2013 Online Math Open Problems

Part of Online Math Open Problems

Subcontests

(50)

1

2012-2013 Winter OMO #47

f(x,y)

be a function from ordered pairs of positive integers to real numbers such that f(1,x) = f(x,1) = \frac{1}{x} \text{and} f(x+1,y+1)f(x,y)-f(x,y+1)f(x+1,y) = 1 for all ordered pairs of positive integers

(x,y)

f(100,100) = \frac{m}{n}

for two relatively prime positive integers

m,n

m+n

1

2012-2013 Winter OMO #44

Suppose tetrahedron

PABC

420

AB = 13

BC = 14

CA = 15

. The minimum possible surface area of

PABC

can be written as

m+n\sqrt{k}

m,n,k

are positive integers and

k

is not divisible by the square of any prime. Compute

m+n+k

1

2012-2013 Winter OMO #43

In a tennis tournament, each competitor plays against every other competitor, and there are no draws. Call a group of four tennis players ``ordered'' if there is a clear winner and a clear loser (i.e., one person who beat the other three, and one person who lost to the other three.) Find the smallest integer

n

for which any tennis tournament with

n

people has a group of four tennis players that is ordered.
Ray Li

1

2012-2013 Winter OMO #42

Find the remainder when

\prod_{i=0}^{100}(1-i^2+i^4)

101

1

2012-2013 Winter OMO #41

While there do not exist pairwise distinct real numbers

a,b,c

a^2+b^2+c^2 = ab+bc+ca

, there do exist complex numbers with that property. Let

a,b,c

be complex numbers such that

a^2+b^2+c^2 = ab+bc+ca

|a+b+c| = 21

|a-b| = 2\sqrt{3}

|a| = 3\sqrt{3}

|b|^2+|c|^2

. [hide="Clarifications"]
[*] The problem should read

|a+b+c| = 21

. An earlier version of the test read

|a+b+c| = 7

; that value is incorrect. [*]

|b|^2+|c|^2

should be a positive integer, not a fraction; an earlier version of the test read ``... for relatively prime positive integers

m

n

m+n

1

2012-2013 Winter OMO #40

ABC

be a triangle with

AB=13

BC=14

AC=15

M

be the midpoint of

BC

\Gamma

be the circle passing through

A

and tangent to line

BC

M

\Gamma

intersect lines

AB

AC

D

E

, respectively, and let

N

be the midpoint of

DE

MN

intersects lines

AB

AC

P

O

, respectively. If the ratio

MN:NO:OP

can be written in the form

a:b:c

a,b,c

positive integers satisfying

\gcd(a,b,c)=1

a+b+c

1

2012-2013 Winter OMO #39

Find the number of 8-digit base-6 positive integers

(a_1a_2a_3a_4a_5a_6a_7a_8)_6

(with leading zeros permitted) such that

(a_1a_2\ldots a_8)_6\mid(a_{i+1}a_{i+2}\ldots a_{i+8})_6

i=1,2,\ldots,7

, where indices are taken modulo

8

a_9=a_1

a_{10}=a_2

, and so on).
Victor Wang

1

2012-2013 Winter OMO #38

ABC

AB = 25

BC = 30

CA=20

P,Q

be the points on segments

AB,AC

, respectively, such that

AP=5

AQ=4

. Suppose lines

BQ

CP

R

and the circumcircles of

\triangle{BPR}

\triangle{CQR}

intersect at a second point

S\ne R

. If the length of segment

SA

can be expressed in the form

\frac{m}{\sqrt{n}}

for positive integers

m,n

n

is not divisible by the square of any prime, find

m+n

1

2012-2013 Winter OMO #37

M

be a positive integer. At a party with 120 people, 30 wear red hats, 40 wear blue hats, and 50 wear green hats. Before the party begins,

M

pairs of people are friends. (Friendship is mutual.) Suppose also that no two friends wear the same colored hat to the party.
During the party,

X

Y

can become friends if and only if the following two conditions hold: [*] There exists a person

Z

X

Y

are both friends with

Z

. (The friendship(s) between

Z,X

Z,Y

could have been formed during the party.) [*]

X

Y

are not wearing the same colored hat.
Suppose the party lasts long enough so that all possible friendships are formed. Let

M_1

be the largest value of

M

such that regardless of which

M

pairs of people are friends before the party, there will always be at least one pair of people

