德国IMO国家队选拔考试2004
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2. (Problem 2) In a triangle ABC, let D be the midpoint of BC, and let E be a point on AC. The lines BE and AD meet at F . Prove: If BF/F E = BC/AB + 1, then the line BE bisects the angle ABC.
9. (Problem 3) Let ABC be a triangle with perimeter 2s and inradius r. Construct the semicircles with diameters BC, CA, AB outwardly (with respect to the triangle). The radius of the circle tangent to these three semicircles will be denoted by t. Prove
√
s 2
≤
t
≤
s 2
+
1−
3 2
r.
2nd TST 2004, 17 February 2004
10. (Problem 1) Each positive integer a is subjected to the following procedure, yielding the number d = d(a): (a) The last digit of a is moved to the first position. The resulting number is called b. (b) The number b is squared. The resulting number is called c. (c) The first digit of c is moved to the last position. The resulting number is called d. (All numbers are considered in the decimal system.) For instance, a = 2003 gives b = 3200, c = 10240000 and d = 02400001 = 2400001 = d(2003). Find all integers a such that d(a) = a2.
2nd pre-TST 2004, 8 December 2003
4. (Problem 1) In the plane, consider n circular disks K1, K2, · · · , Kn with equal radius r. Assume that each point of the plane is contained in not more than 2003 of these circular disks. Show that each circular disk Ki intersects not more than 14020 other circular disks.
aij =
1 if xi + yi ≥ 0 0 if xi + yi < 0
Further, let B be an n×n matrix whose elements are numbers from the set {0, 1} satisfying the following condition: The sum of all elements of each row of B equals the sum of all elements of the corresponding row of A; the sum of all elements of each column of B equals the sum of all elements of the corresponding column of A. Show that in this case, A = B.
Page 2
3rd TST 2004, 14 March 2004 13. (Problem 1) Let ABC be an acute triangle, and let M and N be two points on the line AC such that
the vectors MN and AC are identical. Let X be the orthogonal projection of M on BC, and let Y be the orthogonal projection of N on AB. Finally, let H be the orthocenter of triangle ABC. Show that the points B, X, H, Y lie on one circle. 14. (Problem 2) Find all pairs of positive integers (n; k) such that (n + 1)k − 1 = n! 15. (Problem 3) Let n ≥ 2 be a natural number, and let (a1; a2; · · · ; an) be a permutation of (1; · · · ; n). For any k with 1 ≤ k ≤ n, we place ak raisins on the position k of the real number axis. [The real number axis is the x-axis of a Cartesian coordinate system.] Now, we place three children A, B, C on the positions xA, xB, xC , each of the numbers xA, xB, xC , being an element of {1, 2, · · · , n}. [It is not forbidden to place different children on the same place!] For any k, the ak raisins placed on the position k are equally handed out to those children whose positions are next to k. [So,if there is only one child lying next to k, then he gets the raisin. If there are two children lying next to k (either both on the same position or symmetric with respect to k), then each of them gets one half of the raisin. Etc...] After all raisins are distributed, a child is unhappy if he could have received more raisins than he actually has received if he had moved to another place (while the other children would rest on their places). For which n does there exist a configuration (a1; a2; · · · ; an) and numbers xA, xB, xC , such that all three children are happy?
1st TST 2004, 14 February 2004
7. (Problem 1) Let aij(i, j = 1, 2, 3) be real numbers such that
aij > 0 f or i = j; aij < 0 f or i = j. Prove the existence of positive real numbers c1, c2, c3 such that the numbers a11c1 + a12c2 + a13c3 a21c1 + a22c2 + a23c3 a31c1 + a32c2 + a33c3 are either all negative, or all zero, or all positive.
German IMO TST 2004
1st pre-TST 2004, 1 December 2003
1. (Problem 1) A function f satisfies the equation f (x) + f (1 − 1/x) = 1 + x, for every real number x except for x = 0 and x = 1. Find a formula for f .
6. (Problem 3) Let ABC be an isosceles triangle with AC = BC, whose incenter is I. Let P be a point on the circumcircle of the triangle AIB lying inside the triangle ABC. The lines through P parallel to CA and CB meet AB at D and E respectively. The lines through P parallel to AB meets CA and CB at F and G respectively. Prove that the lines DF and EG intersect on the circumcircle of the triangle ABC.
12. (Problem 3) Regard a plane with a Cartesian coordinate system; for each point with integer coordinates, draw a circular disk centered at this point and having the radius 1/1000. (a) Prove the existence of an equilateral triangle whose vertices lie in the interior of different disks. (b) Show that every equilateral triangle whose vertices lie in the interior of different disks has a sidelength > 96. [The ”>96” in (b) can be strengthened to ”>124”. By the way, part (a) of this problem is the place where I used the well-known ”Dedekind” theorem.]
11. (Problem 2) Given three pairwise distinct points A, B, C lying on one straight line in this order. Let G be a circle through A and C with center not on the line AC. The tangents to G at A and C meet at P ; the segment P B meets G at Q. Show that the point of intersection of the angle bisector of the angle AQC with the line AC does not depend on the choice of the circle G.
5. (Problem 2) Given n real numbers x1, x2, · · · , xn, and n further real numbers y1, y2, · · ·Hale Waihona Puke Baidu, yn. The elements aij (with 1 ≤ i, j ≤ n) of an n × n matrix are defined as follows:
8. (Problem 2) Let n ≥ 5 be an integer. Find the maximal integer k such that there exists a polygon with n vertices (convex or not, but not self-intersecting!) having k internal 90◦ angles.
