IMOmath

Combinatorics

1. (14 p.)
Two students Alice and Bob participated in a two-day math contest. At the end both had attempted questions worth 500 points. Alice scored 160 out of 300 attempted on the first day and 140 out of 200 attempted on the second day, so her two-day success ratio was 300/500 = 3/5. Bob’s scores are different from Alice’s (but with the same two-day total). Bob had a positive integer score on each day. However, for each day Bob’s success ratio was less than Alice’s. Assume that \( p/q \) (\( p \) and \( q \) are relatively prime integers) is the largest possible two-day success ratio that Bob could have achieved. Calculate \( p+q \).

2. (35 p.)
We are given an unfair coin. When the coin is tossed, the probability of heads is 0.4. The coin is tossed 10 times. Let \( a_n \) be the number of heads in the first \( n \) tosses. Let \( P \) be the probability that \( a_n/n \leq 0.4 \) for \( n = 1, 2, \dots , 9 \) and \( a_{10}/10 = 0.4 \). Evaluate \( \frac{P\cdot 10^{10}}{24^4} \).

3. (35 p.)
There are 27 candidates in elections and \( n \) citizens that vote for them. If a candidate gets \( m \) votes, then \( 100m/n \leq m-1 \). What is the smallest possible value of \( n \)?

4. (5 p.)
Let \( S \) be the set of vertices of a unit cube. Find the number of triangles whose vertices belong to \( S \).

5. (8 p.)
Let \( S = \{1, 2, 3, 5, 8, 13, 21, 34\} \). Find the sum \( \sum \max(A) \) where the sum is taken over all 28 two-element subsets \( A \) of \( S \).





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