27th Russian 2001

27th Russian 2001 problems


1. Are there more positive integers under a million for which the nearest square is odd or for which it is even?
2. A monic quartic and a monic quadratic both have real coefficients. The quartic is negative iff the quadratic is negative and the set of values for which they are negative is an interval of length more than 2. Show that at some point the quartic has a smaller value than the quadratic.
3. ABCD is parallelogram and P a point inside, such that the midpoint of AD is equidistant from P and C, and the midpoint of CD is equidistant from P and A. Let Q be the midpoint of PB. Show that ∠PAQ = ∠PCQ.
4. No three diagonals of a convex 2000-gon meet at a point. The diagonals (but not the sides) are each colored with one of 999 colors. Show that there is a triangle whose sides are on three diagonals of the same color.
5. 2001 coins, each value 1, 2 or 3 are arranged in a row. Between any two coins of value 1 there is at least one coin, between any two of value 2 there are at least two coins, and between any two of value 3 there are at least three coins. What is the largest number of value 3 coins that could be in the row?
6. Given a graph of 2n+1 points, given any set of n points, there is another point joined to each point in the set. Show that there is a point joined to all the other points.
7. N is any point on AC is the longest side of the triangle ABC, such that the perpendicular bisector of AN meets the side AB at K and the perpendicular bisector of NC meets the side BC at M. Prove that BKOM is cyclic, where O is the circumcenter of ABC.
8. Find all odd positive integers n > 1 such that if a and b are relatively prime divisors of n, then a + b - 1 divides n.
9. Let A₁, A₂, ... , A₁₀₀ be subsets of a line, each a union of 100 disjoint closed segments. Prove that the intersection of all hundred sets is a union of at most 9901 disjoint closed segments. [A single point is considered to be a closed segment.]
10. The circle C' is inside the circle C and touches it at N. A tangent at the point X of C' meets C at A and B. M is the midpoint of the arc AB which does not contain N. Show that the circumradius of BMX is independent of the position of X.
11. Some pairs of towns in a country are joined by roads, so that there is a unique route from any town to another which does not pass through any town twice. Exactly 100 of the towns have only one road. Show that it is possible to construct 50 new roads so that there will still be a route between any two towns even if any one of the roads (old or new) is closed for maintenance.
12. x³ + ax² + bx + c has three distinct real roots, but (x² + x + 2001)³ + a(x² + x + 2001)² + b(x² + x + 2001) + c has no real roots. Show that 2001³ + a 2001² + b 2001 + c > 1/64.
13. An n x n Latin square has the numbers from 1 to n² arranged in its cells (one per cell) so that the sum of every row and column is the same. For every pair of cells in a Latin square the centers of the cells are joined by an arrow pointing to the cell with the larger number. Show that the sum of these vectors is zero.
14. The altitudes AD, BE, CF of the triangle ABC meet at H. Points P, Q, R are taken on the segments AD, BE, CF respectively, so that the sum of the areas of triangles ABR, AQC and PBC equals the area of ABC. Show that P, Q, R, H are cyclic.
15. S is a set of 100 stones. f(S) is the set of integers n such that we can find n stones in the collection weighing half the total weight of the set. What is the maximum possible number of integers in f(S)?
16. There are two families of convex polygons in the plane. Each family has a pair of disjoint polygons. Any polygon from one family intersects any polygon from the other family. Show that there is a line which intersects all the polygons.
17. N contestants answered an exam with n questions. a_i points are awarded for a correct answer to question i and nil for an incorrect answer. After the questions had been marked it was noticed that by a suitable choice of positive numbers a_i any desired ranking of the contestants could be achieved. What is the largest possible value of N?
18. The quadratics x² + ax + b and x² + cx + d have real coefficients and take negative values on disjoint intervals. Show that there are real numbers h, k such that h(x² + ax + b) + k(x² + cx + d) > 0 for all x.
19. m > n are positive integers such that m² + mn + n² divides mn(m + n). Show that (m - n)³ > mn.
20. A country has 2001 towns. Each town has a road to at least one other town. If a subset of the towns is such that any other town has a road to at least one member of the subset, then it has at least k > 1 towns. Show that the country may be partitioned into 2001 - k republics so that no two towns in the same republic are joined by a road.
21. ABCD is a tetrahedron. O is the circumcenter of ABC. The sphere center O through A, B, C meets the edges DA, DB, DC again at A', B', C'. Show that the tangent planes to the sphere at A', B', C' pass through the center of the sphere through A', B', C', D.