problem stringlengths 10 5.15k | answer stringlengths 0 1.23k |
|---|---|
Consider a triangle $DEF$ where the angles of the triangle satisfy
\[ \cos 3D + \cos 3E + \cos 3F = 1. \]
Two sides of this triangle have lengths 12 and 14. Find the maximum possible length of the third side. | 2\sqrt{127} |
A sphere passes through two adjacent vertices of a unit cube and touches the planes of the faces that do not contain these vertices. What is the radius of this sphere? | 2 - \frac{\sqrt{7}}{2} |
The lengths of two sides of a triangle are 33 units and 42 units. The third side also has an integral length. What is the least possible number of units in the perimeter of the triangle? | 85 |
A particle is placed on the curve $y = x^3 - 3x^2 - x + 3$ at a point $P$ whose $y$-coordinate is $5$. It is allowed to roll along the curve until it reaches the nearest point $Q$ whose $y$-coordinate is $-2$. Compute the horizontal distance traveled by the particle.
A) $|\sqrt{6} - \sqrt{3}|$
B) $\sqrt{3}$
C) $\sqrt{6... | |\sqrt{6} - \sqrt{3}| |
If $a$ and $b$ are two distinct numbers with $\frac{a+b}{a-b}=3$, what is the value of $\frac{a}{b}$? | 2 |
The product of three consecutive positive integers is $8$ times their sum. What is the sum of their squares? | 77 |
Michel starts with the string $H M M T$. An operation consists of either replacing an occurrence of $H$ with $H M$, replacing an occurrence of $M M$ with $M O M$, or replacing an occurrence of $T$ with $M T$. For example, the two strings that can be reached after one operation are $H M M M T$ and $H M O M T$. Compute t... | 144 |
Let $a$ and $b$ be real numbers. One of the roots of
\[x^3 + ax^2 - x + b = 0\]is $1 - 2i.$ Enter the ordered pair $(a,b).$ | (1,15) |
In triangle $ABC$, $AB=3$, $AC=4$, and $\angle BAC=60^{\circ}$. If $P$ is a point in the plane of $\triangle ABC$ and $AP=2$, calculate the maximum value of $\vec{PB} \cdot \vec{PC}$. | 10 + 2 \sqrt{37} |
Given a defect rate of 3%, products are drawn from the batch without replacement until a non-defective product is found or a maximum of three draws have been made. Let $X$ represent the number of products drawn, and calculate $P(X=3)$. | (0.03)^2 \times 0.97 + (0.03)^3 |
In $\triangle ABC$, $\sqrt {2}csinAcosB=asinC$.
(I) Find the measure of $\angle B$;
(II) If the area of $\triangle ABC$ is $a^2$, find the value of $cosA$. | \frac {3 \sqrt {10}}{10} |
Let $S$ be the set of integers between $1$ and $2^{40}$ whose binary expansions have exactly two $1$'s. If a number is chosen at random from $S,$ the probability that it is divisible by $15$ is $p/q,$ where $p$ and $q$ are relatively prime positive integers. Find $p+q.$ | 49 |
Square $ABCD$ has side length $1$ unit. Points $E$ and $F$ are on sides $AB$ and $CB$, respectively, with $AE = CF$. When the square is folded along the lines $DE$ and $DF$, sides $AD$ and $CD$ coincide and lie on diagonal $BD$. The length of segment $AE$ can be expressed in the form $\sqrt{k}-m$ units. What is the ... | 3 |
A traffic light runs repeatedly through the following cycle: green for 45 seconds, then yellow for 5 seconds, and then red for 50 seconds. Mark picks a random five-second time interval to watch the light. What is the probability that the color changes while he is watching? | \frac{3}{20} |
Let $T = \{9^k : k ~ \mbox{is an integer}, 0 \le k \le 4000\}$. Given that $9^{4000}$ has 3817 digits and that its first (leftmost) digit is 9, how many elements of $T$ have 9 as their leftmost digit?
| 184 |
In a given triangle, for $\angle P$ to be the largest angle of the triangle, it must be that $a < y < b$. The side lengths are given by $y+6$, $2y+1$, and $5y-10$. What is the least possible value of $b-a$, expressed as a common fraction? | 4.5 |
In four years, Peter will be twice as old as Harriet. If Peter's age is currently half of his mother's age, who is 60, how old is Harriet now? | If Peter's age is currently half of his mother's age, who is 60, Peter is 1/2*60 = <<30=30>>30 years old.
