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[TẶNG BẠN] TRỌN BỘ Bí kíp học tốt 08 môn
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1. Show that for each n we can find an n-digit number with all its digits odd which is divisible by 5n.
2. A convex polygon has all its sides and diagonals with rational length. It is dissected into smaller polygons by drawing all its diagonals. Show that the small polygons have all sides rational.
3. Given a sequence S1 of n+1 non-negative integers, a0, a1, ... , an we derive another sequence S2 with terms b0, b1, ... , bn, where bi is the number of terms preceding ai in S1 which are different from ai (so b0 = 0). Similarly, we derive S2 from S1 and so on. Show that if ai ≤ i for each i, then Sn = Sn+1.
4. ABC is a triangle. A circle through A and B meets the sides AC, BC at D, E respectively. The lines AB and DE meet at F. The lines BD and CF meet at M. Show that M is the midpoint of CF iff MB·MD = MC2.
5. Prove that for any positive reals x, y, z we have (2x+y+z)2/(2x2 + (y+z)2) + (2y+z+x)2/(2y2 + (z+x)2) + (2z+x+y)2/(2z2 + (x+y)2) ≤ 8.
6. A positive integer is written at each vertex of a hexagon. A move is to replace a number by the (non-negative) difference between the two numbers at the adjacent vertices. If the starting numbers sum to 20032003, show that it is always possible to make a sequence of moves ending with zeros at every vertex.
7. Find all real-valued functions f on the reals such that f(x2 - y2) = x f(x) - y f(y) for all x, y.
8. Show that we can link any two integers m, n greater than 2 by a chain of positive integers m = a1, a2, ... , ak+1 = n, so that the product of any two consecutive members of the chain is divisible by their sum. [For example, 7, 42, 21, 28, 70, 30, 6, 3 links 7 and 3.]
9. A tromino is a 1 x 3 rectangle. Trominoes are placed on an n x n board. Each tromino must line up with the squares on the board, so that it covers exactly three squares. Let f(n) be the smallest number of trominoes required to stop any more being placed. Show that for all n > 0, n2/7 + hn ≤ f(n) ≤ n2/5 + kn for some reals h and k.
10. Non-negative reals x, y, z satisfy x2 + y2 + z2 + xyz = 4. Show that xyz ≤ xy + yz + zx <= xyz + 2.
11. ABC is a triangle and X is a point in the same plane. The three lengths XA, XB, XC can be used to form an obtuse-angled triangle. Show that if XA is the longest length, then ∠BAC is acute.
12. A set of integers is such that if a and b belong to it, then so do a2 - a, and a2 - b. Also, there are two members a, b whose greatest common divisor is 1 and such that a - 2 and b - 2 also have greatest common divisor 1. Show that the set contains all the integers.
13. Every point in the plane is assigned a real number, so that for any three points which are not collinear, the number assigned to the incenter is the mean of the numbers assigned to the three points. Show that the same number is assigned to every point.
14. What is the smallest number of colors needed to color 8 boxes of 6 balls (one color for each ball), so that the balls in each box are all different colors and any pair of colors occurs in at most one box.
2. A convex polygon has all its sides and diagonals with rational length. It is dissected into smaller polygons by drawing all its diagonals. Show that the small polygons have all sides rational.
3. Given a sequence S1 of n+1 non-negative integers, a0, a1, ... , an we derive another sequence S2 with terms b0, b1, ... , bn, where bi is the number of terms preceding ai in S1 which are different from ai (so b0 = 0). Similarly, we derive S2 from S1 and so on. Show that if ai ≤ i for each i, then Sn = Sn+1.
4. ABC is a triangle. A circle through A and B meets the sides AC, BC at D, E respectively. The lines AB and DE meet at F. The lines BD and CF meet at M. Show that M is the midpoint of CF iff MB·MD = MC2.
5. Prove that for any positive reals x, y, z we have (2x+y+z)2/(2x2 + (y+z)2) + (2y+z+x)2/(2y2 + (z+x)2) + (2z+x+y)2/(2z2 + (x+y)2) ≤ 8.
6. A positive integer is written at each vertex of a hexagon. A move is to replace a number by the (non-negative) difference between the two numbers at the adjacent vertices. If the starting numbers sum to 20032003, show that it is always possible to make a sequence of moves ending with zeros at every vertex.
7. Find all real-valued functions f on the reals such that f(x2 - y2) = x f(x) - y f(y) for all x, y.
8. Show that we can link any two integers m, n greater than 2 by a chain of positive integers m = a1, a2, ... , ak+1 = n, so that the product of any two consecutive members of the chain is divisible by their sum. [For example, 7, 42, 21, 28, 70, 30, 6, 3 links 7 and 3.]
9. A tromino is a 1 x 3 rectangle. Trominoes are placed on an n x n board. Each tromino must line up with the squares on the board, so that it covers exactly three squares. Let f(n) be the smallest number of trominoes required to stop any more being placed. Show that for all n > 0, n2/7 + hn ≤ f(n) ≤ n2/5 + kn for some reals h and k.
10. Non-negative reals x, y, z satisfy x2 + y2 + z2 + xyz = 4. Show that xyz ≤ xy + yz + zx <= xyz + 2.
11. ABC is a triangle and X is a point in the same plane. The three lengths XA, XB, XC can be used to form an obtuse-angled triangle. Show that if XA is the longest length, then ∠BAC is acute.
12. A set of integers is such that if a and b belong to it, then so do a2 - a, and a2 - b. Also, there are two members a, b whose greatest common divisor is 1 and such that a - 2 and b - 2 also have greatest common divisor 1. Show that the set contains all the integers.
13. Every point in the plane is assigned a real number, so that for any three points which are not collinear, the number assigned to the incenter is the mean of the numbers assigned to the three points. Show that the same number is assigned to every point.
14. What is the smallest number of colors needed to color 8 boxes of 6 balls (one color for each ball), so that the balls in each box are all different colors and any pair of colors occurs in at most one box.