Đề olympic toán quốc tế international tại trung quốc

Prove that: a There exists positive integer k for which Ak+1= Ak.. Given a fixed acute angle θ and a pair of internally tangent circles, let the line l which passesthrough the point of t

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A collection of IMO Team Selection Tests

September 22, 2013

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Contents

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Due to problems with the pdf generator at AoPS I decided to make my own collection of math Olympiadproblems in pdf format with links to problems at AoPS The idea is also that the list may be printed,and therefore a secondary aim is to not waste as much space as the AoPS pdf’s, but rather to start anew page only for the next country’s Olympiads Of course, when the document it printed, one will not

be able to follow links Therefore I include the link location in as minimalist a form I could think of Thelinks are all of the form http://www.artofproblemsolving.com/Forum/viewtopic.php?p=***,where *** is a number (the post number on AoPS), mostly consisting of 7 digits, but earlier posts mayhave less Thus after each problem that appears on AoPS (to my knowledge) I include the post number,which also links to the problem there

Of course, this is a work in progress, since there is quite a lot to do and there are constantly newcontests being written Therefore I start with the most popular contest, and the ones I have mostcomplete collections of Also, in stead of defining common terms over and over in problems that refer tothem, I include a glossary at the end, where undefined terms can be looked up

I also changed the margins in which the text is written This is not something I normally do, since theappearance turns out to be quite strange However, for printing purposes this saves a lot of pages This

is only a private collection and not something professional That is why I feel this change is warranted.There are also places where I have slightly altered the text This is mostly removing superfluousdefinitions like “where R is the set of real numbers” or “where bxc denotes the greatest integer ”, etc

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China Team Selection Tests

China Team Selection Test 1986

1 If ABCD is a cyclic quadrilateral, then prove that the incentres of the triangles ABC, BCD, CDA,DAB are the vertices of a rectangle

3 Given a positive integer A written in decimal expansion: (an, an−1, , a0) and let f (A) denote

Pn

k=02n−k· ak Define A1= f (A), A2= f (A1) Prove that:

(a) There exists positive integer k for which Ak+1= Ak

(b) Find such Ak for 1986

(a) area EF G ≤ max(area ABC, area ABD, area ACD, area BCD)

(b) The same as above replacing “area” with “perimeter”

AoPS:234069

7 Let xi, 1 ≤ i ≤ n be real numbers with n ≥ 3 Let p and q be their symmetric sum of degree 1 and

2 respectively Prove that:

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8 Mark 4 · k points in a circle and number them arbitrarily with numbers from 1 to 4 · k The chordscannot share common endpoints, also, the endpoints of these chords should be among the 4 · kpoints

(a) Prove that 2 · k pairwise non-intersecting chords can be drawn for each of whom its endpointsdiffer in at most 3 · k − 1

(b) Prove that the 3 · k − 1 cannot be improved

AoPS:234076

1 For all positive integer k find the smallest positive integer f (k) such that 5 sets s1, s2, , s5 existsatisfying:

(a) i each has k elements;

ii si and si+1 are disjoint for i = 1, 2, , 5 (s6= s1)

iii the union of the 5 sets has exactly f (k) elements

(b) Generalisation: Consider n ≥ 3 sets instead of 5

6 Let G be a simple graph with 2 · n vertices and n2+ 1 edges Show that this graph G contains a

K4-one edge, that is, two triangles with a common edge

AoPS:234099

1 Suppose real numbers A, B, C such that for all real numbers x, y, z the following inequality holds:

A(x − y)(x − z) + B(y − z)(y − x) + C(z − x)(z − y) ≥ 0

Find the necessary and sufficient condition A, B, C must satisfy (expressed by means of an equality

or an inequality)

AoPS:269050

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2 Find all functions f : Q 7→ C satisfying

(a) For any x1, x2, , x1988∈ Q, f(x1+ x2+ + x1988) = f (x1)f (x2) f (x1988)

undistin-A of Sk such that the elements of A are pairwise distinguishing Let rk be the maximum possiblenumber of elements of A

6 Let ABCD be a trapezium AB//CD, M and N are fixed points on AB, P is a variable point on

CD E = DN ∩ AP , F = DN ∩ M C, G = M C ∩ P B, DP = λ · CD Find the value of λ for whichthe area of quadrilateral P EF G is maximum

