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|---|---|---|---|---|---|
Rectangle \(ABCD\) has length 9 and width 5. Diagonal \(AC\) is divided into 5 equal parts at \(W, X, Y\), and \(Z\). Determine the area of the shaded region.
|
18
|
deepscale
| 20,734
| ||
Let $f : \mathbb{R} \to \mathbb{R}$ be a function such that $f(1) = 1$ and
\[f(xy + f(x)) = xf(y) + f(x)\]for all real numbers $x$ and $y.$
Let $n$ be the number of possible values of $f \left( \frac{1}{2} \right),$ and let $s$ be the sum of all possible values of $f \left( \frac{1}{2} \right).$ Find $n \times s.$
|
\frac{1}{2}
|
deepscale
| 36,512
| ||
A large supermarket has recorded the number of customers during eight holidays (unit: hundreds of people) as $29$, $30$, $38$, $25$, $37$, $40$, $42$, $32$. What is the $75$th percentile of this data set?
|
39
|
deepscale
| 31,286
| ||
What is the smallest possible sum of two consecutive integers whose product is greater than 420?
|
43
|
deepscale
| 27,759
| ||
What is the nonnegative difference between the roots for the equation $x^2+30x+180=-36$?
|
6
|
deepscale
| 34,073
| ||
A right triangular prism \( ABC-A_{1}B_{1}C_{1} \) has 9 edges of equal length. Point \( P \) is the midpoint of \( CC_{1} \). The dihedral angle \( B-A_{1}P-B_{1} \) is \( \alpha \). What is \( \sin \alpha \)?
|
\frac{\sqrt{10}}{4}
|
deepscale
| 14,735
| ||
Calculate the volumes of the bodies bounded by the surfaces.
$$
z = 2x^2 + 18y^2, \quad z = 6
$$
|
6\pi
|
deepscale
| 24,706
| ||
Given that point $A(-2,3)$ lies on the axis of parabola $C$: $y^{2}=2px$, and the line passing through point $A$ is tangent to $C$ at point $B$ in the first quadrant. Let $F$ be the focus of $C$. Then, $|BF|=$ _____ .
|
10
|
deepscale
| 28,719
| ||
Given the parametric equation of curve $C\_1$ is $\begin{cases} x=3\cos \alpha \ y=\sin \alpha \end{cases} (\alpha \text{ is the parameter})$, and the polar coordinate equation of curve $C\_2$ is $\rho \cos \left( \theta +\frac{\pi }{4} \right)=\sqrt{2}$.
(I) Find the rectangular coordinate equation of curve $C\_2$ and the maximum value of the distance $|OP|$ between the moving point $P$ on curve $C\_1$ and the coordinate origin $O$;
(II) If curve $C\_2$ intersects with curve $C\_1$ at points $A$ and $B$, and intersects with the $x$-axis at point $E$, find the value of $|EA|+|EB|$.
|
\frac{6 \sqrt{3}}{5}
|
deepscale
| 32,191
| ||
Given that $abc$ represents a three-digit number, if it satisfies $a \lt b$ and $b \gt c$, then we call this three-digit number a "convex number". The number of three-digit "convex" numbers without repeated digits is ______.
|
204
|
deepscale
| 19,914
| ||
Given a line $l$ passing through point $A(1,1)$ with a slope of $-m$ ($m>0$) intersects the x-axis and y-axis at points $P$ and $Q$, respectively. Perpendicular lines are drawn from $P$ and $Q$ to the line $2x+y=0$, and the feet of the perpendiculars are $R$ and $S$. Find the minimum value of the area of quadrilateral $PRSQ$.
|
3.6
|
deepscale
| 13,447
| ||
The probability that three friends, Al, Bob, and Carol, will be assigned to the same lunch group is approximately what fraction.
|
\frac{1}{9}
|
deepscale
| 22,485
| ||
I want to choose a license plate which is 3 characters long, where the first character is a letter, the last character is a digit, and the middle is either a letter or a digit. I also want there to be two characters on my license plate which are the same. How many ways are there for me to choose a license plate with these restrictions?
|
520
|
deepscale
| 34,765
| ||
The Cookie Monster encounters a cookie whose boundary is the equation $x^2+y^2 - 6.5 = x + 3 y$ and is very confused. He wants to know if this cookie is a lunch-sized cookie or a snack-sized cookie. What is the radius of this cookie?
|
3
|
deepscale
| 34,246
| ||
In each cell of a $5 \times 5$ board, there is either an X or an O, and no three Xs are consecutive horizontally, vertically, or diagonally. What is the maximum number of Xs that can be on the board?
|
16
|
deepscale
| 27,627
| ||
Vasya has a stick that is 22 cm long. He wants to break it into three pieces with integer lengths such that the pieces can form a triangle. In how many ways can he do this? (Ways that result in identical triangles are considered the same).
|
10
|
deepscale
| 8,413
| ||
Given that the positive integer \( a \) has 15 factors and the positive integer \( b \) has 20 factors, and \( a + b \) is a perfect square, find the smallest possible value of \( a + b \) that meets these conditions.
|
576
|
deepscale
| 8,005
| ||
What is the greatest possible three-digit number that is divisible by 3 and divisible by 6?
|
996
|
deepscale
| 37,875
| ||
What is the sum of the interior numbers of the eighth row of Pascal's Triangle?
|
126
|
deepscale
| 30,151
| ||
Evaluate $|(12-9i)(8+15i)|$.
|
255
|
deepscale
| 39,690
| ||
Find the maximum and minimum values of the function $f(x)=1+x-x^{2}$ in the interval $[-2,4]$.
|
-11
|
deepscale
| 30,450
| ||
A tetrahedron with four equilateral triangular faces has a sphere inscribed within it and a sphere circumscribed about it. For each of the four faces, there is a sphere tangent externally to the face at its center and to the circumscribed sphere. A point $P$ is selected at random inside the circumscribed sphere. The probability that $P$ lies inside one of the five small spheres is closest to
$\mathrm{(A) \ }0 \qquad \mathrm{(B) \ }0.1 \qquad \mathrm{(C) \ }0.2 \qquad \mathrm{(D) \ }0.3 \qquad \mathrm{(E) \ }0.4$
|
.2
|
deepscale
| 36,044
| ||
The coefficient sum of the expansion of the binomial ${{\left(\frac{1}{x}-2x^2\right)}^9}$, excluding the constant term, is $671$.
|
671
|
deepscale
| 18,173
| ||
Consider two solid spherical balls, one centered at $\left( 0, 0, \frac{21}{2} \right),$ with radius 6, and the other centered at $(0,0,1)$ with radius $\frac{9}{2}.$ How many points $(x,y,z)$ with only integer coefficients are there in the intersection of the balls?
|
13
|
deepscale
| 39,909
| ||
What is the largest factor of $130000$ that does not contain the digit $0$ or $5$ ?
|
26
|
deepscale
| 14,789
| ||
Let $S$ be the set of positive real numbers. Let $f : S \to \mathbb{R}$ be a function such that
\[ f(x)f(y) = f(xy) + 2023 \left( \frac{1}{x} + \frac{1}{y} + 2022 \right) \]
for all $x, y > 0.$
Let $n$ be the number of possible values of $f(2)$, and let $s$ be the sum of all possible values of $f(2)$. Find $n \times s$.
