What is the value of

?
Problem 2
For what value of

does

?
Problem 3
The remainder can be defined for all real numbers

and

with

by
![\[\text{rem} (x ,y)=x-y\left \lfloor \frac{x}{y} \right \rfloor\]](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_tUh0UvBSrtZ5Udoz-Y34rmTFRfTrJ2uwbQ55ZUM_cYfSAntEzbzFyDGcJePNoCXmwWay-kdPpvCmB_5B_z9caYrqExrdIIgSlyb8gkP5zzD75MuAb4beHA83odj3BI8LJc84oyQUtbnuDyvhm-TrOWOYdP-jv2J6Me=s0-d)
where

denotes the greatest integer less than or equal to

. What is the value of

?
Problem 4
The mean, median, and mode of the

data values

are all equal to

. What is the value of

?
Problem 5
Goldbach's conjecture states that every even integer greater than 2 can be
written as the sum of two prime numbers (for example,

). So far, no one has been able to prove that the conjecture
is true, and no one has found a counterexample to show that the conjecture is
false. What would a counterexample consist of?
Problem 6
A triangular array of

coins has

coin in the first row,

coins in the second row,

coins in the third row, and so on up to

coins in the

th row. What is the sum of the digits of

?
Problem 7
Which of these describes the graph of

?
Problem 8
What is the area of the shaded region of the given

rectangle?
Problem 9
The five small shaded squares inside this unit square are congruent and have
disjoint interiors. The midpoint of each side of the middle square coincides
with one of the vertices of the other four small squares as shown. The common
side length is

, where

and

are positive integers. What is

?
Problem 10
Five friends sat in a movie theater in a row containing

seats, numbered

to

from left to right. (The directions "left" and "right" are
from the point of view of the people as they sit in the seats.) During the movie
Ada went to the lobby to get some popcorn. When she returned, she found that Bea
had moved two seats to the right, Ceci had moved one seat to the left, and Dee
and Edie had switched seats, leaving an end seat for Ada. In which seat had Ada
been sitting before she got up?
Problem 11
Each of the

students in a certain summer camp can either sing, dance, or
act. Some students have more than one talent, but no student has all three
talents. There are

students who cannot sing,

students who cannot dance, and

students who cannot act. How many students have two of these
talents?
Problem 12
In

,

,

, and

. Point

lies on

, and

bisects

. Point

lies on

, and

bisects

. The bisectors intersect at

. What is the ratio

:

?
Problem 13
Let

be a positive multiple of

. one red ball and

green balls are arranged in a line in random order. Let

be the probability that at least

of the green balls are on the same side of the red ball.
Observe that

and that

approaches

as

grows large. What is the sum of the digits of the least
value of

such that

?
Problem 14
Each vertex of a cube is to be labeled with an integer from

through

, with each integer being used once, in such a way that the
sum of the four numbers on the vertices of a face is the same for each face.
Arrangements that can be obtained from each other through rotations of the cube
are considered to be the same. How many different arrangements are possible?
Problem 15
Circles with centers

and

, having radii

and

, respectively, lie on the same side of line

and are tangent to

at

and

, respectively, with

between

and

. The circle with center

is externally tangent to each of the other two circles. What
is the area of triangle

?
Problem 16
The graphs of

and

are plotted on the same set of axes. How many points in the
plane with positive

-coordinates lie on two or more of the graphs?
Problem 17
Let

be a square. Let

and

be the centers, respectively, of equilateral triangles with
bases

and

each exterior to the square. What is the ratio of the area
of square

to the area of square

?
Problem 18
For some positive integer

the number

has

positive integer divisors, including

and the number

How many positive integer divisors does the number

have?
Problem 19
Jerry starts at

on the real number line. He tosses a fair coin

times. When he gets heads, he moves

unit in the positive direction; when he gets tails, he moves

unit in the negative direction. The probability that he
reaches

at some time during this process is

where

and

are relatively prime positive integers. What is

(For example, he succeeds if his sequence of tosses is

)
n
Problem 20
A binary operation

has the properties that

and that

for all nonzero real numbers

and

(Here the dot

represents the usual multiplication operation.) The solution
to the equation

can be written as

where

and

are relatively prime positive integers. What is
Problem 21
A quadrilateral is inscribed in a circle of radius

Three of the sides of this quadrilateral have length

What is the length of its fourth side?
Problem 22
How many ordered triples

of positive integers satisfy

and

?
Problem 23
Three numbers in the interval
![$\left[0,1\right]$](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_vHna5yt_9pyUjKyU_0o3EKItNsz5kLD3EFZPpA6GIGZ4Icumu_Nou7ugBSomsL75TJC9NjiNzqcb7Fr4GLCLnumfxZ-Imc7guicb3lfc4aosxwTIFLNfG42PfqGC2QHwD7PYCHAVm5i2iFurtImUNGA5OeolSTdX1fBg=s0-d)
are chosen independently and at random. What is the
probability that the chosen numbers are the side lengths of a triangle with
positive area?
Problem 24
There is a smallest positive real number

such that there exists a positive real number

such that all the roots of the polynomial

are real. In fact, for this value of

the value of

is unique. What is the value of
Problem 25
Let

be a positive integer. Bernardo and Silvia take turns writing
and erasing numbers on a blackboard as follows: Bernardo starts by writing the
smallest perfect square with

digits. Every time Bernardo writes a number, Silvia erases
the last

digits of it. Bernardo then writes the next perfect square,
Silvia erases the last

digits of it, and this process continues until the last two
numbers that remain on the board differ by at least 2. Let

be the smallest positive integer not written on the board.
For example, if

, then the numbers that Bernardo writes are

, and the numbers showing on the board after Silvia erases
are

and

, and thus

. What is the sum of the digits of

?
Aops
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