Unnamed: 0 int64 0 40.3k | problem stringlengths 10 5.15k | ground_truth stringlengths 1 1.22k | solved_percentage float64 0 100 |
|---|---|---|---|
16,200 |
A general gathers his troops. When he arranges them in groups of 2, one soldier is left over. When he arranges them in groups of 3, two soldiers are left over. When he arranges them in groups of 5, three soldiers are left over. If the general arranges his soldiers in groups of 30, how many soldiers will be left over? | 23 | 82.03125 |
16,201 | In the Cartesian coordinate system $xOy$, the parametric equations of curve $C_{1}$ are $\left\{\begin{array}{l}{x=2+2\cos\varphi}\\{y=2\sin\varphi}\end{array}\right.$ ($\varphi$ is the parameter). Taking the origin $O$ as the pole and the positive half of the $x$-axis as the polar axis, the polar equation of curve $C_... | \frac{3\pi}{4} | 56.25 |
16,202 | You are given a number composed of three different non-zero digits, 7, 8, and a third digit which is not 7 or 8. Find the minimum value of the quotient of this number divided by the sum of its digits. | 11.125 | 0.78125 |
16,203 | Given $tanA=\frac{2}{3}$, find the value of $\cos A$. | \frac{3\sqrt{13}}{13} | 36.71875 |
16,204 | Let $\triangle ABC$ be a triangle with a right angle $\angle ABC$ . Let $D$ be the midpoint of $\overline{BC}$ , let $E$ be the midpoint of $\overline{AC}$ , and let $F$ be the midpoint of $\overline{AB}$ . Let $G$ be the midpoint of $\overline{EC}$ . One of the angles of $\triangle DFG$ is a right ... | 2/3 | 5.46875 |
16,205 | Given that the terminal side of angle $α$ passes through point $P(\frac{4}{5},-\frac{3}{5})$,
(1) Find the value of $\sin α$;
(2) Find the value of $\frac{\sin (\frac{π}{2}-α)}{\sin (α+π)}-\frac{\tan (α-π)}{\cos (3π-α)}$. | \frac{19}{48} | 68.75 |
16,206 | A clock currently shows the time $10:10$ . The obtuse angle between the hands measures $x$ degrees. What is the next time that the angle between the hands will be $x$ degrees? Round your answer to the nearest minute. | 11:15 | 7.03125 |
16,207 | In the geometric sequence ${a_n}$, $a_3$ and $a_{15}$ are the roots of the equation $x^2 + 6x + 2 = 0$, calculate the value of $$\frac{a_{2}a_{16}}{a_{9}}.$$ | \sqrt{2} | 47.65625 |
16,208 | 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... | 1080 | 83.59375 |
16,209 | Three balls are randomly placed into three boxes. Let the random variable $\xi$ denote the maximum number of balls in any one box. Determine the mathematical expectation $E(\xi)$ of $\xi$. | \frac{17}{9} | 35.15625 |
16,210 | A natural number \( 1 \leq n \leq 221 \) is called lucky if, when dividing 221 by \( n \), the remainder is wholly divisible by the incomplete quotient (the remainder can be equal to 0). How many lucky numbers are there? | 115 | 24.21875 |
16,211 | Let \( A = (2, 0) \) and \( B = (8, 6) \). Let \( P \) be a point on the circle \( x^2 + y^2 = 8x \). Find the smallest possible value of \( AP + BP \). | 6\sqrt{2} | 5.46875 |
16,212 | In the rectangular coordinate system $(xOy)$, if the initial side of angle $\alpha$ is the non-negative semi-axis of $x$, and the terminal side is the ray $l$: $y=2x(x\leqslant 0)$.
