MathDB

3

Part of 2009 Postal Coaching

Problems(6)

lines UD, VE, WF , OI are concurrent.

Source: Indian Postal Coaching 2009 set 1 p3

5/26/2020
Let ABCABC be a triangle with circumcentre OO and incentre II such that OO is different from II. Let AK,BL,CMAK, BL, CM be the altitudes of ABCABC, let U,V,WU, V , W be the mid-points of AK,BL,CMAK, BL, CM respectively. Let D,E,FD, E, F be the points at which the in-circle of ABCABC respectively touches the sides BC,CA,ABBC, CA, AB. Prove that the lines UD,VE,WFUD, VE, WF and OIOI are concurrent.
geometryConcyclicCircumcenteraltitudesincenter
PP_1 x PP_2 x ... x PP_n >= 2

Source: Indian Postal Coaching 2009 set 2 p3

5/26/2020
Let Ω\Omega be an nn-gon inscribed in the unit circle, with vertices P1,P2,...,PnP_1, P_2, ..., P_n.
(a) Show that there exists a point PP on the unit circle such that PP1⋅PP2⋅...⋅PPn≥2PP_1 \cdot PP_2\cdot ... \cdot PP_n \ge 2.
(b) Suppose for each PP on the unit circle, the inequality PP1⋅PP2⋅...⋅PPn≤2PP_1 \cdot PP_2\cdot ... \cdot PP_n \le 2 holds. Prove that Ω\Omega is regular.
inequalitiespolygonCyclicGeometric Inequalities
(sin x) = f(cos x) polynomial

Source: Indian Postal Coaching 2009 set 4 p3

5/26/2020
Find all real polynomial functions f:R→Rf : R \to R such that f(sin⁡x)=f(cos⁡x)f(\sin x) = f(\cos x).
polynomialtrigonometryalgebra
sum of integer weights that come with a two pan balance scale

Source: Indian Postal Coaching 2009 set 3 p3

5/26/2020
Let SS be the sum of integer weights that come with a two pan balance Scale, say ω1≤ω2≤ω3≤...≤ωn\omega_1 \le \omega_2 \le \omega_3 \le ... \le\omega_n. Show that all integer-weighted objects in the range 11 to SS can be weighed exactly if and only if ω1=1\omega_1=1 and ωj+1≤2(∑l=1jωl)+1\omega_{j+1} \le 2 \left( \sum_{l=1}^{j} \omega_l\right) +1
weightscombinatorics
sum_{k=0}^{2009 \choose 2} f(2008, k), if f(n, k) = f(n -1, k) + f(n- 1, k - 2n)

Source: Indian Postal Coaching 2009 set 5 p3

5/26/2020
Let N0N_0 denote the set of nonnegative integers and ZZ the set of all integers. Let a function f:N0×Z→Zf : N_0 \times Z \to Z satisfy the conditions (i) f(0,0)=1f(0, 0) = 1, f(0,1)=1f(0, 1) = 1 (ii) for all k,k≠0,k≠1k, k \ne 0, k \ne 1, f(0,k)=0f(0, k) = 0 and (iii) for all n≥1n \ge 1 and k,f(n,k)=f(n−1,k)+f(n−1,k−2n)k, f(n, k) = f(n -1, k) + f(n- 1, k - 2n). Find the value of
∑k=0(20092)f(2008,k)\sum_{k=0}^{2009 \choose 2} f(2008, k)
functionSumCombinationsrecurrence relation
f_k(x)f_{k+1}(x) = f_{k+1}(f_{k+2}(x))

Source: Indian Postal Coaching 2009 set 6 p3

5/26/2020
Let n≥3n \ge 3 be a positive integer. Find all nonconstant real polynomials f1(x),f2(x),...,fn(x)f_1(x), f_2(x), ..., f_n(x) such that fk(x)fk+1(x)=fk+1(fk+2(x))f_k(x)f_{k+1}(x) = f_{k+1}(f_{k+2}(x)), 1≤k≤n1 \le k \le n for all real x. [All suffixes are taken modulo nn.]
polynomialalgebra