MathDB

1

Part of 2014 Taiwan TST Round 1

Problems(4)

Simple cube root inequality [Taiwan 2014 Quizzes]

Source:

7/18/2014
Prove that for positive reals aa, bb, cc we have 3(a+b+c)≥8abc3+a3+b3+c333. 3(a+b+c) \ge 8\sqrt[3]{abc} + \sqrt[3]{\frac{a^3+b^3+c^3}{3}}.
inequalitiesHi
f(2) = 7, find all integer functions [Taiwan 2014 Quizzes]

Source:

7/18/2014
Find all increasing functions ff from the nonnegative integers to the integers satisfying f(2)=7f(2)=7 and f(mn)=f(m)+f(n)+f(m)f(n) f(mn) = f(m) + f(n) + f(m)f(n) for all nonnegative integers mm and nn.
functionlogarithmsintegrationTaiwan
Bounded modulus in real-root polynom [Taiwan 2014 TST1 P1]

Source:

7/18/2014
Let f(x)=xn+an−2xn−2+an−3xn−3+⋯+a1x+a0f(x) = x^n + a_{n-2} x^{n-2} + a_{n-3}x^{n-3} + \dots + a_1x + a_0 be a polynomial with real coefficients (n≥2)(n \ge 2). Suppose all roots of ff are real. Prove that the absolute value of each root is at most 2(1−n)nan−2\sqrt{\frac{2(1-n)}n a_{n-2}}.
algebrapolynomialinequalitiesabsolute value
Find AD when area of ABC is maximal [Taiwan 2014 Quizzes]

Source:

7/18/2014
Let O1O_1, O2O_2 be two circles with radii R1R_1 and R2R_2, and suppose the circles meet at AA and DD. Draw a line LL through DD intersecting O1O_1, O2O_2 at BB and CC. Now allow the distance between the centers as well as the choice of LL to vary. Find the length of ADAD when the area of ABCABC is maximized.
geometrytrigonometrygeometric transformation