X

Y

with different colored hats who are not friends after the party. Let

M_2

be the smallest value of

M

such that regardless of which

M

pairs of people are friends before the party, every pair of people

X

Y

with different colored hats are friends after the party. Find

M_1+M_2

. [hide="Clarifications"]
[*] The definition of

M_2

should read, ``Let

M_2

be the smallest value of

M

such that...''. An earlier version of the test read ``largest value of

M

''.
Victor Wang

1

2012-2013 Winter OMO #36

ABCD

be a nondegenerate isosceles trapezoid with integer side lengths such that

BC \parallel AD

AB=BC=CD

. Given that the distance between the incenters of triangles

ABD

ACD

8!

, determine the number of possible lengths of segment

AD

1

2012-2013 Winter OMO #35

The rows and columns of a

7\times 7

grid are each numbered

1,2,\ldots, 7

. In how many ways can one choose 8 cells of this grid such that for every two chosen cells

X

Y

, either the positive difference of their row numbers is at least

3

, or the positive difference of their column numbers is at least

3

? [hide="Clarifications"]
[*] The ``or'' here is inclusive (as by convention, despite the ``either''), i.e.

X

Y

are permitted if and only if they satisfy the row condition, the column condition, or both.
Ray Li

1

2012-2013 Winter OMO #34

For positive integers

n

s(n)

denote the sum of the squares of the positive integers less than or equal to

n

that are relatively prime to

n

. Find the greatest integer less than or equal to

\sum_{n\mid 2013} \frac{s(n)}{n^2},

where the summation runs over all positive integers

n

2013

1

2012-2013 Winter OMO #33

n

be a positive integer. E. Chen and E. Chen play a game on the

n^2

n \times n

lattice grid. They alternately mark points on the grid such that no player marks a point that is on or inside a non-degenerate triangle formed by three marked points. Each point can be marked only once. The game ends when no player can make a move, and the last player to make a move wins. Determine the number of values of

n

1

2013

(inclusive) for which the first player can guarantee a win, regardless of the moves that the second player makes.
Ray Li

1

2012-2013 Winter OMO #32

\triangle ABC

I

AB = 61

AC = 51

BC=71

. The circumcircles of triangles

AIB

AIC

BC

D

D \neq B

E

E \neq C

), respectively. Determine the length of segment

DE

1

2012-2013 Winter OMO #31

Beyond the Point of No Return is a large lake containing 2013 islands arranged at the vertices of a regular

2013

-gon. Adjacent islands are joined with exactly two bridges. Christine starts on one of the islands with the intention of burning all the bridges. Each minute, if the island she is on has at least one bridge still joined to it, she randomly selects one such bridge, crosses it, and immediately burns it. Otherwise, she stops.
If the probability Christine burns all the bridges before she stops can be written as

\frac{m}{n}

for relatively prime positive integers

m

n

, find the remainder when

m+n

1000

2

2012-2013 Winter OMO #30

Pairwise distinct points

P_1,P_2,\ldots, P_{16}

lie on the perimeter of a square with side length

4

O

\lvert P_iP_{i+1} \rvert = 1

i=1,2,\ldots, 16

P_{17}

to be the point

P_1

.) We construct points

Q_1,Q_2,\ldots,Q_{16}

as follows: for each

i

, a fair coin is flipped. If it lands heads, we define

Q_i

P_i

; otherwise, we define

Q_i

to be the reflection of

P_i

O

. (So, it is possible for some of the

Q_i

to coincide.) Let

D

be the length of the vector

\overrightarrow{OQ_1} + \overrightarrow{OQ_2} + \cdots + \overrightarrow{OQ_{16}}

. Compute the expected value of

D^2

2013-2014 Fall OMO #30

P(t) = t^3+27t^2+199t+432

a

b

c

x

are distinct positive reals such that

P(-a)=P(-b)=P(-c)=0

\sqrt{\frac{a+b+c}{x}} = \sqrt{\frac{b+c+x}{a}} + \sqrt{\frac{c+a+x}{b}} + \sqrt{\frac{a+b+x}{c}}.