In four years, Peter will be 30+4 = <<30+4=34>>34 years old.
Since Peter's age in four years will be twice Harriet's age, Harriet will be 34/2 = <<34/2=17>>17 years old in four years.
Currently, Harriet is 17-4 = <... |
The probability of an event occurring in each of 900 independent trials is 0.5. Find the probability that the relative frequency of the event will deviate from its probability by no more than 0.02. | 0.7698 |
What is the coefficient of $x^2y^6$ in the expansion of $\left(\frac{3}{5}x-\frac{y}{2}\right)^8$? Express your answer as a common fraction. | \frac{63}{400} |
The square root of $x$ is greater than 2 and less than 4. How many integer values of $x$ satisfy this condition? | 11 |
In a Martian civilization, all logarithms whose bases are not specified as assumed to be base $b$, for some fixed $b\ge2$. A Martian student writes down \[3\log(\sqrt{x}\log x)=56\] \[\log_{\log x}(x)=54\] and finds that this system of equations has a single real number solution $x>1$. Find $b$. | 216 |
Cinderella and her fairy godmother released a collection of seven new models of crystal slippers. The storybook heroines held a presentation of the collection for some guests: the audience had to state which slippers they liked. The guests wrote in a survey which models they considered the best. It is known that no two... | 128 |
Musa is the class teacher of a class of 45 students. He wants to split them into three groups by age. If a third of the class is under 11 years, and two-fifths are above 11 but under 13, how many students will be in the third group (13 years and above)? | The first group is a third of the class which is (1/3)*45 = <<1/3*45=15>>15 students
There are 15 students in the first group, so there are 45-15 = <<45-15=30>>30 students
The second group is two-fifths of the class which is (2/5)*45 = <<2/5*45=18>>18 students
There are 18 in the second group so there are 30-18 = <<30-... |
Lara is trying to solve the following equation by completing the square: $$100x^2 + 60x - 49 = 0.$$ She seeks to rewrite the above equation in the form: $$(ax + b)^2 = c,$$ where $a$, $b$, and $c$ are integers and $a > 0$. What is the value of $a + b + c$? | 71 |
The average age of 33 fifth-graders is 11. The average age of 55 of their parents is 33. What is the average age of all of these parents and fifth-graders? | 24.75 |
Let $r_{k}$ denote the remainder when $\binom{127}{k}$ is divided by 8. Compute $r_{1}+2 r_{2}+3 r_{3}+\cdots+63 r_{63}$. | 8096 |
Find the integer $n$, $0 \le n \le 5$, such that \[n \equiv -3736 \pmod{6}.\] | 2 |
On a blackboard a stranger writes the values of $s_{7}(n)^{2}$ for $n=0,1, \ldots, 7^{20}-1$, where $s_{7}(n)$ denotes the sum of digits of $n$ in base 7 . Compute the average value of all the numbers on the board. | 3680 |
Calculate $\frac{1}{2} \cdot \frac{3}{5} \cdot \frac{7}{11}$. | \frac{21}{110} |
Mary chose an even $4$-digit number $n$. She wrote down all the divisors of $n$ in increasing order from left to right: $1,2,...,\dfrac{n}{2},n$. At some moment Mary wrote $323$ as a divisor of $n$. What is the smallest possible value of the next divisor written to the right of $323$?
$\textbf{(A) } 324 \qquad \textbf{... | 340 |
If line $l_1: (2m+1)x - 4y + 3m = 0$ is parallel to line $l_2: x + (m+5)y - 3m = 0$, determine the value of $m$. | -\frac{9}{2} |
Find the number of odd digits in the base-4 representation of $233_{10}$. | 2 |
In how many ways can a President and a Vice-President be chosen from a group of 5 people (assuming that the President and the Vice-President cannot be the same person)? | 20 |
Evaluate $\lfloor17.2\rfloor+\lfloor-17.2\rfloor$. | -1 |
The spacecraft Gibraltar is a mountain-sized intergalactic vehicle for transporting equipment, building materials, and families to establish colonies on far-away planets. At full capacity, the vehicle can carry 300 family units with four people per family. The space flight is expected to take years, and it is expecte... | The full capacity of the ship is 300 * 4 = <<300*4=1200>>1200 people.
