Unnamed: 0 int64 0 40.3k | problem stringlengths 10 5.15k | ground_truth stringlengths 1 1.22k | solved_percentage float64 0 100 |
|---|---|---|---|
16,700 | Calculate the sum $\frac{3}{50} + \frac{5}{500} + \frac{7}{5000}$.
A) $0.0714$
B) $0.00714$
C) $0.714$
D) $0.0357$
E) $0.00143$ | 0.0714 | 96.875 |
16,701 | Six positive integers are written on the faces of a cube. Each vertex is labeled with the product of the numbers on the three faces adjacent to that vertex. If the sum of the numbers on the vertices is $1512$, and the sum of the numbers on one pair of opposite faces is $8$, what is the sum of the numbers on all the fac... | 38 | 21.09375 |
16,702 | The positive integers $A, B$, and $C$ form an arithmetic sequence, while the integers $B, C$, and $D$ form a geometric sequence. If $\frac{C}{B} = \frac{7}{3},$ what is the smallest possible value of $A + B + C + D$? | 76 | 17.1875 |
16,703 | If four distinct positive integers $m$, $n$, $p$, $q$ satisfy $(6-m)(6-n)(6-p)(6-q)=4$, then $m+n+p+q=$ ? | 24 | 94.53125 |
16,704 | Given $f(x+1) = x^2 - 1$,
(1) Find $f(x)$.
(2) Find the maximum or minimum value of $f(x)$ and the corresponding value of $x$. | -1 | 12.5 |
16,705 | Let \( S = \{8^k : k \text{ is an integer}, 0 \le k \le 3000\} \). Given that \( 8^{3000} \) has 2712 digits and that its first (leftmost) digit is 8, how many elements of \( S \) have 8 as their leftmost digit? | 153 | 11.71875 |
16,706 | If real numbers \( x \) and \( y \) satisfy \( x^{3} + y^{3} + 3xy = 1 \), then the minimum value of \( x^{2} + y^{2} \) is ____. | \frac{1}{2} | 33.59375 |
16,707 | Given a sequence $\{a_n\}$ satisfying $a_1=0$, for any $k\in N^*$, $a_{2k-1}$, $a_{2k}$, $a_{2k+1}$ form an arithmetic sequence with a common difference of $k$. If $b_n= \dfrac {(2n+1)^{2}}{a_{2n+1}}$, calculate the sum of the first $10$ terms of the sequence $\{b_n\}$. | \dfrac {450}{11} | 13.28125 |
16,708 | The graph of the function $f(x)=\cos(\omega x+\frac{π}{4})$ $(\omega>0)$ is transformed to an odd function by shifting it to the left by $\frac{π}{3}$ units. Determine the minimum value of the real number $\omega$. | \frac{3}{4} | 71.09375 |
16,709 | Given the function $f(x)=\frac{ax^{2}+bx+c}{e^{x}} (a > 0)$ whose derivative $y=f′(x)$ has two zeros at $-3$ and $0$.
1. Find the monotonic intervals of $f(x)$;
2. If the minimum value of $f(x)$ is $-e^{3}$, find the maximum value of $f(x)$ on the interval $[-5,+\infty)$. | 5e^{5} | 0 |
16,710 | A rectangular floor is covered with congruent square tiles. If the total number of tiles that lie on the two diagonals is 57, how many tiles cover the floor. | 841 | 27.34375 |
16,711 | Starting with $10,000,000$, Esha forms a sequence by alternatively dividing by 2 and multiplying by 3. If she continues this process, what is the form of her sequence after 8 steps? Express your answer in the form $a^b$, where $a$ and $b$ are integers and $a$ is as small as possible. | (2^3)(3^4)(5^7) | 0 |
16,712 | In the Cartesian coordinate system \(xOy\), there is a point \(P(0, \sqrt{3})\) and a line \(l\) with the parametric equations \(\begin{cases} x = \dfrac{1}{2}t \\ y = \sqrt{3} + \dfrac{\sqrt{3}}{2}t \end{cases}\) (where \(t\) is the parameter). Using the origin as the pole and the non-negative half-axis of \(x\) to es... | \sqrt{14} | 20.3125 |
16,713 | Among the following propositions, the true one is __________
(1) In a plane, the locus of points whose sum of distances from two fixed points $F_{1}$ and $F_{2}$ is a constant is an ellipse;
(2) If vectors $\overrightarrow{e_{1}}$, $\overrightarrow{e_{2}}$, $\overrightarrow{e_{3}}$ are three non-collinear vectors... | (3) | 0 |
16,714 | Two circles \( C_{1} \) and \( C_{2} \) have their centers at the point \( (3, 4) \) and touch a third circle, \( C_{3} \). The center of \( C_{3} \) is at the point \( (0, 0) \) and its radius is 2. What is the sum of the radii of the two circles \( C_{1} \) and \( C_{2} \)? | 10 | 4.6875 |
16,715 | Given that the terminal side of angle $\alpha$ passes through point P $\left(\frac{4}{5}, -\frac{3}{5}\right)$.
