IMO 1991
Problem sheet
International mathematics competition for pre-university students, held annually since 1959. Six problems over two days.
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Search results for “natural logarithm”
29 results
Determinant via Permutations
Theory
The direct definition of a determinant via permutations: inversions, parity, and a sum of n! terms.
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An example of a non-reduced Picard scheme
Theory
An example, due to Serre, of a smooth projective surface in positive characteristic whose Picard scheme fails to be reduced, worked out via the two governing dimension inequalities.
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Finite flat commutative group schemes embed locally into abelian schemes
Theory
A proof of Raynaud's theorem: every finite flat commutative group scheme over a locally noetherian base embeds, Zariski-locally, as a closed subgroup of an abelian scheme.
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IMO 1966
Problem sheet
International mathematics competition for pre-university students, held annually since 1959. Six problems over two days.
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A curiosity: “supersmooth” varieties
Theory
A rigidity condition on schemes, strictly stronger than smoothness, introduced with examples and a look at where it fails to be geometric — no known application, just a curiosity.
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Some examples of algebraic groups
Theory
Two pathological phenomena for algebraic groups over general bases — a group degenerating between the multiplicative and additive group across a DVR, and a non-affine identity component — built as centralizers in SL_n.
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