Found problems: 105
2015 ASDAN Math Tournament, 2
Compute
$$\sum_{n=0}^\infty\frac{n+1}{2^n}.$$
2015 ASDAN Math Tournament, 3
Place points $A$, $B$, $C$, $D$, $E$, and $F$ evenly spaced on a unit circle. Compute the area of the shaded $12$-sided region, where the region is bounded by line segments $AD$, $DF$, $FB$, $BE$, $EC$, and $CA$.
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2015 ASDAN Math Tournament, 4
Compute the number of positive integers less than or equal to $2015$ that are divisible by $5$ or $13$, but not both.
2015 ASDAN Math Tournament, 6
You, your friend, and two strangers are sitting at a table. A standard $52$-card deck is randomly dealt into $4$ piles of $13$ cards each, and each person at the table takes a pile. You look through your hand and see that you have one ace. Compute the probability that your friend’s hand contains the three remaining aces.
2015 ASDAN Math Tournament, 1
Rio likes fruit, and one day she decides to pick persimmons. She picks a total of $12$ persimmons from the first $5$ trees she sees. Rio has $5$ more trees to pick persimmons from. If she wants to pick an average of $4$ persimmons per tree overall, what is the average number of persimmons that she must pick from each of the last $5$ trees for her goal?
2015 ASDAN Math Tournament, 2
Let $ABCD$ be a square with side length $5$, and let $E$ be the midpoint of $CD$. Let $F$ be the point on $AE$ such that $CF=5$. Compute $AF$.
2015 ASDAN Math Tournament, 8
Lynnelle and Moor love toy cars, and together, they have $27$ red cars, $27$ purple cars, and $27$ green cars. The number of red cars Lynnelle has individually is the same as the number of green cars Moor has individually. In addition, Lynnelle has $17$ more cars of any color than Moor has of any color. How many purple cars does Lynnelle have?
2015 ASDAN Math Tournament, 18
Andrew takes a square sheet of paper $ABCD$ of side length $1$ and folds a kite shape. To do this, he takes the corners at $B$ and $D$ and folds the paper such that both corners now rest at a point $E$ on $AC$. This fold results in two creases $CF$ and $CG$, respectively, where $F$ lies on $AB$ and $G$ lies on $AD$. Compute the length of $FG$.
2015 Turkey Team Selection Test, 1
Let $l, m, n$ be positive integers and $p$ be prime. If $p^{2l-1}m(mn+1)^2 + m^2$ is a perfect square, prove that $m$ is also a perfect square.
2015 ASDAN Math Tournament, 11
Let $ABCDEF$ be a regular hexagon, and let $G$, $H$, $I$, $J$, $K$, and $L$ be the midpoints of sides $AB$, $BC$, $CD$, $DE$, $EF$, and $FA$, respectively. The intersection of lines $\overline{AH}$, $\overline{BI}$, $\overline{CJ}$, $\overline{DK}$, $\overline{EL}$, and $\overline{FG}$ bound a smaller regular hexagon. Find the ratio of the area of the smaller hexagon to the area of $ABCDEF$.
2015 FYROM JBMO Team Selection Test, 5
$A$ and $B$ are two identical convex polygons, each with an area of $2015$. The polygon $A$ is divided into polygons $A_1, A_2,...,A_{2015}$, while $B$ is divided into polygons $B_1, B_2,...,B_{2015}$. Each of these smaller polygons has a positive area. Furthermore, $A_1, A_2,...,A_{2015}$ and $B_1, B_2,...,B_{2015}$ are colored in $2015$ distinct colors, such that $A_i$ and $A_j$ are differently colored for every distinct $i$ and $j$ and $B_i$ and $B_j$ are also differently colored for every distinct $i$ and $j$. After $A$ and $B$ overlap, we calculate the sum of the areas with the same colors. Prove that we can color the polygons such that this sum is at least $1$.
2015 ASDAN Math Tournament, 9
A coin is tossed $10$ times. Compute the probability that two heads will turn up in succession somewhere in the sequence of throws.
2015 ASDAN Math Tournament, 16
Find the maximum value of $c$ such that
\begin{align*}
1&=-cx+y\\
-7&=x^2+y^2+8y
\end{align*}
has a unique real solution $(x,y)$.
