CU�C THI TOÁN QU�C T� GI�A CÁC THÀNH PH� L�N TH� 37 Kh�i Trung h�c Cơ s� c�p đ� A, Mùa thu 2015 Hà N�i, ngày 25/10/2015 (K�t qu� đư�c tính b�ng t�ng đi�m c�a ba có đi�m cao nh�t, đi�m c�a có nhi�u ý b�ng t�ng đi�m c�a ý thành ph�n.) đi�m đ� M�t đa giác đơn (hai c�nh b�t kỳ ho�c không c�t nhau, ho�c c�t t�i đ�nh) v�i c�nh n�m đư�ng k� c�a m�t lư�i ô vuông đư�c g�i kỳ di�u n�u khơng ph�i hình ch� nh�t ta có th� ghép m�t s� (l�n 1) b�n c�a đ� thu đư�c m�t đa giác đ�ng d�ng v�i Ví d�, đa giác d�ng ch� L g�m ô vuông đơn v� kỳ di�u (xem hình v� bên ph�i) a) Tìm m�t đa giác kỳ di�u g�m vng đơn v� b) Xác đ�nh t�t c� n > cho t�n t�i đa giác kỳ di�u g�m n ô vuông đơn v� 2 a) b) M�t t�p h�p g�m có s� nguyên t� đ�n 100 b� k s� nguyên H�i có ph�i ch�n đư�c k s� phân bi�t t� t�p có t�ng b�ng 100 n�u k = 9; k = 8? a) b) Ch�ng minh r�ng t�ng đ� dài hai đư�ng trung tuy�n c�a m�t tam giác ln khơng vư�t q 3P/4 P chu vi c�a tam giác; không bé 3p/4 p n�a chu vi c�a tam giác Lư�i vng kích thư�c × đư�c x�p t� que diêm cho m�i có c�nh que diêm hai ô chung c�nh chung m�t que diêm Pete Basil l�n lư�t l�y t�ng que diêm Ngư�i th�ng ngư�i mà sau lư�t c�a s� khơng cịn l�i vng × H�i s� hai ngư�i chơi có chi�n thu�t đ� đ�m b�o ln giành đư�c chi�n th�ng? Tam giác ABC có đư�ng trung tuy�n AA0 , BB0 CC0 c�t t�i đi�m M G�i P , Q, R, T l�n lư�t tâm đư�ng tròn ngo�i ti�p tam giác M A0 B0 , M CB0 , M A0 C0 , M BC0 Ch�ng minh r�ng đi�m P , Q, R, T , M n�m m�t đư�ng tròn 8 Trên b�ng đen cho trư�c m�t s� s� th�c phân bi�t Peter mu�n vi�t m�t bi�u th�c có t�p giá tr� t�p s� b�ng Peter có th� s� d�ng s� th�c b�t kỳ kèm theo d�u ngo�c phép tốn +, −, × Peter có th� s� d�ng phép tốn đ�c bi�t ± đ� kí hi�u phép + ho�c − Ví d�, bi�u th�c ± có t�p giá tr� {4, 6}, bi�u th�c (2 ± 0,5) ± 0,5 có t�p giá tr� {1, 2, 3} H�i Peter có th� vi�t đư�c bi�u th�c hay khơng n�u: a) s� b�ng 1, 2, 4; b) s� b�ng 100 s� th�c phân bi�t b�t kỳ? 10 Ơng già Nơ-en có n lo�i k�o, m�i lo�i có k chi�c Ơng chia chúng ng�u nhiên vào k túi quà, m�i túi có n chi�c t�ng cho k đ�a tr�, m�i đ�a m�t túi Lũ tr� nhanh chóng khám phá nh�ng có túi quy�t đ�nh trao đ�i k�o M�i lư�t trao đ�i g�m có hai đ�a tr�, m�i đ�a l�y m�t chi�c túi đ�i l�y chi�c k�o thu�c lo�i mà chưa có H�i có ph�i ln t�n t�i m�t cách s�p x�p lư�t ThuVienDeThi.com trao đ�i đ� m�i đ�a tr� đ�u có t�t c� lo�i k�o? INTERNATIONAL MATHEMATICS TOURNAMENT OF TOWNS Junior A-Level Paper, Fall 2015 Hanoi, 25/10/2015 (The result is computed from the three problems with the highest scores, the scores for the individual parts of a single problem are summed up.) points problems A grid polygon is called amazing if it is not a rectangle and several its copies can form a polygon similar to it For instance, a corner consisting of three cells is an amazing polygon (see the figure on the right) a) Find an amazing polygon consisting of cells b) Determine all n > such that there exists an amazing polygon consisting of n cells 2 A set consists of all integers from to 100 except some k integers Is it always possible to choose k distinct integers in this set so that their sum equals 100 if a) k = 9; b) k = 8? a) b) Prove that the sum of lengths of any two medians in an arbitrary triangle is not greater than 3P/4 where P is the perimeter of this triangle; not less than 3p/4 where p is the semiperimeter of this triangle A × grid square is made of matches, every cell side consists of a single match, and any two adjacent cells share exactly