|
\frac{4047}{2}
|
deepscale
| 17,923
| ||
Given that $f(x)$ and $g(x)$ are functions defined on $\mathbb{R}$, with $g(x) \neq 0$, $f(x)g'(x) > f'(x)g(x)$, and $f(x) = a^{x}g(x)$ ($a > 0$ and $a \neq 1$), $\frac{f(1)}{g(1)} + \frac{f(-1)}{g(-1)} = \frac{5}{2}$. For the finite sequence $\frac{f(n)}{g(n)} = (n = 1, 2, \ldots, 0)$, find the probability that the sum of the first $k$ terms is greater than $\frac{15}{16}$ for any positive integer $k$ ($1 \leq k \leq 10$).
|
\frac{3}{5}
|
deepscale
| 19,390
| ||
Starting with an equilateral triangle as shown in diagram a, each side of the triangle is divided into three equal parts, and at the middle segment, new equilateral triangles are constructed outward, as shown in diagram b, forming a "snowflake hexagon." Next, each of the 12 sides of the "snowflake hexagon" is divided into three equal parts, and new equilateral triangles are constructed outward at the middle segments, as shown in diagram c, forming a new "snowflake shape." What is the ratio of the area of the shape in diagram c to the area of the triangle in diagram a?
|
40/27
|
deepscale
| 16,423
| ||
The maximum and minimum values of the function y=2x^3-3x^2-12x+5 on the interval [0,3] need to be determined.
|
-15
|
deepscale
| 29,146
| ||
If $x-y=15$ and $xy=4$, what is the value of $x^2+y^2$?
|
233
|
deepscale
| 33,081
| ||
A path of length $n$ is a sequence of points $\left(x_{1}, y_{1}\right),\left(x_{2}, y_{2}\right), \ldots,\left(x_{n}, y_{n}\right)$ with integer coordinates such that for all $i$ between 1 and $n-1$ inclusive, either (1) $x_{i+1}=x_{i}+1$ and $y_{i+1}=y_{i}$ (in which case we say the $i$th step is rightward) or (2) $x_{i+1}=x_{i}$ and $y_{i+1}=y_{i}+1$ (in which case we say that the $i$th step is upward). This path is said to start at $\left(x_{1}, y_{1}\right)$ and end at $\left(x_{n}, y_{n}\right)$. Let $P(a, b)$, for $a$ and $b$ nonnegative integers, be the number of paths that start at $(0,0)$ and end at $(a, b)$. Find $\sum_{i=0}^{10} P(i, 10-i)$.
|
This is just the number of paths of length 10. The $i$th step can be either upward or rightward, so there are $2^{10}=1024$ such paths.
|
1024
|
deepscale
| 3,564
| |
Given real numbers $x$, $y$, $z$ satisfying $2x-y-2z-6=0$, and $x^2+y^2+z^2\leq4$, calculate the value of $2x+y+z$.
|
\frac{2}{3}
|
deepscale
| 30,257
| ||
Let $b_n$ be the number obtained by writing the integers 1 to $n$ from left to right, where each integer is squared. For example, $b_3 = 149$ (since $1^2 = 1$, $2^2 = 4$, $3^2 = 9$), and $b_5 = 1491625$. For $1 \le k \le 100$, determine how many $b_k$ are divisible by 4.
|
50
|
deepscale
| 12,109
| ||
Find the greatest negative value of the expression \( x - y \) for all pairs of numbers \((x, y)\) satisfying the equation
$$
(\sin x + \sin y)(\cos x - \cos y) = \frac{1}{2} + \sin(x - y) \cos(x + y)
$$
|
-\frac{\pi}{6}
|
deepscale
| 11,870
| ||
How many of the first $500$ positive integers can be expressed in the form
\[\lfloor 3x \rfloor + \lfloor 6x \rfloor + \lfloor 9x \rfloor + \lfloor 12x \rfloor\]
where \( x \) is a real number?
|
300
|
deepscale
| 26,974
| ||
An apartment and an office are sold for $15,000 each. The apartment was sold at a loss of 25% and the office at a gain of 25%. Determine the net effect of the transactions.
|
2000
|
deepscale
| 8,037
| ||
The expression $\cos 2x + \cos 6x + \cos 10x + \cos 14x$ can be written in the equivalent form
\[a \cos bx \cos cx \cos dx\] for some positive integers $a,$ $b,$ $c,$ and $d.$ Find $a + b + c + d.$
|
18
|
deepscale
| 32,758
| ||
Determine all integers $k$ such that there exists infinitely many positive integers $n$ [b]not[/b] satisfying
\[n+k |\binom{2n}{n}\]
|
Determine all integers \( k \) such that there exist infinitely many positive integers \( n \) not satisfying
\[
n + k \mid \binom{2n}{n}.
\]
We claim that all integers \( k \neq 1 \) satisfy the desired property.
First, recall that \(\frac{1}{n + 1} \binom{2n}{n}\) is the \( n \)-th Catalan number. Since the Catalan numbers are a sequence of integers, it follows that \( n + 1 \mid \binom{2n}{n} \) for all \( n \). Hence, \( k = 1 \) certainly cannot satisfy the problem statement.
Now, we consider two cases:
**Case 1: \( k \neq 2 \).**
Suppose that \( p \) is a prime divisor of \( k \) and let \( n = p^\alpha \) for any \( \alpha \in \mathbb{N} \). Then, since \( p \mid n + k \), in order to prove that \( n + k \nmid \binom{2n}{n} \), it suffices to show that
\[
p \nmid \binom{2n}{n} = \frac{(n + 1)(n + 2) \cdots (2n)}{1 \cdot 2 \cdots n}.
\]
Note that the greatest power of \( p \) that divides any term in the numerator or denominator of \(\frac{(n + 1)(n + 2) \cdots (2n - 1)}{1 \cdot 2 \cdots (n - 1)}\) is less than \( p^{\alpha} \). Since the sets \(\{1, 2, \cdots, n - 1\}\) and \(\{n + 1, n + 2, \cdots, 2n - 1\}\) are congruent modulo \( p^{\alpha} \), the numerator and denominator of the fraction \(\frac{(n + 1)(n + 2) \cdots (2n - 1)}{1 \cdot 2 \cdots (n - 1)}\) both contain the same number of factors of \( p \). Therefore, \( p \nmid \frac{(n + 1)(n + 2) \cdots (2n - 1)}{1 \cdot 2 \cdots (n - 1)} \).
Now, if we can show that \( p \nmid 2 \), we will be able to conclude that \( p \nmid \binom{2n}{n} \), as desired. Indeed, if \( p \neq 2 \), then trivially \( p \nmid 2 \). Meanwhile, if \( p = 2 \), then let us take \( \alpha \geq 2 \) so that \( 2^2 \mid n + k \). Hence, we wish to show that \( 2^2 \nmid \binom{2n}{n} \). But since \( 2 \nmid \frac{(n + 1)(n + 2) \cdots (2n - 1)}{1 \cdot 2 \cdots (n - 1)} = \frac{\binom{2n}{n}}{2} \), we need only show that \( 2^2 \nmid 2 \), which is obvious. This concludes Case 1.
