China Team Selection Test 1986 China Team Selection Test 1986 Day 1 1 Let ABCD be a cyclic quadrilateral Prove that the incenters of the triangles ABC, BCD, CDA and DAB form a rectangle 2 Let ai , 1 ≤[.]
China Team Selection Test 1986 Day 1 Let ABCD be a cyclic quadrilateral Prove that the incenters of the triangles ABC, BCD, CDA and DAB form a rectangle Let , ≤ i ≤ n , and , ≤ i ≤ n be · n real numbers Prove that for xi , ≤ i ≤ n satisfying x1 ≤ x2 ≤ xn the following statemenst are equivalent: n n X X i) ak · xk ≤ bk · xk , k=1 s X ii.) ak ≤ k=1 k=1 s X n X k=1 k=1 bk for s = 1, 2, , n − and ak = n X bk k=1 Given a positive integer A written in decimal expansion: (an , an−1 , , a0 ) and let f (A) n X denote 2n−k · ak Define A1 = f (A), A2 = f (A1 ) Prove that: k=0 I There exists positive integer k for which Ak+1 = Ak II Find such Ak for 1986 Given a triangle ABC for which C = 90 degrees, prove that given n points inside it, we can name them P1 , P2 , , Pn in some way such that: n−1 X (PK Pk+1 )2 ≤ AB (the sum is over the consecutive square of the segments from up to k=1 n − 1) China Team Selection Test 1986 Day Given a square ABCD whose side length is 1, P and Q are points on the sides AB and AD If the perimeter of AP Q is find the angle P CQ Given a tetrahedron ABCD, E, F , G, are on the respectively on the segments AB, AC and AD Prove that: i) area EF G ≤ maxarea ABC,area ABD,area ACD,area BCD ii) The same as above replacing ”area” for ”perimeter” Let xi , ≤ i ≤ n be real numbers with n ≥ Let p and q be their symmetric sum of degree and respectively Prove that: n−1 − 2q ≥ i) p2 · n r p 2nq n − ii) xi − ≤ p2 − · for every meaningful i n n−1 n Mark · k points in a circle and number them arbitrarily with numbers from to · k The chords cannot share common endpoints, also, the endpoints of these chords should be among the · k points I Prove that · k pairwisely non-intersecting chords can be drawn for each of whom its endpoints differ in at most · k − II Prove that the · k − cannot be improved China Team Selection Test 1987 Day 1 a.) For all positive integer k find the smallest positive integer f (k) such that sets s1 , s2 , , s5 exist satisfying: I each has k elements; II si and si+1 are disjoint for i = 1, 2, , (s6 = s1 ) III the union of the sets has exactly f (k) elements b.) Generalisation: Consider n ≥ sets instead of A closed recticular polygon with 100 sides (may be concave) is given such that it’s vertices have integer coordinates, it’s sides are parallel to the axis and all it’s sides have odd length Prove that it’s area is odd Let r1 = and rn = n−1 Y k=1 ri + 1, n ≥ Prove that among all sets of positive integers such that n X < 1, the partial sequences r1 , r2 , , rn are the one that gets nearer to k=1 China Team Selection Test 1987 Day Given a convex figure in the Cartesian plane that is symmetric with respect of both axis, we construct a rectangle A inside it with maximum area (over all posible rectangles) Then we enlarge it with center in the center of the rectangle and ratio lamda such that is covers the convex figure Find the smallest lamda such that it works for all convex figures Find all positive integer n such that the equation x3 + y + z = n · x2 · y · z has positive integer solutions Let G be a simple graph with · n vertices and n2 + edges, then there is a K4 -one edge, that is two triangles with a common edge China Team Selection Test 1996 Day 1 Let side BC of 4ABC be the diameter of a semicircle which cuts AB and AC at D and E respectively F and G are the feet of the perpendiculars from D and E to BC respectively DG and EF intersect at M Prove that AM ⊥ BC S is the set of functions f : N → R that satisfy the following conditions: n f (2n) for n = 1, 2, I f (1) = II f (n + 1) ≥ f (n) ≥ n+1 Find the smallest M ∈ N such that for any f ∈ S and any n ∈ N, f (n) < M Let M = {2, 3, 4, 1000} Find the smallest n ∈ N such that any n-element subset of