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(a) 28 integers are chosen from the interval $$[104, 208]$$. Show that there exist two of them having a common prime divisor.
(b) AB is a line segment. C is a point on AB. ACPQ and CBRS are squares drawn on the same side of AB. Prove the S is the orthocentre of the triangle APB.
For part (a), there are 19 primes in the interval $$[104,208]$$, and only 6 more integers can be found (one multiple of each of 2, 3, 5, 7, 11 and 13) that share no common prime factor with any other chosen number, giving a maximum of $$19+6=25$$ pairwise coprime-in-this-sense numbers; choosing 28 numbers forces at least two to share a common prime factor by the pigeonhole principle. For part (b), drawing a line from B through S to meet AP at M, since PC (the diagonal of square ACPQ) and BM both pass through S, tracking the angles around S using the square's $$45^\circ$$ diagonal angles shows $$\angle PMS = 90^\circ$$, meaning BM is perpendicular to AP; since PC and BM are both altitudes of triangle APB meeting at S, S is the orthocenter.
(a) a, b, c are distinct real numbers such that $$a^3 = 3(b^2+c^2)-25$$, $$b^3=3(c^2+a^2)-25$$, $$c^3=3(a^2+b^2)-25$$. Find the numerical value of $$abc$$.
(b) $$a = 1+\frac{1}{2^2}+\frac{1}{3^2}+\frac{1}{4^2}+\cdots+\frac{1}{2015^2}$$,
find $$[a]$$, where $$[a]$$ denotes the integer part of a.
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The arithmetic mean of a number of pairwise distinct prime numbers is 27. Determine the biggest prime among them.
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65 bugs are placed at different squares of a 9x9 square board. A bug in each move moves to a horizontal or vertical adjacent square. No bug makes two horizontal or two vertical moves in succession. Show that after some moves, there will be at least two bugs in the same square.
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f(x) is a fifth degree polynomial. It is given that $$f(x)+1$$ is divisible by $$(x-1)^3$$ and $$f(x)-1$$ is divisible by $$(x+1)^3$$. Find f(x).
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ABC and DBC are two equilateral triangles on the same base BC. A point P is taken on the circle with centre D, radius BD. Show that PA, PB, PC are the sides of a right triangle.
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a, b, c are real numbers such that $$a+b+c=0$$ and $$a^2+b^2+c^2=1$$. Prove that $$a^2b^2c^2 \le \frac{1}{54}$$. When does the equality hold?
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