2015 ASDAN Math Tournament, 3
For a math tournament, each person is assigned an ID which consists of two uppercase letters followed by two digits. All IDs have the property that either the letters are the same, the digits are the same, or both the letters are the same and the digits are the same. Compute the number of possible IDs that the tournament can generate.
2015 ASDAN Math Tournament, 17
How many ways are there to write $91$ as the sum of at least $2$ consecutive positive integers?
2015 ASDAN Math Tournament, 7
Compute the minimum value of
$$\frac{x^4+2x^3+3x^2+2x+10}{x^2+x+1}$$
where $x$ can be any real number.
2015 ASDAN Math Tournament, 6
Let $ABC$ be a triangle and let $D$ be a point on $AC$. The angle bisector of $\angle BAC$ intersects $BD$ at $E$ and $BC$ at $F$. Suppose that $\tfrac{CF}{DE}=\tfrac{5}{4}$ and that $\tfrac{BE}{BF}=\tfrac{3}{2}$. What is $\tfrac{CD}{AD}$?
2015 ASDAN Math Tournament, 10
Triangle $ABC$ has $\angle BAC=90^\circ$. A semicircle with diameter $XY$ is inscribed inside $\triangle ABC$ such that it is tangent to a point $D$ on side $BC$, with $X$ on $AB$ and $Y$ on $AC$. Let $O$ be the midpoint of $XY$. Given that $AB=3$, $AC=4$, and $AX=\tfrac{9}{4}$, compute the length of $AO$.
2015 ASDAN Math Tournament, 7
In a rectangle $ABCD$, two segments $EG$ and $FH$ divide it into four smaller rectangles. $BH$ intersects $EG$ at $X$, $CX$ intersects $HF$ and $Y$, $DY$ intersects $EG$ at $Z$. Given that $AH=4$, $HD=6$, $AE=4$, and $EB=5$, find the area of quadrilateral $HXYZ$.
2015 ASDAN Math Tournament, 1
In his spare time, Thomas likes making rectangular windows. He builds windows by taking four $30\text{ cm}\times20\text{ cm}$ rectangles of glass and arranging them in a larger rectangle in wood. The window has an $x\text{ cm}$ wide strip of wood between adjacent glass pieces and an $x\text{ cm}$ wide strip of wood between each glass piece and the adjacent edge of the window. Given that the total area of the glass is equivalent to the total area of the wood, what is $x$?
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2015 Turkey Team Selection Test, 6
Prove that there are infinitely many positive integers $n$ such that $(n!)^{n+2015}$ divides $(n^{2})!$.
2015 ASDAN Math Tournament, 7
The Yamaimo family is moving to a new house, so they’ve packed their belongings into boxes, which weigh $100\text{ kg}$ in total. Mr. Yamaimo realizes that $99\%$ of the weight of the boxes is due to books. Later, the family unpacks some of the books (and nothing else). Mr. Yamaimo notices that now only $95\%$ of the weight of the boxes is due to books. How much do the boxes weigh now in kilograms?
2015 ASDAN Math Tournament, 5
The Fibonacci numbers are a sequence of numbers defined recursively as follows: $F_1=1$, $F_2=1$, and $F_n=F_{n-1}+F_{n-2}$. Using this definition, compute the sum
$$\sum_{k=1}^{10}\frac{F_k}{F_{k+1}F_{k+2}}.$$
2015 ASDAN Math Tournament, 21
Parallelogram $ABCD$ has $AB=CD=6$ and $BC=AD=10$, where $\angle ABC$ is obtuse. The circumcircle of $\triangle ABD$ intersects $BC$ at $E$ such that $CE=4$. Compute $BD$.
2015 ASDAN Math Tournament, 15
In a given acute triangle $\triangle ABC$ with the values of angles given (known as $a$, $b$, and $c$), the inscribed circle has points of tangency $D,E,F$ where $D$ is on $BC$, $E$ is on $AB$, and $F$ is on $AC$. Circle $\gamma$ has diameter $BC$, and intersects $\overline{EF}$ at points $X$ and $Y$. Find $\tfrac{XY}{BC}$ in terms of the angles $a$, $b$, and $c$.