(1) Find the value of $\tan \alpha$;
(2) Find the value of $\frac{\cos \left(\alpha-\pi\right)-2\cos \left( \frac{\pi}{2}+\alpha\right)}... | -3 | 59.375 |
16,213 | Compute the length of the segment tangent from the point $(1,1)$ to the circle that passes through the points $(4,5),$ $(7,9),$ and $(6,14).$ | 5\sqrt{2} | 5.46875 |
16,214 | Given complex numbers ${z_1}=2i$, ${z_2}=1-i$, where $i$ is the imaginary unit,
(1) Find the conjugate of the complex number $\frac{z_1}{z_2}$;
(2) In the complex plane, let points $Z_1$, $Z_2$ correspond to ${z_1}$, ${z_2}$ respectively, and $O$ be the origin. Form a parallelogram with $\overrightarrow{OZ_1}$, $\ove... | \sqrt{10} | 69.53125 |
16,215 | The area of rectangle PRTV is divided into four rectangles, PQXW, QRSX, XSTU, and WXUV. Given that the area of PQXW is 9, the area of QRSX is 10, and the area of XSTU is 15, find the area of rectangle WXUV. | \frac{27}{2} | 2.34375 |
16,216 | Given that point $P$ is a moving point on the parabola $y=\frac{1}{4}x^2$, determine the minimum value of the sum of the distance from point $P$ to the line $x+2y+4=0$ and the $x$-axis. | \frac{6\sqrt{5}}{5}-1 | 7.8125 |
16,217 | A wholesaler purchased $50$ packs of shirts of size $L$ from a clothing manufacturer. Due to the negligence of the packaging workers, some packs were mixed with shirts of size $M$. The number of $M$ shirts mixed in (pieces) and the corresponding number of packs (packs) are shown in the table below:
|M shirts (pieces)|... | \frac{3}{25} | 46.09375 |
16,218 | Let $\{b_k\}$ be a sequence of integers such that $b_1 = 2$ and $b_{m+n} = b_m + b_n + mn + 1$, for all positive integers $m$ and $n$. Find $b_{12}$. | 101 | 25 |
16,219 | Given that in triangle $\triangle ABC$, the sides opposite to the internal angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. Angle $B$ is obtuse. Let the area of $\triangle ABC$ be $S$. If $4bS=a(b^{2}+c^{2}-a^{2})$, then the maximum value of $\sin A + \sin C$ is ____. | \frac{9}{8} | 39.84375 |
16,220 | Given $a_{1}+a_{2}=1$, $a_{2}+a_{3}=2$, $a_{3}+a_{4}=-3$, $a_{4}+a_{5}=-4$, $a_{5}+a_{6}=5$, $a_{6}+a_{7}=6$, $a_{7}+a_{8}=-7$, $a_{8}+a_{9}=-8$, $\ldots $, $a_{99}+a_{100}=-99$, $a_{100}+a_{1}=-100$, calculate the value of $a_{1}+a_{2}+a_{3}+\ldots +a_{100}$. | -50 | 79.6875 |
16,221 | Let $P$ be the least common multiple of all integers from $10$ to $40$, inclusive. Let $Q$ be the least common multiple of $P$ and the integers $41, 42, 43, 44, 45, 46, 47, 48, 49,$ and $50$. What is the value of $\frac{Q}{P}?$
A) 1763
B) 82861
C) 9261
D) 14756 | 82861 | 13.28125 |
16,222 | Calculate \(\frac{2}{3} \cdot \frac{4}{7} \cdot \frac{5}{9} \cdot \frac{11}{13}\). | \frac{440}{2457} | 77.34375 |
16,223 | 13. Given that $a$, $b$, $c$, are the lengths of the sides opposite to angles $A$, $B$, $C$ in $\triangle ABC$ respectively, with $a=2$, and $(2+b)(\sin A-\sin B)=(c-b)\sin C$, find the maximum area of $\triangle ABC$. | \sqrt{3} | 1.5625 |
16,224 | Given $sinα-cosα=\frac{1}{5},0≤α≤π$, calculate $sin(2α-\frac{π}{4})$. | \frac{31\sqrt{2}}{50} | 57.8125 |
16,225 | Find the integer $n$ such that $-150 < n < 150$ and $\tan n^\circ = \tan 286^\circ$. | -74 | 27.34375 |
16,226 | Given that $α∈\left( \frac{π}{2},π\right) $, and $\sin \left(π-α\right)+\cos \left(2π+α\right)= \frac{ \sqrt{2}}{3} $. Find the values of:
$(1)\sin {α} -\cos {α} .$
$(2)\tan {α} $. | - \frac{9+4 \sqrt{2}}{7} | 81.25 |
