IMOmath

Combinatorics

1. (29 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 \( n \)?

2. (17 p.)
Given a convex polyhedron with 26 vertices, 60 edges and 36 faces, 24 of the faces are triangular and 12 are quadrilaterals. A space diagonal is a line segment connecting two vertices which do not belong to the same face. How many space diagonals does the polyhedron have?

3. (4 p.)
Let \( S \) be the set of vertices of a unit cube. Find the number of triangles whose vertices belong to \( S \).

4. (12 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 \( p+q \).

5. (36 p.)
At the basement of a building with 5 floors, Adam, Bob, Cindy, Diana and Ernest entered the elevator. The elevator goes only up and doesn’t come back, and each person gets out of the elevator at one of the five floors. In how many ways can the five people leave the elevator in such a way that at no time are there a male and a female alone in the elevator?





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