x=\frac{m}{n}

for relatively prime positive integers

m

n

m+n

.
Proposed by Evan Chen

2

2012-2013 Winter OMO #29

\phi(n)

denote the number of positive integers less than or equal to

n

that are relatively prime to

n

d(n)

denote the number of positive integer divisors of

n

\phi(6) = 2

d(6) = 4

. Find the sum of all odd integers

n \le 5000

n \mid \phi(n) d(n)

2013-2014 Fall OMO #29

255

cookies, each labeled with a unique nonempty subset of

\{1,2,3,4,5,6,7,8\}

. Each day, he chooses one cookie uniformly at random out of the cookies not yet eaten. Then, he eats that cookie, and all remaining cookies that are labeled with a subset of that cookie (for example, if he chooses the cookie labeled with

\{1,2\}

, he eats that cookie as well as the cookies with

\{1\}

\{2\}

). The expected value of the number of days that Kevin eats a cookie before all cookies are gone can be expressed in the form

\frac{m}{n}

m

n

are relatively prime positive integers. Find

m + n

.
Proposed by Ray Li

2

2012-2013 Winter OMO #28

S

be the set of all lattice points

(x, y)

in the plane satisfying

|x|+|y|\le 10

P_1,P_2,\ldots,P_{2013}

be a sequence of 2013 (not necessarily distinct) points such that for every point

Q

S

, there exists at least one index

i

1\le i\le 2013

P_i = Q

. Suppose that the minimum possible value of

|P_1P_2|+|P_2P_3|+\cdots+|P_{2012}P_{2013}|

can be expressed in the form

a+b\sqrt{c}

a,b,c

are positive integers and

c

is not divisible by the square of any prime. Find

a+b+c

. (A lattice point is a point with all integer coordinates.) [hide="Clarifications"]
[*]

k = 2013

, i.e. the problem should read, ``... there exists at least one index

i

1\le i\le 2013

...''. An earlier version of the test read

1 \le i \le k

.
Anderson Wang

2013-2014 Fall OMO #28

n

denote the product of the first

2013

primes. Find the sum of all primes

p

20 \le p \le 150

\frac{p+1}{2}

is even but is not a power of

2

, and (ii) there exist pairwise distinct positive integers

a,b,c

a^n(a-b)(a-c) + b^n(b-c)(b-a) + c^n(c-a)(c-b)

is divisible by

p

p^2

.
Proposed by Evan Chen

2

2012-2013 Winter OMO #27

Geodude wants to assign one of the integers

1,2,3,\ldots,11

to each lattice point

(x,y,z)

in a 3D Cartesian coordinate system. In how many ways can Geodude do this if for every lattice parallelogram

ABCD

, the positive difference between the sum of the numbers assigned to

A

C

and the sum of the numbers assigned to

B

D

must be a multiple of

11

? (A lattice point is a point with all integer coordinates. A lattice parallelogram is a parallelogram with all four vertices lying on lattice points. Here, we say four not necessarily distinct points

A,B,C,D

form a parallelogram

ABCD

if and only if the midpoint of segment

AC

coincides with the midpoint of segment

BD

.) [hide="Clarifications"]
[*] The ``positive difference'' between two real numbers

x

y

is the quantity

|x-y|

. Note that this may be zero. [*] The last sentence was added to remove confusion about ``degenerate parallelograms.''
Victor Wang

2013-2014 Fall OMO #27

Ben has a big blackboard, initially empty, and Francisco has a fair coin. Francisco flips the coin

2013

n^{\text{th}}

n=1,2,\dots,2013

), Ben does the following if the coin flips heads: (i) If the blackboard is empty, Ben writes

n

on the blackboard. (ii) If the blackboard is not empty, let

m

denote the largest number on the blackboard. If

m^2+2n^2

is divisible by

3

m

from the blackboard; otherwise, he writes the number

n

. No action is taken when the coin flips tails. If probability that the blackboard is empty after all