16,227 | Given sets $A=\{1, a, b\}$ and $B=\{a, a^2, ab\}$. If $A=B$, find the value of $a+b$. | -1 | 52.34375 |
16,228 | In $\triangle ABC$, points $D$ and $E$ lie on $\overline{BC}$ and $\overline{AC}$ respectively. Lines $\overline{AD}$ and $\overline{BE}$ intersect at point $T$ such that $AT/DT=2$ and $BT/ET=3$. Determine the ratio $CD/BD$. | \frac{3}{5} | 17.1875 |
16,229 | In the Cartesian coordinate system $xOy$, the parametric equations of curve $C$ are given by $$\begin{cases} x=2\cos\theta \\ y=\sqrt{3}\sin\theta \end{cases}$$ (where $\theta$ is the parameter). With the origin $O$ as the pole and the positive $x$-axis as the polar axis, establish a polar coordinate system. Point $P$ ... | \frac{24}{7} | 73.4375 |
16,230 | Let point G be the centroid of triangle ABC. If $\angle A=120^\circ$ and $\overrightarrow {AB} \cdot \overrightarrow {AC}=-1$, find the minimum value of $|\overrightarrow {AG}|$. | \frac{\sqrt{2}}{3} | 71.875 |
16,231 | Consider a chess board, with the numbers $1$ through $64$ placed in the squares as in the diagram below.
\[\begin{tabular}{| c | c | c | c | c | c | c | c |}
\hline
1 & 2 & 3 & 4 & 5 & 6 & 7 & 8
\hline
9 & 10 & 11 & 12 & 13 & 14 & 15 & 16
\hline
17 & 18 & 19 & 20 & 21 & 22 & 23 & 24
\hline
25 & 26 & 27 & 28 & 2... | 1056 | 18.75 |
16,232 | Given the hyperbola $\dfrac {x^{2}}{a^{2}}- \dfrac {y^{2}}{b^{2}}=1$ ($a > 0$, $b > 0$), its left and right vertices are $A$ and $B$, respectively. The right focus is $F$, and the line $l$ passing through point $F$ and perpendicular to the $x$-axis intersects the hyperbola at points $M$ and $N$. $P$ is a point on line ... | \sqrt{2} | 84.375 |
16,233 | Find the set of values of the parameter \(a\) for which the sum of the cubes of the roots of the equation \(x^{2}-a x+a+2=0\) is equal to -8. | -2 | 0.78125 |
16,234 | Given the function $f(x)=x^{2}+ax+4$, if for any $x \in (0,2]$, $f(x) \leqslant 6$ always holds, then find the maximum value of the real number $a$. | -1 | 96.875 |
16,235 | Let \( x, y, z \) be complex numbers such that
\[
xy + 5y = -25, \\
yz + 5z = -25, \\
zx + 5x = -25.
\]
Find all possible values of \( xyz \). | 125 | 60.9375 |
16,236 | In triangle $XYZ$, $\angle Y = 90^\circ$, $YZ = 4$, and $XY = \sqrt{34}$. What is $\tan X$? | \frac{2\sqrt{2}}{3} | 82.03125 |
16,237 | Given the sequence of numbers with only even digits in their decimal representation, determine the $2014^\text{th}$ number in the sequence. | 62048 | 0 |
16,238 | If $g(x) = \frac{x - 3}{x^2 + cx + d}$, and $g(x)$ has vertical asymptotes at $x = 2$ and $x = -1$, find the sum of $c$ and $d$. | -3 | 81.25 |
16,239 | Given that the perimeter of each of the nine small equilateral triangles is $6 \mathrm{cm}$, calculate the perimeter of $\triangle A B C$. | 18 | 83.59375 |
16,240 | Let $\mathcal{P}$ be the parabola given by the equation \( y = x^2 \). Suppose a circle $\mathcal{C}$ intersects $\mathcal{P}$ at four distinct points. If three of these points are \((-4,16)\), \((1,1)\), and \((6,36)\), find the sum of the distances from the directrix of the parabola to all four intersection points. | 63 | 58.59375 |
16,241 | How many distinct arrangements of the letters in the word "balloon" are there? | 1260 | 46.875 |
16,242 | Let $a$ , $b$ , $c$ be positive reals for which
\begin{align*}
(a+b)(a+c) &= bc + 2
(b+c)(b+a) &= ca + 5
(c+a)(c+b) &= ab + 9
\end{align*}
If $abc = \frac{m}{n}$ for relatively prime positive integers $m$ and $n$ , compute $100m+n$ .
