MathDB

1

Part of 2008 Postal Coaching

Problems(6)

E(x)={[nx]: n\in N}, a> 1, b>0,E ( b) \subset E(a),. then b/a in N

Source: Indian Postal Coaching 2008 set 1 p1

5/25/2020
For each positive x∈R x \in \mathbb{R}, define
E(x)={[nx]:n∈N} E(x)=\{[nx]: n\in \mathbb{N}\}
Find all irrational α>1 \alpha >1 with the following property: If a positive real β \beta satisfies E(β)⊂E(α) E(\beta) \subset E(\alpha). then βα \frac{\beta}{\alpha} is a natural number.
floor functionnumber theoryFractionnaturalalgebrairrational
every integer occurs exactly once in the sequence.

Source: Indian Postal Coaching 2008 set 2 p1

5/25/2020
Define a sequence <xn><x_n> by x0=0x_0 = 0 and xn={xn−1+3r−12if  n=3r−1(3k+1)xn−1−3r+12if  n=3r−1(3k+2)\large x_n = \left\{ \begin{array}{ll} x_{n-1} + \frac{3^r-1}{2} & if \,\,n = 3^{r-1}(3k + 1)\\ & \\ x_{n-1} - \frac{3^r+1}{2} & if \,\, n = 3^{r-1}(3k + 2)\\ \end{array} \right. where k,rk, r are integers. Prove that every integer occurs exactly once in the sequence.
SequencealgebraIntegerrecurrence relation
concurrency wated, trapezium and 2 circles related

Source: Indian Postal Coaching 2008 set 4 p1

5/25/2020
Let ABCDABCD be a trapezium in which ABAB is parallel to CDCD. The circles on ADAD and BCBC as diameters intersect at two distinct points PP and QQ. Prove that the lines PQ,AC,BDPQ,AC,BD are concurrent.
geometrytrapezoidcirclesconcurrent
LCM identity with combinations

Source: Indian Postal Coaching 2008 set 3 p1

5/25/2020
Prove that for any n≥1n \ge 1,
LCM0≤k≤n{LCM _{0\le k\le n} \big \{ (nk)n \choose k }=1n+1LCM{1,2,3,...,n+1}\big\} = \frac{1}{n + 1} LCM \{1, 2,3,...,n + 1\}
number theoryleast common multipleLCMCombinations
angle chasing, arc midpoint, incenter, cicumcenter related

Source: Indian Postal Coaching 2008 set 5 p1

5/25/2020
In triangle ABC,∠B>∠C,TABC,\angle B > \angle C, T is the midpoint of arc BACBAC of the circumcicle of ABCABC, and II is the incentre of ABCABC. Let EE be point such that ∠AEI=900\angle AEI = 90^0 and AEAE is parallel to BCBC. If TETE intersects the circumcircle of ABCABC at P(≠T)P(\ne T) and ∠B=∠IPB\angle B = \angle IPB, determine ∠A\angle A.
geometryincentercircumcircleright anglearc midpointanglesAngle Chasing
points D,E, F,Q lie on the same circle.

Source: Indian Postal Coaching 2008 set 6 p1

5/25/2020
Let ABCDABCD be a quadrilateral that can be inscribed in a circle. Denote by PP the intersection point of lines ADAD and BCBC, and by QQ the intersection point of lines ABAB and DCDC. Let EE be the fourth vertex of the parallelogram ABCEABCE, and FF the intersection of lines CECE is PQPQ. Prove that the points D,E,FD,E, F, and QQ lie on the same circle.
geometryConcyclicparallelogramcyclic quadrilateral