one match Pete and Basil in turn remove matches one by one A player wins if there remains no entire × square after his move Who of the players has a winning strategy? In triangle ABC, medians AA0 , BB0 and CC0 intersect at point M Let P , Q, R, and T be the circumcenters of triangles M A0 B0 , M CB0 , M A0 C0 , M BC0 respectively Prove that points P , Q, R, T , M are concyclic Several distinct real numbers are written on a blackboard Peter wants to make an expression such that its values are exactly these numbers To make such an expression, he may use any real numbers, brackets, and usual signs +, − and × He may also use a special sign ±: computing the values of the resulting expression, he chooses values + or − for every ± in all possible combinations For instance, the expression ± results in {4, 6}, and (2 ± 0.5) ± 0.5 results in {1, 2, 3} Can Pete construct such an expression: if the numbers on the blackboard are 1, 2, 4; for any collection of 100 distinct real numbers on a blackboard? 8 a) b) 10 Santa Claus had n sorts of candies, k candies of each sort He distributed them at random between k gift bags, n candies in each, and gave a bag to each of k children The children learned what they had in the bags and decided to trade Two children can trade one candy for one candy in case if each of them gets a candy of the sort that he/she lacks Is it true that a sequence of trades can always be arranged so that in the end every child has candies of each sort? ThuVienDeThi.com CU�C THI TOÁN QU�C T� GI�A CÁC THÀNH PH� L�N TH� 37 Kh�i Trung h�c Ph� thông c�p đ� A, Mùa thu 2015 Hà N�i, ngày 25/10/2015 (K�t qu� đư�c tính b�ng t�ng đi�m c�a ba có đi�m cao nh�t, đi�m c�a có nhi�u ý b�ng t�ng đi�m c�a ý thành ph�n.) đi�m đ� M�t c�p s� nhân g�m 37 s� nguyên dương Bi�t r�ng, s� h�ng đ�u s� h�ng cu�i c�a c�p s� nguyên t� Ch�ng minh r�ng s� h�ng th� 19 c�a c�p s� lũy th�a b�c 18 c�a m�t s� nguyên dương M�t b�ng k� vng kích thư�c 10 × 10 đư�c chia b�i 80 đo�n th�ng đ� dài đơn v� thành 20 đa giác v�i di�n tích b�ng (các đo�n th�ng n�m đư�ng k� khơng n�m c�nh ngồi c�a b�ng vuông) Ch�ng minh r�ng t�t c� 20 đa giác b�ng Cho m�t đa th�c khác h�ng s� v�i h� s� s� nguyên có giá tr� tuy�t đ�i khơng vư�t q 2015 Ch�ng minh r�ng nghi�m dương c�a đa th�c đ�u l�n 1/2016 Cho t� giác n�i ti�p ABCD v�i K N trung đi�m c�a đư�ng chéo AC BD Các đư�ng kéo dài c�a c�p c�nh đ�i c�t t�i hai đi�m P Q Ch�ng minh r�ng ∠P KQ + ∠P N Q = 180◦ 6 Trên b�ng đen cho trư�c m�t s� s� th�c phân bi�t Peter mu�n vi�t m�t bi�u th�c có t�p giá tr� t�p s� b�ng Peter có th� s� d�ng s� th�c b�t kỳ kèm theo d�u ngo�c phép tốn +, −, × Peter có th� s� d�ng phép tốn đ�c bi�t ± đ� kí hi�u phép + ho�c − Ví d�, bi�u th�c ± có t�p giá tr� {4, 6}, bi�u th�c (2 ± 0,5) ± 0,5 có t�p giá tr� {1, 2, 3} H�i Peter có th� vi�t đư�c bi�u th�c hay khơng n�u: a) s� b�ng 1, 2, 4; b) s� b�ng 100 s� th�c phân bi�t b�t kỳ? 6 a) b) 12 Basil có m�t qu� dưa h�u m�t hình c�u đư�ng kính 20cm S� d�ng m�t dao dài, Basil th�c hi�n ba nhát c�t đơi m�t vng góc v�i Bi�t r�ng m�i nhát c�t có đ� sâu h (nhát c�t t�o m�t cung tròn v�i đ� cao h m�t ph�ng c�t) H�i có ph�i qu� dưa h�u ln đư�c chia thành nh�t hai ph�n r�i n�u h = 17 cm; h = 18 cm? Có N b�n h�c sinh đ�ng x�p thành m�t hàng th�ng Bi�t r�ng, s� khơng có hai b�n có chi�u cao Ta đư�c phép th�c hi�n m�t s� l�n chuy�n ch� c�a b�n h�c sinh sau M�i l�n chuy�n ch�, trư�c h�t b�n h�c sinh đư�c chia thành nhóm v�i chi�u cao tăng d�n t� trái