

1. There are an odd number of soldiers on an exercise. The distance between every pair of soldiers is different. Each soldier watches his nearest neighbour. Prove that at least one soldier is not being watched.


2. (a) B and C are on the segment AD with AB = CD. Prove that for any point P in the plane: PA + PD ≥ PB + PC.
(b) Given four points A, B, C, D on the plane such that for any point P on the plane we have PA + PD ≥ PB + PC. Prove that B and C are on the segment AD with AB = CD.


3. Can both x^{2} + y and x + y^{2} be squares for x and y natural numbers?


4. A group of children are arranged into two equal rows. Every child in the back row is taller than the child standing in front of him in the other row. Prove that this remains true if each row is rearranged so that the children increase in height from left to right.


5. A rectangle ABCD is drawn on squared paper with its vertices at lattice points and its sides lying along the gridlines. AD = k AB with k an integer. Prove that the number of shortest paths from A to C starting out along AD is k times the number starting out along AB.


6. Given nonnegative real numbers a_{1}, a_{2}, ... , a_{n}, such that a_{i1} ≤ a_{i} ≤ 2a_{i1} for i = 2, 3, ... , n. Show that you can form a sum s = b_{1}a_{1} + ... + b_{n}a_{n} with each b_{i} +1 or 1, so that 0 ≤ s ≤ a_{1}.


7. Prove that you can always draw a circle radius A/P inside a convex polygon with area A and perimeter P.


8. A graph has at least three vertices. Given any three vertices A, B, C of the graph we can find a path from A to B which does not go through C. Prove that we can find two disjoint paths from A to B.
[A graph is a finite set of vertices such that each pair of distinct vertices has either zero or one edges joining the vertices. A path from A to B is a sequence of vertices A_{1}, A_{2}, ... , A_{n} such that A=A_{1}, B=A_{n} and there is an edge between A_{i} and A_{i+1} for i = 1, 2, ... , n1. Two paths from A to B are disjoint if the only vertices they have in common are A and B.]


9. Given a triangle ABC. Suppose the point P in space is such that PH is the smallest of the four altitudes of the tetrahedron PABC. What is the locus of H for all possible P?


10. Given 100 points on the plane. Prove that you can cover them with a collection of circles whose diameters total less than 100 and the distance between any two of which is more than 1. [The distance between circles radii r and s with centers a distance d apart is the greater of 0 and d  r  s.]


11. The distance from A to B is d kilometers. A plane P is flying with constant speed, height and direction from A to B. Over a period of 1 second the angle PAB changes by α degrees and the angle PBA by β degrees. What is the minimal speed of the plane?


12. Two players alternately choose the sign for one of the numbers 1, 2, ... , 20. Once a sign has been chosen it cannot be changed. The first player tries to minimize the final absolute value of the total and the second player to maximize it. What is the outcome (assuming both players play perfectly)? Example: the players might play successively: 1, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2. Then the outcome is 12. However, in this example the second player played badly!