One-third of capacity is 1200/3 = <<1200/3=400>>400 people.
100 less than one-third capacity is 400 - 100 = 300 people.
#### 300 |
Given the piecewise function $f(x)= \begin{cases} x+2 & (x\leq-1) \\ x^{2} & (-1<x<2) \\ 2x & (x\geq2)\end{cases}$, if $f(x)=3$, determine the value of $x$. | \sqrt{3} |
The equation of the asymptotes of the hyperbola \\(x^{2}- \frac {y^{2}}{2}=1\\) is \_\_\_\_\_\_; the eccentricity equals \_\_\_\_\_\_. | \sqrt {3} |
Anna Lisa bought two dozen apples for $\$$15.60. At that same rate, how much would three dozen of these apples cost? | \$23.40 |
Compute: $\displaystyle \frac{66,\!666^4}{22,\!222^4}$. | 81 |
A school is planning a community outreach program. Each classroom must raise $200 for this activity. Classroom A has raised $20 from each of two families, $10 from each of eight families, and $5 from each of ten families. How much more money does Classroom A need in order to reach the goal? | Classroom A raised 2 x $20 = $<<2*20=40>>40 from two families.
They raised 8 x $10 = $<<8*10=80>>80 from eight families.
They raised 10 x $5 = $<<10*5=50>>50 from ten families.
Altogether, they raised $40 + $80 + $50 = $<<40+80+50=170>>170.
Therefore, they need $200 - $170 = $<<200-170=30>>30 more to reach the goal.
##... |
The calculation result of the expression \(143 \times 21 \times 4 \times 37 \times 2\) is $\qquad$. | 888888 |
How many non-empty subsets $S$ of $\{1, 2, 3, \ldots, 10\}$ satisfy the following two conditions?
1. No two consecutive integers belong to $S$.
2. If $S$ contains $k$ elements, then $S$ contains no number less than $k$. | 59 |
Mady has an infinite number of balls and boxes available to her. The empty boxes, each capable of holding sixteen 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 t... | 30 |
Out of the 80 students who took the biology exam, only 2/5 of them managed to score 100%. If a score below 80% qualified as failing, and 50 percent of the remaining students who didn't score 100% managed to score over 80%, calculate the number of students who failed the exam. | The number of students who scored 100% is 2/5*80 = <<2/5*80=32>>32
Out of the 80 students, 80-32 = <<80-32=48>>48 did not score 100%.
If 50% of the students who did not score 100% managed to score over 80%, then 50/100*48 =<<50/100*48=24>>24 students scored over 80%
The number of students who failed is 48-24 = <<48-24=... |
Convert $199_{10}$ to base 2. Let $x$ be the number of zeros and $y$ be the number of ones in base 2. What is the value of $y-x?$ | 2 |
In the cells of an $8 \times 8$ chessboard, there are 8 white and 8 black pieces arranged such that no two pieces are in the same cell. Additionally, no pieces of the same color are in the same row or column. For each white piece, the distance to the black piece in the same column is calculated. What is the maximum val... | 32 |
Let $G$ be the centroid of triangle $ABC.$ If $GA^2 + GB^2 + GC^2 = 58,$ then find $AB^2 + AC^2 + BC^2.$ | 174 |
Given $\cos x + \cos y = \frac{1}{2}$ and $\sin x + \sin y = \frac{1}{3}$, find the value of $\cos (x - y)$. | $-\frac{59}{72}$ |
Let $AB$ be a segment of length $2$ . The locus of points $P$ such that the $P$ -median of triangle $ABP$ and its reflection over the $P$ -angle bisector of triangle $ABP$ are perpendicular determines some region $R$ . Find the area of $R$ . | 2\pi |
A positive integer $n$ is picante if $n$ ! ends in the same number of zeroes whether written in base 7 or in base 8 . How many of the numbers $1,2, \ldots, 2004$ are picante? | 4 |
A sphere is inscribed in the tetrahedron whose vertices are $A = (6,0,0), B = (0,4,0), C = (0,0,2),$ and $D = (0,0,0).$ The radius of the sphere is $m/n,$ where $m$ and $n$ are relatively prime positive integers. Find $m + n.$ | 5 |
For how many values of $a$ is it true that the line $y=x+a$ passes through the vertex of parabola $y=x^2+a^2$? | 2 |
Sandra wants to buy some sweets. She saved $10 for this purpose. Her mother gave her an additional $4, and her father twice as much as her mother. One candy costs $0.5, and one jelly bean $0.2. She wants to buy 14 candies and 20 jelly beans. How much money will she be left with after the purchase? | Sandra's father gave her $4 * 2 = $<<4*2=8>>8.
