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IMO Shortlist 1966 problem 58

In a mathematical contest, three problems, $A,B,C$ were posed. Among the participants ther were 25 students who solved at least one problem each. Of all the contestants who did not solve problem $A$ , the number who solved $B$ was twice the number who solved $C$ . The number of students who solved only problem $A$ was one more than the number of students who solved $A$ and at least one other problem. Of all students who solved just one problem, half did not solve problem $A$ . How many students solved only problem $B$ ?

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IMO Shortlist 1979 problem 7

If $p$ and $q$ are natural numbers so that $\frac{p}{q}=1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+ \ldots -\frac{1}{1318}+\frac{1}{1319},$ prove that $p$ is divisible with $1979$ .

IMO Shortlist 1984 problem 5

Prove that $0\le yz+zx+xy-2xyz\le{7\over27}$ , where $x,y$ and $z$ are non-negative real numbers satisfying $x+y+z=1$ .

IMO Shortlist 1986 problem 5

Let $d$ be any positive integer not equal to $2, 5$ or $13$ . Show that one can find distinct $a,b$ in the set $\{2,5,13,d\}$ such that $ab-1$ is not a perfect square.

IMO Shortlist 1987 problem 16

Let $p_n(k)$ be the number of permutations of the set $\{1,2,3,\ldots,n\}$ which have exactly $k$ fixed points. Prove that $\sum_{k=0}^nk p_n(k)=n!$ .(IMO Problem 1)

Original formulation

Let $S$ be a set of $n$ elements. We denote the number of all permutations of $S$ that have exactly $k$ fixed points by $p_n(k).$ Prove:

(a) $\sum_{k=0}^{n} kp_n(k)=n! \ ;$

(b) $\sum_{k=0}^{n} (k-1)^2 p_n(k) =n!$

Proposed by Germany, FR

IMO Shortlist 1989 problem 22

Prove that in the set $\{1,2, \ldots, 1989\}$ can be expressed as the disjoint union of subsets $A_i, \{i = 1,2, \ldots, 117\}$ such that

i.) each $A_i$ contains 17 elements

ii.) the sum of all the elements in each $A_i$ is the same.

IMO Shortlist 1992 problem 13

Find all integers $\,a,b,c\,$ with $\,1<a<b<c\,$ such that $(a-1)(b-1)(c-1)$ is a divisor of $abc-1.$