cgmo10test
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9 th Chinese Girls’ Mathematics Olympiad Shijiazhuang, China Day I 8:00 AM - 12:00 PM August 10, 2010 1. Let n be an integer greater than two, and let A 1 ,A 2 ,...,A 2n be pairwise distinct nonempty subsets of {1, 2,...,n}. Determine the maximum value of 2n X i=1 |A i ∩ A i+1 | |A i |·|A i+1 | . (Here we set A 2n+1 = A 1 . For a set X , let |X | denote the number of elements in X .) 2. In triangle ABC , AB = AC . Point D is the midpoint of side BC . Point E lies outside the triangle ABC such that CE ⊥ AB and BE = BD. Let M be the midpoint of segment BE. Point F lies on the minor arc d AD of the circumcircle of triangle ABD such that MF ⊥ BE. Prove that ED ⊥ FD. A B C D E F M 3. Prove that for every given positive integer n, there exists a prime p and an integer m such that (a) p ≡ 5 (mod 6); (b) p 6 | n; (c) n ≡ m 3 (mod p). 4. Let x 1 ,x 2 ,...,x n (where n ≥ 2) be real numbers with x 2 1 + x 2 2 + ··· + x 2 n = 1. Prove that n X k=1 1 - k ∑ n i=1 ix 2 i ¶ 2 · x 2 k k ≤ n - 1 n +1 ¶ 2 n X k=1 x 2 k k . Determine when does the equality hold. Copyright c Committee on the Chinese Mathematics Olympiad, Mathematical Association of China
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Transcript of cgmo10test
9th Chinese Girls’ Mathematics Olympiad
Shijiazhuang, China
Day I 8:00 AM - 12:00 PM
August 10, 2010
1. Let n be an integer greater than two, and let A1, A2, . . . , A2n be pairwise distinct nonempty subsetsof {1, 2, . . . , n}. Determine the maximum value of
2n∑
i=1
|Ai ∩Ai+1||Ai| · |Ai+1| .
(Here we set A2n+1 = A1. For a set X, let |X| denote the number of elements in X.)
2. In triangle ABC, AB = AC. Point D is the midpoint of side BC. Point E lies outside the triangleABC such that CE ⊥ AB and BE = BD. Let M be the midpoint of segment BE. Point F lies onthe minor arc AD of the circumcircle of triangle ABD such that MF ⊥ BE. Prove that ED ⊥ FD.
A
B CD
E
F
M
3. Prove that for every given positive integer n, there exists a prime p and an integer m such that
(a) p ≡ 5 (mod 6);
(b) p 6 | n;
(c) n ≡ m3 (mod p).
4. Let x1, x2, . . . , xn (where n ≥ 2) be real numbers with x21 + x2
2 + · · ·+ x2n = 1. Prove that
n∑
k=1
(1− k∑n
i=1 ix2i
)2
· x2k
k≤
(n− 1n + 1
)2 n∑
k=1
x2k
k.
Determine when does the equality hold.
Copyright c© Committee on the Chinese Mathematics Olympiad,Mathematical Association of China
9th Chinese Girls’ Mathematics Olympiad
Shijiazhuang, China
Day II 8:00 AM - 12:00 PM
August 11, 2010
5. Let f(x) and g(x) be strictly increasing linear functions from R to R such that f(x) is an integer if and onlyif g(x) is an integer. Prove that for any real number x, f(x)− g(x) is an integer.
6. In acute triangle ABC, AB > AC. Let M be the midpoint of side BC. The exterior angle bisector of ∠BACmeets ray BC at P . Points K and F lie on the line PA such that MF ⊥ BC and MK ⊥ PA. Prove thatBC2 = 4PF ·AK.
A
B C
F
MP
K
7. Let n be an integer greater than or equal to 3. For a permutation p = (x1, x2, . . . , xn) of (1, 2, . . . , n), we saythat xj lies in between xi and xk if i < j < k. (For example, in the permutation (1, 3, 2, 4), 3 lies in between 1and 4, and 4 does not lie in between 1 and 2.) Set S = {p1, p2, . . . , pm} consists of (distinct) permutations pi
of (1, 2, . . . , n). Suppose that among every three distinct numbers in {1, 2, . . . , n}, one of these number doesnot lie in between the other two numbers in every permutations pi ∈ S. Determine the maximum value of m.
8. Determine the least odd number a > 5 satisfying the following conditions: There are positive integersm1,m2, n1, n2 such that a = m2
1 + n21, a2 = m2
2 + n22, and m1 − n1 = m2 − n2.
Copyright c© Committee on the Chinese Mathematics Olympiad,Mathematical Association of China