2013

\frac{2u+1}{2^k(2v+1)}

u

v

k

are nonnegative integers, compute

k

.
Proposed by Evan Chen

2

2012-2013 Winter OMO #26

ABC

F

AB

CF

\angle ACB

D

E

CF

such that lines

AD,BE

are perpendicular to

CF

M

is the midpoint of

AB

ME=13

AD=15

BE=25

AC+CB

2013-2014 Fall OMO #26

ABC

be a triangle with

AB=13

AC=25

\tan A = \frac{3}{4}

. Denote the reflections of

B,C

\overline{AC},\overline{AB}

D,E

, respectively, and let

O

be the circumcenter of triangle

ABC

P

be a point such that

\triangle DPO\sim\triangle PEO

X

Y

be the midpoints of the major and minor arcs

\widehat{BC}

of the circumcircle of triangle

ABC

PX \cdot PY

.
Proposed by Michael Kural

2

2012-2013 Winter OMO #25

Positive integers

x,y,z \le 100

satisfy \begin{align*} 1099x+901y+1110z &= 59800 \\ 109x+991y+101z &= 44556 \end{align*} Compute

10000x+100y+z

2013-2014 Fall OMO #25

ABCD

be a quadrilateral with

AD = 20

BC = 13

\triangle ABC

338

and the area of

\triangle DBC

212

. Compute the smallest possible perimeter of

ABCD

.
Proposed by Evan Chen

2

2012-2013 Winter OMO #24

For a permutation

\pi

of the integers from 1 to 10, define

S(\pi) = \sum_{i=1}^{9} (\pi(i) - \pi(i+1))\cdot (4 + \pi(i) + \pi(i+1)),

\pi (i)

i

th element of the permutation. Suppose that

M

is the maximum possible value of

S(\pi)

over all permutations

\pi

of the integers from 1 to 10. Determine the number of permutations

\pi

S(\pi) = M

2013-2014 Fall OMO #24

The real numbers

a_0, a_1, \dots, a_{2013}

b_0, b_1, \dots, b_{2013}

a_{n} = \frac{1}{63} \sqrt{2n+2} + a_{n-1}

b_{n} = \frac{1}{96} \sqrt{2n+2} - b_{n-1}

for every integer

n = 1, 2, \dots, 2013

a_0 = b_{2013}

b_0 = a_{2013}

\sum_{k=1}^{2013} \left( a_kb_{k-1} - a_{k-1}b_k \right).

Proposed by Evan Chen

2

2012-2013 Winter OMO #23

A set of 10 distinct integers

S

M

be the number of nonempty subsets of

S

whose elements have an even sum. What is the minimum possible value of

M

? [hide="Clarifications"]
[*]

S

is the ``set of 10 distinct integers'' from the first sentence.
Ray Li

2013-2014 Fall OMO #23

ABCDE

be a regular pentagon, and let

F

\overline{AB}

\angle CDF=55^\circ

\overline{FC}

\overline{BE}

G

H

on the extension of

\overline{CE}

E

\angle DHE=\angle FDG

. Find the measure of

\angle GHD

, in degrees.
Proposed by David Stoner

2

2012-2013 Winter OMO #22

ABC

AB = 28

AC = 36

BC = 32

D

be the point on segment

BC

\angle BAD = \angle DAC

E

be the unique point such that

DE \parallel AB

AE

is tangent to the circumcircle of

ABC

. Find the length of segment

AE

2013-2014 Fall OMO #22

Find the sum of all integers

m

1 \le m \le 300

such that for any integer

n

n \ge 2

2013m

n^n-1

2013m

n-1

.
Proposed by Evan Chen

2

2012-2013 Winter OMO #21

Dirock has a very neat rectangular backyard that can be represented as a

32\times 32

grid of unit squares. The rows and columns are each numbered

1,2,\ldots, 32

. Dirock is very fond of rocks, and places a rock in every grid square whose row and column number are both divisible by

3

. Dirock would like to build a rectangular fence with vertices at the centers of grid squares and sides parallel to the sides of the yard such that [*] The fence does not pass through any grid squares containing rocks; [*] The interior of the fence contains exactly 5 rocks. In how many ways can this be done?
Ray Li

2013-2014 Fall OMO #21

ABC

be a triangle with

AB = 5

AC = 8

BC = 7

D

AC

AD = 5

CD = 3

I

be the incenter of triangle

ABC

E

be the intersection of the perpendicular bisectors of

\overline{ID}

\overline{BC}

DE = \frac{a\sqrt{b}}{c}

a

c

are relatively prime positive integers, and

b

is a positive integer not divisible by the square of any prime. Find

a+b+c

.
Proposed by Ray Li

2

2012-2013 Winter OMO #20

a_1,a_2,\ldots, a_{2013}

be a permutation of the numbers from

1

2013

A_n = \frac{a_1 + a_2 + \cdots + a_n} {n}

n = 1,2,\ldots, 2013

. If the smallest possible difference between the largest and smallest values of

A_1,A_2,\ldots, A_{2013}

\frac{m}{n}

m

n

are relatively prime positive integers, find

m+n

2013-2014 Fall OMO #20

A positive integer

n

is called mythical if every divisor of

n

is two less than a prime. Find the unique mythical number with the largest number of divisors.
Proposed by Evan Chen

