5th Irish 1992


A1. Give a geometric description for the set of points (x, y) such that t² + yt + x ≥ 0 for all real t satisfying \|t\| ≤ 1.
A2. How many (x, y, z) satisfy x² + y² + z² = 9, x⁴ + y⁴ + z⁴ = 33, xyz = -4?
A3. A has n elements. How many (B, C) are such that ∅ ≠ B ⊆ C ⊆ A?
A4. ABC is a triangle with circumradius R. A', B', C' are points on BC, CA, AB such that AA', BB', CC' are concurrent. Show that the AB'·BC'·CA'/area A'B'C' = 2R.
A5. A triangle has two vertices with rational coordinates. Show that the third vertex has rational coordinates iff each angle X of the triangle has X = 90^o or tan X rational.
B1. Let m = ∑ k³, where the sum is taken over 1 ≤ k < n such that k is relatively prime to n. Show that m is a multiple of n.
B2. The digital root of a positive integer is obtained by repeatedly taking the product of the digits until we get a single-digit number. For example 24378 → 1344 → 48 → 32 → 6. Show that if n has digital root 1, then all its digits are 1.
B3. All three roots of az³ + bz² + cz + d have negative real part. Show that ab > 0, bc > ad > 0.
B4. Each diagonal of a convex pentagon divides the pentagon into a quadrilateral and a triangle of unit area. Find the area of the pentagon.
B5. Show that for any positive reals a_i, b_i, we have (a₁a₂ ... a_n)^1/n + (b₁b₂ ... b_n)^1/n ≤ ( (a₁+b₁)(a₂+b₂) ... (a_n+b_n) )^1/n with equality iff a₁/b₁ = a₂/b₂ = ... = a_n/b_n.

To avoid possible copyright problems, I have changed the wording, but not the substance, of the problems.