*Proposed by Evan Chen* | 4532 | 50 |
16,243 | Find the volume of a cylinder formed by rotating a square with side length 10 centimeters about its horizontal line of symmetry. Express your answer in terms of $\pi$. | 250\pi | 96.09375 |
16,244 | What is the smallest positive value of $x$ such that $x + 4321$ results in a palindrome? | 13 | 85.15625 |
16,245 | If $\left( r + \frac{1}{r} \right)^2 = 5,$ then find $r^3 + \frac{1}{r^3}.$ | 2\sqrt{5} | 57.03125 |
16,246 | The left and right foci of a hyperbola are $F_{1}$ and $F_{2}$, respectively. A line passing through $F_{2}$ intersects the right branch of the hyperbola at points $A$ and $B$. If $\triangle F_{1} A B$ is an equilateral triangle, what is the eccentricity of the hyperbola? | \sqrt{3} | 51.5625 |
16,247 | A point $(x,y)$ is randomly picked from inside the rectangle with vertices $(0,0)$, $(2,0)$, $(2,2)$, and $(0,2)$. What is the probability that $x^2 + y^2 < y$? | \frac{\pi}{32} | 0.78125 |
16,248 | Find the value of $f(2017)$ for a function $f(x)$ defined on $\mathbb{R}$ that satisfies $f(x) \cdot f(x+2) = 13$ given that $f(3) = 4$. | \frac{13}{4} | 91.40625 |
16,249 | A right triangle is inscribed in the ellipse given by the equation $x^2 + 9y^2 = 9$. One vertex of the triangle is at the point $(0,1)$, and one leg of the triangle is fully contained within the x-axis. Find the squared length of the hypotenuse of the inscribed right triangle, expressed as the ratio $\frac{m}{n}$ with ... | 11 | 28.90625 |
16,250 | Chewbacca has 25 pieces of orange gum and 35 pieces of apple gum. Some of the pieces are in complete packs, while others are loose. Each complete pack has exactly $y$ pieces of gum. If Chewbacca loses two packs of orange gum, then the ratio of the number of pieces of orange gum he has to the number of pieces of apple g... | \frac{15}{4} | 0 |
16,251 | A right circular cone and a sphere possess the same radius, $r$. If the volume of the cone is one-third of the volume of the sphere, determine the ratio of the height of the cone to the radius $r$. | \frac{4}{3} | 88.28125 |
16,252 | Five consecutive two-digit positive integers, each less than 50, are not prime. What is the largest of these five integers? | 36 | 0 |
16,253 | Two diagonals of a regular decagon (a 10-sided polygon) are chosen. What is the probability that their intersection lies inside the decagon and forms a convex quadrilateral? | \dfrac{42}{119} | 0 |
16,254 | Mr. Ambulando is at the intersection of $5^{\text{th}}$ and $\text{A St}$ , and needs to walk to the intersection of $1^{\text{st}}$ and $\text{F St}$ . There's an accident at the intersection of $4^{\text{th}}$ and $\text{B St}$ , which he'd like to avoid.
[center]<see attached>[/center]
Given that Mr. Ambu... | 56 | 60.15625 |
16,255 | Given that $\overset{⇀}{m}=(2,1)$, $\overset{⇀}{n}=(\sin θ,\cos θ)$, where $θ∈(0, \dfrac{π}{2})$ is the inclination angle of line $l$ passing through point $A(1,4)$, if $\overset{⇀}{m}· \overset{⇀}{n}$ is at its maximum when line $l$ is tangent to the circle $(x+1)^{2}+(y-2)^{2}={r}^{2}(r > 0)$, then $r=$ . | \dfrac{2 \sqrt{5}}{5} | 91.40625 |
16,256 | In the convex quadrilateral $ABCD$ angle $\angle{BAD}=90$ , $\angle{BAC}=2\cdot\angle{BDC}$ and $\angle{DBA}+\angle{DCB}=180$ . Then find the angle $\angle{DBA}$ | 45 | 10.15625 |
16,257 | Given non-negative real numbers $a$, $b$, $c$ satisfy $\frac{a-1}{2}=\frac{b-2}{3}=\frac{3-c}{4}$, let the maximum value of $S=a+2b+c$ be $m$, and the minimum value be $n$. Then the value of $\frac{n}{m}$ is ______. | \frac{6}{11} | 78.90625 |
16,258 | Insert a digit in the middle of a two-digit number to form a three-digit number. For some two-digit numbers, the resulting three-digit number can be $k$ times the original two-digit number (where $k$ is a positive integer). What is the maximum value of $k$? | 19 | 30.46875 |
16,259 | In triangle $ΔABC$, $BC=a$, $AC=b$, where $a$ and $b$ are the two roots of the equation $x^2-2\sqrt{3}x+2=0$, and $2\cos(A+B)=1$.
