________ What is the sum of all positive two-digit integers with exactly 12 positive factors? $$ $$
________ What is the sum of the three numbers less than 1000 that have exactly five positive integer divisors? $$ $$
________ The sequence 3, 4, 13, 15, 30, 33, 54, 58, 85, 90, …. contains, in increasing order, all the positive integers that yield a triangle number when multiplied by 7. What is the $100^{th}$ term of this sequence? $$ $$
________ How many whole numbers $n$, such that $100 \le n \le 1000$, have the same number of odd factors as even factors? $$ $$
________ How many positive integer factors of $2020$ have more than $3$ factors? (As an example, $12$ has $6$ factors, namely $1,2,3,4,6,$ and $12$.$$ $$
________ Let $(a\times b\times c)\div(a+b+c)=341$ to be an equation where $a$, $b$ and $c$ are consecutive positive integers. What is the least possible value of $a$? $$ $$
________ The letters $A$, $B$, $C$, $D$, $E$ and $F$ represent digits and $ABC,DEF$ represents a positive six-dight integer. What is the number $ABC,DEF$ if $$4(ABC,DEF)=3(DEF,ABC)$$ $$ $$
________ Jan is thinking of a positive integer. Her integer has exactly 16 positive divisors, two of which are 12 and 15. What is Jan’s number? $$ $$
________ Mady has an infinite number of balls and empty boxes available to her. The empty boxes, each capable of holding four balls, are arranged in a row from left to right. At the first step, she places a ball in the first box (the leftmost box) of the row. At each subsequent step, she places a ball in the first box of the row that still has room for a ball and empties any boxes to its left. How many balls in total are in the boxes as a result of Mady’s $2010^{th}$ step? $$ $$
________ The product of a set of positive integers is 144. What is the least possible sum of this set of positive integers? $$ $$
________ What is the greatest positive integer that must divide the sum of the first ten terms of any arithmetic sequence whose terms are positive integers?$$ $$
________ A digit can be placed in each of the boxes for hundreds and units digits to form the least positive five-digit number divisible by 96. What is the ratio between the smaller digit to the larger digit? Express your answer as a common fraction. $$21\boxed{\phantom{8} }7\boxed{\phantom{8}}$$
________ In order, the first four terms of a sequence are 2, 6, 12 and 72, where each team, beginning with the third term, is the product of the two proceeding terms. If the ninth term is $2^a3^b$, what is the value of $a+b$? $$ $$
________ What is the sum of the numbers less than 200 that have exactly 9 divisors? $$ $$
________ Given that $A=\{\dfrac{n}{24}\}$, $n$ is a natural number, $gcd(n,24)=1$, and $\dfrac{n}{24}\lt 2$, what is the sum of the elements in $A$? $$ $$
________ What is the sum of all integer values of $n$ such that $\dfrac{20}{2n-1}$ is an integer?
________ What is the units digit of the sum of the sum of all the integers from 100 to 202 inclusive?
________ How many of the divisors of $8!$ are larger than $7!$?
________ If $n$ is a prime number, what is the smallest composite number produced by $n^2-n-1$?
________ $A$, $B$, $C$, and $D$ are distinct positive integers such that the product $AB=60$, the product $CD=60$ and $A-B=C+D$. What is the value of $A$?
Two dices are thrown simultaneously. The probability that the sum of the two numbers on the top faces of the dices is a prime number is __________.
Suppose $ABCD$ is a regular triangular pyramid, with each face as a unit equilateral triangle. Let S be the midpoint of the edge $AB$ and T the midpoint of the edge of $CD$. The length of $ST$ is _________.
The value of $\dfrac{(2^3-1)(3^3-1)(4^3-1)…(100^3-1)}{(2^3+1)(3^3+1)(4^3+1)…(100^3+1)}$ is __________.
If $x^2+\dfrac{1}{x^2}=47$, the value of $\sqrt{x}+\dfrac{1}{\sqrt{x}}$ is __________.
4 shoes are randomly selected from 4 pair of shoes in different colors. The probability of 4 shoes containing at least one pair in same color selected is __________.
For all real number $x$, $f(x+1)=\dfrac{1+f(x)}{1-f(x)}$. If $f(2)=2$, then $f(2020)=$ __________.
Positive integer $k$, $m$, and $n$ satisfy the equation $\dfrac{k}{m}+\dfrac{m}{4n}=\dfrac{1}{6}$. The smallest value of $m$ is __________.
The value of $100^2-99^2+98^2-97^2+96^2-95^2+…+4^2-3^2+2^2-1^2$ is __________.
Let $ABCD$ be a quadrilateral. Let $E$,$F$,$G$, and $H$ be the midpoints of sides $AB$,$BC$,$CD$, and $DA$, respectively. The ratio of the area of the quadrilateral $EFGH$ to that of the quadrilateral $ABCD$ is __________.
A convex quadrilateral is formed by randomly selecting four distinct points on the circumference of a circle. The probability of the quadrilateral is inside half of the circle is __________.
The total number of ways to distribute 6 distinguish gifts to 3 children with each child having at least one gift is __________.
Let $P$ be a point inside an equilateral $\triangle{ABC}$, with $PA = 4$, $PB = 3$, and $PC = 5$. The side length of $\triangle{ABC}$ is __________.
The number of factors of $2^5\times 3^6\times 5^2$ are perfect squares is __________.
There are 4 oranges, 5 apples and 6 mangoes in a basket. The number of ways that a person make a selection of fruits among the fruits in the basket is __________.
5 adults and 3 kids are seated in a round table. The number of ways that all 3 kids are not sitting together is __________.
In a chess competition involving some men and women, every player needs to play exactly one game with every other player. It was found that in 45 games, both the players were women and in 190 games, both players were men. The number of games in which one person was a man and other person was a woman is __________.
