2017 China Western Mathematical Olympiad

Part of China Western Mathematical Olympiad

Subcontests

(8)

1

Number of solutions to sum of squares

n=2^{\alpha} \cdot q

be a positive integer, where

\alpha

is a nonnegative integer and

q

is an odd number. Show that for any positive integer

m

, the number of integer solutions to the equation

x_1^2+x_2^2+\cdots +x_n^2=m

is divisible by

2^{\alpha +1}

1

Orthocenter condition

In acute triangle

ABC

D

E

be points on sides

AB

AC

respectively. Let segments

BE

DC

H

M

N

be the midpoints of segments

BD

CE

respectively. Show that

H

is the orthocenter of triangle

AMN

B,C,E,D

are concyclic and

BE\perp CD

1

Find number of 9-tuples

a_1,a_2,\cdots ,a_9

9

positive integers (not necessarily distinct) satisfying: for all

1\le i<j<k\le 9

l (1\le l\le 9)

i,j

j

a_i+a_j+a_k+a_l=100

. Find the number of

9

(a_1,a_2,\cdots ,a_9)

satisfying the above conditions.

1

Taking turns coloring cells

n

k

be given integers such that

n\ge k\ge 2

. Alice and Bob play a game on an

n

n

table with white cells. They take turns to pick a white cell and color it black. Alice moves first. The game ends as soon as there is at least one black cell in every

k

k

square after a player moves, who is declared the winner of the game. Who has the winning strategy?

1

Find greatest value of n

n

be a positive integer such that there exist positive integers

x_1,x_2,\cdots ,x_n

x_1x_2\cdots x_n(x_1 + x_2 + \cdots + x_n)=100n.

Find the greatest possible value of

n

1

Inequality from divisibility condition

p

n

be a positive integer such that

p^2

\prod_{k=1}^n (k^2+1)

p<2n

1

2017 China Western Mathematical Olympiad Q8

a_1,a_2,\cdots,a_n>0

(n\geq 2)

\sum_{i=1}^n max\{a_1,a_2,\cdots,a_i \} \cdot min \{a_i,a_{i+1},\cdots,a_n\}\leq \frac{n}{2\sqrt{n-1}}\sum_{i=1}^n a^2_i

1

Prove vertical

D is the a point on BC，I1 is the heart of a triangle ABD, I2 is the heart of a triangle ACD，O1 is the Circumcenter of triangle AI1D, O2 is the Circumcenter of the triangle AI2D，P is the intersection point of O1I2 and O2I1，Prove: PD is perpendicular to BC.