Practice (EndingDigits,TheDivideByNineMethod,MODBasic)
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Three circles of radius $s$ are drawn in the first quadrant of the $xy$-plane. The first circle is tangent to both axes, the second is tangent to the first circle and the $x$-axis, and the third is tangent to the first circle and the $y$-axis. A circle of radius $r > s$ is tangent to both axes and to the second and third circles. What is $\frac{r}{s}$?
A unit cube is cut twice to form three triangular prisms, two of which are congruent, as shown in Figure 1. The cube is then cut in the same manner along the dashed lines shown in Figure 2. This creates nine pieces. What is the volume of the piece that contains vertex $W$?
Call a number prime-looking if it is composite but not divisible by $2, 3,$ or $5.$ The three smallest prime-looking numbers are $49, 77$, and $91$. There are $168$ prime numbers less than $1000$. How many prime-looking numbers are there less than $1000$?
A faulty car odometer proceeds from digit 3 to digit 5, always skipping the digit 4, regardless of position. If the odometer now reads 002005, how many miles has the car actually traveled?
For each $x$ in $[0,1]$, define \[\begin{array}{clr} f(x) & = 2x, & \text { if } 0 \leq x \leq \frac {1}{2}; \\ f(x) & = 2 - 2x, & \text { if } \frac {1}{2} < x \leq 1. \end{array}\] Let $f^{[2]}(x) = f(f(x))$, and $f^{[n + 1]}(x) = f^{[n]}(f(x))$ for each integer $n \geq 2$. For how many values of $x$ in $[0,1]$ is $f^{[2005]}(x) = \frac {1}{2}$?
How many ordered triples of integers $(a,b,c)$, with $a \ge 2$, $b\ge 1$, and $c \ge 0$, satisfy both $\log_a b = c^{2005}$ and $a + b + c = 2005$?
A rectangular box $P$ is inscribed in a sphere of radius $r$. The surface area of $P$ is 384, and the sum of the lengths of its 12 edges is 112. What is $r$?
Two distinct numbers a and b are chosen randomly from the set $\{2, 2^2, 2^3, ..., 2^{25}\}$. What is the probability that $\mathrm{log}_a b$ is an integer?
Let $P(x)=(x-1)(x-2)(x-3)$. For how many polynomials $Q(x)$ does there exist a polynomial $R(x)$ of degree 3 such that $P(Q(x))=P(x)* R(x)$?
Let $S$ be the set of all points with coordinates $(x,y,z)$, where $x$, $y$, and $z$ are each chosen from the set $\{0,1,2\}$. How many equilateral triangles all have their vertices in $S$?
An $8$-foot by $10$-foot floor is tiles with square tiles of size $1$ foot by $1$ foot. Each tile has a pattern consisting of four white quarter circles of radius $\frac{1}{2}$ foot centered at each corner of the tile. The remaining portion of the tile is shaded. How many square feet of the floor are shaded?
What is the area enclosed by the graph of $|3x|+|4y|=12$?
For how many values of $a$ is it true that the line $y = x + a$ passes through the vertex of the parabola $y = x^2 + a^2$ ?
The quadratic equation $x^2+mx+n$ has roots twice those of $x^2+px+m$, and none of $m,n,$ and $p$ is zero. What is the value of $\frac{n}{p}$?
Suppose that $4^{x_1}=5$, $5^{x_2}=6$, $6^{x_3}=7$, ... , $127^{x_{124}}=128$. What is $x_1x_2...x_{124}$?
A circle having center $(0,k)$, with $k>6$, is tangent to the lines $y=x$, $y=-x$ and $y=6$. What is the radius of this circle?
The sum of four two-digit numbers is $221$. None of the eight digits is $0$ and no two of them are the same. Which of the following is not included among the eight digits?
Eight spheres of radius 1, one per octant, are each tangent to the coordinate planes. What is the radius of the smallest sphere, centered at the origin, that contains these eight spheres?
How many distinct four-tuples $(a,b,c,d)$ of rational numbers are there with
\[a\cdot\log_{10}2+b\cdot\log_{10}3+c\cdot\log_{10}5+d\cdot\log_{10}7=2005?\]
Let $A(2,2)$ and $B(7,7)$ be points in the plane. Define $R$ as the region in the first quadrant consisting of those points $C$ such that $\triangle ABC$ is an acute triangle. What is the closest integer to the area of the region $R$?
Let $x$ and $y$ be two-digit integers such that $y$ is obtained by reversing the digits of $x$. The integers $x$ and $y$ satisfy $x^{2}-y^{2}=m^{2}$ for some positive integer $m$. What is $x+y+m$?
Let $a,b,c,d,e,f,g$ and $h$ be distinct elements in the set $\{-7,-5,-3,-2,2,4,6,13\}.$
What is the minimum possible value of $(a+b+c+d)^{2}+(e+f+g+h)^{2}?$
A positive integer $n$ has $60$ divisors and $7n$ has $80$ divisors. What is the greatest integer $k$ such that $7^k$ divides $n$?
A sequence of complex numbers $z_{0}, z_{1}, z_{2}, ...$ is defined by the rule
\[z_{n+1} = \frac {iz_{n}}{\overline {z_{n}}},\]
where $\overline {z_{n}}$ is the complex conjugate of $z_{n}$ and $i^{2}=-1$. Suppose that $|z_{0}|=1$ and $z_{2005}=1$. How many possible values are there for $z_{0}$?
Let $S$ be the set of ordered triples $(x,y,z)$ of real numbers for which
\[\log_{10}(x+y) = z \text{ and } \log_{10}(x^{2}+y^{2}) = z+1.\] There are real numbers $a$ and $b$ such that for all ordered triples $(x,y.z)$ in $S$ we have $x^{3}+y^{3}=a \cdot 10^{3z} + b \cdot 10^{2z}.$ What is the value of $a+b?$