MathDB

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2013 Japan Mathematical Olympiad Finals Problem 5

n

be a positive integer. Given are points

P_1,\ P_2,\ \cdots,\ P_{4n}

of which any three points are not collinear. For

i=1,\ 2,\ \cdots,\ 4n

, rotating half-line

P_iP_{i-1}

clockwise in

90^\circ

about the pivot

P_i

gives half-line

P_iP_{i+1}.

Find the maximum value of the number of the pairs of

(i,\ j)

such that line segments

P_iP_{i+1}

and

P_jP_{j+1}

intersect at except endpoints. Note that :

P_0=P_{4n},\ P_{4n+1}=P_1

and

1\leq i<j\leq 4n.

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2013 Japan Mathematical Olympiad Finals Problem 1

Let

n,\ k

be positive integers with

n\geq k

. There are

n

persons, each person belongs to exactly one of group

1

, group

2,\ \cdots

, group

k

and more than or equal to one person belong to any groups. Show that

n^2

sweets can be delivered to

n

persons in such way that all of the following condition are satisfied.

\bullet

At least one sweet are delivered to each person.

\bullet

a_i

sweet are delivered to each person belonging to group

i\ (1\leq i\leq k).

\bullet

If

1\leq i<j\leq k

, then

a_i>a_j.

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2013 Japan Mathematical Olympiad Finals Problem 3

Let

n\geq 2

be a positive integer. Find the minimum value of positive integer

m

for which there exist positive integers

a_1,\ a_2,\ \cdots, a_n

such that :

\bullet\ a_1<a_2<\cdots <a_n=m

\bullet \ \frac{a_1^2+a_2^2}{2},\ \frac{a_2^2+a_3^2}{2},\ \cdots,\ \frac{a_{n-1}^2+a_n^2}{2}

are all square numbers.

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2013 Japan Mathematical Olympiad Finals Problem 4

Given an acute-angled triangle ABC, let

H

be the orthocenter. A cirlcle passing through the points

B,\ C

and a cirlcle with a diameter

AH

intersect at two distinct points

X,\ Y

. Let

D

be the foot of the perpendicular drawn from

A

to line

BC

, and let

K

be the foot of the perpendicular drawn from

D

to line

XY

. Show that

\angle{BKD}=\angle{CKD}

.

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2013 Japan Mathematical Olympiad Finals Problem 2

Find all functions

f:\mathbb{Z}\rightarrow\mathbb{R}

such that the equality

f(m)+f(n)=f(mn)+f(m+n+mn)

holds for all

m,n\in\mathbb{Z}.

2013 Japan MO Finals

Subcontests

2013 Japan Mathematical Olympiad Finals Problem 5

2013 Japan Mathematical Olympiad Finals Problem 1

2013 Japan Mathematical Olympiad Finals Problem 3

2013 Japan Mathematical Olympiad Finals Problem 4

2013 Japan Mathematical Olympiad Finals Problem 2