MathDB

2

Part of 2013 China Team Selection Test

Problems(6)

2013 China IMO Team Selection Test 1 Day 1 Q2

Source: 13 Mar 2013

3/29/2013
For the positive integer nn, define f(n)=min⁡m∈Z∣2−mn∣f(n)=\min\limits_{m\in\Bbb Z}\left|\sqrt2-\frac mn\right|. Let {ni}\{n_i\} be a strictly increasing sequence of positive integers. CC is a constant such that f(ni)<Cni2f(n_i)<\dfrac C{n_i^2} for all i∈{1,2,…}i\in\{1,2,\ldots\}. Show that there exists a real number q>1q>1 such that ni⩾qi−1n_i\geqslant q^{i-1} for all i∈{1,2,…}i\in\{1,2,\ldots \}.
pigeonhole principlenumber theory proposednumber theory
Circumcentres of 6 circles are concyclic

Source: Chinese TST 1 2013 Day 2 Q2

4/1/2013
Let PP be a given point inside the triangle ABCABC. Suppose L,M,NL,M,N are the midpoints of BC,CA,ABBC, CA, AB respectively and PL:PM:PN=BC:CA:AB.PL: PM: PN= BC: CA: AB. The extensions of AP,BP,CPAP, BP, CP meet the circumcircle of ABCABC at D,E,FD,E,F respectively. Prove that the circumcentres of APF,APE,BPF,BPD,CPD,CPEAPF, APE, BPF, BPD, CPD, CPE are concyclic.
geometrycircumcirclegeometric transformationhomothetytrigonometryEulergeometry proposed
a_n&lt;K(1.01)^n

Source: 2013 China TST Quiz 2 Day 1 P2

3/31/2013
Prove that: there exists a positive constant KK, and an integer series {an}\{a_n\}, satisfying: (1)(1) 0<a1<a2<⋯<an<⋯0<a_1<a_2<\cdots <a_n<\cdots ; (2)(2) For any positive integer nn, an<1.01nKa_n<1.01^n K; (3)(3) For any finite number of distinct terms in {an}\{a_n\}, their sum is not a perfect square.
limitmodular arithmeticquadraticsnumber theory proposednumber theory
Arithmetic series with an odd common difference

Source: 2013 China TST Quiz 2 Day 2 P2

3/31/2013
Find the greatest positive integer mm with the following property: For every permutation a1,a2,⋯ ,an,⋯a_1, a_2, \cdots, a_n,\cdots of the set of positive integers, there exists positive integers i1<i2<⋯<imi_1<i_2<\cdots <i_m such that ai1,ai2,⋯ ,aima_{i_1}, a_{i_2}, \cdots, a_{i_m} is an arithmetic progression with an odd common difference.
arithmetic sequencenumber theoryChina TSTalgebra proposed
2013 China IMO Team Selection Test 3 Day 1 Q2

Source: Mar 24

4/1/2013
The circumcircle of triangle ABCABC has centre OO. PP is the midpoint of BAC^\widehat{BAC} and QPQP is the diameter. Let II be the incentre of △ABC\triangle ABC and let DD be the intersection of PIPI and BCBC. The circumcircle of △AID\triangle AID and the extension of PAPA meet at FF. The point EE lies on the line segment PDPD such that DE=DQDE=DQ. Let R,rR,r be the radius of the inscribed circle and circumcircle of △ABC\triangle ABC, respectively. Show that if ∠AEF=∠APE\angle AEF=\angle APE, then sin⁡2∠BAC=2rR\sin^2\angle BAC=\dfrac{2r}R
geometrycircumcircleincentertrigonometrygeometry proposed
China Team Selection Test 2013 TST 3 Day 2 Q2

Source: Nanjing high School , Jiangsu 25 Mar 2013

3/25/2013
Let k≥2k\ge 2 be an integer and let a1,a2,⋯ ,an,b1,b2,⋯ ,bna_1 ,a_2 ,\cdots ,a_n,b_1 ,b_2 ,\cdots ,b_n be non-negative real numbers. Prove that\left(\frac{n}{n-1}\right)^{n-1}\left(\frac{1}{n} \sum_{i\equal{}1}^{n} a_i^2\right)+\left(\frac{1}{n} \sum_{i\equal{}1}^{n} b_i\right)^2\ge\prod_{i=1}^{n}(a_i^{2}+b_i^{2})^{\frac{1}{n}}.
inequalities proposedinequalitiesChina TST