MathDB

4

Part of 2014 China Team Selection Test

Problems(3)

China Team Selection Test 2014 TST 1 Day 2 Q4

Source: China Nanjing , 13 Mar 2014

3/13/2014
For any real numbers sequence {xn}\{x_n\} ,suppose that {yn}\{y_n\} is a sequence such that: y1=x1,yn+1=xn+1−(∑i=1nxi2)12y_1=x_1, y_{n+1}=x_{n+1}-(\sum\limits_{i = 1}^{n} {x^2_i})^{ \frac{1}{2}} (n≥1){(n \ge 1}) . Find the smallest positive number λ\lambda such that for any real numbers sequence {xn}\{x_n\} and all positive integers mm , have 1m∑i=1mxi2≤∑i=1mλm−iyi2.\frac{1}{m}\sum\limits_{i = 1}^{m} {x^2_i}\le\sum\limits_{i = 1}^{m} {\lambda^{m-i}y^2_i} . (High School Affiliated to Nanjing Normal University )
inductioninequalities proposedinequalitiesChina TST
Circumradius and heights

Source: 2014 China TST 2 Day 2 Q4

3/20/2014
Given circle OO with radius RR, the inscribed triangle ABCABC is an acute scalene triangle, where ABAB is the largest side. AHA,BHB,CHCAH_A, BH_B,CH_C are heights on BC,CA,ABBC,CA,AB. Let DD be the symmetric point of HAH_A with respect to HBHCH_BH_C, EE be the symmetric point of HBH_B with respect to HAHCH_AH_C. PP is the intersection of AD,BEAD,BE, HH is the orthocentre of △ABC\triangle ABC. Prove: OP⋅OHOP\cdot OH is fixed, and find this value in terms of RR.
(Edited)
geometrycircumcirclegeometric transformationreflectiontrigonometrytrig identitiesLaw of Sines
Sum of 2 divisors of n^2+1/2

Source: 2014 China TST 3 Day 2 Q4

4/5/2014
Let kk be a fixed odd integer, k>3k>3. Prove: There exist infinitely many positive integers nn, such that there are two positive integers d1,d2d_1, d_2 satisfying d1,d2d_1,d_2 each dividing n2+12\frac{n^2+1}{2}, and d1+d2=n+kd_1+d_2=n+k.
number theory proposednumber theoryVieta Jumping