MathDB

2

Part of 2018 India IMO Training Camp

Problems(4)

Count numbers divisible by 11111

Source: IMOTC PT1 2018 P2, India

7/18/2018
A 1010 digit number is called interesting if its digits are distinct and is divisible by 1111111111. Then find the number of interesting numbers.
combinatoricscounting
Find sequences with given sum and sum-of-squares

Source: IMOTC PT2 2018 P2, India

7/18/2018
For an integer n≥2n\ge 2 find all a1,a2,⋯ ,an,b1,b2,⋯ ,bna_1,a_2,\cdots ,a_n, b_1,b_2,\cdots , b_n so that (a) 0≤a1≤a2≤⋯≤an≤1≤b1≤b2≤⋯≤bn;0\le a_1\le a_2\le \cdots \le a_n\le 1\le b_1\le b_2\le \cdots \le b_n; (b) ∑k=1n(ak+bk)=2n;\sum_{k=1}^n (a_k+b_k)=2n; (c) ∑k=1n(ak2+bk2)=n2+3n.\sum_{k=1}^n (a_k^2+b_k^2)=n^2+3n.
algebrainequalities
Tangent semicircles and concyclic points

Source: India TST 2018, D1 P2

7/18/2018
Let A,B,CA,B,C be three points in that order on a line ℓ\ell in the plane, and suppose AB>BCAB>BC. Draw semicircles Γ1\Gamma_1 and Γ2\Gamma_2 respectively with ABAB and BCBC as diameters, both on the same side of ℓ\ell. Let the common tangent to Γ1\Gamma_1 and Γ2\Gamma_2 touch them respectively at PP and QQ, P≠QP\ne Q. Let DD and EE be points on the segment PQPQ such that the semicircle Γ3\Gamma_3 with DEDE as diameter touches Γ2\Gamma_2 in SS and Γ1\Gamma_1 in TT. [*]Prove that A,C,S,TA,C,S,T are concyclic. [*]Prove that A,C,D,EA,C,D,E are concyclic.
geometry
Sequence becomes positive halfway through

Source: India TST 2018, D4 P2

7/18/2018
Let n≥2n\ge 2 be a natural number. Let a1≤a2≤a3≤⋯≤ana_1\le a_2\le a_3\le \cdots \le a_n be real numbers such that a1+a2+⋯+an>0a_1+a_2+\cdots +a_n>0 and n(a12+a22+⋯+an2)=2(a1+a2+⋯+an)2.n(a_1^2+a_2^2+\cdots +a_n^2)=2(a_1+a_2+\cdots +a_n)^2. If m=⌊n/2⌋+1m=\lfloor n/2\rfloor+1, the smallest integer larger than n/2n/2, then show that am>0.a_m>0.
algebraSequences