(1) Find the angle $C$;
(2) Find the length of $AB$. | \sqrt{10} | 84.375 |
16,260 | I am dining at a Mexican restaurant with a friend who is vegan and allergic to nuts. The restaurant menu lists 8 dishes that are vegan. These vegan options constitute one-fourth of the entire menu. However, 5 of these vegan dishes contain nuts. What fraction of the menu can my friend eat? | \frac{3}{32} | 84.375 |
16,261 | Points $E$ and $F$ lie on $\overline{GH}$. The length of $\overline{GE}$ is $3$ times the length of $\overline{EH}$, and the length of $\overline{GF}$ is $5$ times the length of $\overline{FH}$. Determine the length of $\overline{EF}$ as a fraction of the length of $\overline{GH}$.
A) $\frac{1}{10}$
B) $\frac{1}{12}$... | \frac{1}{12} | 92.1875 |
16,262 | A circle with a radius of 3 units has its center at $(0, 0)$. A circle with a radius of 5 units has its center at $(12, 0)$. A line tangent to both circles intersects the $x$-axis at $(x, 0)$ to the right of the origin. What is the value of $x$? Express your answer as a common fraction. | \frac{9}{2} | 64.0625 |
16,263 | At a hypothetical school, there are three departments in the faculty of sciences: biology, physics and chemistry. Each department has three male and one female professor. A committee of six professors is to be formed containing three men and three women, and each department must be represented by two of its members. Ev... | 27 | 5.46875 |
16,264 | The volume of the parallelepiped determined by vectors $\mathbf{a}$, $\mathbf{b}$, and $\mathbf{c}$ is 6. Find the volume of the parallelepiped determined by $\mathbf{a} + 2\mathbf{b},$ $\mathbf{b} + \mathbf{c},$ and $2\mathbf{c} - 5\mathbf{a}.$ | 48 | 38.28125 |
16,265 | For a finite sequence \(P = \left(p_1, p_2, \cdots, p_n\right)\), the Cesaro sum (named after the mathematician Cesaro) is defined as \(\frac{1}{n}(S_1 + S_2 + \cdots + S_n)\), where \(S_k = p_1 + p_2 + \cdots + p_k\) for \(1 \leq k \leq n\). If a sequence \(\left(p_1, p_2, \cdots, p_{99}\right)\) of 99 terms has a Ces... | 991 | 86.71875 |
16,266 | Find the number of solutions to
\[\sin x = \left( \frac{1}{3} \right)^x\] on the interval $(0,150 \pi).$ | 75 | 17.96875 |
16,267 | Given that $F_1$ and $F_2$ are the two foci of the hyperbola $x^2 - \frac{y^2}{24} = 1$, and $P$ is a common point of the hyperbola and the ellipse $\frac{x^2}{49} + \frac{y^2}{24} = 1$, find the area of the triangle $PF_1F_2$. | 24 | 70.3125 |
16,268 | A square with a side length of 2 rotates around one of its sides, which is the axis of rotation. What is the volume of the cylinder obtained from this rotation? | 8\pi | 80.46875 |
16,269 | Convert the quadratic equation $3x=x^{2}-2$ into general form and determine the coefficients of the quadratic term, linear term, and constant term. | -2 | 11.71875 |
16,270 | Given that $\cos \alpha = -\frac{4}{5}$, and $\alpha$ is an angle in the third quadrant, find the values of $\sin \alpha$ and $\tan \alpha$. | \frac{3}{4} | 22.65625 |
16,271 | When three standard dice are tossed, the numbers $a, b, c$ are obtained. Find the probability that the product of these three numbers, $abc$, equals 8. | \frac{7}{216} | 33.59375 |
16,272 | (1) Consider the function $f(x) = |x - \frac{5}{2}| + |x - a|$, where $x \in \mathbb{R}$. If the inequality $f(x) \geq a$ holds true for all $x \in \mathbb{R}$, find the maximum value of the real number $a$.
