Practice (EndingDigits,TheDivideByNineMethod,MODBasic,EulerFermatTheorem,CRT,TheModMethod,SquareNumber)
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Cities $A$, $B$, $C$, $D$, and $E$ are connected by roads $\widetilde{AB}$, $\widetilde{AD}$, $\widetilde{AE}$, $\widetilde{BC}$, $\widetilde{BD}$, $\widetilde{CD}$, and $\widetilde{DE}$. How many different routes are there from $A$ to $B$ that use each road exactly once? (Such a route will necessarily visit some cities more than once.)
The internal angles of quadrilateral $ABCD$ form an arithmetic progression. Triangles $ABD$ and $DCB$ are similar with $\angle DBA = \angle DCB$ and $\angle ADB = \angle CBD$. Moreover, the angles in each of these two triangles also form an arithemetic progression. In degrees, what is the largest possible sum of the two largest angles of $ABCD$?
The number $2013$ is expressed in the form
$2013 = \frac {a_1!a_2!...a_m!}{b_1!b_2!...b_n!}$,
where $a_1 \ge a_2 \ge \cdots \ge a_m$ and $b_1 \ge b_2 \ge \cdots \ge b_n$ are positive integers and $a_1 + b_1$ is as small as possible. What is $|a_1 - b_1|$?
Let $ABCDE$ be an equiangular convex pentagon of perimeter $1$. The pairwise intersections of the lines that extend the sides of the pentagon determine a five-pointed star polygon. Let $s$ be the perimeter of this star. What is the difference between the maximum and the minimum possible values of $s$.
Let $a,b,$ and $c$ be real numbers such that
\[a+b+c=2, \text{ and}\]\[a^2+b^2+c^2=12\]
What is the difference between the maximum and minimum possible values of $c$?
Barbara and Jenna play the following game, in which they take turns. A number of coins lie on a table. When it is Barbara's turn, she must remove $2$ or $4$ coins, unless only one coin remains, in which case she loses her turn. When it is Jenna's turn, she must remove $1$ or $3$ coins. A coin flip determines who goes first. Whoever removes the last coin wins the game. Assume both players use their best strategy. Who will win when the game starts with $2013$ coins and when the game starts with $2014$ coins?
For $135^\circ < x < 180^\circ$, points $P=(\cos x, \cos^2 x), Q=(\cot x, \cot^2 x), R=(\sin x, \sin^2 x)$ and $S =(\tan x, \tan^2 x)$ are the vertices of a trapezoid. What is $\sin(2x)$?
Consider the set of 30 parabolas defined as follows: all parabolas have as focus the point (0,0) and the directrix lines have the form $y=ax+b$ with a and b integers such that $a\in \{-2,-1,0,1,2\}$ and $b\in \{-3,-2,-1,1,2,3\}$. No three of these parabolas have a common point. How many points in the plane are on two of these parabolas?
Let $m>1$ and $n>1$ be integers. Suppose that the product of the solutions for $x$ of the equation \[8(\log_n x)(\log_m x)-7\log_n x-6 \log_m x-2013 = 0\] is the smallest possible integer. What is $m+n$?
Let $ABC$ be a triangle where $M$ is the midpoint of $\overline{AC}$, and $\overline{CN}$ is the angle bisector of $\angle{ACB}$ with $N$ on $\overline{AB}$. Let $X$ be the intersection of the median $\overline{BM}$ and the bisector $\overline{CN}$. In addition $\triangle BXN$ is equilateral with $AC=2$. What is $BN^2$?
Let $G$ be the set of polynomials of the form \[P(z)=z^n+c_{n-1}z^{n-1}+\cdots+c_2z^2+c_1z+50,\] where $c_1,c_2,\cdots, c_{n-1}$ are integers and $P(z)$ has distinct roots of the form $a+ib$ with $a$ and $b$ integers. How many polynomials are in $G$?
Julia's age is a two-digit multiple of $5$, and when Julia's age is divided by $2$, $3$, $4$, $6$ or $8$, the remainder is always $1$. If Julia is five times as old as Bart, how old is Bart?
If $p$ is the maximum number of points of intersection possible of $n$ distinct lines, and the ratio $p:n = 6:1$, what is the value of n?
If $p$ is the greatest prime whose digits are distinct prime numbers, what is the units digit of $p^2$?
If $\frac{a}{4 - a}=\frac{b}{5 - b} =\frac{c}{7 - c}= 3$, what is the value of $a + b + c$?
A building constructed in January of the year 2000 will celebrate its 1-month anniversary in February of 2000 and its 12-month anniversary in January of 2001. If during the year $n$ this building will celebrate its n month anniversary, what is the value of $n$?
The shape below can be folded along the dashed lines and taped together along the edges to form a three-dimensional polyhedron. All lengths in the diagram are given in inches. What is the volume of the resulting polyhedron? Express your answer in simplest radical form.
When trying to recall some facts about the ages of his three aunts, Josh made the following claims:
- Alice is fifteen years younger than twice Catherine's age.
- Beatrice is twelve years older than half of Alice's age.
- Catherine is eight years younger than Beatrice.
- The three women\u2019s ages add to exactly one-hundred years.
However, Josh's memory is not perfect, and in fact only three of these four claims are true. If each aunt's age is an integer number of years, how old is Beatrice?
Circle $O$ is tangent to two sides of equilateral triangle $XYZ$. If the two shaded regions have areas 50 $cm^2$ and 100 $cm^2$ as indicated, what is the ratio of the area of triangle $XYZ$ to the area of circle $O$? Express your answer as a decimal to the nearest hundredth.
When the integers 1 through 7 are written in base two, what fraction of the digits are 1s? Express your answer as a common fraction.
Using the figure of 15 circles shown, how many sets of three distinct circles A, B and C are there such that circle A encloses circle B, and circle B encloses circle C?
Equilateral triangle ABC with side-length 12 cm is inscribed in a circle. What is the area of the largest equilateral triangle that can be drawn with two vertices on segment AB and the third vertex on minor arc AB of the circle? Express your answer in simplest radical form.
Sue and Kara run along the perimeter of a 120-yard by 40-yard rectangular field. In the time it takes Sue to run 120 yards along one side at a rate of 10 minutes per mile, Kara runs the lengths of the other three sides. At Kara\u2's 019s current rate, how many minutes would it take her to run a mile?
An arithmetic sequence has first term $a$ and common difference $d$. If the sum of the first ten terms is half the sum of the next ten terms, what is the ratio
$\frac{a}{d}$ ? Express your answer as a common fraction.
In one math class, all of the students wrote the amount of change in their pockets on the board. Then they computed some properties of these numbers: the median was 50 cents, the mean was 40 cents and there was a unique mode of 40 cents. What is the fewest number of students that can be in this class?