AoPS Community 2013 China Team Selection Test

China Team Selection Test 2013

www.artofproblemsolving.com/community/c4969
by Uagu, s372102, sqing, iarnab kundu

Day 1 March 13th

1 The quadrilateral ABCD is inscribed in circle ω. F is the intersection point of AC and BD. BA
and CD meet at E. Let the projection of F on AB and CD be G and H, respectively. Let M
and N be the midpoints of BC and EF , respectively. If the circumcircle of 4M N G only meets
segment BF at P , and the circumcircle of 4M N H only meets segment CF at Q, prove that
P Q is parallel to BC.
√
For the positive integer n, define f (n) = min  2 − n .
m
Let {ni } be a strictly increasing se-

2
m∈Z
C
quence of positive integers. C is a constant such that f (ni ) < for all i ∈ {1, 2, . . .}. Show
n2i
that there exists a real number q > 1 such that ni > q i−1 for all i ∈ {1, 2, . . .}.

3 There aren balls numbered 1, 2, · · · , n, respectively. They are painted with 4 colours, red, yellow,
blue, and green, according to the following rules:
First, randomly line them on a circle.
Then let any three clockwise consecutive balls numbered i, j, k, in order.
1) If i > j > k, then the ball j is painted in red;
2) If i < j < k, then the ball j is painted in yellow;
3) If i < j, k < j, then the ball j is painted in blue;
4) If i > j, k > j, then the ball j is painted in green.
And now each permutation of the balls determine a painting method.
We call two painting methods distinct, if there exists a ball, which is painted with two different
colours in that two methods.
Find out the number of all distinct painting methods.

Day 2 March 14th

1 Let n and k be two integers which are greater than 1. Let a1 , a2 , . . . , an , c1 , c2 , . . . , cm be non-
negative real numbers such that
i) a1 ≥ a2 ≥ . . . ≥ an and a1 + a2 + . . . + an = 1;
ii) For any integer m ∈ {1, 2, . . . , n}, we have that c1 + c2 + . . . + cm ≤ mk .
Find the maximum of c1 ak1 + c2 ak2 + . . . + cn akn .

2 Let P be a given point inside the triangle ABC. Suppose L, M, N are the midpoints of BC, CA, AB

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AoPS Community 2013 China Team Selection Test

respectively and
P L : P M : P N = BC : CA : AB.
The extensions of AP, BP, CP meet the circumcircle of ABC at D, E, F respectively. Prove
that the circumcentres of AP F, AP E, BP F, BP D, CP D, CP E are concyclic.

3 Find all positive real numbers r < 1 such that there exists a set S with the given properties:
i) For any real number t, exactly one of t, t + r and t + 1 belongs to S;
ii) For any real number t, exactly one of t, t − r and t − 1 belongs to S.

Day 3 March 18th

1 For a positive integer k ≥ 2 define Tk = {(x, y) | x, y = 0, 1, . . . , k − 1} to be a collection of

k 2 lattice points on the cartesian coordinate plane. Let d1 (k) > d2 (k) > · · · be the decreasing
sequence of the distinct distances between any two points in Tk . Suppose Si (k) be the number
of distances equal to di (k).
Prove that for any three positive integers m > n > i we have Si (m) = Si (n).

2 Prove that: there exists a positive constant K, and an integer series {an }, satisfying: (1) 0 <
a1 < a2 < · · · < an < · · · ; (2) For any positive integer n, an < 1.01n K; (3) For any finite number
of distinct terms in {an }, their sum is not a perfect square.

3 Let A be a set consisting of 6 points in the plane. denoted n(A) as the number of the unit
circles which meet at least three points of A. Find the maximum of n(A)

Day 4 March 19th

1 For a positive integer N > 1 with unique factorization N = pα1 1 pα2 2 · · · pαk k , we define

Ω(N ) = α1 + α2 + · · · + αk .

Let a1 , a2 , . . . , an be positive integers and p(x) = (x + a1 )(x + a2 ) · · · (x + an ) such that for all
positive integers k, Ω(P (k)) is even. Show that n is an even number.

2 Find the greatest positive integer m with the following property:

For every permutation a1 , a2 , · · · , an , · · · of the set of positive integers, there exists positive
integers i1 < i2 < · · · < im such that ai1 , ai2 , · · · , aim is an arithmetic progression with an odd
common difference.

3 Let
Pk n >
1 be an integer and let a0 , a1 , . . . , an be non-negative real numbers. Definite Sk =
i=0 i ai for k = 0, 1, . . . , n. Prove that
k

n−1 n
!2
1X 2 1 X 4
Sk − 2 Sk ≤ (Sn − S0 )2 .
n n 45
k=0 k=0

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AoPS Community 2013 China Team Selection Test

Day 5 March 24th

1 Let n ≥ 2 be an integer. a1 , a2 , . . . , an are arbitrarily chosen positive integers with (a1 , a2 , . . . , an ) =

1. Let A = a1 + a2 + · · · + an and (A, ai ) = di . Let (a2 , a3 , . . . , an ) = D1 , (a1 , a3 , . . . , an ) =
D2 , . . . , (a1 , a2 , . . . , an−1 ) = Dn .
n A−a
i
Find the minimum of
Q
i=1 d D
i i

2 The circumcircle of triangle ABC has centre O. P is the midpoint of BAC

\ and QP is the diame-
ter. Let I be the incentre of 4ABC and let D be the intersection of P I and BC. The circumcircle
of 4AID and the extension of P A meet at F . The point E lies on the line segment P D such
that DE = DQ. Let R, r be the radius of the inscribed circle and circumcircle of 4ABC, re-
spectively.
2r
Show that if ∠AEF = ∠AP E, then sin2 ∠BAC =
R

3 101 people, sitting at a round table in any order, had 1, 2, ..., 101 cards, respectively.
A transfer is someone give one card to one of the two people adjacent to him.
Find the smallest positive integer k such that there always can through no more than k times
transfer, each person hold cards of the same number, regardless of the sitting order.

Day 6 March 25th

1 Let p be a prime number and a, k be positive integers such that pa < k < 2pa . Prove that there
exists a positive integer n such that

n < p2a , Cnk ≡ n ≡ k (mod pa ).

2 Let k ≥ 2 be an integer and let a1 , a2 , · · · , an , b1 , b2 , · · · , bn be non-negative real numbers. Prove

that !2
n n n
 n−1 !
n 1X 2 1X Y 1
ai + bi ≥ (a2i + b2i ) n .
n−1 n n
i=1 i=1 i=1

3 A point (x, y) is a lattice point if x, y ∈ Z. Let E = {(x, y) : x, y ∈ Z}. In the coordinate plane,
P and Q are both sets of points in and on the boundary of a convex polygon with vertices on
lattice points. Let T = P ∩ Q. Prove that if T 6= ∅ and T ∩ E = ∅, then T is a non-degenerate
convex quadrilateral region.

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