AoPS:269056

7 A polygonQ is given in the OXY plane and its area exceeds n Prove that there exist n + 1 points

P1(x1, y1), P2(x2, y2), , Pn+1(xn+1, yn+1) in Q such that ∀i, j ∈ {1, 2, , n + 1}, xj− xi and

yj− yi are all integers

AoPS:269057

8 There is a broken computer such that only three primitive data c, 1 and −1 are reserved Onlyallowed operation may take u and v and output u · v + v At the beginning, u, v ∈ {c, 1, −1} Afterthen, it can also take the value of the previous step (only one step back) besides {c, 1, −1} Provethat for any polynomial Pn(x) = a0· xn+ a1· xn−1+ + an with integer coefficients, the value of

Pn(c) can be computed using this computer after only finite operation

AoPS:269058

1 A triangle of sides 32,

√ 5

2 ,√

2 is folded along a variable line perpendicular to the side of 32 Find themaximum value of the coincident area

AoPS:269059

2 Let v0= 0, v1= 1 and vn+1= 8 · vn− vn−1, n = 1, 2, Prove that in the sequence {vn} there is

no term of the form 3α· 5β

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4 Given triangle ABC, squares ABEF, BCGH, CAIJ are constructed externally on side AB, BC, CA,respectively Let AH ∩ BJ = P1, BJ ∩ CF = Q1, CF ∩ AH = R1, AG ∩ CE = P2, BI ∩ AG = Q2,

CE ∩ BI = R2 Prove that triangle P1Q1R1is congruent to triangle P2Q2R2

AoPS:269065

8 ∀n ∈ N, P (n) denotes the number of the partition of n as the sum of positive integers (disregardingthe order of the parts), e.g since 4 = 1 + 1 + 1 + 1 = 1 + 1 + 2 = 1 + 3 = 2 + 2 = 4, so P (4) = 5.The dispersion of a partition denotes the number of different parts in that partition And denoteq(n) is the sum of all the dispersions, e.g q(4) = 1 + 2 + 2 + 1 + 1 = 7 n ≥ 1 Prove that

1 In a wagon, every m ≥ 3 people have exactly one common friend (When A is B’s friend, B is alsoA’s friend No one was considered as his own friend.) Find the number of friends of the person whohas the most friends

AoPS:269067

2 Finitely many polygons are placed in the plane If for any two polygons of them, there exists aline through origin O that cuts them both, then these polygons are called properly placed Find theleast m ∈ N, such that for any group of properly placed polygons, m lines can drawn through Oand every polygon is cut by at least one of these m lines

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5 Given a triangle ABC with angle C ≥ 60 Prove that:

1 Let real coefficient polynomial f (x) = xn+ a1· xn−1+ + an has real roots b1, b2, , bn, n ≥ 2,prove that ∀x ≥ max{b1, b2, , bn}, we have

f (x + 1) ≥ 1 2·n2

x−b1+1 x−b2+ +

1 x−bn

AoPS:269077

2 For i = 1, 2, , 1991, we choose ni points and write number i on them, every point are writtenonly one number A set of chords are drawn such that:

(a) They are pairwise non-intersecting

(b) The endpoints of each chord have distinct numbers

If for all possible assignments of numbers the operation can always be done, find the necessary andsufficient condition the numbers n1, n2, , n1991must satisfy for this to be possible

AoPS:269078

3 Five points are given in the plane Any three of them are non-collinear Any four are non-cyclic

If three points determine a circle that has one of the remaining points inside it and the other oneoutside it, then the circle is said to be good Let the number of good circles be n, find all possiblevalues of n

AoPS:269082

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1 16 students took part in a competition All problems were multiple choice style Each problemhad four choices It was said that any two students had at most one answer in common, find themaximum number of problems

3 For any prime p, prove that there exists integer x0 such that p|(x2− x0+ 3) ⇔ there exists integer

y0such that p|(y2− y0+ 25)

1 For all primes p ≥ 3, define F (p) =Pp−12

k=1k120 and f (p) = 12 −nF (p)p o, where {x} = x − [x], findthe value of f (p)

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6 Let ABC be a triangle and its bisector at A cuts its circumcircle at D Let I be the incentre oftriangle ABC, M be the midpoint of BC, P is the symmetric to I with respect to M (Assuming

P is in the circumcircle) Extend DP until it cuts the circumcircle again at N Prove that amongsegments AN, BN, CN , there is a segment that is the sum of the other two

AoPS:269099

1 Find all sets comprising of 4 natural numbers such that the product of any 3 numbers in the setleaves a remainder of 1 when divided by the remaining number

