China Girls Math Olympiad - 01

8/3/2019 China Girls Math Olympiad - 01

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China

China Gids Math Olympiad

2 0 0 2

Day J l

O J Find all positive integers n such 20n + 2 can divide 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 day.

(l)\\"hen n= 3, is there any arrangement satisfying the requirement above. Prove yor

conclusion.

(2) Prove that 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 and 02 interest at two points Band C , and BC I: S the diameter of circle O J. Construct

a tangent line of circle 01 at C and interesting circle O2 at another point ,A . ...,Veoin AB to

intersect circle O J at point E, then 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 point 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

2 0 0 2

Day 2

O J There are n :::: permutations PI, P 2, ... ,Pn each being an arbitrary permutation 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) such 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


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

(1) S6BDF = (] - X)ySl:>.ABG and Sl:>.CEF = x(1 - y){1 - Z)S6ABC ;

(2 ) {lS6BDF + \lS6CEF :::;{lS6.4.BC.

[I] There are 47 students in a classroom with seats. arranged in 6 IfOWS x 8 columns, and the seat

in the i-tib row and j-th column is denoted by (i,j). Now, an adjustment ismade for students

seats in the new school te:rm. For a student with the original seat U,j), if his/her new seat

is (m .,n ) , we say that the student is moved by [a,b] = [i - rn ,j - n ] and define the position

value of the student as a.+ b. Let S denote the SHm of the position values of all the students.

Determine the difference between the greatest and smallest possible values of S.

[1] As shown ill the figure, quadrilateral ABeD is inscribed in a circle with AC as its diameter,

BD J L AG, and E the intersection ofAC and BD. Extend line segment DA and BA through

A to F and G respectively, such that DC I I BF. Extend CP to H such that GH j_ OH.

Prove that points B, E, F and H lie on one circle.



China

China Girls Matih Olympiad

2003

m (1) Prove that there exist fivenonnegacive real nurnbers a, b, c, d and e with their sum equal

to 1 such that for any arrangement of these numbers around a circle, there are always twn

neighboring numbers with their product not less than ~.

(2) Prove that for any five nonnegative real numbers with their sum. equal to 1 , it is always

possible to arrange them around a circle such that there axe two neighboring numbers with

their product not greater than ~.

China

Cihina Girls Math Olympiad

2003

Day 2

[I]Let [a,,}'f be a sequence of real numbers such that al= 2, and

Un-:t-l =a ; - an + 1,\In E N.

Prove that

~ Let n. ::::2be an integer. Find the largest real number A such that the inequality

n-]

u.~2 : ALaj + 2 . an·

i=l

holds for rutlypositive integers al, U2,.... an satisfying al < a2 < ... < an.

[l]Let the sides of a scalene triangle l:::,.ABC be AB = c, Be = a, CA = II, and D, E, F be

points on BC, CA, AB SIlC'B that AD, BE, CF are angle bisectors ofthe triangle, respectively.

Assume that DE =DF. Prove that

if) ~ __b_ + - - - - " - - ,~ b+c ._ c+a a+b·

(2)LBAC > 900.

[!] Let n be a positive integer, and Sn, be the set of all positive integer diivisursof n. (including

1 and itself). Prove that at most half of the elements in Sn have their last digit.'>equal to 3.



China

Cihina Girls Math Olympiad

2 0 0 4

Day 1.

[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

squares On the board.)



China

China Girls Matih Olympiad

2005

Day 1

[I] As shown in the followingfigure, point P lies on the circumcicle of triangle ABC. Lines AB

and CP meet at E, and Iines AC and BP meet at F. The perpendicular bisector of line

segment AB meets line segment AC at K, and the perpendicular bisector of line segment AC

meets line segment AB at J. Prove that

I ( C . E ) 2BF

AJ· JE

AK·KFIII Find all ordered triples. (x , y, z) of real numbers such that

5 ( X + ~ ) = 1 2 ( Y + ~ ) = 1 3 ( Z + ~ ) ,and

xy + yz + .zy =I.

[[] Determine if there exists aconvex polyhedron SI1!C ' l i t that

(1) it has J1.2edges, (; faces and 8 vertices;

(2) it has 4 faces with each pair of them sharing a common edge of the polyhedron.

[!] Determine all] positive real numbers a. such that there exists a positive integer n and sets

Al,A2, ... , An satisfyi!t1lgthe following conditions:

(1) every set A. has infinitely manyelements; (2) every pair of distinct sets Ai and Aj db not

share any common dement (3) the union of sets A]., A2, ... , An is the set of all integers; (4)

for every set Ai, the pOB:i~ivedifference of any pair of elements i~ Ai is at least a'.

Day 2

IITJLet x and y be positive real numbers with x3 + y3= z - y. Prove that

I[I] An iJIl!tegern is called good if there are n :;::3 lattice points Pl, P2, .... ,p...m the coordinate

plane satisfying the following conditions: H tine segment P~Pjhaaa rational length, thea

there is Pk such that both Iine segments PiPk and PjPk have irrational lengl;hs.; and if line

segment PiPj has an irrational length, then there is Pk such that Doth line segments PiPk

and PjPk have rational lengths.

(1) Determine the mimmum good number,

(2) Determine iJ f 2005 is a . good number. (A point iJj]the coordinate plane is a lattice point if

both of itsc~~:di~ate are integers.)



China


2 0 0 6

Day 2

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:

n 1 .n n x"+1 n 1

L 1+x' . EXi:S: L] ~1;' •L xki=1 ,. i=1 ;.=1 • i=1 •

r n Let p be a prime number that is greater than. 3. Show that there exist some integers

a.t, 112, . •• ak that satisfy:

making the product:

equals to 3mwhere m is a positive integer.

China Girls Math Olympiad - 01

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