MathDB

5

Part of 2004 Estonia National Olympiad

Problems(4)

a + b + c =? when a^2 + b^2 + c^2 = 1, a^3 + b^3 + c^3 = 1

Source: 2004 Estonia National Olympiad Final Round grade 10 p5

3/25/2020
Real numbers a,ba, b and cc satisfy {a2+b2+c2=1a3+b3+c3=1.\begin{cases} a^2 + b^2 + c^2 = 1 \\ a^3 + b^3 + c^3 = 1. \end{cases} Find a+b+ca + b + c.
algebrasystem of equations
circle tangent to 3 equal intersecting circles

Source: 2004 Estonia National Olympiad Final Round grade 9 p5

3/25/2020
Three different circles of equal radii intersect in point QQ. The circle CC touches all of them. Prove that QQ is the center of CC.
circlestangent circlesintersecting circlesgeometryCenter
alphabet of language BAU consists of letters B, A, and U

Source: 2004 Estonia National Olympiad Final Round grade 11 p5

3/25/2020
The alphabet of language BAUBAU consists of letters B,AB, A, and UU. Independently of the choice of the BAUBAU word of length n from which to start, one can construct all the BAUBAU words with length n using iteratively the following rules: (1) invert the order of the letters in the word; (2) replace two consecutive letters: BA→UU,AU→BB,UB→AA,UU→BA,BB→AUBA \to UU, AU \to BB, UB \to AA, UU \to BA, BB \to AU or AA→UBAA \to UB. Given that BBAUABAUUABAUUUABAUUUUABBBBAUABAUUABAUUUABAUUUUABB is a BAUBAU word, does BAUBAU have a) the word BUABUABUABUABAUBAUBAUBAUBBUABUABUABUABAUBAUBAUBAUB ? b) the word ABUABUABUABUAUBAUBAUBAUBAABUABUABUABUAUBAUBAUBAUBA ?
alphabetcombinatorics
A_iA_j //A_kA_i then A_{i'}A_{j'} // A_{k'}A_{i'}

Source: 2004 Estonia National Olympiad Final Round grade 12 p5

3/25/2020
Let nn and cc be coprime positive integers. For any integer ii, denote by i′i' the remainder of division of product cici by nn. Let Ao.A1,A2,...,An−1A_o.A_1,A_2,...,A_{n-1} be a regular nn-gon. Prove that a) if AiAj∥AkAiA_iA_j \parallel A_kA_i then Ai′Aj′∥Ak′Ai′A_{i'}A_{j'} \parallel A_{k'}A_{i'} b) if AiAj⊥AkAlA_iA_j \perp A_kA_l then Ai′Aj′⊥Ak′Al′A_{i'}A_{j'} \perp A_{k'}A_{l'}
polygonparallelperpendiculargeometrycombinatorial geometry