MathDB

2

Part of 1961 All-Soviet Union Olympiad

Problems(3)

Outer tangents form a circumscribed quadrangle

Source: 1961 All-Soviet Union Olympiad

8/4/2015
Consider a rectangle A1A2A3A4A_1A_2A_3A_4 and a circle Ci\mathcal{C}_i centered at AiA_i with radius rir_i for i=1,2,3,4i=1,2,3,4. Suppose that r1+r3=r2+r4<dr_1+r_3=r_2+r_4<d, where dd is the diagonal of the rectangle. The two pairs of common outer tangents of C1\mathcal{C}_1 and C3\mathcal{C}_3, and of C2\mathcal{C}_2 and C4\mathcal{C}_4 form a quadrangle. Prove that this quadrangle has an inscribed circle.
geometryrectanglecircumscribed quadrilateral
Classic - sum of rows and columns nonnegative

Source: 1961 All-Soviet Union Olympiad

8/4/2015
Consider a table with one real number in each cell. In one step, one may switch the sign of the numbers in one row or one column simultaneously. Prove that one can obtain a table with non-negative sums in each row and each column.
combinatoricsinvariantmonovariantextremal principle
Another circle fits

Source: 1961 All-Soviet Union Olympiad

8/4/2015
Consider 120120 unit squares arbitrarily situated in a 20×2520\times 25 rectangle. Prove that one can place a circle with unit diameter in the rectangle without intersecting any of the squares.
geometryrectanglecirclecombinatorial geometry