M contains pairwise disjoint 4-element subsets S, T, U such that I For any elements in S, the larger number is a multiple of the smaller number The same applies for T and U II For any s ∈ S and t ∈ T , (s, t) = 1 For any s ∈ S and u ∈ U , (s, u) > China Team Selection Test 1996 Day countries A, B, C participate in a competition where each country has representatives The rules are as follows: every round of competition is between competitor each from countries The winner plays in the next round, while the loser is knocked out The remaining country will then send 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 countries until another country is knocked out The remaining team is the champion I At least how many games does the champion team win? II If the champion team won 11 matches, at least how many matches were played? Let α1 , α2 , , αn , β1 , β2 , , βn (n ≥ 4) be sets of real numbers such that 1, n X i=1 βi2 < Let A2 = − n X i=1 αi2 , B = − n X i=1 n X αi2 < i=1 n X βi2 , W = (1 − αi βi )2 Find all i=1 real numbers λ such that xn + λ(xn−1 + · · · + x3 + W x2 + ABx + 1) = only has real roots Corrected due to the courtesy of [url=http://www.mathlinks.ro/Forum/profile.php?mode=viewprofileu=2616]zhaoli.[/url] Does there exist non-zero complex numbers a, b, c and natural number h such that if integers k, l, m satisfy |k| + |l| + |m| ≥ 1996, then |ka + lb + mc| > is true? h China Team Selection Test 1997 Day 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 Q such that U Q = λU V Find the locus of point Q There are n football teams in a round-robin competition where every teams meet once The winner of each match receives points while the loser receives points In the case of a draw, both teams receive point each Let k be as follows: ≤ k ≤ n − At least how many points must a certain team get in the competition so as to ensure that there are at most k − teams whose scores are not less than that particular team’s score? Corrected due to the courtesy of [url=http://www.mathlinks.ro/Forum/profile.php?mode=viewprofileu=2616]zhaoli.[/url] Prove that there exists m ∈ N such that there exists an integral sequence {an } which satisfies: I a0 = 1, a1 = 337; II (an+1 an−1 − a2n ) + (an+1 + an−1 − 2an ) = m, ∀ n ≥ 1; III (an + 1)(2an + 1) is a perfect square ∀ n ≥ China Team Selection Test 1997 Day Find all real-coefficient polynomials f (x) which satisfy the following conditions: n X 2n 2n 2n−2 a2j a2n−2j ≤ a0 a2n ; I f (x) = a0 x + a2 x + · · · + a2n−2 x + a2n , a0 > 0; II n j=0 III All the roots of f (x) are imaginary numbers with no real part Corrected due to the courtesy of [url=http://www.mathlinks.ro/Forum/profile.php?mode=viewprofileu=2616]zhaoli.[/url] Let n be a natural number greater than X is a set such that |X| = n A1 , A2 , , Am are n(n − 1)(n − 2)(n − 3)(4n − 15) , prove that there distinct 5-element subsets of X If m > 600 [ exists Ai1 , Ai2 , , Ai6 (1 ≤ i1 < i2 < · · · , i6 ≤ m), such that Aik = k=1 There are 1997 pieces of medicine Three bottles A, B, C can contain at most 1997, 97, 17 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? China Team Selection Test 1998 Day 1 Find k ∈ N such that a.) For any n∈ N, there doesnot exist j ∈ Z which satisfies the conditions ≤ j ≤ n − k + n n n and , , , forms an arithmetic progression j j+1 j+k−1 b.) exists j which satisfies ≤ j ≤ n − k + 2, and There existsn ∈ N such that there n n n , , , forms an arithmetic progression j j+1 j+k−2 Find all n which satisfies part b.) Corrected due to the courtesy of [url=http://www.mathlinks.ro/Forum/profile.php?mode=viewprofileu=2616]zhaoli.