(2) Given positive numbers $x$, $y$, and $z$ satisfying $x + 2y + 3z = 1$, find the minimum value of $\frac{3}{x... | 16 + 8\sqrt{3} | 9.375 |
16,273 | Let \( N \) be the smallest positive integer whose digits have a product of 2000. The sum of the digits of \( N \) is | 25 | 55.46875 |
16,274 | A basketball is dropped from 150 feet and rebounds two-fifths of the distance it falls each time it bounces. How many feet will the basketball have traveled when it hits the ground the sixth time? | 347.952 | 0.78125 |
16,275 | The greatest common divisor of two positive integers is $(x+6)$ and their least common multiple is $x(x+6)$, where $x$ is a positive integer. If one of the integers is 36, what is the smallest possible value of the other one? | 24 | 64.0625 |
16,276 | Determine the fourth-largest divisor of $1,234,560,000$. | 154,320,000 | 0 |
16,277 | Given triangle $\triangle ABC$, the sides opposite angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively, and the area is $S$. It is given that $6S=a^{2}\sin A+b^{2}\sin B$. Express $\cos C$ in terms of $a$, $b$, and $c$ when the maximum value of $\frac{{a+b}}{c}$ is achieved. | \frac{7}{9} | 73.4375 |
16,278 | What is the area of the polygon whose vertices are the points of intersection of the curves $x^2 + y^2 = 16$ and $(x-5)^2 + 4y^2 = 64$?
A) $\frac{5\sqrt{110}}{6}$
B) $\frac{5\sqrt{119}}{6}$
C) $\frac{10\sqrt{119}}{6}$
D) $\frac{5\sqrt{125}}{6}$ | \frac{5\sqrt{119}}{6} | 15.625 |
16,279 | If the polynomial $x^{2}+x^{10}=a_{0}+a_{1}(x+1)+\cdots+a_{9}(x+1)^{9}+a_{10}(x+1)^{10}$, find the value of $a_{9}$. | -10 | 80.46875 |
16,280 | Given $x^3y = k$ for a positive constant $k$, find the percentage decrease in $y$ when $x$ increases by $20\%$. | 42.13\% | 30.46875 |
16,281 | For a positive integer $n$ , let $v(n)$ denote the largest integer $j$ such that $n$ is divisible by $2^j$ . Let $a$ and $b$ be chosen uniformly and independently at random from among the integers between 1 and 32, inclusive. What is the probability that $v(a) > v(b)$ ? | 341/1024 | 20.3125 |
16,282 | Given $a_1 + a_2 = 1$, $a_2 + a_3 = 2$, $a_3 + a_4 = 3$, ..., $a_{99} + a_{100} = 99$, $a_{100} + a_1 = 100$, find the value of $a_1 + a_2 + a_3 + \ldots + a_{100}$. | 2525 | 100 |
16,283 | The minimum distance from a point on the parabola $y=x^2$ to the line $2x-y-10=0$ is what? | \frac{9\sqrt{5}}{5} | 96.875 |
16,284 | Given a circle C with its center C on the positive x-axis and a radius of 5, the chord intercepted by the line $x-y+3=0$ has a length of $2\sqrt{17}$.
