Hanoi Open Mathematics Competition 2013 Senior

How many three-digit perfect squares are there such that, if each digit is increased by one, the resulting number is also a perfect square?

Find the smallest value of \(f(x)=|x|+\left|\frac{1-2013x}{2013-x}\right|\), where \(x\in[-1,1]\).

What is the largest integer not exceeding \(8x^3+6x-1\), where \(x=\tfrac12\big(\sqrt[3]{2+\sqrt5}+\sqrt[3]{2-\sqrt5}\big)\)?

Let \(x_0=[\alpha]\), \(x_1=[2\alpha]-[\alpha]\), \(x_2=[3\alpha]-[2\alpha]\), \(x_4=[5\alpha]-[4\alpha]\), \(x_5=[6\alpha]-[5\alpha],\ldots\), where \(\alpha=\sqrt{2013}/\sqrt{2014}\). The value of \(x_9\) is: Here \([t]=\lfloor t\rfloor\). Editorial interpretation: the omitted index \(x_3\) follows the displayed pattern \(x_k=\lfloor(k+1)\alpha\rfloor-\lfloor k\alpha\rfloor\).

A composite number is a number \(n=ab\), where \(a,b\) are integers greater than 1. Write 2013 as a sum of \(m\) composite numbers. What is the largest value of \(m\)?

Given \(a\in\{0,1,\ldots,1006\}\), find all \(n\in\{0,1,\ldots,2013\}\) such that \(C_{2013}^n\gt C_{2013}^a\), where \(C_m^k=\frac{m!}{k!(m-k)!}\).

Let \(ABC\) be equilateral, and let \(M\) lie inside it with \(MA^2=MB^2+MC^2\). Draw equilateral \(ACD\), where \(D\ne B\). Let \(N\) lie inside \(ACD\) so that \(AMN\) is equilateral. Determine \(\angle BMC\).

Let \(ABCDE\) be a convex pentagon with \([ABC]=[BCD]=[CDE]=[DEA]=[EAB]\). Given \([ABCDE]=2\), evaluate \([ABC]\).

A polynomial \(P(t)=t^3+at^2+bt+c\) has three distinct real roots. If \((x^2+x+2013)^3+a(x^2+x+2013)^2+b(x^2+x+2013)+c=0\) has no real roots, prove that \(P(2013)\gt 1/64\).

Among all rectangles with given area \(S\), find the largest value of \(M=\frac{16S-p}{p^2+2p}\), where \(p\) is the perimeter.

Positive numbers \(a,b,c,d,p,q\) satisfy \((x+a)(x+b)(x+c)(x+d)=x^4+4px^3+6x^2+4qx+1\) for all real \(x\). Find the smallest value of \(p\) or the largest value of \(q\).

Let \(f(x)=ax^2+bx+c\) satisfy \(f(\sqrt2)=3\) and \(|f(x)|\le1\) for all \(x\in[-1,1]\). Evaluate \(f(\sqrt{2013})\).

Solve \(xy=1\), \(\frac{x}{x^4+y^2}+\frac{y}{x^2+y^4}=1\).

Solve the system \(x^3+\tfrac13y=x^2+x-\tfrac43\), \(y^3+\tfrac14z=y^2+y-\tfrac54\), \(z^3+\tfrac15x=z^2+z-\tfrac65\).

Let \(\mathbb Q\) and \(\mathbb N^*\) denote the rational numbers and positive integers. Suppose \(\frac{ax+b}{cx+d}\in\mathbb Q\) for every \(x\in\mathbb N^*\). Prove that there exist integers \(A,B,C,D\) such that \(\frac{ax+b}{cx+d}=\frac{Ax+B}{Cx+D}\) for every \(x\in\mathbb N^*\).