MathDB

1

Part of 1996 All-Russian Olympiad

Problems(3)

the sum of the square and cube

Source: All-Russian olympiad 1996, Grade 9, First Day, Problem 1

4/18/2013
Which are there more of among the natural numbers from 1 to 1000000, inclusive: numbers that can be represented as the sum of a perfect square and a (positive) perfect cube, or numbers that cannot be?
A. Golovanov
geometry3D geometrynumber theory proposednumber theory
Prove that $\angle FAC = \angle EDB$

Source: All-Russian Olympiad 1996, Grade 10, First Day, Problem 1

4/18/2013
Points EE and FF are given on side BCBC of convex quadrilateral ABCDABCD (with EE closer than FF to BB). It is known that ∠BAE=∠CDF\angle BAE = \angle CDF and ∠EAF=∠FDE\angle EAF = \angle FDE. Prove that ∠FAC=∠EDB\angle FAC = \angle EDB.
M. Smurov
geometry proposedgeometry
read left-to-right and right-to-left

Source: All-Russian Olympiad 1996, Grade 11, First Day, Problem 1

4/19/2013
Can the number obtained by writing the numbers from 1 to nn in order (n>1n > 1) be the same when read left-to-right and right-to-left?
N. Agakhanov
geometrygeometric transformationreflectionextremal principlenumber theory proposednumber theory