(1) Find the equation of circle C;
(2) Suppose the line $ax-y+5=0$ intersects circle C at points A and B, find the range of the real number $a$;
(3) Under the condition ... | \frac{3}{4} | 60.15625 |
16,285 | Given the geometric sequence $(-1, x, y, z, -2)$, find the value of $xyz$. | -2\sqrt{2} | 64.84375 |
16,286 | In $\triangle ABC$, $a$, $b$, $c$ are the sides opposite to angles $A$, $B$, $C$ respectively, and $b\sin A=(3b-c)\sin B$
(1) If $2\sin A=3\sin B$, and the perimeter of $\triangle ABC$ is $8$, find $c$
(2) If $\triangle ABC$ is an isosceles triangle, find $\cos 2B$. | \frac{17}{81} | 15.625 |
16,287 | An artist wants to completely cover a rectangle with identically sized squares which do not overlap and do not extend beyond the edges of the rectangle. If the rectangle is \(60 \frac{1}{2} \mathrm{~cm}\) long and \(47 \frac{2}{3} \mathrm{~cm}\) wide, what is the minimum number of squares required? | 858 | 70.3125 |
16,288 | The sum of the coefficients of the expanded form of $(x+ \frac{a}{x})(2x- \frac{1}{x})^{5}$ is 2. Find the constant term in the expanded form. | 40 | 50.78125 |
16,289 | Given that points $P$ and $Q$ are moving points on the curve $y=xe^{-2x}$ and the line $y=x+2$ respectively, find the minimum distance between points $P$ and $Q$. | \sqrt{2} | 22.65625 |
16,290 | 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} | 17.96875 |
16,291 | A quartic (4th degree) polynomial \( p(x) \) satisfies:
\[ p(n) = \frac{1}{n^2} \] for \( n = 1, 2, 3, 4, \) and \( 5 \). Find \( p(6) \). | \frac{1}{18} | 7.03125 |
16,292 | A $150$-gon $Q_1$ is drawn in the Cartesian plane. The sum of the $x$-coordinates of the $150$ vertices equals $3000$. The midpoints of the sides of $Q_1$ form a second $150$-gon, $Q_2$. Finally, the midpoints of the sides of $Q_2$ form a third $150$-gon, $Q_3$. Find the sum of the $x$-coordinates of the vertices of $Q... | 3000 | 95.3125 |
16,293 | Juan wants to calculate the area of a large circular garden, whose actual diameter is 50 meters. Unfortunately, his measurement device has an accuracy error of up to 30%. Compute the largest possible percent error, in percent, in Juan’s computed area of the circle in square meters. | 69\% | 71.875 |
16,294 | For natural numbers $m$ greater than or equal to 2, the decomposition of their cube powers can be represented as follows: $2^3 = 3 + 5$, $3^3 = 7 + 9 + 11$, $4^3 = 13 + 15 + 17 + 19$. Then,
(1) The smallest number in the decomposition of $8^3$ is;
(2) Following the above pattern, the $n$-th equation can be represen... | 57 | 28.90625 |
16,295 | If $f(x)$ is a function defined on $R$, and $f(x) - x^2$ is an odd function, and $f(x) + 2^x$ is an even function, then the minimum value of $f(x)$ on the interval $\left[-2,-1\right]$ is ______. | \frac{7}{4} | 42.1875 |
16,296 | Given a sequence of positive terms $\{a\_n\}$, where $a\_2=6$, and $\frac{1}{a\_1+1}$, $\frac{1}{a\_2+2}$, $\frac{1}{a\_3+3}$ form an arithmetic sequence, find the minimum value of $a\_1a\_3$. | 19+8\sqrt{3} | 11.71875 |
16,297 | Masha and the bear ate a basket of raspberries and 40 pies, starting and finishing at the same time. Initially, Masha ate raspberries while the bear ate pies, and then (at some moment) they switched. The bear ate both raspberries and pies 3 times faster than Masha. How many pies did Masha eat if they ate an equal amoun... | 10 | 42.1875 |
16,298 | If $x \cdot \log_{27} 64 = 1$, then $4^x + 4^{-x} =$ \_\_\_\_\_\_\_\_. | \dfrac{10}{3} | 83.59375 |
16,299 | Let \\(\alpha\\) be an acute angle, and \\(\cos (\alpha+ \frac {\pi}{6})= \frac {3}{5}\\).
\\((1)\\) Find the value of \\(\cos (\alpha- \frac {\pi}{3})\\);
\\((2)\\) Find the value of \\(\cos (2\alpha- \frac {\pi}{6})\\). | \frac {24}{25} | 46.09375 |
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