In $\triangle{ABC}$, point $D$, $E$, $F$, $G$, and $H$ are on side $BC$, point $I$, $J$, $K$ are on side $AC$. Draw line $AD$, $AE$, $AF$, $AG$, $AH$, $BI$, $BJ$, and $BK$. The number of unique triangles formed by various lines are __________.
Answer: 134
One corner of a cube is cut off by a plane. The minimum number of corners in the remaining part of the cube is __________.
The smallest natural number with $12$ factors is ________.
The remainder of 6666666666….6666666 divided by 7 is __________ with 6 repeated 2020 times.
Adam, Bob, Carl, Dave and Ed are playing chess against each other. Each pair play one match. After awhile, if Adam has played 4 matches, Bob 3 matches, Carl 2 matches, and Dave 1 match, the number of matches Ed has played is __________.
A truck is moving from City A to City B. If the truck speed is increased 20%, the truck can arrive at City B 1 hour earlier. After the truck has traveled 120 miles at its normal speed, its speeds was increased 25% and arrived at City B 40 minutes earlier. The distance between City A and City B is ________ miles.
Answer: 270
Solution: Set up some equations that will help us solve for the variables \(t\) for time, \(r\) for the normal truck speed, and \(d\) for the distance between A and B. Using the first information given, we can set up two equations: $$d=rt$$ $$t-1=\frac{d}{1.2r}$$ With the last equation, here is where it gets tricky.
40 minutes is equivalent to $\frac{2}{3}$ hours, so our left side of the equation is $t-\frac{2}{3}$. The right side is more difficult. The first part of the trip is 120 miles long and at the speed of $r$, so the time taken there is $\frac{120}{r}$. The second part is $d-120$ miles long and is at the speed of $1.25r$, giving the expression $\frac{d-120}{1.25r}$. Therefore, we get the equation $$t-\frac{2}{3}=\frac{120}{r}+\frac{d-120}{1.25r}$$ Combining all three equations together, we get $$d=rt$$ $$t-1=\frac{d}{1.2r}$$ $$t-\frac{2}{3}=\frac{120}{r}+\frac{d-120}{1.25r}$$
First, taking the first 2 equations together, we can get $d=1.2rt-1.2r$. Substituting, we obtain the equation $rt=1.2rt-1.2r$. Therefore, $1.2r=0.2rt$, which simplifies to $6r=rt$. Dividing by $r$ gives $t=6$.
Substituting all $t$’s for 6’s and all $d$’s with $rt$’s, we get $$6-\frac{2}{3}=\frac{120}{r}+\frac{6r-120}{1.25r}$$ This becomes $$\frac{16}{3}=\frac{120+24r}{5r}$$ Solving the proportion gives $r=45$. Since $d=rt$, and $t=6$, substituting $r$ gives $d=(45)(6)=\boxed{270}$.
In parallelogram $ABCD$, $DE$ is perpendicular to $AB$. intersecting $AB$ at point $E$ and diagonal $AC$ at point $F$. If $AD:FC=1:2$, then $\angle{CAB}:\angle{CAD}=$__________.
An isosceles trapezoid is tangent to a unit circle on all its sides. If the area of the trapezoid is 4, the perimeter if the trapezoid is __________.
In $\triangle{ABC}$, $\angle{ACB}=90^\circ$, $AC=8$, $BC=15$, $D$ is a point on $AB$. If $CD=AC$, then $AD=$__________.
In trapezoid $ABCD$, $AB \parallel CD$. Diagonal $AC$ and $BD$ intersect at point $E$. If the area of $\triangle{AEB}$ is $m^2$, and that of $\triangle{CED}$ is $n^2$, then, the area of trapeziod $ABCD$ is ________.
In trapezoid $ABCD$, $AB \parallel CD$. Diagonal $AC$ and $BD$ intersect at point $O$. Line $EF$ pass thru $O$ and $EF \parallel AB$, intersecting $AD$ at point $E$, $BC$ at point $F$. If $AB=a$, $CD=d$, then $EF=$ ________.
In $\triangle{ABC}$, $\overline{AB}=1$. Point $D$ is on $\overline{AB}$ and point $E$ is on $\overline{AC}$ such that $\overline{DE} \parallel \overline{BC}$. If $\dfrac{[ADE]}{[BCD]}=k$, then what is $\dfrac{[ADE]}{[ABC]}$ in terms of $k$?
First, draw the altitudes for triangles $ADE$ and $BCD$, each intersecting $DE$ and $BC$ at points $F$ and $G$, respectively. Let $x=AD$. Therefore, $BD=1-x$. We can use similar triangles to get $\dfrac{1-x}{x}=\dfrac{AF}{DG}$ and $\dfrac{1}{x}=\dfrac{BC}{DE}$. Using the triangles’ area properties, we get $DE\cdot AF = k\cdot BC \cdot DG$.
With our similarity properties discussed above, we can simplify the area ratio to a ratio between $AF$ and $DG$: $$AF=\dfrac{k}{x} DG$$ Plugging this back into the equation, we get $\dfrac{1-x}{x}=\dfrac{x}{k}$. Cross multiplying obtains $x^2=k-kx$, creating a quadratic $x^2+kx-k=0$. Solving for the roots, we get $$x=\dfrac{-k \pm \sqrt{k^2+4k}}{2}$$ Noticing that $\sqrt{k^2+nk}\geq k$, we can determine that $$x = \dfrac{-k + \sqrt{k^2+4k}}{2}$$ Therefore, $$\dfrac{[ADE]}{[ABC]} = x^2 = \boxed{\dfrac{2k+k^2-k\sqrt{k^2+4k}}{2}}$$