2

2012-2013 Winter OMO #19

A,B,C

are points in the plane such that

\angle ABC=90^\circ

. Circles with diameters

BA

BC

D

BA=20

BC=21

, then the length of segment

BD

can be expressed in the form

\frac{m}{n}

m

n

are relatively prime positive integers. What is

m+n

2013-2014 Fall OMO #19

\sigma(n)

be the number of positive divisors of

n

\operatorname{rad} n

be the product of the distinct prime divisors of

n

. By convention,

\operatorname{rad} 1 = 1

. Find the greatest integer not exceeding

100\left(\sum_{n=1}^{\infty}\frac{\sigma(n)\sigma(n \operatorname{rad} n)}{n^2\sigma(\operatorname{rad} n)}\right)^{\frac{1}{3}}.

Proposed by Michael Kural

2

2012-2013 Winter OMO #18

Determine the absolute value of the sum

\lfloor 2013\sin{0^\circ} \rfloor + \lfloor 2013\sin{1^\circ} \rfloor + \cdots + \lfloor 2013\sin{359^\circ} \rfloor,

\lfloor x \rfloor

denotes the greatest integer less than or equal to

x

.
(You may use the fact that

\sin{n^\circ}

is irrational for positive integers

n

not divisible by

30

2013-2014 Fall OMO #18

n\times n

grid of dots, let

f(n)

be the largest number of segments between adjacent dots which can be drawn such that (i) at most one segment is drawn between each pair of dots, and (ii) each dot has

1

3

segments coming from it. (For example,

f(4)=16

f(2000)

.
Proposed by David Stoner

2

2012-2013 Winter OMO #17

Determine the number of ordered pairs of positive integers

(x,y)

y < x \le 100

x^2-y^2

x^3 - y^3

are relatively prime. (Two numbers are relatively prime if they have no common factor other than

1

2013-2014 Fall OMO #17

ABXC

be a parallelogram. Points

K,P,Q

\overline{BC}

in this order such that

BK = \frac{1}{3} KC

BP = PQ = QC = \frac{1}{3} BC

XP

XQ

\overline{AB}

\overline{AC}

D

E

, respectively. Suppose that

\overline{AK} \perp \overline{BC}

EK-DK=9

BC=60

AB+AC

.
Proposed by Evan Chen

2

2012-2013 Winter OMO #16

S_1

S_2

be two circles intersecting at points

A

B

C

D

S_1

S_2

respectively such that line

CD

is tangent to both circles and

A

is closer to line

CD

B

\angle BCA = 52^\circ

\angle BDA = 32^\circ

, determine the degree measure of

\angle CBD

2013-2014 Fall OMO #16

Al has the cards

1,2,\dots,10

in a row in increasing order. He first chooses the cards labeled

1

2

3

, and rearranges them among their positions in the row in one of six ways (he can leave the positions unchanged). He then chooses the cards labeled

2

3

4

, and rearranges them among their positions in the row in one of six ways. (For example, his first move could have made the sequence

3,2,1,4,5,\dots,

and his second move could have rearranged that to

2,4,1,3,5,\dots

.) He continues this process until he has rearranged the cards with labels

8

9

10

. Determine the number of possible orderings of cards he can end up with.
Proposed by Ray Li

2

2012-2013 Winter OMO #15

a_1, a_2, ..., a_{13}

of the numbers from

1

13

is given such that

a_i > 5

i=1,2,3,4,5

. Determine the maximum possible value of

a_{a_1} + a_{a_2} + a_{a_3} + a_{a_4} + a_{a_5}.