**Case 2: \( k = 2 \).**
Seeking a nice expression for \( n + k \), we choose to set \( n = 2^{\alpha} - 2 \) for any \( \alpha \in \mathbb{N} \) with \( \alpha \geq 2 \). Then, since \( n + k = 2^{\alpha} \), we wish to show that
\[
2^{\alpha} \nmid \binom{2n}{n} = \frac{(n + 1)(n + 2) \cdots (2n)}{1 \cdot 2 \cdots n}.
\]
Notice that since \( 2n < 2^{\alpha + 1} \), the greatest power of \( 2 \) that divides any term in the numerator or denominator of \(\frac{(n + 1)(n + 2) \cdots (2n)}{1 \cdot 2 \cdots n}\) is \( 2^{\alpha} \). Then, because the sets \(\{1, 2, \cdots, n - 2\}\) and \(\{n + 3, n + 4, \cdots, 2n\}\) are congruent modulo \( 2^{\alpha} \), we deduce that \( 2 \nmid \frac{(n + 3)(n + 4) \cdots (2n)}{1 \cdot 2 \cdots (n - 2)} \). Removing this fraction from the fraction \(\binom{2n}{n} = \frac{(n + 1)(n + 2) \cdots (2n)}{1 \cdot 2 \cdots n}\), it suffices to show that \( 2^{\alpha} \nmid \frac{(n + 1)(n + 2)}{(n - 1)n} \). Keeping in mind that \( n + 2 = 2^{\alpha} \), we see that the largest power of \( 2 \) that divides the numerator is \( 2^{\alpha} \), while the largest power of \( 2 \) that divides the denominator is \( 2^1 \) (since \( 2 \mid n \)). Therefore, \( 2^{\alpha - 1} \) is the largest power of \( 2 \) that divides \(\frac{(n + 1)(n + 2)}{(n - 1)n}\), so
\[
2^{\alpha} \nmid \frac{(n + 1)(n + 2)}{(n - 1)n} \implies n + k \nmid \binom{2n}{n},
\]
as desired.
Thus, the integers \( k \) that satisfy the condition are all integers \( k \neq 1 \).
The answer is: \boxed{k \neq 1}.
|
k \neq 1
|
deepscale
| 2,979
| |
Steve says to Jon, "I am thinking of a polynomial whose roots are all positive integers. The polynomial has the form $P(x) = 2x^3-2ax^2+(a^2-81)x-c$ for some positive integers $a$ and $c$. Can you tell me the values of $a$ and $c$?"
After some calculations, Jon says, "There is more than one such polynomial."
Steve says, "You're right. Here is the value of $a$." He writes down a positive integer and asks, "Can you tell me the value of $c$?"
Jon says, "There are still two possible values of $c$."
Find the sum of the two possible values of $c$.
|
Since each of the roots is positive, the local maximum of the function must occur at a positive value of $x$. Taking $\frac{d}{dx}$ of the polynomial yields $6x^2-4ax+a^2-81$, which is equal to $0$ at the local maximum. Since this is a quadratic in $a$, we can find an expression for $a$ in terms of $x$. The quadratic formula gives $a=\frac{4x\pm\sqrt{324-8x^2}}{2}$, which simplifies to $a=2x\pm\sqrt{81-2x^2}$. We know that $a$ is a positive integer, and testing small positive integer values of $x$ yields $a=15$ or $a=1$ when $x=4$, and $a=15$ or $a=9$ when $x=6$. Because the value of $a$ alone does not determine the polynomial, $a$, $a$ must equal $15$.
Now our polynomial equals $2x^3-30x^2+144x-c$. Because one root is less than (or equal to) the $x$ value at the local maximum (picture the graph of a cubic equation), it suffices to synthetically divide by small integer values of $x$ to see if the resulting quadratic also has positive integer roots. Dividing by $x=3$ leaves a quotient of $2x^2-24x+72=2(x-6)^2$, and dividing by $x=4$ leaves a quotient of $2x^2-22x+56=2(x-4)(x-7)$. Thus, $c=2\cdot 3\cdot 6\cdot 6=216$, or $c=2\cdot 4\cdot 4\cdot 7=224$. Our answer is $216+224=\boxed{440}$ ~bad_at_mathcounts
|
440
|
deepscale
| 7,141
| |
Calculate the value of the polynomial $f(x) = 3x^6 + 4x^5 + 5x^4 + 6x^3 + 7x^2 + 8x + 1$ at $x=0.4$ using Horner's method, and then determine the value of $v_1$.
|
5.2
|
deepscale
| 16,623
| ||
What is the least common multiple of 105 and 360?
|
2520
|
deepscale
| 22,400
| ||
When $n$ standard 8-sided dice are rolled, the probability of obtaining a sum of 3000 is greater than zero and is the same as the probability of obtaining a sum of S. Find the smallest possible value of S.
|
375
|
deepscale
| 8,694
| ||
Calculate the value of $\sin 135^{\circ}\cos (-15^{\circ}) + \cos 225^{\circ}\sin 15^{\circ}$.
|
\frac{1}{2}
|
deepscale
| 21,159
| ||
The solutions to the system of equations
$\log_{225}x+\log_{64}y=4$
$\log_{x}225-\log_{y}64=1$
are $(x_1,y_1)$ and $(x_2,y_2)$. Find $\log_{30}\left(x_1y_1x_2y_2\right)$.
|
Let $A=\log_{225}x$ and let $B=\log_{64}y$.
From the first equation: $A+B=4 \Rightarrow B = 4-A$.
Plugging this into the second equation yields $\frac{1}{A}-\frac{1}{B}=\frac{1}{A}-\frac{1}{4-A}=1 \Rightarrow A = 3\pm\sqrt{5}$ and thus, $B=1\pm\sqrt{5}$.
So, $\log_{225}(x_1x_2)=\log_{225}(x_1)+\log_{225}(x_2)=(3+\sqrt{5})+(3-\sqrt{5})=6$ $\Rightarrow x_1x_2=225^6=15^{12}$.
And $\log_{64}(y_1y_2)=\log_{64}(y_1)+\log_{64}(y_2)=(1+\sqrt{5})+(1-\sqrt{5})=2$ $\Rightarrow y_1y_2=64^2=2^{12}$.
Thus, $\log_{30}\left(x_1y_1x_2y_2\right) = \log_{30}\left(15^{12}\cdot2^{12} \right) = \log_{30}\left(30^{12} \right) = \boxed{012}$.
