2016 Japan TST

Part of Japan TST

Subcontests

(6)

1

IMO fever

2015

people. Initially, only one of these people is infected with IMO fever, but we don't know who it is. When two people shake hands, if one of them is infected with IMO fever, the other person get infected with it. Determine the minimum number of handshakes needed to infect all

2015

people, regardless of who the first infected person is.

1

Cycle in a grid

Find the maximum possible value of

n

, such that the integers

1,2,\ldots n

can be filled once each in distinct cells of a

2015\times 2015

grid and satisfies the following conditions:
[*] For all

1\le i \le n-1

, the cells with

i

i+1

share an edge. Cells with

1

n

also share an edge. In addition, no other pair of numbers share an edge. [*] If two cells with

i<j

in them share a vertex, then

\min\{j-i,n+i-j\}=2

1

Difference between prime powers

4

-tuples of primes

p<q<r<s

p^s-r^q=3.

1

Inequality with sum of reciprocals

Find the smallest positive real

k

such that for any positive integer

n\ge 2

and positive reals

a_0,a_1,\ldots ,a_n

\frac{1}{a_0+a_1} +\frac{1}{a_0+a_1+a_2} +\ldots +\frac{1}{a_0+a_1+\ldots +a_n} < k\left(\frac{1}{a_0}+\frac{1}{a_1}+\ldots +\frac{1}{a_n}\right).

1

Pair of rational functions

Find all pairs of functions

f,g:\mathbb{Q}\rightarrow\mathbb{Q}

such that for all rational

x,y

f(f(x)+yg(x))=(x+1)g(y)+f(y).

1

Prove CPM=ACB

In acute triangle

\triangle ABC

M

is the midpoint of

BC

\ell

be the angle bisector of

\angle BAC

. The tangent at

C

to the circumcircle of

\triangle ABC

\ell

D

H

\ell

\angle ABH=90^{\circ}

. The circumcircles of

\triangle CDH

\triangle ABC

intersect again at

P

\angle CPM=\angle ACB