This website contains problems from math contests. Problems and corresponding tags were obtained from the Art of Problem Solving website.

Tags were heavily modified to better represent problems.

AND:
OR:
NO:

Found problems: 5

1997 Moscow Mathematical Olympiad, 4
Tags: Grade 10 , 1997
Given real numbers $a_1\leq{a_2}\leq{a_3}$ and $b_1\leq{b_2}\leq{b_3}$ such that $$a_1+a_2+a_3=b_1+b_2+b_3,$$ $$a_1a_2+a_2a_3+a_1a_3=b_1b_2+b_2b_3+b_1b_3.$$ Prove that if $a_1\leq{b_1},$ then $a_3\leq{b_3}$

1997 Moscow Mathematical Olympiad, 6
Tags: Grade 10 , 1997
Consider the sequence formed by the first digits of the powers of $5$:$$1,5,2,1,6,...$$ Prove any segment in this sequence, when written in reversed order, will be encountered in the sequence of the first digits of the powers of $2:$ $$1,2,4,8,1,3,6,1...$$

1997 Moscow Mathematical Olympiad, 3
Tags: Grade 10 , 1997
A quadrilateral is rotated clockwise, and the sides are extended its length in the direction of the movement. It turns out the endpoints of the segments constructed form a square. Prove the initial quadrilateral must also be a square. [b]Generalization:[/b] Prove that if the same process is applied to any $n$-gon and the result is a regular $n$-gon, then the intial $n$-gon must also be regular.

1997 Moscow Mathematical Olympiad, 1
Tags: Grade 10 , 1997
Is there a convex body distinct from ball whose three orthogonal projections on three pairwise perpendicular planes are discs?

1997 Moscow Mathematical Olympiad, 2
Tags: Grade 10 , 1997
Prove that among the quadrilaterals with given lengths of the diagonals and the angle between them, the parallelogram has the least perimeter.