One may simplify the work by applying Vieta's formulas to directly find that $\log x_1 + \log x_2 = 6 \log 225, \log y_1 + \log y_2 = 2 \log 64$.
|
12
|
deepscale
| 6,736
| |
What is the smallest integer $x$ for which $x<2x-7$ ?
|
8
|
deepscale
| 38,592
| ||
The first three stages of a pattern are shown below, where each line segment represents a matchstick. If the pattern continues such that at each successive stage, four matchsticks are added to the previous arrangement, how many matchsticks are necessary to create the arrangement for the 100th stage?
|
400
|
deepscale
| 23,800
| ||
If $a$,$b$, and $c$ are positive real numbers such that $a(b+c) = 152$, $b(c+a) = 162$, and $c(a+b) = 170$, then find $abc.$
|
720
|
deepscale
| 37,316
| ||
Given the right triangles ABC and ABD, what is the length of segment BC, in units? [asy]
size(150);
pair A, B, C, D, X;
A=(0,0);
B=(0,12);
C=(-16,0);
D=(-35,0);
draw(A--B--D--A);
draw(B--C);
draw((0,1.5)--(-1.5,1.5)--(-1.5,0));
label("$37$", (B+D)/2, NW);
label("$19$", (C+D)/2, S);
label("$16$", (A+C)/2, S);
label("A", A, SE);
label("B", B, NE);
label("D", D, SW);
label("C", C, S);
[/asy]
|
20
|
deepscale
| 38,569
| ||
Let the equation $x^3 - 6x^2 + 11x - 6 = 0$ have three real roots $a$, $b$, and $c$. Find $\frac{1}{a^3} + \frac{1}{b^3} + \frac{1}{c^3}$.
|
\frac{251}{216}
|
deepscale
| 16,888
| ||
Determine the value of \(x\) if \(x\) is positive and \(x \cdot \lfloor x \rfloor = 90\). Express your answer as a decimal.
|
10
|
deepscale
| 26,738
| ||
Given $\vec{a}=(\cos \alpha, \sin \alpha), \vec{b}=(\cos \beta, \sin \beta)$, and $|\vec{a}-\vec{b}|=\frac{2 \sqrt{5}}{5}$. If $0 < \alpha < \frac{\pi}{2}$, $-\frac{\pi}{2} < \beta < 0$, and $\sin \beta=-\frac{5}{13}$, then $\sin \alpha=$
|
$\frac{33}{65}$
|
deepscale
| 17,501
| ||
An infinite geometric series has a first term of \( 512 \) and its sum is \( 8000 \). What is its common ratio?
|
0.936
|
deepscale
| 8,099
| ||
The measure of the angles of a pentagon are in the ratio of 3:3:3:4:5. What is the number of degrees in the measure of the largest angle?
|
150^\circ
|
deepscale
| 39,538
| ||
The value of \(\frac{1^{2}-3^{2}+5^{2}-7^{2}+\cdots+97^{2}-99^{2}}{1-3+5-7+\cdots+97-99}\) is:
|
100
|
deepscale
| 17,114
| ||
On a circle, 103 natural numbers are written. It is known that among any 5 consecutive numbers, there will be at least two even numbers. What is the minimum number of even numbers that can be in the entire circle?
|
42
|
deepscale
| 7,668
| ||
All the roots of the polynomial $z^6-10z^5+Az^4+Bz^3+Cz^2+Dz+16$ are positive integers, possibly repeated. What is the value of $B$?
|
1. **Identify the Roots**: Given that all roots of the polynomial $z^6 - 10z^5 + Az^4 + Bz^3 + Cz^2 + Dz + 16$ are positive integers, and their sum (as coefficients of $z^5$) is 10, we consider possible sets of roots that sum to 10. The roots provided in the solution are $2, 2, 2, 2, 1, 1$.
2. **Symmetric Sums and Polynomial Coefficients**: The coefficient $B$ corresponds to the negation of the third elementary symmetric sum of the roots. The third elementary symmetric sum $s_3$ is given by the sum of all products of the roots taken three at a time.
3. **Calculate the Third Symmetric Sum**:
- The number of ways to choose three roots from four 2's (and no 1's) is $\binom{4}{3} = 4$. Each product is $2^3 = 8$. Thus, this contributes $4 \times 8 = 32$.
- The number of ways to choose two 2's from four and one 1 from two is $\binom{4}{2} \binom{2}{1} = 6 \times 2 = 12$. Each product is $2^2 \times 1 = 4$. Thus, this contributes $12 \times 4 = 48$.
- The number of ways to choose one 2 from four and two 1's from two is $\binom{4}{1} \binom{2}{2} = 4 \times 1 = 4$. Each product is $2 \times 1^2 = 2$. Thus, this contributes $4 \times 2 = 8$.
4. **Summing the Contributions**:
- The total third symmetric sum is $32 + 48 + 8 = 88$.
5. **Determine $B$**:
- Since $B$ is the negation of the third symmetric sum, we have $B = -88$.
Thus, the value of $B$ is $\boxed{\textbf{(A) }{-}88}$.
|
-88
|
deepscale
| 1,897
| |
One million bucks (i.e. one million male deer) are in different cells of a $1000 \times 1000$ grid. The left and right edges of the grid are then glued together, and the top and bottom edges of the grid are glued together, so that the grid forms a doughnut-shaped torus. Furthermore, some of the bucks are honest bucks, who always tell the truth, and the remaining bucks are dishonest bucks, who never tell the truth. Each of the million bucks claims that "at most one of my neighboring bucks is an honest buck." A pair of neighboring bucks is said to be buckaroo if exactly one of them is an honest buck. What is the minimum possible number of buckaroo pairs in the grid?
|
Note that each honest buck has at most one honest neighbor, and each dishonest buck has at least two honest neighbors. The connected components of honest bucks are singles and pairs. Then if there are $K$ honest bucks and $B$ buckaroo pairs, we get $B \geq 3 K$. From the dishonest buck condition we get $B \geq 2(1000000-K)$, so we conclude that $B \geq 1200000$. To find equality, partition the grid into five different parts with side $\sqrt{5}$, and put honest bucks on every cell in two of the parts.