3. (Problem 3) Given the six real numbers a, b, c, x, y, z such that a) 0 < b − c < a < b + c b) ax + by + cz = 0 What is the sign of the sum ayz + bzx + cxy? Additional question by DG: Prove using triangle geometry.
9. (Problem 3) Let ABC be a triangle with perimeter 2s and inradius r. Construct the semicircles with diameters BC, CA, AB outwardly (with respect to the triangle). The radius of the circle tangent to these three semicircles will be denoted by t. Prove
√
s 2
≤
t
≤
s 2
+
1−
3 2
r.
2nd TST 2004, 17 February 2004
10. (Problem 1) Each positive integer a is subjected to the following procedure, yielding the number d = d(a): (a) The last digit of a is moved to the first position. The resulting number is called b. (b) The number b is squared. The resulting number is called c. (c) The first digit of c is moved to the last position. The resulting number is called d. (All numbers are considered in the decimal system.) For instance, a = 2003 gives b = 3200, c = 10240000 and d = 02400001 = 2400001 = d(2003). Find all integers a such that d(a) = a2.
2nd pre-TST 2004, 8 December 2003
4. (Problem 1) In the plane, consider n circular disks K1, K2, · · · , Kn with equal radius r. Assume that each point of the plane is contained in not more than 2003 of these circular disks. Show that each circular disk Ki intersects not more than 14020 other circular disks.
aij =
1 if xi + yi ≥ 0 0 if xi + yi < 0
Further, let B be an n×n matrix whose elements are numbers from the set {0, 1} satisfying the following condition: The sum of all elements of each row of B equals the sum of all elements of the corresponding row of A; the sum of all elements of each column of B equals the sum of all elements of the corresponding column of A. Show that in this case, A = B.
Page 2
3rd TST 2004, 14 March 2004 13. (Problem 1) Let ABC be an acute triangle, and let M and N be two points on the line AC such that
the vectors MN and AC are identical. Let X be the orthogonal projection of M on BC, and let Y be the orthogonal projection of N on AB. Finally, let H be the orthocenter of triangle ABC. Show that the points B, X, H, Y lie on one circle. 14. (Problem 2) Find all pairs of positive integers (n; k) such that (n + 1)k − 1 = n! 15. (Problem 3) Let n ≥ 2 be a natural number, and let (a1; a2; · · · ; an) be a permutation of (1; · · · ; n). For any k with 1 ≤ k ≤ n, we place ak raisins on the position k of the real number axis. [The real number axis is the x-axis of a Cartesian coordinate system.] Now, we place three children A, B, C on the positions xA, xB, xC , each of the numbers xA, xB, xC , being an element of {1, 2, · · · , n}. [It is not forbidden to place different children on the same place!] For any k, the ak raisins placed on the position k are equally handed out to those children whose positions are next to k. [So,if there is only one child lying next to k, then he gets the raisin. If there are two children lying next to k (either both on the same position or symmetric with respect to k), then each of them gets one half of the raisin. Etc...] After all raisins are distributed, a child is unhappy if he could have received more raisins than he actually has received if he had moved to another place (while the other children would rest on their places). For which n does there exist a configuration (a1; a2; · · · ; an) and numbers xA, xB, xC , such that all three children are happy?
1st TST 2004, 14 February 2004
7. (Problem 1) Let aij(i, j = 1, 2, 3) be real numbers such that
aij > 0 f or i = j; aij < 0 f or i = j. Prove the existence of positive real numbers c1, c2, c3 such that the numbers a11c1 + a12c2 + a13c3 a21c1 + a22c2 + a23c3 a31c1 + a32c2 + a33c3 are either all negative, or all zero, or all positive.
German IMO TST 2004
1st pre-TST 2004, 1 December 2003
1. (Problem 1) A function f satisfies the equation f (x) + f (1 − 1/x) = 1 + x, for every real number x except for x = 0 and x = 1. Find a formula for f .
6. (Problem 3) Let ABC be an isosceles triangle with AC = BC, whose incenter is I. Let P be a point on the circumcircle of the triangle AIB lying inside the triangle ABC. The lines through P parallel to CA and CB meet AB at D and E respectively. The lines through P parallel to AB meets CA and CB at F and G respectively. Prove that the lines DF and EG intersect on the circumcircle of the triangle ABC.
12. (Problem 3) Regard a plane with a Cartesian coordinate system; for each point with integer coordinates, draw a circular disk centered at this point and having the radius 1/1000. (a) Prove the existence of an equilateral triangle whose vertices lie in the interior of different disks. (b) Show that every equilateral triangle whose vertices lie in the interior of different disks has a sidelength > 96. [The ”>96” in (b) can be strengthened to ”>124”. By the way, part (a) of this problem is the place where I used the well-known ”Dedekind” theorem.]
11. (Problem 2) Given three pairwise distinct points A, B, C lying on one straight line in this order. Let G be a circle through A and C with center not on the line AC. The tangents to G at A and C meet at P ; the segment P B meets G at Q. Show that the point of intersection of the angle bisector of the angle AQC with the line AC does not depend on the choice of the circle G.
5. (Problem 2) Given n real numbers x1, x2, · · · , xn, and n further real numbers y1, y2, · · ·Hale Waihona Puke Baidu, yn. The elements aij (with 1 ≤ i, j ≤ n) of an n × n matrix are defined as follows:
8. (Problem 2) Let n ≥ 5 be an integer. Find the maximal integer k such that there exists a polygon with n vertices (convex or not, but not self-intersecting!) having k internal 90◦ angles.
3. (Problem 3) Given the six real numbers a, b, c, x, y, z such that a) 0 < b − c < a < b + c b) ax + by + cz = 0 What is the sign of the sum ayz + bzx + cxy? Additional question by DG: Prove using triangle geometry.