So Sandra has in total $8 + $4 + $10 = $<<8+4+10=22>>22.
She wants 14 candies, so she is going to pay 14 candies * $0.50/candy = $<<14*0.5=7>>7 for them.
She wants also 20 jellybeans, and they're going to cost 20 jellybeans * $0.20/jellybean = $<<20*0.2=4>>4.
So after the ... |
(1) Find the constant term in the expansion of ${\left( \frac{1}{x}- \sqrt{\frac{x}{2}}\right)}^{9}$;
(2) Given that ${x}^{10}={a}_{0}+{a}_{1}\left( x+2 \right)+{a}_{2}{\left( x+2 \right)}^{2}+... +{a}_{10}{\left( x+2 \right)}^{10}$, find the value of ${{a}_{1}+{a}_{2}+{a}_{3}+... +{a}_{10}}$. | -1023 |
Let $min|a, b|$ denote the minimum value between $a$ and $b$. When positive numbers $x$ and $y$ vary, let $t = min|2x+y, \frac{2y}{x^2+2y^2}|$, then the maximum value of $t$ is ______. | \sqrt{2} |
The sum other than $11$ which occurs with the same probability when all 8 dice are rolled is equal to what value. | 45 |
From the numbers 0, 1, 2, 3, 4, select three different digits to form a three-digit number. What is the sum of the units digit of all these three-digit numbers? | 90 |
In $\triangle ABC$, $2\sin^2 \frac{A+B}{2}-\cos 2C=1$, and the radius of the circumcircle $R=2$.
$(1)$ Find $C$;
$(2)$ Find the maximum value of $S_{\triangle ABC}$. | \sqrt{3} |
Let $S=\{1,2, \ldots, 9\}$. Compute the number of functions $f: S \rightarrow S$ such that, for all $s \in S, f(f(f(s)))=s$ and $f(s)-s$ is not divisible by 3. | 288 |
The sum of the digits of the integer equal to \( 777777777777777^2 - 222222222222223^2 \) is | 74 |
The matrix for reflecting over a certain line $\ell,$ which passes through the origin, is given by
\[\begin{pmatrix} \frac{7}{25} & -\frac{24}{25} \\ -\frac{24}{25} & -\frac{7}{25} \end{pmatrix}.\]Find the direction vector of line $\ell.$ Enter your answer in the form $\begin{pmatrix} a \\ b \end{pmatrix},$ where $a,$... | \begin{pmatrix} 4 \\ -3 \end{pmatrix} |
Evaluate the expression $\frac{2020^3 - 3 \cdot 2020^2 \cdot 2021 + 5 \cdot 2020 \cdot 2021^2 - 2021^3 + 4}{2020 \cdot 2021}$. | 4042 + \frac{3}{4080420} |
What is \( \frac{1}{4} \) more than 32.5? | 32.75 |
For positive integers $n$, define $S_n$ to be the minimum value of the sum
\[\sum_{k=1}^n \sqrt{(2k-1)^2+a_k^2},\]where $a_1,a_2,\ldots,a_n$ are positive real numbers whose sum is $17$. Find the unique positive integer $n$ for which $S_n$ is also an integer. | 12 |
The dilation, centered at $2 + 3i,$ with scale factor 3, takes $-1 - i$ to which complex number? | -7 - 9i |
Compute the sum of the squares of cosine for the angles progressing by 10 degrees starting from 0 degrees up to 180 degrees:
\[\cos^2 0^\circ + \cos^2 10^\circ + \cos^2 20^\circ + \dots + \cos^2 180^\circ.\] | \frac{19}{2} |
(1) Calculate $\dfrac{2A_{8}^{5}+7A_{8}^{4}}{A_{8}^{8}-A_{9}^{5}}$,
(2) Calculate $C_{200}^{198}+C_{200}^{196}+2C_{200}^{197}$. | 67331650 |
An equilateral triangle $ABC$ is inscribed in the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ such that vertex $B$ is at $(0,b)$, and side $\overline{AC}$ is parallel to the $x$-axis. The foci $F_1$ and $F_2$ of the ellipse lie on sides $\overline{BC}$ and $\overline{AB}$, respectively. Determine the ratio $\frac{A... | \frac{8}{5} |
Find the maximum value of $S$ such that any finite number of small squares with a total area of $S$ can be placed inside a unit square $T$ with side length 1, in such a way that no two squares overlap. | \frac{1}{2} |