2013-2014 Fall OMO #15

Find the positive integer

n

\underbrace{f(f(\cdots f}_{2013 \ f\text{'s}}(n)\cdots ))=2014^2+1

f(n)

n

th positive integer which is not a perfect square.
Proposed by David Stoner

2

2012-2013 Winter OMO #14

What is the smallest perfect square larger than

1

with a perfect square number of positive integer factors?
Ray Li

2013-2014 Fall OMO #14

In the universe of Pi Zone, points are labeled with

2 \times 2

arrays of positive reals. One can teleport from point

M

M'

M

can be obtained from

M'

by multiplying either a row or column by some positive real. For example, one can teleport from

\left( \begin{array}{cc} 1 & 2 \\ 3 & 4 \end{array} \right)

\left( \begin{array}{cc} 1 & 20 \\ 3 & 40 \end{array} \right)

\left( \begin{array}{cc} 1 & 20 \\ 6 & 80 \end{array} \right)

.
A tourist attraction is a point where each of the entries of the associated array is either

1

2

4

8

16

. A company wishes to build a hotel on each of several points so that at least one hotel is accessible from every tourist attraction by teleporting, possibly multiple times. What is the minimum number of hotels necessary?
Proposed by Michael Kural

2

2012-2013 Winter OMO #13

There are three flies of negligible size that start at the same position on a circular track with circumference 1000 meters. They fly clockwise at speeds of 2, 6, and

k

meters per second, respectively, where

k

is some positive integer with

7\le k \le 2013

. Suppose that at some point in time, all three flies meet at a location different from their starting point. How many possible values of

k

are there?
Ray Li

2013-2014 Fall OMO #13

In the rectangular table shown below, the number

1

is written in the upper-left hand corner, and every number is the sum of the any numbers directly to its left and above. The table extends infinitely downwards and to the right.

\begin{array}{cccccc} 1 & 1 & 1 & 1 & 1 & \cdots \\ 1 & 2 & 3 & 4 & 5 & \cdots \\ 1 & 3 & 6 & 10 & 15 & \cdots \\ 1 & 4 & 10 & 20 & 35 & \cdots \\ 1 & 5 & 15 & 35 & 70 & \cdots \\ \vdots & \vdots & \vdots & \vdots & \vdots & \ddots \end{array}

Wanda the Worm, who is on a diet after a feast two years ago, wants to eat

n

numbers (not necessarily distinct in value) from the table such that the sum of the numbers is less than one million. However, she cannot eat two numbers in the same row or column (or both). What is the largest possible value of

n

?
Proposed by Evan Chen

2

2012-2013 Winter OMO #12

There are 25 ants on a number line; five at each of the coordinates

1

2

3

4

5

. Each minute, one ant moves from its current position to a position one unit away. What is the minimum number of minutes that must pass before it is possible for no two ants to be on the same coordinate?
Ray Li

2013-2014 Fall OMO #12

a_n

denote the remainder when

(n+1)^3

n^3

; in particular,

a_1=0

. Compute the remainder when

a_1+a_2+\dots+a_{2013}

1000

.
Proposed by Evan Chen

2

2012-2013 Winter OMO #11

A

B

C

be distinct points on a line with

AB=AC=1

ABDE

and equilateral triangle

ACF

are drawn on the same side of line

BC

. What is the degree measure of the acute angle formed by lines

EC

BF

2013-2014 Fall OMO #11

Four orange lights are located at the points

(2,0)

(4,0)

(6,0)

(8,0)

xy

-plane. Four yellow lights are located at the points

(1,0)

(3,0)

(5,0)

(7,0)

. Sparky chooses one or more of the lights to turn on. In how many ways can he do this such that the collection of illuminated lights is symmetric around some line parallel to the

y

-axis?
Proposed by Evan Chen

2

2012-2013 Winter OMO #10

At certain store, a package of 3 apples and 12 oranges costs 5 dollars, and a package of 20 apples and 5 oranges costs 13 dollars. Given that apples and oranges can only be bought in these two packages, what is the minimum nonzero amount of dollars that must be spent to have an equal number of apples and oranges?
Ray Li

2013-2014 Fall OMO #10

In convex quadrilateral

AEBC

\angle BEA = \angle CAE = 90^{\circ}

AB = 15

BC = 14

CA = 13

D

be the foot of the altitude from

C

\overline{AB}

CD

\overline{AE}

F

AE \cdot AF

.
Proposed by David Stoner

2

2012-2013 Winter OMO #9

David has a collection of 40 rocks, 30 stones, 20 minerals and 10 gemstones. An operation consists of removing three objects, no two of the same type. What is the maximum number of operations he can possibly perform?
Ray Li

