largest real number that can be expressed as

\dfrac{1}{a+\dfrac{1}{b+\dfrac{1}{c}}}

, where

a

b

, and c are all real numbers (not necessarily distinct) between

1

and

10

(inclusive). Similarly, let

y

be the smallest real number that can be expressed in the same way. Find

x - y

.
p12. Daisy finds a chalkboard with the number

4

on it. She may write more numbers on the chalkboard as follows: if any number x is on the chalkboard, she may write

x + 3

and/or

x^2 + 2

, and she may repeat this process as many times as she wants. What is the largest whole number that Daisy is not able to write on the chalkboard?
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here.

Problem Statement