MathDB

10.5

Part of I Soros Olympiad 1994-95 (Rus + Ukr)

Problems(3)

| a_1+a_2+...+a_n - nS | <= 997 (I Soros Olympiad 1994-95 R1 10.5)

Source:

7/31/2021
Let a1,a2,...,a1994a_1,a_2,...,a_{1994} be real numbers in the interval [−1,1][-1,1], S=a1+a2+...+a19941994.S=\frac{a_1+a_2+...+a_{1994}}{1994}. Prove that for an arbitrary natural , 1≤n≤19941\le n \le 1994, holds the inequality ∣a1+a2+...+an−nS∣≤997.| a_1+a_2+...+a_n - nS | \le 997.
algebrainequalities
LP bisects BD

Source: I Soros Olympiad 1994-95 Round 2 10.5 https://artofproblemsolving.com/community/c2416727_soros_olympiad_in_mathematics

5/25/2024
A circle can be drawn around the quadrilateral ABCDABCD. Let straight lines ABAB and CDCD intersect at point MM, and straight lines BCBC and ADAD intersect at point KK. (Points BB and PP lie on segments AMAM and AKAK, respectively.) Let PP be the projection of point MM onto straight line AKAK, LL be the projection of point KK on the straight line AMAM. Prove that the straight line LPLP divides the diagonal BDBD in half.
geometrybisect
product sin \frac{k'\pi}{2n} (I Soros Olympiad 1994-95 Ukraine R2 10.5)

Source:

6/6/2024
For an arbitrary natural nn, prove the equality sin⁡π2nsin⁡3π2nsin⁡5π2n...sin⁡n′π2n=21−n2\sin \frac{\pi}{2n}\sin \frac{3\pi}{2n}\sin \frac{5\pi}{2n}...\sin \frac{n'\pi}{2n}=2^{\dfrac{1-n}{2}} where n′n' is the largest odd number not exceeding nn.
trigonometryalgebraProduct