2013-2014 Fall OMO #9

AXYZB

be a regular pentagon with area

5

inscribed in a circle with center

O

Y'

denote the reflection of

Y

\overline{AB}

C

is the center of a circle passing through

A

Y'

B

. Compute the area of triangle

ABC

.
Proposed by Evan Chen

2

2012-2013 Winter OMO #8

How many ways are there to choose (not necessarily distinct) integers

a,b,c

\{1,2,3,4\}

a^{(b^c)}

is divisible by

4

2013-2014 Fall OMO #8

x_1 < x_2 < \dots < x_n

is a sequence of positive integers such that

x_k

x_{k+2}

k = 1, 2, \dots, n-2

x_n = 1000

, what is the largest possible value of

n

?
Proposed by Evan Chen

2

2012-2013 Winter OMO #7

Jacob's analog clock has 12 equally spaced tick marks on the perimeter, but all the digits have been erased, so he doesn't know which tick mark corresponds to which hour. Jacob takes an arbitrary tick mark and measures clockwise to the hour hand and minute hand. He measures that the minute hand is 300 degrees clockwise of the tick mark, and that the hour hand is 70 degrees clockwise of the same tick mark. If it is currently morning, how many minutes past midnight is it?
Ray Li

2013-2014 Fall OMO #7

M

N

P

are selected on sides

\overline{AB}

\overline{AC}

\overline{BC}

, respectively, of triangle

ABC

. Find the area of triangle

MNP

AM=MB=BP=15

AN=NC=CP=25

.
Proposed by Evan Chen

2

2012-2013 Winter OMO #6

S_1

5

S_2

7

and has its center lying on

S_1

S_3

has an integer radius and has its center lying on

S_2

. If the center of

S_1

S_3

, how many possible values are there for the radius of

S_3

2013-2014 Fall OMO #6

Find the number of integers

n

n \ge 2

such that the remainder when

2013

n

is equal to the remainder when

n

3

.
Proposed by Michael Kural

2

2012-2013 Winter OMO #5

At the Mountain School, Micchell is assigned a submissiveness rating of

3.0

4.0

for each class he takes. His college potential is then defined as the average of his submissiveness ratings over all classes taken. After taking 40 classes, Micchell has a college potential of

3.975

. Unfortunately, he needs a college potential of at least

3.995

to get into the [url=http://en.wikipedia.org/wiki/Accepted#Plot]South Harmon Institute of Technology. Otherwise, he becomes a rock. Assuming he receives a submissiveness rating of

4.0

in every class he takes from now on, how many more classes does he need to take in order to get into the South Harmon Institute of Technology?
Victor Wang

2013-2014 Fall OMO #5

A wishing well is located at the point

(11,11)

xy

-plane. Rachelle randomly selects an integer

y

\left\{ 0, 1, \dots, 10 \right\}

. Then she randomly selects, with replacement, two integers

a,b

\left\{ 1,2,\dots,10 \right\}

. The probability the line through

(0,y)

(a,b)

passes through the well can be expressed as

\frac mn

m

n

are relatively prime positive integers. Compute

m+n

.
Proposed by Evan Chen

2

2012-2013 Winter OMO #4

For how many ordered pairs of positive integers

(a,b)

a,b<1000

is it true that

a

b

b^2

a

3

9

9^2

3

2013-2014 Fall OMO #4

a_1, a_2, a_3, \dots

is an increasing arithmetic progression of positive integers. Given that

a_3 = 13

, compute the maximum possible value of

a_{a_1} + a_{a_2} + a_{a_3} + a_{a_4} + a_{a_5}.

Proposed by Evan Chen

2

2012-2013 Winter OMO #3

m

n

\ell

lie in a plane such that no two are parallel. Lines

m

n

meet at an acute angle of

14^{\circ}

m

\ell

meet at an acute angle of

20^{\circ}

. Find, in degrees, the sum of all possible acute angles formed by lines

n

\ell

2013-2014 Fall OMO #3

A palindromic table is a

3 \times 3

array of letters such that the words in each row and column read the same forwards and backwards. An example of such a table is shown below. \begin{array}[h]{ccc} O & M & O \\ N & M & N \\ O & M & O \end{array} How many palindromic tables are there that use only the letters

