www.mathandarte.blogspot.com China China Gids Math Olympiad 2002 Day J l O J Fi nd al l pos it ive inte ger s n such 20n + 2 can di vide 2003n + 2002. [2 ] There are 3n, n E Z+ girl students who took part in a summer ,camp. There were three girl students to be on duty every day. When the summer camp ended, it was found that any two of the 3n students had just one time to he on duty on the same ay. (l)\\"hen n= 3, is there any arrangement satisfying the requirement above. Prove yor conclusion. (2) Pro ve th at n is an odd number. @ ] Find all positive integers k such that for any positive numbers a, band c satisfying the inequality there must exist a triangle with u, band c as the length of its three sides respectively. [i] Circles 01 an d 02 i nt er est at t wo poi nt s Band C , an d BC I: S th e diameter of circle O J. Construct a tangent line of circle 01 at C and interesting circle O 2 at another point ,A . ...,Veoin AB to intersect circle O J at poi nt E, th en join CE and extend it to intersect circle 02 at point F. Assume H is an arbitrary point on line segment AF . We join HE and extend it to intersect circle 0, at poi nt G, and then join BG and! extend it to intersect the extend line of AC at point D. Prove: AH AC HF CD' China China Girls Math Olympiad 2002 Day 2 O J There are n :::: permutations PI, P2, ... ,P n each bein g an ar bi tr ar y pe rmut at ion of {1, ... , n}. Prove that 00-1 1 n - 1 8 Pi + Pi+1 > n+2' r n Find all pairs of positive integers (x, y) su ch that Albania
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[!] An acute triangle ABC has three heights AD, BE and CF respectively, Prove that the
perimeter of triangle DEF is not over half of the perimeter of triangle ABC.
m Assume that A1, A2, ... , A I ' ! are eight points. taken arbitrarily on a plane. For a directed liine
l taken arbitrarily on the plane, assume that projections. of Al,A2, ... ,A s on the line arePI, P2, ... , Ps respectively. If the eight projections are pairwise disjoint, they can be arranged!
as HI' Pi2, .... ,P;,~ a.ccon:llingto the direction of line 1. Thus we get one permutation for
1,2, ... ,8, namely, ·il, i 2, ... , is. In the figure, this permutation is 2,1,8,3,7,4,6, 5. Assume
that after these eight points are projected to every directed line em the plane, we get thenumber of different permutations as Ns= N(Ali, A2, ... ,As). Find the maximal value of Na.
China
China Girls Math Olympiad
2 0 0 3
Day 1
I T ] let ABC be a triangle. Points. D and E are on sides AB and AC, respectively, and. point P. I" t DE L • AD AE DF P 'h tlH on me seglnen . u , ·eo .4.B=.X, AC = y ., DE =Z.. rove u, a
[I]\Ve say a positive integer n is good if there exists a permutation a'l, 112: .... , an of 1: 2, .... , n
such that k + 11k is perfect square for all 1 :::;k$ n. Determine all the good numbers in the
set (n,iL3:15: 17, ]_'9}.
[I Let !1, b , c be positive teals, Find the smallest value of
a+3c 4b 8c~~~~+ - .a+2b+c' a+b+2c a+b+3c
[l] Let ABC be an obtuse inscribed! in a circle of radius 1.. Prove that 6.ABC can be covered
by an isosceles right-angled. triangle with hypotenuse of length "j2+ 1.
[i] A deck of 32 cards has 2 ,different jokers each of which is numbered O. There are 10 red!canis
nambered 1 through 10 and similarly for blue and green cards. One chooses a number of
cards from the deck. If a eard in hand is numbered k; then the value of the card is 21 0 , and
the value of the hand! is sum of the values of the cards in hand, Determine the number of
hands having the value 2004.
China
China GirIs Math. Olympiad
2 0 0 4
Day 2
[!] Let : 1 1 . , :u,W bc posi.tlve real numbers such that 'iL"fiiW + :u,fiiiU + wy1iI j ' 2 : i l l . . Find the smallest
value of 1' £ +V + ui.
[1] Given an acute triangle ABC with 0 fl."! its cireumecntcr. Line AO intersects Be at D.Points E, Fare on AB, AC respectively such thi'IltA, E, D, F are eoncyclic, Prove that the
length of the projection of line segment EF on side Be docs not. depend on the pcsicions fJf
E and F.
m Let 1 ) and If be twn coprime positive integers, and n be a non-negative integer. Determine the
Ilutnbcr of intcgers that can be writ.ten in the form 'ip + jq, where i illid j are non-negative
intcgers with i + i:5 n.
G O When the unit squares at the four comers arc removed from a three by three sqnares, the
resulting shape is called across. "that. is the maxirnum Humber of non-overlapping crosses
placed within the boundary of a lli.Ox H chessboard? (Eadl CH~S myers exactly five unit
I T J The set 8 = {(11, b ) II 1 ::;.11, b S 5, n, 1 1 E Z} be a set of points in the plane with ineegeral
coordinates. 1" is another set ofpcrinta with integeralcoordinates in the plane. If for anypoim
P 'E S, there is always another point Q E T, P f : - Q, such that there is no other ineegeralpoints on segment PQ. Find! the least value of the number of elements of T.
~ let M = {], 2,," ,19} and A = {ai, a2 ,'" ,a ,,} ~ AI. Find. the least k so that for amy
b EAt, there exist a.i.,aj E A , satisfying 1 1=a;. or b = ,l1.i± a . ; . (ai and aj do not have Ito be
different) .
[!]Given that Xi > 0, i=1,2, ... ,n, k 2 : : 1. Show that: