MathDB

1

Number of functions

p\ge 5

be a prime number and

n

be a natural number. Let

f

be a function

f: \mathbb{Z_{ \neq }}_0 \rightarrow \mathbb{ N }_0

satisfy the following conditions:
i) For all sequences of integers satisfy

a_i \not\in \{0, 1\}

, and

p

\not |

a_i-1

,

\forall

1 \le i \le p-2

,\\

\displaystyle \sum^{p-2}_{i=1}f(a_i)=f(a_1a_2 \cdots a_{p-2})

ii) For all coprime integers

a

and

b

,

a \equiv b \pmod p \Rightarrow f(a)=f(b)

iii) There exist

k \in \mathbb{Z}_{\neq 0}

that satisfy

f(k)=n

Prove that the number of such functions is

d(n)

, where

d(n)

denotes the number of divisors of

n

.

N7

1

Function

Find all functions

f:\mathbb{N} \rightarrow \mathbb{ N }_0

satisfy the following conditions:
i)

f(ab)=f(a)+f(b)-f(\gcd(a,b)), \forall a,b \in \mathbb{N}

ii) For all primes

p

and natural numbers

a

,

f(a)\ge f(ap) \Rightarrow f(a)+f(p) \ge f(a)f(p)+1

N6

1

Sum of Reciprocals of Products of Pairs of Primes

Let

p_i

denote the

i

-th prime number. Let

n = \lfloor\alpha^{2015}\rfloor

, where

\alpha

is a positive real number such that

2 < \alpha < 2.7

. Prove that

\displaystyle\sum_{2 \le p_i \le p_j \le n}\frac{1}{p_ip_j} < 2017

N5

1

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

Let

a,b,c

be pairwise coprime positive integers. Find all positive integer values of

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

N4

1

Diophantine Equation

Determine all triplet of non-negative integers

(x,y,z)

satisfy

2^x3^y+1=7^z

N2

1

Two sets

Let

\mathbb{A} \subset \mathbb{N}

such that all elements in

\mathbb{A}

can be representable in the form of

x^2+2y^2

,

x,y \in \mathbb{N}

, and

x>y

. Let

\mathbb{B} \subset \mathbb{N}

such that all elements in

\mathbb{B}

can be representable in the form of

\displaystyle \frac{a^3+b^3+c^3}{a+b+c}

,

a,b,c \in \mathbb{N}

, and

a,b,c

are distinct.
a) Prove that

\mathbb{A} \subset \mathbb{B}

.
b) Prove that there exist infinitely many positive integers

n

satisfy

n \in \mathbb{B}

and

n \not \in \mathbb{A}

N1

1

Infinite sequence with weird conditions

Prove that there exists an infinite sequence of positive integers

a_1, a_2, ...

such that for all positive integers

i

, \\ i)

a_{i + 1}

is divisible by

a_{i}

.\\ ii)

a_i

is not divisible by

3

.\\ iii)

a_i

is divisible by

2^{i + 2}

but not

2^{i + 3}

.\\ iv)

6a_i + 1

is a prime power.\\ v)

a_i

can be written as the sum of the two perfect squares.

G8

1

Circumcircles tangent

Let

ABCDE

be a convex pentagon such that

BC

and

DE

are tangent to the circumcircle of

ACD

. Prove that if the circumcircles of

ABC

and

ADE

intersect at the midpoint of

CD

, then the circumcircles

ABE

and

ACD

are tangent to each other.

G7

1

Prove that there exists $6$ pairs of tangent circles

Let

ABC

be an acute triangle. Let

H_A,H_B,H_C

be points on

BC,AC,AB

respectively such that

AH_A\perp BC, BH_B\perp AC, CH_C\perp AB

. Let the circumcircles

AH_BH_C,BH_AH_C,CH_AH_B

be

\omega_A,\omega_B,\omega_C

with circumcenters

O_A,O_B,O_C

respectively and define

O_AB\cap \omega_B=P_{AB}\neq B

. Define

P_{AC},P_{BA},P_{BC},P_{CA},P_{CB}

similarly. Define circles

\omega_{AB},\omega_{AC}

to be

O_AP_{AB}H_C,O_AP_{AC}H_B

respectively. Define circles

\omega_{BA},\omega_{BC},\omega_{CA},\omega_{CB}

similarly.
Prove that there are

6

pairs of tangent circles in the

6

circles of the form

\omega_{xy}

.

G6

1

BE, CF, PR concurrent

Let

ABC

be a triangle. Let

\omega_1

be circle tangent to

BC

at

B

and passes through

A

. Let

\omega_2

be circle tangent to

BC

at

C

and passes through

A

. Let

\omega_1

and

\omega_2

intersect again at

P \neq A

. Let

\omega_1

intersect

AC

again at

E\neq A

, and let

\omega_2

intersect

AB

again at

F\neq A

. Let

R

be the reflection of

A

about

BC

, Prove that lines

BE, CF, PR

are concurrent.

