Question bank HOMC 2006

What is the last two digits of the number $(11+12+13+\cdots+2006)^2$?

Find the last two digits of the sum $2005^{11}+2005^{12}+\cdots+2005^{2006}$.

Find the number of different positive integer triples $(x,y,z)$ satisfying $x^2+y-z=100$ and $x+y^2-z=124$.

Suppose $x$ and $y$ are two real numbers such that $x+y-xy=155$ and $x^2+y^2=325$. Find the value of $|x^3-y^3|$.

Suppose $n$ is a positive integer and 3 arbitrary numbers are choosen from the set $\{1,2,3,\ldots,3n+1\}$ with their sum equal to $3n+1$. What is the largest possible product of those 3 numbers?

Editorial convention for this imported question: the three values may repeat. This is an editor-selected interpretation, not a condition confirmed in the source. The distinct-elements interpretation is also proved in the solution.

The figure $ABCDEF$ is a regular hexagon. Find all points $M$ belonging to the hexagon such that Area of triangle $MAC$ = Area of triangle $MCD$.

Editorial clarification: M belongs to the closed hexagonal region, including its boundary; zero-area triangles are allowed.

On the circle $(O)$ of radius $15\text{ cm}$ are given 2 points $A,B$. The altitude $OH$ of the triangle $OAB$ intersect $(O)$ at $C$. What is $AC$ if $AB=16\text{ cm}$?

Editorial convention: C is the intersection of the circle with the ray from O through H. The other intersection of the full altitude line is discussed separately in the solution.

In $\triangle ABC$, $PQ\parallel BC$ where $P$ and $Q$ are points on $AB$ and $AC$ respectively. The lines $PC$ and $QB$ intersect at $G$. It is also given $EF\parallel BC$, where $G\in EF$, $E\in AB$ and $F\in AC$ with $PQ=a$ and $EF=b$. Find value of $BC$.

Editorial clarification: use a nondegenerate triangle and P,Q in the interiors of the stated sides, so the intersection G is well-defined.

What is the smallest possible value of $x^2+y^2-x-y-xy$?

What is the last three digits of the sum $11!+12!+13!+\cdots+2006!$?

Find the last three digits of the sum $2005^{11}+2005^{12}+\cdots+2005^{2006}$.

Suppose that $a^{\log_b c}+b^{\log_c a}=m$. Find the value of $c^{\log_b a}+a^{\log_c b}$.

Which is larger: $2^{\sqrt2}$, $2^{1+\frac1{\sqrt2}}$ and $3$?

The figure $ABCDEF$ is a regular hexagon. Find all points $M$ belonging to the hexagon such that Area of triangle $MAC$ = Area of triangle $MCD$.

Editorial clarification: M belongs to the closed hexagonal region, including its boundary; zero-area triangles are allowed.

On the circle of radius $30\text{ cm}$ are given 2 points $A,B$ with $AB=16\text{ cm}$ and $C$ is a midpoint of $AB$. What is the perpendicular distance from $C$ to the circle?

In $\triangle ABC$, $PQ\parallel BC$ where $P$ and $Q$ are points on $AB$ and $AC$ respectively. The lines $PC$ and $QB$ intersect at $G$. It is also given $EF\parallel BC$, where $G\in EF$, $E\in AB$ and $F\in AC$ with $PQ=a$ and $EF=b$. Find value of $BC$.

Editorial clarification: use a nondegenerate triangle and P,Q in the interiors of the stated sides, so the intersection G is well-defined.

Find all polynomials $P(x)$ such that $P(x)+\left(\frac1x\right)=x+\frac1x$, $\forall x\ne0$.

Let $x,y,z$ be real numbers such that $x^2+y^2+z^2=1$. Find the largest possible value of $|x^3+y^3+z^3-xyz|$.