qua ph�i (m�t nhóm có th� g�m m�t b�n) cho s� nhóm nh�t Sau đó, th� t� c�a b�n m�i nhóm đư�c đ�o ngư�c, có nghĩa m�i nhóm, b�n s� đ�ng theo chi�u cao gi�m d�n t� trái qua ph�i Ch�ng minh r�ng sau N − l�n chuy�n ch� v�y, b�n h�c sinh s� đ�ng theo chi�u cao gi�m d�n t� trái qua ph�i ThuVienDeThi.com INTERNATIONAL MATHEMATICS TOURNAMENT OF TOWNS Senior A-Level Paper, Fall 2015 Hanoi, 25/10/2015 (The result is computed from the three problems with the highest scores, the scores for the individual parts of a single problem are summed up.) points problems A geometrical progression consists of 37 positive integers The first and the last terms are relatively prime numbers Prove that the 19th term of the progression is the 18th power of a positive integer A 10 × 10 grid square is split by 80 unit grid segments (lying inside the square) into 20 polygons of equal area Prove that all these polygons are congruent Each coefficient of a non-constant polynomial is an integer of absolute value not exceeding 2015 Prove that every positive root of this polynomial is greater than 1/2016 Suppose that a quadrilateral ABCD is cyclic Let extensions of the opposite sides intersect at points P and Q, and let K and N be the midpoints of the diagonals Prove that ∠P KQ + ∠P N Q = 180◦ Several distinct real numbers are written on a blackboard Peter wants to make an expression such that its values are exactly these numbers To make such an expression, he may use any real numbers, brackets, and usual signs +, − and × He may also use a special sign ±: computing the values of the resulting expression, he chooses values + or − for every ± in all possible combinations For instance, the expression ± results in {4, 6}, and (2 ± 0.5) ± 0.5 results in {1, 2, 3} Can Pete construct such an expression: if the numbers on the blackboard are 1, 2, 4; for any collection of 100 distinct real numbers on a blackboard? 6 a) b) 6 a) b) 12 Basil has a watermelon in a shape of a ball with diameter 20 cm Using a long knife, Basil makes three pairwise perpendicular cuts, each cut is of depth h (a cut produces a circular segment with height h in the plane of the cut) Does it necessarily follow that the watermelon is divided into two or more pieces if h = 17 cm; h = 18 cm? N children, no two of the same height, stand in a line in some order The following two-step procedure is applied repeatedly: firstly, the line is split into the least possible number of groups so that in each group all children are arranged from the left to the right in ascending order of the height (a group may consist of a single child) Secondly, the order of children in each group is changed to the opposite one (so now in each group the children stand in descending order) Prove that after N − rearrangements the children in the line will stand in descending order from the left to the right ThuVienDeThi.com ... end every child has candies of each sort? ThuVienDeThi.com CU�C THI TOÁN QU�C T� GI�A CÁC THÀNH PH� L�N TH� 37 Kh�i Trung h�c Ph� thông c�p đ� A, Mùa thu 2015 Hà N�i, ngày 25/10/2015 (K�t qu�... triangle is not greater than 3P/4 where P is the perimeter of this triangle; not less than 3p/4 where p is the semiperimeter of this triangle A × grid square is made of matches, every cell side... t�ng đi�m c�a ba có đi�m cao nh�t, đi�m c�a có nhi�u ý b�ng t�ng đi�m c�a ý thành ph�n.) đi�m đ� M�t c�p s� nhân g�m 37 s� nguyên dương Bi�t r�ng, s� h�ng đ�u s� h�ng cu�i c�a c�p s� nguyên t