BAMO 2000: Practice Problems
ZVEZDELINA STANKOVA-FRENKEL
MILLS COLLEGE


Note: You have 4 hours to solve as many problems as you can from the following list of 5 problems. Each solution should be written clearly and in detail on a separate sheet of paper. Each problem is worth $ 7$ points. Partial credit will be awarded for partial solutions.


Problem 1. Alice plays the following game: she writes in one row all numbers 1,2,3,...,2001. She then starts placing the signs $ +$ or $ -$ between any two adjacent numbers. In the end, she calculates the value of the resulting expression and wins if this value is 0. Prove that Alice can never win.


Problem 2. Let $ a_1,\,\,a_2,...,a_n$ ($ n\geq 2$) be a sequence of integers. Prove that we can choose several of its members so that the sum of their squares is divisible by $ n$.


Problem 3. On circle $ k$ we have 4 points $ P$, $ A$, $ Q$ and $ B$. We draw the circles $ k_1$ and $ k_2$ so that they have diameters $ AQ$ and $ BQ$, correspondingly. Let $ k_1$ and $ k_2$ intersect in point $ L$ (and $ Q$). Line $ PA$ and circle $ k_1$ intersect in point $ M$ (and $ A$). Line $ PB$ and circle $ k_2$ intersect in point $ N$ (and $ B$). Prove that the three points $ N$, $ L$ and $ M$ lie on a line (i.e. they are collinear.)

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Problem 4. Prove that if a is a positive integer such that $ 3^{a}-2^{a}$ is a power of a prime number, then a is also a prime number.


Problem 5. A lock has 25 keys arranged in a 5x5 array, each key oriented either horizontally or vertically. In order to open it, all the keys must be vertically oriented. When a key is switched to another position, all the other keys in the same row and column automatically switch their positions too (see diagram). Show that there exists an initial position of all keys, which cannot be unlocked. (Only one key at a time can be switched. Compare with Problem 3 from BAMO 1999.)