MathDB

2

Part of 2015 Caucasus Mathematical Olympiad

Problems(5)

max no of no 1-9 so that for any two adjacent, one is divided by the other

Source: Caucasus 2015 7.2

4/26/2019
There are 99 cards with the numbers 1,2,3,4,5,6,7,81, 2, 3, 4, 5, 6, 7, 8 and 99. What is the largest number of these cards can be decomposed in a certain order in a row, so that in any two adjacent cards, one of the numbers is divided by the other?
number theorydivisiblemaximum
if KS=LS,NS=MS then <KSN=<MSL (Caucasus 2015 geometry for 8th grade)

Source: I Caucasus 2015 8.2

9/6/2018
In the convex quadrilateral ABCDABCD, point KK is the midpoint of ABAB, point LL is the midpoint of BCBC, point MM is the midpoint of CD, and point NN is the midpoint of DADA. Let SS be a point lying inside the quadrilateral ABCDABCD such that KS=LSKS = LS and NS=MSNS = MS .Prove that ∠KSN=∠MSL\angle KSN = \angle MSL.
geometryequal anglesequal segmentsconvex quadrilateral
two roots for (x +a) (x+b)=2x+a+b , when a \ne b

Source: Caucasus 2015 9.2

4/26/2019
Let aa and bb be arbitrary distinct numbers. Prove that the equation (x+a)(x+b)=2x+a+b(x +a) (x+b)=2x+a+b has two different roots.
algebrarootstrinomial
a_n^2 = a_{n-1}a_{n+1} if a_2^2 = a_1a_3 where a_n=1+x^{n+1}+x^{n+2}

Source: Caucasus 2015 10.2

4/26/2019
Vasya chose a certain number xx and calculated the following: a1=1+x2+x3,a2=1+x3+x4,a3=1+x4+x5,...,an=1+xn+1+xn+2,...a_1=1+x^2+x^3, a_2=1+x^3+x^4, a_3=1+x^4+x^5, ..., a_n=1+x^{n+1}+x^{n+2} ,... It turned out that a22=a1a3a_2^2 = a_1a_3. Prove that for all n≥3n\ge 3, the equality an2=an−1an+1a_n^2 = a_{n-1}a_{n+1} holds.
algebraidentityalgebraic identities
if (x+a) (x+b) = 9 has a root a+b, prove ab <= 1

Source: Caucasus 2015 11.2

4/26/2019
The equation (x+a)(x+b)=9(x+a) (x+b) = 9 has a root a+ba+b. Prove that ab≤1ab\le 1.
algebrainequalitiestrinomial