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Find the number of sets of 4 positive integers less than or equal to 25, such that the difference between any two elements in the set is at least 3.
Solve the system of equations
$$(1+4^{2x-y})5^{1-2x+y} = 1+2^{2x-y+1}$$ and $$y^3+4x+1+\log(y^2+2x)=0$$.
a, b, c, d, e, f are real numbers such that the polynomial $$P(x) = x^8 - 4x^7 + 7x^6 + ax^5 + bx^4 + cx^3 + dx^2 + ex + f$$ factorises into eight linear factors $$(x-x_i)$$ with $$x_i>0$$ for $$i=1,2,3,\ldots,8$$. Find the possible values of the constant f.
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An acute angled triangle PQR is inscribed in a circle. Given $$\angle P = 60^\circ$$ and $$\angle R > \angle Q$$. Let H and I be the orthocentre and incentre of $$\triangle PQR$$ respectively. Find the ratio of $$\angle PHI$$ to $$\angle PQR$$.
ABC is a triangle in which $$\angle A$$ is obtuse. AD, BE and CF are the altitudes from A, B and C respectively to BC, CA and AB. $$A_1$$, $$B_1$$ and $$C_1$$ are arbitrary points on BC, CA and AB respectively. The circles on $$AA_1$$, $$BB_1$$ and $$CC_1$$ as diameters are drawn. Show that the lengths of the tangents from the orthocentre of $$\triangle ABC$$ to these circles are equal.
Find all natural number solutions of the equation $$3^x - 5^y = z^2$$.
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a, b, c are positive reals such that $$ab+bc+ca = 3abc$$. Show that $$a^2b+b^2c+c^2a \ge 2(a+b+c) - 3$$. When will the equality hold?
A regular polygon with 100 vertices is given. To each vertex, a natural number from the set $$\{1, 2, 3, \ldots, 49\}$$ is assigned. Prove that there are four vertices A, B, C, D such that if the numbers a, b, c, d are assigned to them respectively then $$a+b=c+d$$ and ABCD is a parallelogram.
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