|
1200000
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deepscale
| 4,786
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Let $A$ be an acute angle such that $\tan A = 2 \cos A.$ Find the number of positive integers $n$ less than or equal to $1000$ such that $\sec^n A + \tan^n A$ is a positive integer whose units digit is $9.$
|
\[\tan A = 2 \cos A \implies \sin A = 2 \cos^2 A \implies \sin^2 A + \cos^2 A = 4 \cos^4 A + \cos^2 A = 1\] \[\implies \cos^2 A = \frac {\sqrt {17} - 1}{8}.\] \[c_n = \sec^n A + \tan^n A = \frac {1}{\cos^n A} + 2^n \cos^n A = (4\cos^2 A +1)^{\frac {n}{2}}+(4 \cos^2 A)^{\frac {n}{2}} =\] \[= \left(\frac {\sqrt {17} + 1}{2}\right)^{\frac {n}{2}}+ \left(\frac {\sqrt {17} - 1}{2}\right)^{\frac {n}{2}}.\]
It is clear, that $c_n$ is not integer if $n \ne 4k, k > 0.$
Denote $x = \frac {\sqrt {17} + 1}{2}, y = \frac {\sqrt {17} - 1}{2} \implies$ \[x \cdot y = 4, x + y = \sqrt{17}, x - y = 1 \implies x^2 + y^2 = (x - y)^2 + 2xy = 9 = c_4.\]
\[c_8 = x^4 + y^4 = (x^2 + y^2)^2 - 2x^2 y^2 = 9^2 - 2 \cdot 16 = 49.\] \[c_{4k+4} = x^{4k+4} + y^{4k+4} = (x^{4k} + y^{4k})(x^2+y^2)- (x^2 y^2)(x^{4k-2}+y^{4k-2}) = 9 c_{4k}- 16 c_{4k – 4} \implies\] \[c_{12} = 9 c_8 - 16 c_4 = 9 \cdot 49 - 16 \cdot 9 = 9 \cdot 33 = 297.\] \[c_{16} = 9 c_{12} - 16 c_8 = 9 \cdot 297 - 16 \cdot 49 = 1889.\] \[c_{12m + 4} \pmod{10} = 9 \cdot c_{12m} \pmod{10} - 16 \pmod{10} \cdot c_{12m - 4} \pmod{10} =\] \[= (9 \cdot 7 - 6 \cdot 9) \pmod{10} = (3 - 4) \pmod{10} = 9.\] \[c_{12m + 8}\pmod{10} = 9 \cdot c_{12m+4} \pmod{10} - 16 \pmod{10} \cdot c_{12m } \pmod{10} =\] \[= (9 \cdot 9 - 6 \cdot 7) \pmod{10} = (1 - 2)\pmod{10} = 9.\] \[c_{12m + 12} \pmod{10} = 9 \cdot c_{12m + 8} \pmod{10} - 16 \pmod{10} \cdot c_{12m + 4} \pmod{10} =\] \[= (9 \cdot 9 - 6 \cdot 9) \pmod{10} = (1 - 4) \pmod{10} = 7 \implies\]
The condition is satisfied iff $n = 12 k + 4$ or $n = 12k + 8.$
If $n \le N$ then the number of possible n is $\left\lfloor \frac{N}{4} \right\rfloor - \left\lfloor \frac{N}{12} \right\rfloor.$
For $N = 1000$ we get $\left\lfloor \frac{1000}{4} \right\rfloor - \left\lfloor \frac{1000}{12} \right\rfloor = 250 - 83 = \boxed{167}.$
vladimir.shelomovskii@gmail.com, vvsss
~MathProblemSolvingSkills.com
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167
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deepscale
| 7,388
| |
There are several white rabbits and gray rabbits. When 6 white rabbits and 4 gray rabbits are placed in a cage, there are still 9 more white rabbits remaining, and all the gray rabbits are placed. When 9 white rabbits and 4 gray rabbits are placed in a cage, all the white rabbits are placed, and there are still 16 gray rabbits remaining. How many white rabbits and gray rabbits are there in total?
|
159
|
deepscale
| 14,715
| ||
Find the coefficient of $x^{90}$ in the expansion of
\[(x - 1)(x^2 - 2)(x^3 - 3) \dotsm (x^{12} - 12)(x^{13} - 13).\]
|
-1
|
deepscale
| 30,837
| ||
Mattis is hosting a badminton tournament for $40$ players on $20$ courts numbered from $1$ to $20$. The players are distributed with $2$ players on each court. In each round a winner is determined on each court. Afterwards, the player who lost on court $1$, and the player who won on court $20$ stay in place. For the remaining $38$ players, the winner on court $i$ moves to court $i + 1$ and the loser moves to court $i - 1$. The tournament continues until every player has played every other player at least once. What is the minimal number of rounds the tournament can last?
|
Mattis is organizing a badminton tournament with the following setup: there are \(40\) players distributed evenly across \(20\) courts, with \(2\) players on each court. In each round of the tournament, a match is played on each court, and a winner and a loser are determined. Following the match results, the player who lost on court \(1\) and the player who won on court \(20\) remain on their respective courts. Meanwhile, the winner on court \(i\) moves to court \(i + 1\), and the loser moves to court \(i - 1\), for \(1 \leq i \leq 19\).
The goal is to determine the minimal number of rounds required for every player to have played against every other player at least once.
### Analysis
- Initially, each court has two players. A round of matches determines winners and losers on each court.
- The player movement rules ensure that players circulate among the courts: winners move to higher-numbered courts, and losers move to lower-numbered courts, except for the special rules applying on courts \(1\) and \(20\).
### Finding the Minimum Number of Rounds
Here's a step-by-step explanation to find the minimal number of rounds:
1. **Player Circulation**: Notably, we need to ensure that each player has the opportunity to play against every other player at least once.
2. **Court Movement**: Each round shifts winners up one court and losers down one court, meaning players need to cycle through all other players.
3. **Special Court Behavior**: Since the player on court \(1\) who loses and the player on court \(20\) who wins remain on their courts, player circulation between these two courts requires special tracking.
4. **Closed Cycle**: Since there are \(40\) players, the arrangement of courts and movement should form a closed cycle allowing for all pairings to occur at least once.
5. **Estimation and Calculation**:
- Realizing that each player remains on their court in the first instance requires rotating fully through as the circulation scheme allows players to shift \(1\) position left or right depending on their court performance, focusing primarily on \(38\) players moving.
- Therefore, for completeness, all players must essentially have the opportunity to progress through a full cycle of opponents due to movement constraints.
6. **Determine Minimum Rounds**:
- Each player must encounter all others. This is feasible in \(39\) rounds.
- This can be calculated as \(2 \times (20 - 1)\), since every arrangement in \(39\) rounds exhausts possible matchups considering player interactions and the specialized movement rules. It's assured each player will have played every other player across \(2 \times (n - 1)\) iterations, where \(n\) is the number of courts, covering all necessary combinations through optimal match shuffling.
Thus, the minimal number of rounds that ensures every player plays against every other player at least once is:
\[
\boxed{39}
\]
|
39
|
deepscale
| 6,010
| |
Let the function
$$
f(x) = A \sin(\omega x + \varphi) \quad (A>0, \omega>0).
$$
If \( f(x) \) is monotonic on the interval \(\left[\frac{\pi}{6}, \frac{\pi}{2}\right]\) and
$$
f\left(\frac{\pi}{2}\right) = f\left(\frac{2\pi}{3}\right) = -f\left(\frac{\pi}{6}\right),
$$
then the smallest positive period of \( f(x) \) is ______.
|
\pi
|
deepscale
| 15,938
| ||
Find the maximum positive integer $r$ that satisfies the following condition: For any five 500-element subsets of the set $\{1,2, \cdots, 1000\}$, there exist two subsets that have at least $r$ common elements.
|
200
|
deepscale
| 13,211
| ||
Add $101_2 + 11_2 + 1100_2 + 11101_2.$ Express your answer in base $2.$
|
110001_2
|
deepscale
| 37,878
| ||
In a certain sequence the first term is $a_1 = 2007$ and the second term is $a_2 = 2008.$ Furthermore, the values of the remaining terms are chosen so that
\[a_n + a_{n + 1} + a_{n + 2} = n\]for all $n \ge 1.$ Determine $a_{1000}.$
|
2340
|
deepscale
| 36,422
| ||
The circumference of a circle has 50 numbers written on it, each of which is either +1 or -1. We want to find the product of these numbers. What is the minimum number of questions needed to determine this product if we can ask about the product of three consecutive numbers at a time?
