Question bank HOMC 2007

What is the last two digits of the number $(3+7+11+\cdots+2007)^2$?

What is largest positive integer $n$ satisfying $n^{2006}\lt 7^{2007}$?

Which of the following is a possible number of diagonals of a convex polygon?

Editorial note: In this imported version, select all valid options.

Let $m$ and $n$ denote the number of digits in $2^{2007}$ and $5^{2007}$ when expressed in base 10. What is the sum $m+n$?

Let be given an open interval $(\alpha;eta)$ with $eta-\alpha=\frac1{2007}$. Determine the maximum number of irreducible fractions $\frac ab$ in $(\alpha;eta)$ with $1\le b\le2007$.

In triangle $ABC$, $\angle BAC=60^\circ$, $\angle ACB=90^\circ$ and $D$ is on $BC$. If $AD$ bisects $\angle BAC$ and $CD=3\text{ cm}$, then $DB$ is

Nine points, no three of which lie on the same straight line, are located inside an equilateral triangle of side 4. Prove that some three of these points are vertices of a triangle whose area is not greater than $\sqrt3$.

Let $a,b,c$ be positive integers. Prove that $\frac{(b+c-a)^2}{(b+c)^2+a^2}+\frac{(c+a-b)^2}{(c+a)^2+b^2}+\frac{(a+b-c)^2}{(a+b)^2+c^2}\ge\frac35$.

A triangle is said to be the Heron triangle if it has integer sides and integer area. In a Heron triangle, the sides $a,b,c$ satisfy the equation $b=a(a-c)$. Prove that the triangle is isosceles.

Let $a,b,c$ be positive real numbers such that $\frac1{bc}+\frac1{ca}+\frac1{ab}\ge1$. Prove that $\frac a{bc}+\frac b{ca}+\frac c{ab}\ge1$.

How many possible values are there for the sum $a+b+c+d$ if $a,b,c,d$ are positive integers and $abcd=2007$?

Calculate the sum $\frac5{2\cdot7}+\frac5{7\cdot12}+\cdots+\frac5{2002\cdot2007}$.

Let be given triangle $ABC$. Find all points $M$ such that area of $\triangle MAB$ = area of $\triangle MAC$.

How many ordered pairs of integers $(x,y)$ satisfy $2x^2+y^2+xy=2(x+y)$?

Let $p=\overline{abc}$ be the 3-digit prime number. Prove that the equation $ax^2+bx+c=0$ has no rational roots.

What is the last two digits of the number $(11^2+15^2+19^2+\cdots+2007^2)^2$?

Which is largest positive integer $n$ satisfying $n^{2007}\gt (2007)^n$?

Find the number of different positive integer triples $(x,y,z)$ satisfying $x+y-z=1$ and $x^2+y^2-z^2=1$.

List the numbers $\sqrt2$, $\sqrt[3]3$, $\sqrt[4]4$, $\sqrt[5]5$ and $\sqrt[6]6$ in order from greatest to least.

Suppose that $A,B,C,D$ are points on a circle, $AB$ is the diameter, $CD$ is perpendicular to $AB$ and meets $AB$ at $E$, $AB$ and $CD$ are integers and $AE-EB=\sqrt3$. Find $AE$.