G4

1

Inequality with concurrent cevians

Let

ABC

be a triangle and let

AD, BE, CF

be cevians of the triangle which are concurrent at

G

. Prove that if

CF \cdot BE \ge AF \cdot EC + AE \cdot BF + BC \cdot FE

then

AG \le GD

.

G3

1

A, P, Q are collinear

Let

ABC

a triangle. Let

D

on

AB

and

E

on

AC

such that

DE||BC

. Let line

DE

intersect circumcircle of

ABC

at two distinct points

F

and

G

so that line segments

BF

and

CG

intersect at P. Let circumcircle of

GDP

and

FEP

intersect again at

Q

. Prove that

A, P, Q

are collinear.

G2

1

Intersection lies on AM

Let

ABC

be a triangle, and let

M

be midpoint of

BC

. Let

I_b

and

I_c

be incenters of

AMB

and

AMC

. Prove that the second intersection of circumcircles of

ABI_b

and

ACI_c

distinct from

A

lies on line

AM

.

G1

1

OEHF is parallelogram

Given a triangle

ABC

, and let

E

and

F

be the feet of altitudes from vertices

B

and

C

to the opposite sides. Denote

O

and

H

be the circumcenter and orthocenter of triangle

ABC

. Given that

FA=FC

, prove that

OEHF

is a parallelogram.

C8

1

Permutations

Let

a

be a permutation on

\{0,1,\ldots ,2015\}

and

b,c

are also permutations on

\{1,2,\ldots ,2015\}

. For all

x\in \{1,2,\ldots ,2015\}

, the following conditions are satisfied:
(i)

a(x)-a(x-1)\neq 1

,\\ (ii) if

b(x)\neq x

, then

c(x)=x

,\\
Prove that the number of

a

's is equal to the number of ordered pairs of

(b,c)

.

C6

1

Students and Subjects

In a massive school which has

m

students, and each student took at least one subject. Let

p

be an odd prime. Given that:
(i) each student took at most

p+1

subjects. \\ (ii) each subject is taken by at most

p

students. \\ (iii) any pair of students has at least

1

subject in common. \\
Find the maximum possible value of

m

.

C5

1

Switching Points

Let

G

be a simple connected graph. Each edge has two phases, which is either blue or red. Each vertex are switches that change the colour of every edge that connects the vertex. All edges are initially red. Find all ordered pairs

(n,k)

,

n\ge 3

, such that:
a) For all graph

G

with

n

vertex and

k

edges, it is always possible to perform a series of switching process so that all edges are eventually blue.
b) There exist a graph

G

with

n

vertex and

k

edges and it is possible to perform a series of switching process so that all edges are eventually blue.

C4

1

Coloring on a plane

Nikees has a set

S

of

n

points on a plane and decides to colour them. All

\dbinom{n}{2}

line segments are drawn and they have distinct lengths. Find the maximum number of colours that are used at least once, given that:
(a) For each point

P

, the two endpoints of the longest line segment connecting

P

must be of the same colour.
(b) For each point

P

, the two endpoints of the shortest line segment connecting

P

must be of the same colour.

C3

1

2 Functions

Let

n\ge 2

be a positive integer and

S= \{1,2,\cdots ,n\}

. Let two functions

f:S \rightarrow \{1,-1\}

and

g:S \rightarrow S

satisfy:
i)

f(x)f(y)=f(x+y) , \forall x,y \in S

\\ ii)

f(g(x))=f(x) , \forall x \in S

\\ iii)

f(x+n)=f(x) ,\forall x \in S

\\ iv)

g

is bijective.\\
Find the number of pair of such functions

(f,g)

for every

n

.

C2

1

Cauchy the magician

Cauchy the magician has a new card trick. He takes a standard deck(which consists of 52 cards with 13 denominations in each 4 suits) and let Schwartz to shuffle randomly. Schwartz is told to take

m

cards not more than

\frac{1}{3}

form the top of the deck. Then, Cauchy take

18

cards one by one from the top of the remaining deck and show it to Schwartz with the second card is placed in front of the first card (from Schwartz view) and so on. He ask Schwartz to memorize the

m-th

card when showing the cards. Let it be

C_1

. After that, Cauchy places the

18

cards and the

m

cards on the bottom of the deck with the

m

cards are placed lower than the

18

cards. Now, Cauchy distributes and flip the cards on the table from the top of the deck while shouting the numbers

10

until

1

with the following operation:
a) When a card flipped has the number which is same as the number shouted by Cauchy, stop the distribution and continue with another set.\\ b) When

10

cards are flipped and none of the cards flipped has the number which is same as the number shouted by Cauchy, take a card from the top of the deck and place it on top of the set with backside(the site which has no value) facing up. Then continue with another set.\\
Cauchy stops when 3 sets of cards are placed. Then, he adds up all the numbers on top of each sets of cards( backside is consider

0

). Let

k

be the sum. He placed another

k

cards to the table from the top of the remaining deck. Finally, he shows the first card on top of the remaining deck to Schwartz. Let it be

C_2

.
Show that

C_1 = C_2

.

