Question bank HOMC 2016

How many 10-digit numbers can be composed using only the digits 1, 2, 3, if every two neighbouring digits differ by 1?

The numbers \(A=(672,673,674,\ldots,2016)\) are on a table. At each step, three arbitrary numbers \(a,b,c\) currently on the table are replaced by \(\frac13\min(a,b,c)\). After 672 steps, only one number \(m\) remains. Which assertion holds?

Positive numbers \(a,b\) satisfy \(a^3+b^3=a^5+b^5\). What is the greatest value of \(M=a^2+b^2-ab\)?

In a zoo, a monkey becomes lucky if it eats three different fruits. What is the largest number of monkeys that can be made lucky with 20 oranges, 30 bananas, 40 peaches and 50 tangerines? Justify your answer.

Positive integers \(x,y\) satisfy \(3x^2+x=4y^2+y\). Which of the following can equal \(x-y\)?

Let \(A\) consist of 16 elements of \(\{1,2,3,\ldots,106\}\), such that the difference of any two elements of \(A\) is different from 6, 9, 12, 15, 18, 21. Prove that two elements of \(A\) have difference 3.

Nine points form a \(3\times3\) grid. How many triangles have all three vertices among these points?

Determine all three-digit numbers equal to the cube of the sum of their digits.

Rational numbers \(a,b,c\) satisfy \(a+b+c=a^2+b^2+c^2\in\mathbb Z\). Prove that there exist two relatively prime numbers \(m,n\) such that \(abc=m^2/n^3\).

Natural numbers \(a,b\) satisfy \(2015a^2+a=2016b^2+b\). Prove that \(\sqrt{a-b}\) is a natural number.

Let \(I\) be the incenter of triangle \(ABC\), and let \(\omega\) be its circumcircle. Line \(AI\) meets \(\omega\) again at \(D\ne A\). Let \(F\) and \(E\) lie on side \(BC\) and arc \(BDC\), respectively, such that \(\angle BAF=\angle CAE\lt \tfrac12\angle BAC\). Let \(X\) be the second intersection of line \(EI\) with \(\omega\), and let \(T\) be the intersection of segment \(DX\) with line \(AF\). Prove that \(TF\cdot AD=ID\cdot AT\).

Let \(A\) lie inside the acute angle \(xOy\). An arbitrary circle \(\omega\) through \(O,A\) meets rays \(Ox,Oy\) again at \(B,C\), respectively. Let \(M\) be the midpoint of \(BC\). Prove that \(M\) always lies on a fixed line as \(\omega\) varies through \(O,A\).

Find all real triples \((a,b,c)\) such that \(|2a+b|\ge4\) and \(|ax^2+bx+c|\le1\) for every \(x\in[-1,1]\).

Let \(f(x)=x^2+px+q\), where \(p,q\) are integers. Prove that there is an integer \(m\) such that \(f(m)=f(2015)f(2016)\).

Real numbers \(a,b,c\) satisfy \(18ab+9ca+29bc=1\). Find the minimum of \(T=42a^2+34b^2+43c^2\).