AoPS:234853

2 An n by n grid, where every square contains a number, is called an n-code if the numbers in everyrow and column form an arithmetic progression If it is sufficient to know the numbers in certainsquares of an n-code to obtain the numbers in the entire grid, call these squares a key

(a) Find the smallest s ∈ N such that any s squares in an n−code (n ≥ 4) form a key

(b) Find the smallest t ∈ N such that any t squares along the diagonals of an n-code (n ≥ 4) form

a key

AoPS:234855

3 Find the smallest n ∈ N such that if any 5 vertices of a regular n-gon are coloured red, there exists

a line of symmetry l of the n-gon such that every red point is reflected across l to a non-red point

AoPS:234857

4 Given 5n real numbers ri, si, ti, ui, vi ≥ 1(1 ≤ i ≤ n), let R = 1

n

Pn i=1ri, S = 1

n

Pn i=1si, T =

5 Given distinct prime numbers p and q and a natural number n ≥ 3, find all a ∈ Z such that thepolynomial f (x) = xn+ axn−1+ pq can be factored into 2 integral polynomials of degree at least 1

(b) Find the smallest possible value of m

AoPS:234862

1 Find the smallest prime number p that cannot be represented in the form |3a− 2b|, where a and bare non-negative integers

AoPS:234864

2 Given a fixed acute angle θ and a pair of internally tangent circles, let the line l which passesthrough the point of tangency, A, cut the larger circle again at B (l does not pass through thecentres of the circles) Let M be a point on the major arc AB of the larger circle, N the pointwhere AM intersects the smaller circle, and P the point on ray M B such that ∠M P N = θ Findthe locus of P as M moves on major arc AB of the larger circle

AoPS:234865

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3 21 people take a test with 15 true or false questions It is known that every 2 people have at least

1 correct answer in common What is the minimum number of people that could have correctlyanswered the question which the most people were correct on?

AoPS:234866

4 Let S = {A = (a1, , as) | ai = 0 or 1, i = 1, , 8} For any 2 elements of S, A = {a1, , a8}and B = {b1, , b8} Let d(A, B) =P

i=18|ai− bi| Call d(A, B) the distance between A and B

At most how many elements can S have such that the distance between any 2 sets is at least 5?

AoPS:234867

5 A and B play the following game with a polynomial of degree at least 4:

x2n+ x2n−1+ x2n−2+ + x + 1 = 0

A and B take turns to fill in one of the blanks with a real number until all the blanks are filled up

If the resulting polynomial has no real roots, A wins Otherwise, B wins If A begins, which playerhas a winning strategy?

AoPS:234868

6 Prove that the interval [0, 1] can be split into black and white intervals for any quadratic polynomial

P (x), such that the sum of weights of the black intervals is equal to the sum of weights of the whiteintervals (Define the weight of the interval [a, b] as P (b) − P (a).)

Does the same result hold with a degree 3 or degree 5 polynomial?

AoPS:234870

1 Let side BC of 4ABC be the diameter of a semicircle which cuts AB and AC at D and Erespectively F and G are the feet of the perpendiculars from D and E to BC respectively DGand EF intersect at M Prove that AM ⊥ BC

3 Let M = {2, 3, 4, 1000} Find the smallest n ∈ N such that any n-element subset of M contains

3 pairwise disjoint 4-element subsets S, T, U such that

(a) For any 2 elements in S, the larger number is a multiple of the smaller number The sameapplies for T and U

(b) For any s ∈ S and t ∈ T , (s, t) = 1

(c) For any s ∈ S and u ∈ U , (s, u) > 1

AoPS:234882

4 Three countries A, B, C participate in a competition where each country has 9 representatives Therules are as follows: every round of competition is between 1 competitor each from 2 countries Thewinner plays in the next round, while the loser is knocked out The remaining country will thensend a representative to take on the winner of the previous round The competition begins with

A and B sending a competitor each If all competitors from one country have been knocked out,the competition continues between the remaining 2 countries until another country is knocked out.The remaining team is the champion

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(a) At least how many games does the champion team win?

(b) If the champion team won 11 matches, at least how many matches were played?