[/url] n ≥ football teams participate in a round-robin tournament For every game played, the winner receives points, the loser receives points, and in the event of a draw, both teams receive point The third-from-bottom team has fewer points than all the teams ranked before it, and more points than the last teams; it won more games than all the teams before it, but fewer games than the teams behind it Find the smallest possible n π For a fixed θ ∈ [0, ], find the smallest a ∈ R+ which satisfies the following conditions: √ √ a a I + > cos θ sin θ √ √ p a a II There exists x ∈ [1 − , ] such that [(1 − x) sin θ − a − x2 cos2 θ]2 + [x cos θ − sin θ cos θ q 2 a − (1 − x) sin θ] ≤ a China Team Selection Test 1998 Day In acute-angled 4ABC, H is the orthocenter, O is the circumcenter and I is the incenter Given that ∠C > ∠B > ∠A, prove that I lies within 4BOH Let n be a natural number greater than l is a line on a plane There are n distinct points P1 , P2 , , Pn on l Let the product of distances between Pi and the other n − points be di (i = 1, 2, , n) There exists a point Q, which does not lie on l, on the plane Let the distance n X c2 (−1)n−i i from Q to Pi be Ci (i = 1, 2, , n) Find Sn = di i=1 Corrected due to the courtesy of [url=http://www.mathlinks.ro/Forum/profile.php?mode=viewprofileu=2616]zhaoli.[/url] For any h = 2r (r is a non-negative integer), find all k ∈ N which satisfy the following condition: There exists an odd natural number m > and n ∈ N, such that k | mh − 1, m | n mh −1 k + China Team Selection Test 1999 Day 1 For non-negative real numbers x1 , x2 , , xn which satisfy x1 + x2 + · · · + xn = 1, find the n X (x4j − x5j ) largest possible value of j=1 Find all prime numbers p which satisfy the following condition: For any prime q < p, if p = kq + r, ≤ r < q, there does not exist an integer q > such that a2 | r Let S = {1, 2, , 15} Let A1 , A2 , , An be n subsets of S which satisfy the following conditions: I |Ai | = 7, i = 1, 2, , n; II |Ai ∩ Aj | ≤ 3, ≤ i < j ≤ n III For any 3-element subset M of S, there exists Ak such that M ⊂ Ak Find the smallest possible value of n China Team Selection Test 1999 Day 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? For a fixed natural number m ≥ 2, prove that a.) There exists integers x1 , x2 , , x2m such that xi xm+i = xi+1 xm+i−1 + 1, i = 1, 2, , m (∗) b.) For any set of integers {x1 , x2 , , x2m which fulfils (*), an integral sequence , y−k , , y−1 , y0 , y1 , , yk can be constructed such that yk ym+k = yk+1 ym+k−1 + 1, k = 0, ±1, ±2, such that yi = xi , i = 1, 2, , 2m For every permutation τ of 1, 2, , 10, τ = (x1 , x2 , , x1 0), define S(τ ) = 10 X k=1 Let x11 = x1 Find I The maximum and minimum values of S(τ ) II The number of τ which lets S(τ ) attain its maximum III The number of τ which lets S(τ ) attain its minimum |2xk − 3xk−1 | China Team Selection Test 2000 Day 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 Given positive integers k, m, n such that ≤ k ≤ m ≤ n Evaluate n X i=0 (m + n + i)! · n + k + i i!(n − i)!(m + i)! For positive integer a ≥ 2, denote Na as 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: a.) Na is odd; b.) For every positive integer M , there exist a positive integer a ≥ such that Na ≥ M China Team Selection Test 2000 Day Let F be the set of all polynomials Γ such that all the coefficients of Γ(x) are integers and Γ(x) = has integer roots Given a positive intger k, find the smallest integer m(k) > such that there exist Γ ∈ F for which Γ(x) = m(k) has exactly k distinct integer roots a.) Let a, b be real numbers Define sequence xk and yk such that x0 = 1, y0 = 0, xk+1 = a · xk − b · yl , yk+1 = xk − a · yk for k = 0, 1, 2, Prove that [k/2] xk = X l (−1)l · ak−2·l · a2 + b · λk,l l=0 [k/2] where λk,l = X m=l k m · 2·m l [k/2] b.) Let uk = X λk,l For positive integer m, denote the remainder of uk divided by 2m as l=0 zm,k Prove that zm,k , k = 0, 1, 2, is a periodic function, and find the smallest period Let n be a positive integer Denote M = {(x, y)|x, y are integers , ≤ x, y ≤ n} Define function f on M with the following properties: n X f (x, y) = n − for 1x ≤ n; c.) If a.) f (x, y) takes non-negative integer value; b.) y=1 f (x1 , y1 )f (x2, y2) > 0, then (x1 − x2 )(y1 − y2 ) ≥ Find N (n), the