|
50
|
deepscale
| 16,364
| ||
Line segment $\overline{AB}$ has perpendicular bisector $\overline{CD}$ , where $C$ is the midpoint of $\overline{AB}$ . The segments have lengths $AB = 72$ and $CD = 60$ . Let $R$ be the set of points $P$ that are midpoints of line segments $\overline{XY}$ , where $X$ lies on $\overline{AB}$ and $Y$ lies on $\overline{CD}$ . Find the area of the region $R$ .
|
1080
|
deepscale
| 16,213
| ||
Seven teams play a soccer tournament in which each team plays every other team exactly once. No ties occur, each team has a $50\%$ chance of winning each game it plays, and the outcomes of the games are independent. In each game, the winner is awarded a point and the loser gets 0 points. The total points are accumulated to decide the ranks of the teams. In the first game of the tournament, team $A$ beats team $B.$ The probability that team $A$ finishes with more points than team $B$ is $m/n,$ where $m$ and $n$ are relatively prime positive integers. Find $m+n.$
|
After the first game, there are $10$ games we care about-- those involving $A$ or $B$. There are $3$ cases of these $10$ games: $A$ wins more than $B$, $B$ wins more than $A$, or $A$ and $B$ win the same number of games. Also, there are $2^{10} = 1024$ total outcomes. By symmetry, the first and second cases are equally likely, and the third case occurs $\binom{5}{0}^2+\binom{5}{1}^2+\binom{5}{2}^2+\binom{5}{3}^2+\binom{5}{4}^2+\binom{5}{5}^2 = \binom{10}{5} = 252$ times, by a special case of Vandermonde's Identity. There are therefore $\frac{1024-252}{2} = 386$ possibilities for each of the other two cases.
If $B$ has more wins than $A$ in its $5$ remaining games, then $A$ cannot beat $B$ overall. However, if $A$ has more wins or if $A$ and $B$ are tied, $A$ will beat $B$ overall. Therefore, out of the $1024$ possibilites, $386+252 = 638$ ways where $A$ wins, so the desired probability is $\frac{638}{1024} = \frac{319}{512}$, and $m+n=\boxed{831}$.
|
831
|
deepscale
| 6,875
| |
Given a positive integer $n$, find all $n$-tuples of real number $(x_1,x_2,\ldots,x_n)$ such that
\[ f(x_1,x_2,\cdots,x_n)=\sum_{k_1=0}^{2} \sum_{k_2=0}^{2} \cdots \sum_{k_n=0}^{2} \big| k_1x_1+k_2x_2+\cdots+k_nx_n-1 \big| \]
attains its minimum.
|
Given a positive integer \( n \), we aim to find all \( n \)-tuples of real numbers \( (x_1, x_2, \ldots, x_n) \) such that
\[
f(x_1, x_2, \cdots, x_n) = \sum_{k_1=0}^{2} \sum_{k_2=0}^{2} \cdots \sum_{k_n=0}^{2} \left| k_1 x_1 + k_2 x_2 + \cdots + k_n x_n - 1 \right|
\]
attains its minimum.
To solve this, we first claim that the minimum is achieved when all \( x_i \) are equal. Specifically, we seek to show that the minimum occurs at \( x_1 = x_2 = \cdots = x_n = \frac{1}{n+1} \).
### Proof:
1. **Symmetry Argument:**
Let \( y = \frac{x_1 + x_2}{2} \). We will prove that
\[
f(x_1, x_2, x_3, \cdots, x_n) \geq f(y, y, x_3, \cdots, x_n).
\]
For any \( k, m, c \in \mathbb{R} \),
\[
|kx_1 + mx_2 + c| + |mx_1 + kx_2 + c| \geq |(k+m)(x_1 + x_2) + 2c| = |ky + my + c| + |my + ky + c|
\]
by the triangle inequality.
2. **Application of Inequality:**
Applying the above inequality, let \( c = \sum_{j=3}^n k_j x_j - 1 \) and summing over all \( c \) for all \( (k_3, \cdots, k_n) \in \{0, 1, 2\}^{n-2} \) where \( k_1 \neq k_2 \) gives the desired result.
3. **Reduction to Single Variable:**
Now, let \( x_1 = x_2 = \cdots = x_n = x \), and we need to minimize
\[
g(x) = f(x, x, x, \cdots, x).
\]
Let \( H(n, t) \) be the number of solutions to \( k_1 + \cdots + k_n = t \) where \( k_j \in \{0, 1, 2\} \) for \( j = 1, \cdots, n \). Observe that \( H(n, t) \leq H(n, n) \) and \( H(n, t) = H(n, 2n - t) \).
4. **Minimization:**
Therefore,
\[
g(x) = \sum_{j=0}^{2n} H(n, j) |jx - 1| = \sum_{j=0}^{2n} j H(n, j) \left| x - \frac{1}{j} \right|.
\]
We claim that \( g(x) \) is minimized at \( x = \frac{1}{n+1} \).
5. **Uniqueness:**
It remains to show that \( x_1 = x_2 \) is forced. In the smoothing process, any move that produces a smaller sum is not possible. If \( x_1 = x_2 \), then there exist \( k_1 \neq k_2 \) such that \( k_1 x_1 + \cdots + k_n x_n = 1 \). Now, let \( y_1, y_2 \) such that \( y_1 + y_2 = 2x_1 \). Then,
\[
|k_1 y_1 + k_2 y_2 + k_3 x_3 + \cdots + k_n x_n - 1| + |k_2 y_1 + k_1 y_2 + k_3 x_3 + \cdots + k_n x_n - 1| > 2 |k_1 x_1 + k_2 x_2 + k_3 x_3 + \cdots + k_n x_n - 1|,
\]
so the \( x_i \)'s must be constant.
Thus, the minimum value of the function is attained when \( x_1 = x_2 = \cdots = x_n = \frac{1}{n+1} \).
The answer is: \(\boxed{\left( \frac{1}{n+1}, \frac{1}{n+1}, \ldots, \frac{1}{n+1} \right)}\).
|
\left( \frac{1}{n+1}, \frac{1}{n+1}, \ldots, \frac{1}{n+1} \right)
|
deepscale
| 2,965
| |
A laptop is originally priced at $800. The store offers a $15\%$ discount, followed by another $10\%$ discount on the discounted price. Tom also has a special membership card giving an additional $5\%$ discount on the second discounted price. What single percent discount would give the same final price as these three successive discounts?
|
27.325\%
|
deepscale
| 8,738
| ||
In how many ways can one fill a \(4 \times 4\) grid with a 0 or 1 in each square such that the sum of the entries in each row, column, and long diagonal is even?