1 Given a real number λ > 1, let P be a point on the arc BAC of the circumcircle of 4ABC Extend

BP and CP to U and V respectively such that BU = λBA, CV = λCA Then extend U V to Qsuch that U Q = λU V Find the locus of point Q

AoPS:239173

2 There are n football teams in a round-robin competition where every 2 teams meet once Thewinner of each match receives 3 points while the loser receives 0 points In the case of a draw, bothteams receive 1 point each Let k be as follows: 2 ≤ k ≤ n − 1 At least how many points must acertain team get in the competition so as to ensure that there are at most k − 1 teams whose scoresare not less than that particular team’s score?

a0a2n;(c) All the roots of f (x) are imaginary numbers with no real part

AoPS:239181

5 Let n be a natural number greater than 6 X is a set such that |X| = n A1, A2, , Amare distinct5-element subsets of X If m > n(n−1)(n−2)(n−3)(4n−15)600 , prove that there exists Ai1, Ai2, , Ai6(1 ≤ i1< i2< · · · , i6≤ m), such thatS6

k=1Aik= 6

AoPS:239182

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6 There are 1997 pieces of medicine Three bottles A, B, C can contain at most 1997, 97, 19 pieces

of medicine respectively At first, all 1997 pieces are placed in bottle A, and the three bottles are

closed Each piece of medicine can be split into 100 part When a bottle is opened, all pieces of

medicine in that bottle lose a part each A man wishes to consume all the medicine However, he

can only open each of the bottles at most once each day, consume one piece of medicine, move some

pieces between the bottles, and close them At least how many parts will be lost by the time he

finishes consuming all the medicine?

AoPS:239184

1 Find k ∈ N such that

(a) For any n ∈ N, there does not exist j ∈ Z which satisfies the conditions 0 ≤ j ≤ n − k + 1 and

n

j, n

j+1, , n

j+k−1 forms an arithmetic progression

(b) There exists n ∈ N such that there exists j which satisfies 0 ≤ j ≤ n−k+2, and nj, n

j+1, , n

j+k−2

forms an arithmetic progression

Find all n which satisfies part (b)

AoPS:239188

2 n ≥ 5 football teams participate in a round-robin tournament For every game played, the winner

receives 3 points, the loser receives 0 points, and in the event of a draw, both teams receive 1 point

The third-from-bottom team has fewer points than all the teams ranked before it, and more points

than the last 2 teams; it won more games than all the teams before it, but fewer games than the 2

teams behind it Find the smallest possible n

(b) There exists x ∈ [1−

√ a sin θ,

√ a cos θ] such that [(1−x) sin θ−√

a − x2cos2θ]2+[x cos θ−

q

a − (1 − x)2sin2θ]2≤a

AoPS:239195

4 In acute-angled 4ABC, H is the orthocentre, O is the circumcentre and I is the incentre Given

that ∠C > ∠B > ∠A, prove that I lies within 4BOH

AoPS:239196

5 Let n be a natural number greater than 2 l is a line on a plane There are n distinct points P1,

P2, , Pnon l Let the product of distances between Pi and the other n − 1 points be di (i = 1, 2,

, n) There exists a point Q, which does not lie on l, on the plane Let the distance from Q to

(r is a non-negative integer), find all k ∈ N which satisfy the following condition:

There exists an odd natural number m > 1 and n ∈ N, such that k | mh− 1, m | nmh −1k + 1

AoPS:239202

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1 For non-negative real numbers x1, x2, , xn which satisfy x1+ x2+ · · · + xn= 1, find the largest

2 Find all prime numbers p which satisfy the following condition: For any prime q < p, if p =

kq + r, 0 ≤ r < q, there does not exist an integer q > 1 such that a2| r

AoPS:239205

3 Let S = {1, 2, , 15} Let A1, A2, , An be n subsets of S which satisfy the following conditions:

(a) |Ai| = 7, i = 1, 2, , n;

(b) |Ai∩ Aj| ≤ 3, 1 ≤ i < j ≤ n

(c) For any 3-element subset M of S, there exists Ak such that M ⊂ Ak

Find the smallest possible value of n

AoPS:239209

4 A circle is tangential to sides AB and AD of convex quadrilateral ABCD at G and H respectively,

and cuts diagonal AC at E and F What are the necessary and sufficient conditions such that there

exists another circle which passes through E and F , and is tangential to DA and DC extended?

