2014 APMO

Part of APMO

Subcontests

(5)

1

Discerning Sets

n

b

be positive integers. We say

n

b

-discerning if there exists a set consisting of

n

different positive integers less than

b

that has no two different subsets

U

V

such that the sum of all elements in

U

equals the sum of all elements in

V

.
(a) Prove that

8

100

-discerning. (b) Prove that

9

100

-discerning.
Senior Problems Committee of the Australian Mathematical Olympiad Committee

1

x^3+x is a surjection mod n

Find all positive integers

n

such that for any integer

k

there exists an integer

a

a^3+a-k

is divisible by

n

.
Warut Suksompong, Thailand

1

Show that a circumcircle is tangent to Omega

\omega

\Omega

A

B

M

be the midpoint of the arc

AB

\omega

M

\Omega

MP

\omega

\Omega

Q

Q

\omega

\ell_P

be the tangent line to

\omega

P

\ell_Q

be the tangent line to

\Omega

Q

. Prove that the circumcircle of the triangle formed by the lines

\ell_P

\ell_Q

AB

\Omega

.
Ilya Bogdanov, Russia and Medeubek Kungozhin, Kazakhstan

1

Picking Representatives of Sets

S = \{1,2,\dots,2014\}

. For each non-empty subset

T \subseteq S

, one of its members is chosen as its representative. Find the number of ways to assign representatives to all non-empty subsets of

S

so that if a subset

D \subseteq S

is a disjoint union of non-empty subsets

A, B, C \subseteq S

, then the representative of

D

is also the representative of one of

A

B

C

.
Warut Suksompong, Thailand

1

Increasing Sums of Digits

For a positive integer

m

S(m)

P(m)

the sum and product, respectively, of the digits of

m

. Show that for each positive integer

n

, there exist positive integers

a_1, a_2, \ldots, a_n

satisfying the following conditions: S(a_1) < S(a_2) < \cdots < S(a_n) \text{ and } S(a_i) = P(a_{i+1}) (i=1,2,\ldots,n). (We let

a_{n+1} = a_1

.)
Problem Committee of the Japan Mathematical Olympiad Foundation