AoPS Community 2023 China National Olympiad
2023 China National Olympiad
www.artofproblemsolving.com/community/c3238312
by CHN Lucas, JG666, CANBANKAN, David-Vieta
– Day 1 (December 29th)
1 Define the sequences (an ), (bn ) by
an , bn > 0, ∀n ∈ N+
1
an+1 = an − Pn 1
1 + i=1 ai
1
bn+1 = bn + Pn 1
1 + i=1 bi
1) If a100 b100 = a101 b101 , find the value of a1 − b1 ;
2) If a100 = b99 , determine which is larger between a100 + b100 and a101 + b101 .
2 Let △ABC be an equilateral triangle of side length 1. Let D, E, F be points on BC, AC, AB
respectively, such that DE
20 = 22 = 38 . Let X, Y, Z be on lines BC, CA, AB respectively, such
EF FD
that XY ⊥ DE, Y Z ⊥ EF, ZX ⊥ F D. Find all possible values of [DEF1
] + [XY Z] .
1
3 Given positive integer m, n, color the points of the regular (2m + 2n)-gon in black and white, 2m
in black and 2n in white.
The coloring distance d(B, C) of two black points B, C is defined as the smaller number of white
points in the two paths linking the two black points.
The coloring distance d(W, X) of two white points W, X is defined as the smaller number of
black points in the two paths linking the two white points.
We define the matching of black points B : label the 2m black points with A1 , · · · , Am , B1 , · · · , Bm
satisfying no Ai Bi intersects inside the gon.
We define the matching of white points W : label the 2n white points with C1 , · · · , Cn , D1 , · · · , Dn
satisfying no Ci Di P
intersects inside the gon.P
We define P (B) = m i=1 d(Ai , Bi ), P (W) =
n
j=1 d(Cj , Dj ).
Prove that: maxB P (B) = maxW P (W)
– Day 2 (December 30th)
4 Find the minimum positive integer n ≥ 3, such that there exist n points A1 , A2 , · · · , An satisfying
no three points are collinear and for any 1 ≤ i ≤ n, there exist 1 ≤ j ≤ n(j ̸= i), segment Aj Aj+1
pass through the midpoint of segment Ai Ai+1 , where An+1 = A1
© 2023 AoPS Incorporated 1
AoPS Community 2023 China National Olympiad
5 Prove that there exist C > 0, which satisfies the following conclusion:
For any infinite positive arithmetic integer sequence a1 , a2 , a3 , · · · , if the greatest common divi-
sor of a1 and a2 is squarefree, then there exists a positive integer m ≤ C · a2 2 , such that am is
squarefree.
Note: A positive integer N is squarefree if it is not divisible by any square number greater than
1.
Proposed by Qu Zhenhua
6 There are n(n ≥ 8) airports, some of which have one-way direct routes between them. For
any two airports a and b, there is at most one one-way direct route from a to b (there may be
both one-way direct routes from a to b and from b to a). For any set A composed of airports
(1 ≤ |A| ≤ n − 1), there are at least 4 · min{|A|, n − |A|} one-way direct routes from the airport
in A to the airport not in A. √
Prove that: For any airport x, we can start from x and return to the airport by no more than 2n
one-way direct routes.
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