O

M

? (The table may contain only a single letter.)
Proposed by Evan Chen

2

2012-2013 Winter OMO #2

123454321

is written on a blackboard. Evan walks by and erases some (but not all) of the digits, and notices that the resulting number (when spaces are removed) is divisible by

9

. What is the fewest number of digits he could have erased?
Ray Li

2013-2014 Fall OMO #2

The figure below consists of several unit squares,

M

of which are white and

N

of which are green. Compute

100M+N

. [asy] size(4cm); int N = 4; path square; for (int x=-N; x<=N; ++x) { for (int y=-N+abs(x); y<=N-abs(x); ++y) { square = rotate(9)*((x+0.5,y+0.5)--(x+0.5,y-0.5)--(x-0.5,y-0.5)--(x-0.5,y+0.5)--cycle); if ((x+y) % 2 == 0) { filldraw(square, green, black); } else { filldraw(square, white, black); } } } [/asy]
Proposed by Evan Chen

2

2012-2013 Winter OMO #1

x

be the answer to this problem. For what real number

a

is the answer to this problem also

a-x

2013-2014 Fall OMO #1

Determine the value of

142857 + 285714 + 428571 + 571428.

Proposed by Ray Li

1

2012-2013 Winter OMO #46

ABC

be a triangle with

\angle B - \angle C = 30^{\circ}

D

be the point where the

A

-excircle touches line

BC

O

the circumcenter of triangle

ABC

X,Y

the intersections of the altitude from

A

with the incircle with

X

A

Y

. Suppose points

A

O

D

are collinear. If the ratio

\frac{AO}{AX}

can be expressed in the form

\frac{a+b\sqrt{c}}{d}

for positive integers

a,b,c,d

\gcd(a,b,d)=1

c

not divisible by the square of any prime, find

a+b+c+d

1

2012-2013 Winter OMO #48

\omega

is a complex number such that

\omega^{2013} = 1

\omega^m \neq 1

m=1,2,\ldots,2012

. Find the number of ordered pairs of integers

(a,b)

1 \le a, b \le 2013

\frac{(1 + \omega + \cdots + \omega^a)(1 + \omega + \cdots + \omega^b)}{3}

is the root of some polynomial with integer coefficients and leading coefficient

1

. (Such complex numbers are called algebraic integers.)
Victor Wang

1

2012-2013 Winter OMO #49

\triangle ABC

CA=1960\sqrt{2}

CB=6720

\angle C = 45^{\circ}

K

L

M

BC

CA

AB

AK \perp BC

BL \perp CA

AM=BM

N

O

P

KL

BA

BL

AN=KN

BO=CO

A

NP

H

is the orthocenter of

\triangle MOP

HK^2

. [hide="Clarifications"]
[*] Without further qualification, ``

XY

'' denotes line

XY

1

2012-2013 Winter OMO #45

N

denote the number of ordered 2011-tuples of positive integers

(a_1,a_2,\ldots,a_{2011})

1\le a_1,a_2,\ldots,a_{2011} \le 2011^2

such that there exists a polynomial

f

4019

satisfying the following three properties: [*]

f(n)

is an integer for every integer

n

2011^2 \mid f(i) - a_i

i=1,2,\ldots,2011

2011^2 \mid f(n+2011) - f(n)

for every integer

n

. Find the remainder when

N

1000

1

2012-2013 Winter OMO #50

S

denote the set of words

W = w_1w_2\ldots w_n

n\ge0

(including the empty string

\lambda

), with each letter

w_i

\{x,y,z\}

. Call two words

U,V

similar if we can insert a string

s\in\{xyz,yzx,zxy\}

of three consecutive letters somewhere in

U

(possibly at one of the ends) to obtain

V

or somewhere in

V

(again, possibly at one of the ends) to obtain

U

, and say a word

W

is trivial if for some nonnegative integer

m

, there exists a sequence

W_0,W_1,\ldots,W_m

W_0=\lambda

is the empty string,

W_m=W

W_i,W_{i+1}

are similar for

i=0,1,\ldots,m-1

. Given that for two relatively prime positive integers

p,q

\frac{p}{q} = \sum_{n\ge0} f(n)\left(\frac{225}{8192}\right)^n,

f(n)

denotes the number of trivial words in

S

3n

(in particular,

f(0)=1

p+q