number of functions f that satisfy all the conditions Give the explicit value of N (4) China Team Selection Test 2001 Day 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 a and b are natural numbers such that b > a > 1, and a does not divide b The sequence of ∞ natural numbers {bn }∞ n=1 satisfies bn+1 ≥ 2bn ∀n ∈ N Does there exist a sequence {an }n=1 of natural numbers such that for all n ∈ N, an+1 − an ∈ {a, b}, and for all m, l ∈ N (m may be equal to l), am + al 6∈ {bn }∞ n=1 ? For a given natural number k > 1, find all functions f : R → R such that for all x, y ∈ R, f [xk + f (y)] = y + [f (x)]k China Team Selection Test 2001 Day For a given natural number n > 3, the real numbers x1 , x2 , , xn , xn+1 , xn+2 satisfy the conditions < x1 < x2 < · · · < xn < xn+1 < xn+2 Find the minimum possible value of Pn xj+2 P )( ( ni=1 xxi+1 j=1 xj+1 ) i P P x2 +xl xl+2 x xk+2 )( nl=1 l+1xl xl+1 ) ( nk=1 x2 k+1 +x x k+1 k k+2 and find all (n + 2)-tuplets of real numbers (x1 , x2 , , xn , xn+1 , xn+2 ) which gives this value In the equilateral 4ABC, D is a point on side BC O1 and I1 are the circumcenter and incenter of 4ABD respectively, and O2 and I2 are the circumcenter and incenter of 4ADC respectively O1 I1 intersects O2 I2 at P Find the locus of point P as D moves along BC Let F = max1≤x≤3 |x3 − ax2 − bx − c| When a, b, c run over all the real numbers, find the smallest possible value of F China Team Selection Test 2002 Day 1 Let E and F be the intersections of opposite sides of a convex quadrilateral ABCD The two diagonals meet at P Let O be the foot of the perpendicular from O to EF Show that ∠BOC = ∠AOD 1 Suppose a1 = , an = (1 + an−1 )2 , n ≥ Find the minimum real λ such that for any 4 non-negative reals x1 , x2 , , x2002 , it holds 2002 X Ak ≤ λa2002 , k=1 where Ak = xk − k (xk + · · · + x2002 + k(k−1) + 1)2 , k ≥ Seventeen football fans were planning to go to Korea to watch the World Cup football match They selected 17 matches The conditions of the admission tickets they booked were such that - One person should book at most one admission ticket for one match; - At most one match was same in the tickets booked by every two persons; - There was one person who booked six tickets How many tickets did those football fans book at most? China Team Selection Test 2002 Day Find all natural numbers n(n ≥ 2) such that there exists reals a1 , a2 , , an which satisfy n(n − 1) {|ai − aj | | ≤ i < j ≤ n} = 1, 2, , Let A = {1, 2, 3, 4, 5, 6}, B = {7, 8, 9, , n} Ai (i = 1, 2, , 20) contains eight numbers, three of which are chosen from A and the other five numbers from B |Ai ∩ Aj | ≤ 2, ≤ i < j ≤ 20 Find the minimum possible value of n Given an integer k f (n) is defined on negative integer set and its values are integers f (n) satisfies f (n)f (n + 1) = (f (n) + n − k)2 , for n = −2, −3, · · · Find an expression of f (n) Let f (x1 , x2 , x3 ) = −2 · (x31 + x32 + x33 ) + · (x21 (x2 + x3 ) + x22 · (x1 + x3 ) + x23 · (x1 + x2 ) − 12x1 x2 x3 For any reals r, s, t, we denote g(r, s, t) = Find the minimum value of g(r, s, t) max t≤x3 ≤t+2 |f (r, r + 2, x3 ) + s| China Team Selection Test 2003 Day 1 ABC is an acute-angled triangle Let D be the point on BC such that AD is the bisector of ∠A Let E, F be the feet of perpendiculars from D to AC, AB respectively Suppose the lines BE and CF meet at H The circumcircle of triangle AF H meets BE at G (apart from H) Prove that the triangle constructed from BG, GE and BF is right-angled Suppose A ⊆ {0, 1, , 29} It satisfies that for any integer k and any two members a, b ∈ A(a, b is allowed to be same), a+b+30k is always not the product of two consecutive integers Please find A with largest possible cardinality Suppose A ⊂ {(a1 , a2 , , an ) | ∈ R, i = 1, , n} For any α = (a1 , a2 , , an ) ∈ A and β = (b1 , b2 , , bn ) ∈ A, we define γ(α, β) = (|a1 − b1 |, |a2 − b2 |, , |an − bn |), D(A) = {γ(α, β) | α, β ∈ A} Please show that |D(A)| ≥ |A|