Find the sum of the rational roots of $g(x)=x^3-9x^2+16x-4$. | 2 |
Let $ABC$ be an acute-angled triangle and $P$ be a point in its interior. Let $P_A,P_B$ and $P_c$ be the images of $P$ under reflection in the sides $BC,CA$ , and $AB$ , respectively. If $P$ is the orthocentre of the triangle $P_AP_BP_C$ and if the largest angle of the triangle that can be formed by the... | 120 |
In a convex 1950-sided polygon, all the diagonals are drawn, dividing it into smaller polygons. Consider the polygon with the greatest number of sides among these smaller polygons. What is the maximum number of sides it can have? | 1949 |
Let \( a \) and \( b \) be real numbers such that \( a + b = 1 \). Then, the minimum value of
\[
f(a, b) = 3 \sqrt{1 + 2a^2} + 2 \sqrt{40 + 9b^2}
\]
is ______. | 5 \sqrt{11} |
Let $a,$ $b,$ $c,$ $d$ be real numbers such that
\[\frac{(a - b)(c - d)}{(b - c)(d - a)} = \frac{2}{5}.\]Find the sum of all possible values of
\[\frac{(a - c)(b - d)}{(a - b)(c - d)}.\] | -\frac{3}{2} |
In triangle $ABC,$ $\angle B = 60^\circ$ and $\angle C = 45^\circ.$ The point $D$ divides $\overline{BC}$ in the ratio $1:3$. Find
\[\frac{\sin \angle BAD}{\sin \angle CAD}.\] | \frac{\sqrt{6}}{6} |
Given $m$ points on a plane, where no three points are collinear, and their convex hull is an $n$-gon. Connecting the points appropriately can form a mesh region composed of triangles. Let $f(m, n)$ represent the number of non-overlapping triangles in this region. Find $f(2016, 30)$. | 4000 |
Let $a$ and $b$ be positive integers for which $45a+b=2021$. What is the minimum possible value of $a+b$? | 85 |
If I roll 5 standard 6-sided dice and multiply the number on the face of each die, what is the probability that the result is a composite number? | \frac{485}{486} |
What is the coefficient of $x^3$ when $$x^4-3x^3 + 5x^2-6x + 1$$is multiplied by $$2x^3 - 3x^2 + 4x + 7$$and the like terms are combined? | 19 |
Find the sum of the distinct prime factors of $7^7 - 7^4$. | 24 |
Consider the set $\{45, 52, 87, 90, 112, 143, 154\}$. How many subsets containing three different numbers can be selected from this set so that the sum of the three numbers is odd? | 19 |
Let $$ A=\lim _{n \rightarrow \infty} \sum_{i=0}^{2016}(-1)^{i} \cdot \frac{\binom{n}{i}\binom{n}{i+2}}{\binom{n}{i+1}^{2}} $$ Find the largest integer less than or equal to $\frac{1}{A}$. | 1 |
There are infinitely many positive integers $k$ which satisfy the equation
\[\cos^2 (k^2 + 6^2)^\circ = 1.\]Enter the two smallest solutions, separated by commas. | 18 |
Let $\triangle ABC$ be an acute scalene triangle with circumcircle $\omega$. The tangents to $\omega$ at $B$ and $C$ intersect at $T$. Let $X$ and $Y$ be the projections of $T$ onto lines $AB$ and $AC$, respectively. Suppose $BT = CT = 16$, $BC = 22$, and $TX^2 + TY^2 + XY^2 = 1143$. Find $XY^2$. | 717 |
A positive integer is *happy* if:
1. All its digits are different and not $0$ ,
2. One of its digits is equal to the sum of the other digits.
For example, 253 is a *happy* number. How many *happy* numbers are there? | 32 |
The second hand on the clock pictured below is 6 cm long. How far in centimeters does the tip of this second hand travel during a period of 30 minutes? Express your answer in terms of $\pi$.