|
First we name the elements of the square as follows: \(a_{11}, a_{12}, a_{13}, a_{14}, a_{21}, a_{22}, a_{23}, a_{24}, a_{31}, a_{32}, a_{33}, a_{34}, a_{41}, a_{42}, a_{43}, a_{44}\). We claim that for any given values of \(a_{11}, a_{12}, a_{13}, a_{21}, a_{22}, a_{23}, a_{32}\), and \(a_{33}\) (the + signs in the diagram below), there is a unique way to assign values to the rest of the entries such that all necessary sums are even. Taking additions mod 2, we have \(a_{14}=a_{11}+a_{12}+a_{13}\), \(a_{24}=a_{21}+a_{22}+a_{23}\), \(a_{44}=a_{11}+a_{22}+a_{33}\), \(a_{42}=a_{12}+a_{22}+a_{32}\), \(a_{43}=a_{13}+a_{23}+a_{33}\). Since the 4th column, the 4th row, and the 1st column must have entries that sum to 0, we have \(a_{34}=a_{14}+a_{24}+a_{44}=a_{12}+a_{13}+a_{21}+a_{23}+a_{33}\), \(a_{41}=a_{42}+a_{43}+a_{44}=a_{11}+a_{12}+a_{13}+a_{23}+a_{32}\), \(a_{31}=a_{11}+a_{21}+a_{41}=a_{12}+a_{13}+a_{21}+a_{23}+a_{32}\). It is easy to check that the sum of entries in every row, column, and the main diagonal is even. Since there are \(2^{8}=256\) ways to assign the values to the initial 8 entries, there are exactly 256 ways to fill the board.
|
256
|
deepscale
| 3,253
| |
The integer parts of two finite decimals are 7 and 10, respectively. How many possible values are there for the integer part of the product of these two finite decimals?
|
18
|
deepscale
| 12,290
| ||
A building contractor needs to pay his $108$ workers $\$ 200 $ each. He is carrying $ 122 $ one hundred dollar bills and $ 188 $ fifty dollar bills. Only $ 45 $ workers get paid with two $ \ $100$ bills. Find the number of workers who get paid with four $\$ 50$ bills.
|
31
|
deepscale
| 32,381
| ||
There is a reservoir A and a town B connected by a river. When the reservoir does not release water, the water in the river is stationary; when the reservoir releases water, the water in the river flows at a constant speed. When the reservoir was not releasing water, speedboat M traveled for 50 minutes from A towards B and covered $\frac{1}{3}$ of the river's length. At this moment, the reservoir started releasing water, and the speedboat took only 20 minutes to travel another $\frac{1}{3}$ of the river's length. The driver then turned off the speedboat's engine and allowed it to drift with the current, taking $\quad$ minutes for the speedboat to reach B.
|
100/3
|
deepscale
| 12,243
| ||
What is the largest possible distance between two points, one on the sphere of radius 15 with center $(3, -5, 7),$ and the other on the sphere of radius 95 with center $(-10, 20, -25)$?
|
110 + \sqrt{1818}
|
deepscale
| 17,930
| ||
Let $r,$ $s,$ and $t$ be the roots of the equation $x^3 - 20x^2 + 18x - 7 = 0.$ Find the value of $(1+r)(1+s)(1+t).$
|
46
|
deepscale
| 36,973
| ||
It is known that the probabilities of person A and person B hitting the target in each shot are $\frac{3}{4}$ and $\frac{4}{5}$, respectively. Person A and person B do not affect each other's chances of hitting the target, and each shot is independent. If they take turns shooting in the order of A, B, A, B, ..., until one person hits the target and stops shooting, then the probability that A and B have shot a total of four times when stopping shooting is ____.
|
\frac{1}{100}
|
deepscale
| 17,754
| ||
Given 8 teams, of which 3 are weak teams, they are randomly divided into two groups, A and B, with 4 teams in each group. Find:
(1) The probability that one of the groups A or B has exactly 2 weak teams.
(2) The probability that group A has at least 2 weak teams.
|
\frac{1}{2}
|
deepscale
| 17,852
| ||
The water tank in the diagram below is in the shape of an inverted right circular cone. The radius of its base is 8 feet, and its height is 64 feet. The water in the tank is $40\%$ of the tank's capacity. The height of the water in the tank can be written in the form $a\sqrt[3]{b}$, where $a$ and $b$ are positive integers, and $b$ is not divisible by a perfect cube greater than 1. What is $a+b$?
|
66
|
deepscale
| 28,141
| ||
The function $y= |x-1|+|2x-1|+|3x-1|+ |4x-1|+|5x-1|$ achieves its minimum value when the variable $x$ equals what value?
|
\frac{1}{3}
|
deepscale
| 16,295
| ||
In triangle $ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively, and $\frac{\cos B}{b} + \frac{\cos C}{c} = \frac{2\sqrt{3}\sin A}{3\sin C}$.
(1) Find the value of $b$;
(2) If $B = \frac{\pi}{3}$, find the maximum area of triangle $ABC$.
|
\frac{3\sqrt{3}}{16}
|
deepscale
| 20,986
| ||
The line with equation $y = x$ is an axis of symmetry of the curve with equation
\[y = \frac{px + q}{rx + s},\]where $p,$ $q,$ $r,$ $s$ are all nonzero. Which of the following statements must hold?
(A) $p + q = 0$
(B) $p + r = 0$
(C) $p + s = 0$
(D) $q + r = 0$
(E) $q + s = 0$
(F) $r + s = 0$
|
\text{(C)}
|
deepscale
| 37,456
| ||
The function $f(x)$ satisfies
\[xf(y) = yf(x)\]for all real numbers $x$ and $y.$ If $f(15) = 20,$ find $f(3).$
|
4
|
deepscale
| 36,428
| ||
For each integer $n$ greater than 1, let $G(n)$ be the number of solutions of the equation $\sin x = \sin (n+1)x$ on the interval $[0, 2\pi]$. Calculate $\sum_{n=2}^{100} G(n)$.
|
10296
|
deepscale
| 16,491
| ||
In the Cartesian coordinate system $xOy$, the curve $C$ is given by $\frac{x^2}{4} + \frac{y^2}{3} = 1$. Taking the origin $O$ of the Cartesian coordinate system $xOy$ as the pole and the positive half-axis of $x$ as the polar axis, and using the same unit length, a polar coordinate system is established. It is known that the line $l$ is given by $\rho(\cos\theta - 2\sin\theta) = 6$.
(I) Write the Cartesian coordinate equation of line $l$ and the parametric equation of curve $C$;
(II) Find a point $P$ on curve $C$ such that the distance from point $P$ to line $l$ is maximized, and find this maximum value.
|
2\sqrt{5}
|
deepscale
| 24,024
| ||
Given a four-digit positive integer $\overline{abcd}$, if $a+c=b+d=11$, then this number is called a "Shangmei number". Let $f(\overline{abcd})=\frac{{b-d}}{{a-c}}$ and $G(\overline{abcd})=\overline{ab}-\overline{cd}$. For example, for the four-digit positive integer $3586$, since $3+8=11$ and $5+6=11$, $3586$ is a "Shangmei number". Also, $f(3586)=\frac{{5-6}}{{3-8}}=\frac{1}{5}$ and $G(M)=35-86=-51$. If a "Shangmei number" $M$ has its thousands digit less than its hundreds digit, and $G(M)$ is a multiple of $7$, then the minimum value of $f(M)$ is ______.
|
-3
|
deepscale
| 23,846
| ||
The Hoopers, coached by Coach Loud, have 15 players. George and Alex are the two players who refuse to play together in the same lineup. Additionally, if George plays, another player named Sam refuses to play. How many starting lineups of 6 players can Coach Loud create, provided the lineup does not include both George and Alex?
|
3795
|
deepscale
| 31,826
| ||
Points $P$ and $Q$ lie in a plane with $PQ=8$. How many locations for point $R$ in this plane are there such that the triangle with vertices $P$, $Q$, and $R$ is a right triangle with area $12$ square units?
|
1. **Given Information and Formula for Area**: We know that the area of a triangle can be expressed as \([PQR] = \frac{1}{2} \cdot PQ \cdot h_R\), where \(h_R\) is the perpendicular distance from point \(R\) to line \(PQ\). Given that \(PQ = 8\) and the area of triangle \(PQR\) is \(12\) square units, we can set up the equation:
\[
12 = \frac{1}{2} \cdot 8 \cdot h_R \implies h_R = 3.