AoPS:239211

5 For a fixed natural number m ≥ 2, prove that

(a) There exists integers x1, x2, , x2msuch that {x1, x2, , x2m

(b) For any set of integers {x1, x2, , x2mwhich fulfils (*), an integral sequence , y−k, , y−1, y0, y1, , yk, can be constructed such that ykym+k = yk+1ym+k−1+ 1, k = 0, ±1, ±2, such that yi =

(a) The maximum and minimum values of S(τ )

(b) The number of τ which lets S(τ ) attain its maximum

(c) The number of τ which lets S(τ ) attain its minimum

AoPS:239300

1 Let ABC be a triangle such that AB = AC Let D, E be points on AB, AC respectively such that

DE = AC Let DE meet the circumcircle of triangle ABC at point T Let P be a point on AT

Prove that P D + P E = AT if and only if P lies on the circumcircle of triangle ADE

3 For positive integer a ≥ 2, denote Naas the number of positive integer k with the following property:

the sum of squares of digits of k in base a representation equals k Prove that:

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where λk,l=P[k/2]

m=l k 2·m · m l

(b) Let uk = P[k/2]

l=0 λk,l For positive integer m, denote the remainder of uk divided by 2m as

zm,k Prove that zm,k, k = 0, 1, 2, is a periodic function, and find the smallest period

AoPS:239383

6 Let n be a positive integer Denote M = {(x, y)|x, y are integers , 1 ≤ x, y ≤ n} Define function f

on M with the following properties:

(a) f (x, y) takes non-negative integer value;

(b) Pn

y=1f (x, y) = n − 1 for 1 ≤ x ≤ n;

(c) If f (x1, y1)f (x2, y2) > 0, then (x1− x2)(y1− y2) ≥ 0

Find N (n), the number of functions f that satisfy all the conditions Give the explicit value of

N (4)

AoPS:239398

1 E and F are interior points of convex quadrilateral ABCD such that AE = BE, CE = DE,

∠AEB = ∠CED, AF = DF , BF = CF , ∠AF D = ∠BF C Prove that ∠AF D + ∠AEB = π

4 For a given natural number n > 3, the real numbers x1, x2, , xn, xn+1, xn+2satisfy the conditions

0 < x1< x2< · · · < xn< xn+1< xn+2 Find the minimum possible value of

(n + 2) and find all (n + 2)-tuples of real numbers (x1, x2, , xn, xn+1, xn+2) which gives this value

AoPS:239334

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5 In the equilateral 4ABC, D is a point on side BC O1 and I1 are the circumcentre and incentre

of 4ABD respectively, and O2 and I2 are the circumcentre and incentre of 4ADC respectively

O1I1 intersects O2I2at P Find the locus of point P as D moves along BC

AoPS:239330

6 Let F = max1≤x≤3|x3− ax2− bx − c| When a, b, c run over all the real numbers, find the smallestpossible value of F

AoPS:239342

1 Given n ≥ 3, n is a integer Prove that:

3 .AoPS:1389494

3 The positive integers α, β, γ are the roots of a polynomial f (x) with degree 4 and the coefficient ofthe first term is 1 If there exists an integer such that f (−1) = f2(s)

Prove that αβ is not a perfect square

, bi ∈ R+, i = 0, 1, 2, · · · , n − 2, and there is some i,

1 ≤ i ≤ n − 1, making a2i − 4ai−1ai+1≤ 0

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7 Given triangle ABC and AB = c, AC = b and BC = a satisfying a ≥ b ≥ c, BE and CF are twointerior angle bisectors P is a point inside triangle AEF R and Q are the projections of P onsides AB and AC.

9 Sequence {fn(a)} satisfies fn+1(a) = 2 − a

fn(a), f1(a) = 2, n = 1, 2, · · · If there exists a naturalnumber n, such that fn+k(a) = fk(a), k = 1, 2, · · · , then we call the non-zero real a a periodic point

13 In acute triangle ABC, show that:

sin3A cos2(B − C) + sin3B cos2(C − A) + sin3C cos2(A − B) ≤ 3 sin A sin B sin C

and find out when the equality holds

AoPS:1389548

14 m and n are positive integers In a 8 × 8 chessboard, (m, n) denotes the number of grids a Horse canjump in a chessboard (m horizontal n vertical or n horizontal m vertical ) If a (m, n)Horse startsfrom one grid, passes every grid once and only once, then we call this kind of Horse jump route a

H Route For example, the (1, 2)Horse has its H Route Find the smallest positive integer t,such that from any grid of the chessboard, the (t, t + 1)Horse does not has any H Route

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and there exists subscripts i, j, t, and l (at least 3 of them are distinct) such that xi+ xj= xt+ xl.

Find a expression of the general term of {an}

AoPS:1389560

17 O1 and O2 meet at points P and Q The circle through P , O1 and O2 meets O1and O2 atpoints A and B Prove that the distance from Q to the lines P A, P B and AB are equal

(Prove the following three cases: O1and O2 are in the common space of O1and O2; O1 and O2

are out of the common space of O1and O2; O1 is in the common space of O1 and

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