Question bank HOMC 2013

Write 2013 as a sum of \(m\) prime numbers. The smallest value of \(m\) is:

How many natural numbers \(n\) are there such that \(n^2+2014\) is a perfect square?

The largest integer not exceeding \([(n+1)\alpha]-[n\alpha]\), where \(n\) is a natural number and \(\alpha=\frac{\sqrt{2013}}{\sqrt{2014}}\), is: (Here square brackets denote the greatest-integer function.)

Let \(A\) be an even integer not divisible by 10. The last two digits of \(A^{20}\) are:

The number of integer solutions \(x\) of \((12x-1)(6x-1)(4x-1)(3x-1)=330\) is:

Let \(ABC\) be a triangle of area \(1\text{ cm}^2\). Points \(D,E,F\) lie on \(AB,BC,CA\), respectively. Prove that \(\min\{[ADF],[BED],[CEF]\}\le\tfrac14\text{ cm}^2\), where brackets denote area.

Let \(ABC\) be a triangle with \(\angle A=90^\circ\), \(\angle B=60^\circ\), and \(BC=1\text{ cm}\). Draw outside \(ABC\) the three equilateral triangles \(ABD,ACE,BCF\). Determine \([DEF]\).

Let \(ABCDE\) be a convex pentagon with \([ABC]=[BCD]=[CDE]=[DEA]=[EAB]=2\text{ cm}^2\). Find its area.

Solve in positive numbers: \(x+y\le1\) and \(\frac2{xy}+\frac1{x^2+y^2}=10\).

Among all rectangles with fixed perimeter \(p\), find the largest value of \(M=\frac{S}{2S+p+2}\), where \(S\) denotes the area.

Positive numbers \(a,b,c,d,e\) satisfy \((x+a)(x+b)(x+c)=x^3+3dx^2+3x+e^3\) for every real \(x\). Find the smallest value of \(d\).

If \(f(x)=ax^2+bx+c\) satisfies \(|f(x)|\lt 1\) for every \(x\in[-1,1]\), prove that \(f(x)=2x^2-1\) has two real roots.

Solve \(\frac1x+\frac1y=\frac16\), \(\frac3x+\frac2y=\frac56\).

Solve the system \(x^3+y=x^2+1\), \(2y^3+z=2y^2+1\), \(3z^3+x=3z^2+1\).

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

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 \(m\)?