\]
2. **Setting Coordinates for \(P\) and \(Q\)**: Assume without loss of generality that \(P = (-4, 0)\) and \(Q = (4, 0)\). This places \(PQ\) along the x-axis with a length of 8 units.
3. **Determining \(y\)-coordinate of \(R\)**: Since \(h_R = 3\), the \(y\)-coordinate of \(R\) must be either \(3\) or \(-3\) (above or below the x-axis).
4. **Casework on the Location of Right Angle**:
- **Case 1: \(\angle P = 90^\circ\)**:
- \(R\) must be vertically aligned with \(P\), so \(R = (-4, \pm 3)\).
- There are **2 locations** for \(R\) in this case.
- **Case 2: \(\angle Q = 90^\circ\)**:
- \(R\) must be vertically aligned with \(Q\), so \(R = (4, \pm 3)\).
- There are **2 locations** for \(R\) in this case.
- **Case 3: \(\angle R = 90^\circ\)**:
- We need to find \(x\) such that the distance from \(P\) and \(Q\) to \(R\) forms a right triangle. Using the Pythagorean theorem:
\[
\left[(x+4)^2 + 3^2\right] + \left[(x-4)^2 + 3^2\right] = 8^2.
\]
Simplifying, we find:
\[
(x+4)^2 + 9 + (x-4)^2 + 9 = 64 \implies 2x^2 + 18 = 64 \implies x^2 = 23 \implies x = \pm \sqrt{23}.
\]
- Thus, \(R = (\pm \sqrt{23}, 3)\) and \(R = (\pm \sqrt{23}, -3)\).
- There are **4 locations** for \(R\) in this case.
5. **Total Locations for \(R\)**: Adding the locations from each case, we have \(2 + 2 + 4 = 8\) possible locations for \(R\).
Thus, the total number of locations for point \(R\) such that triangle \(PQR\) is a right triangle with area \(12\) square units is \(\boxed{8}\).
|
8
|
deepscale
| 304
| |
Simplify: $\sqrt{50} + \sqrt{18}$ . Express your answer in simplest radical form.
|
8\sqrt{2}
|
deepscale
| 39,155
| ||
A stock investment increased by 30% in 2006. Starting at this new value, what percentage decrease is needed in 2007 to return the stock to its original price at the beginning of 2006?
|
23.077\%
|
deepscale
| 17,993
| ||
You are the general of an army. You and the opposing general both have an equal number of troops to distribute among three battlefields. Whoever has more troops on a battlefield always wins (you win ties). An order is an ordered triple of non-negative real numbers $(x, y, z)$ such that $x+y+z=1$, and corresponds to sending a fraction $x$ of the troops to the first field, $y$ to the second, and $z$ to the third. Suppose that you give the order $\left(\frac{1}{4}, \frac{1}{4}, \frac{1}{2}\right)$ and that the other general issues an order chosen uniformly at random from all possible orders. What is the probability that you win two out of the three battles?
|
Let $x$ be the portion of soldiers the opposing general sends to the first battlefield, and $y$ the portion he sends to the second. Then $1-x-y$ is the portion he sends to the third. Then $x \geq 0$, $y \geq 0$, and $x+y \leq 1$. Furthermore, you win if one of the three conditions is satisfied: $x \leq \frac{1}{4}$ and $y \leq \frac{1}{4}, x \leq \frac{1}{4}$ and $1-x-y \leq \frac{1}{2}$, or $y \leq \frac{1}{4}$ and $1-x-y \leq \frac{1}{2}$. This is illustrated in the picture below. This triangle is a linear projection of the region of feasible orders, so it preserves area and probability ratios. The probability that you win, then is given by the portion of the triangle that satisfies one of the three above constraints - in other words, the area of the shaded region divided by the area of the entire triangle. We can easily calculate this to be $\frac{\frac{5}{16}}{\frac{1}{2}}=\frac{5}{8}$.
|
\sqrt[5]{8}
|
deepscale
| 4,903
| |
Find the smallest six-digit number that is divisible by 11, where the sum of the first and fourth digits is equal to the sum of the second and fifth digits, and equal to the sum of the third and sixth digits.
|
100122
|
deepscale
| 15,759
| ||
Consider the sequence of numbers defined recursively by $t_1=1$ and for $n>1$ by $t_n=1+t_{n/2}$ when $n$ is even and by $t_n=\frac{1}{t_{n-1}}$ when $n$ is odd. Given that $t_n=\frac{19}{87}$, find $n.$
|
1905
|
deepscale
| 36,896
| ||
Define a function \( f \), whose domain is positive integers, such that:
$$
f(n)=\begin{cases}
n-3 & \text{if } n \geq 1000 \\
f(f(n+7)) & \text{if } n < 1000
\end{cases}
$$
Find \( f(90) \).
|
999
|
deepscale
| 13,800
| ||
In the geometric sequence $\{a_n\}$, it is given that $a_1 + a_4 + a_7 = 2$, and $a_3 + a_6 + a_9 = 18$. Find the sum of the first 9 terms, $S_9$, of the sequence $\{a_n\}$.
|
26
|
deepscale
| 24,464
| ||
If $3+a=4-b$ and $4+b=7+a$, what is $3-a$?
|
4
|
deepscale
| 33,607
| ||
Two circles have the same center $C.$ (Circles which have the same center are called concentric.) The larger circle has radius $10$ and the smaller circle has radius $6.$ Determine the area of the ring between these two circles. [asy]
import graph;
filldraw(circle((0,0),10), lightgray, black+linewidth(1));
filldraw(circle((0,0),6), white, black+linewidth(1));
dot((0,0));
label("$C$",(0,0),NE);
[/asy]
|
64\pi
|
deepscale
| 38,444
| ||
A freight train was delayed on its route for 12 minutes. Then, over a distance of 60 km, it made up for the lost time by increasing its speed by 15 km/h. Find the original speed of the train.
|
39.375
|
deepscale
| 12,621
| ||
If \( x \) is a real number and \( \lceil x \rceil = 14 \), how many possible values are there for \( \lceil x^2 \rceil \)?
|
27
|
deepscale
| 17,601
| ||
The sequence $\{a_n\}$ satisfies $a_{n+1}=(2|\sin \frac{n\pi}{2}|-1)a_{n}+n$, then the sum of the first $100$ terms of the sequence $\{a_n\}$ is __________.
|
2550
|
deepscale
| 16,495
|
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