[asy]
draw(Circle((0,0),20));
label("12",(0,20),S);
label("9",(-20,0),E);
label("6",(0,-20),N);
label("3",(20,0),W);
dot((0,0))... | 360\pi |
Ioana has three ropes whose lengths are 39 inches, 52 inches and 65 inches. She wants to cut the ropes into equal length pieces for magic tricks. No rope is to be wasted. What is the greatest number of inches possible in the length of each piece? | 13 |
Let $ABCD$ and $BCFG$ be two faces of a cube with $AB=12$. A beam of light emanates from vertex $A$ and reflects off face $BCFG$ at point $P$, which is 7 units from $\overline{BG}$ and 5 units from $\overline{BC}$. The beam continues to be reflected off the faces of the cube. The length of the light path from the time ... | 230 |
There are two colors Jessica can use to color this 2 by 2 grid. If non-overlapping regions that share a side cannot be the same color, how many possible ways are there for Jessica to color the grid?
[asy]
size(101);
draw(unitsquare);
draw((0,.5)--(1,.5));
draw((.5,0)--(.5,1));
label("$A$",(0,1),NW); label("$B$",(1,1),... | 2 |
How many nonzero complex numbers $z$ have the property that $0, z,$ and $z^3,$ when represented by points in the complex plane, are the three distinct vertices of an equilateral triangle? | 4 |
Given that $α∈(0, \dfrac {π}{2})$ and $β∈(0, \dfrac {π}{2})$, with $cosα= \dfrac {1}{7}$ and $cos(α+β)=- \dfrac {11}{14}$, find the value of $sinβ$. | \dfrac{\sqrt{3}}{2} |
$ABCDEFGH$ is a cube. Find $\sin \angle HAD$. | \frac{\sqrt{2}}{2} |
The vertices of a regular hexagon are labeled $\cos (\theta), \cos (2 \theta), \ldots, \cos (6 \theta)$. For every pair of vertices, Bob draws a blue line through the vertices if one of these functions can be expressed as a polynomial function of the other (that holds for all real $\theta$ ), and otherwise Roberta draw... | 14 |
A box contains $5$ chips, numbered $1$, $2$, $3$, $4$, and $5$. Chips are drawn randomly one at a time without replacement until the sum of the values drawn exceeds $4$. What is the probability that $3$ draws are required? | \frac{1}{5} |
Given that $P$ is a moving point on the parabola $y^{2}=4x$, and $Q$ is a moving point on the circle $x^{2}+(y-4)^{2}=1$, find the minimum value of the sum of the distance between points $P$ and $Q$ and the distance between point $P$ and the axis of symmetry of the parabola. | \sqrt{17}-1 |
The graph of the quadratic $y = ax^2 + bx + c$ has the following properties: (1) The maximum value of $y = ax^2 + bx + c$ is 5, which occurs at $x = 3$. (2) The graph passes through the point $(0,-13)$. If the graph passes through the point $(4,m)$, then what is the value of $m$? | 3 |
In the right triangle $ABC$, where $\angle B = \angle C$, the length of $AC$ is $8\sqrt{2}$. Calculate the area of triangle $ABC$. | 64 |
Suppose that $a_1, a_2, a_3, \ldots$ is an infinite geometric sequence such that for all $i \ge 1$ , $a_i$ is a positive integer. Suppose furthermore that $a_{20} + a_{21} = 20^{21}$ . If the minimum possible value of $a_1$ can be expressed as $2^a 5^b$ for positive integers $a$ and $b$ , find $a + b$ .
... | 24 |
In triangle \( \triangle ABC \), the sides opposite to the angles \( A \), \( B \), and \( C \) are of lengths \( a \), \( b \), and \( c \) respectively. Point \( G \) satisfies
$$
\overrightarrow{GA} + \overrightarrow{GB} + \overrightarrow{GC} = \mathbf{0}, \quad \overrightarrow{GA} \cdot \overrightarrow{GB} = 0.
$$
... | \frac{1}{2} |
In right triangle $DEF$, where $DE = 15$, $DF = 9$, and $EF = 12$ units, how far is point $F$ from the midpoint of segment $DE$? | 7.5 |
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