A9

1

Inequality + Sequence

Let

2n

positive reals

a_1, a_2, \cdots, a_n, b_1, b_2, \cdots, b_n

satisfy

a_{i+1}\ge 2a_i

and

b_{i+1} \le b_i

for

1\le i\le n-1

. Find the least constant

C

that satisfy:

\displaystyle \sum^{n}_{i=1}{\frac{a_i}{b_i}} \ge \displaystyle \frac{C(a_1+a_2+\cdots+a_n)}{b_1+b_2+\cdots+b_n}

and determine all equality case with that constant

C

.

A8

1

Sequence

Let

a_1,a_2, \cdots ,a_{2015}

be

2015

-tuples of positive integers (not necessary distinct) and let

k

be a positive integers. Denote

\displaystyle f(i)=a_i+\frac{a_1a_2 \cdots a_{2015}}{a_i}

.
a) Prove that if

k=2015^{2015}

, there exist

a_1, a_2, \cdots , a_{2015}

such that

f(i)= k

for all

1\le i\le 2015

.\\ b) Find the maximum

k_0

so that for

k\le k_0

, there are no

k

such that there are at least

2

different

2015

-tuples which fulfill the above condition.

A7

1

$\frac{a^3 + bc}{a^2 + bc}$

Given positive reals

a, b, c

that satisfy

a + b + c = 1

, show that

\displaystyle \sum^{}_{cyc}\frac{a^3+bc}{a^2+bc}\ge 2

A6

1

Two Sequences

Let

(a_{n})_{n\ge 0}

and

(b_{n})_{n\ge 0}

be two sequences with arbitrary real values

a_0, a_1, b_0, b_1

. For

n\ge 1

, let

a_{n+1}, b_{n+1}

be defined in this way:

a_{n+1}=\dfrac{b_{n-1}+b_{n}}{2}, b_{n+1}=\dfrac{a_{n-1}+a_{n}}{2}

Prove that for any constant

c>0

there exists a positive integer

N

s.t. for all

n>N

,

|a_{n}-b_{n}|<c

.

A5

1

ab + bc + ca = 18

Let

a, b, c

be the side length of a triangle, with

ab + bc + ca = 18

and

a, b, c > 1

. Prove that

\sum_{cyc}\frac{1}{(a - 1)^3} > \frac{1}{a + b + c - 3}

A4

1

Sequence

Suppose

2015= a_1 <a_2 < a_3<\cdots <a_k

be a finite sequence of positive integers, and for all

m, n \in \mathbb{N}

and

1\le m,n \le k

,

a_m+a_n\ge a_{m+n}+|m-n|

Determine the largest possible value

k

can obtain.

A3

1

abc = 2, a, b, c <= sqrt{2}

Let

a, b, c

be positive real numbers less than or equal to

\sqrt{2}

such that

abc = 2

, prove that

\sqrt{2}\displaystyle\sum_{cyc}\frac{ab + 3c}{3ab + c} \ge a + b + c

A1

1

Inequality

Let

a, b, c

be the side lengths of a triangle. Prove that

\displaystyle\sum_{cyc} \frac{(a^2 + b^2)(a + c)}{b} \ge 2(a^2 + b^2 + c^2)

JOM 2015 Shortlist

Subcontests

Number of functions

Function

Sum of Reciprocals of Products of Pairs of Primes

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

Diophantine Equation

Two sets

Infinite sequence with weird conditions

Circumcircles tangent

Prove that there exists $6$ pairs of tangent circles

BE, CF, PR concurrent

Inequality with concurrent cevians

A, P, Q are collinear

Intersection lies on AM

OEHF is parallelogram

Permutations

Students and Subjects

Switching Points

Coloring on a plane

2 Functions

Cauchy the magician

Inequality + Sequence

Sequence

$\frac{a^3 + bc}{a^2 + bc}$

Two Sequences

ab + bc + ca = 18

Sequence

abc = 2, a, b, c <= sqrt{2}

Inequality

JOM 2015 Shortlist

Subcontests

Number of functions

Function

Sum of Reciprocals of Products of Pairs of Primes

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

Diophantine Equation

Two sets

Infinite sequence with weird conditions

Circumcircles tangent

Prove that there exists $6$ pairs of tangent circles

BE, CF, PR concurrent

Inequality with concurrent cevians

A, P, Q are collinear

Intersection lies on AM

OEHF is parallelogram

Permutations

Students and Subjects

Switching Points

Coloring on a plane

2 Functions

Cauchy the magician

Inequality + Sequence

Sequence

$\frac{a^3 + bc}{a^2 + bc}$

Two Sequences

ab + bc + ca = 18

Sequence

abc = 2, a, b, c &lt;= sqrt{2}

Inequality

abc = 2, a, b, c <= sqrt{2}