(1) Find the value of $\sin\alpha$.
(2) Calculate the value of $\frac{\sin\left(\frac{\pi}{2}-\alpha\right)}{\sin(\alpha +\pi)} \cdot \frac{\tan(\alpha-\pi)}{\cos(3\pi -\alpha)}$. | \frac{5}{4} | 86.71875 |
16,716 | Let $a,$ $b,$ and $c$ be positive real numbers such that $a + b + c = 3.$ Find the minimum value of
\[\frac{a + b}{abc}.\] | \frac{16}{9} | 17.1875 |
16,717 | In triangle $\triangle XYZ$, the medians $\overline{XM}$ and $\overline{YN}$ are perpendicular. If $XM=12$ and $YN=18$, then what is the area of $\triangle XYZ$? | 144 | 19.53125 |
16,718 | $(1)$ Calculate: $(\frac{1}{2})^{-1}+(\sqrt{2})^{2}-4\times |-\frac{1}{2}|$.
$(2)$ Simplify first, then find the value: $(1+\frac{4}{a-1})÷\frac{{a}^{2}+6a+9}{{a}^{2}-a}$, where $a=2$. | \frac{2}{5} | 64.84375 |
16,719 | Source: 1976 Euclid Part B Problem 1
-----
Triangle $ABC$ has $\angle{B}=30^{\circ}$ , $AB=150$ , and $AC=50\sqrt{3}$ . Determine the length of $BC$ . | 50\sqrt{3} | 6.25 |
16,720 | A teen age boy wrote his own age after his father's. From this new four place number, he subtracted the absolute value of the difference of their ages to get $4,289$ . The sum of their ages was | 59 | 19.53125 |
16,721 | In $\Delta ABC$, the sides opposite to angles $A$, $B$, and $C$ are respectively $a$, $b$, and $c$. It is known that $A=\frac{\pi}{4}$ and $b=\frac{\sqrt{2}}{2}a$.
(Ⅰ) Find the magnitude of $B$;
(Ⅱ) If $a=\sqrt{2}$, find the area of $\Delta ABC$. | \frac{\sqrt{3}+1}{4} | 30.46875 |
16,722 | Given positive numbers $m$ and $n$ that satisfy $m^2 + n^2 = 100$, find the maximum or minimum value of $m + n$. | 10\sqrt{2} | 38.28125 |
16,723 | Using systematic sampling, extract a sample of size 12 from a population of 123 individuals. The sampling interval is ______. | 10 | 98.4375 |
16,724 | Louise is designing a custom dress and needs to provide her hip size in millimeters. If there are $12$ inches in a foot and $305$ millimeters in a foot, and Louise's hip size is $42$ inches, what size should she specify in millimeters? | 1067.5 | 78.90625 |
16,725 | In a chess match between players A and B, the probabilities of A winning, B winning, and a tie are $0.5$, $0.3$, and $0.2$, respectively. Find the probability of B winning at least one match against A after two matches. | 0.51 | 89.84375 |
16,726 | Find the area of the shape enclosed by the curve $y=x^2$ (where $x>0$), the tangent line at point A(2, 4), and the x-axis. | \frac{2}{3} | 35.9375 |
16,727 | Find the greatest constant $N,$ so that
\[\frac{a^2 + b^2 + ab}{c^2} > N\]whenever $a,$ $b,$ and $c$ are the sides of a triangle. | \frac{3}{4} | 54.6875 |
16,728 | Let $\mathcal{S}$ be the set $\{1, 2, 3, \dots, 12\}$. Let $n$ be the number of sets of two non-empty disjoint subsets of $\mathcal{S}$. Calculate the remainder when $n$ is divided by 500. | 125 | 0.78125 |
16,729 | There are 18 teams participating in the opening ceremony of a competition. When entering, the 1st team has 27 members, the 2nd team has 26 members, and the 18th team has 10 members. If they enter in a single file, and all 18 teams' members are assigned numbers from 1 to 333 in the order they enter, then the number of t... | 10 | 9.375 |
16,730 | Given $A=\{x|ax^{2}+bx+c\leqslant 0\left(a \lt b\right)\}$ has one and only one element, then the minimum value of $M=\frac{{a+3b+4c}}{{b-a}}$ is ______. | 2\sqrt{5} + 5 | 0.78125 |
16,731 | Given a biased coin with probabilities of $\frac{3}{4}$ for heads and $\frac{1}{4}$ for tails, determine the difference between the probability of winning Game A, which involves 4 coin tosses and at least three heads, and the probability of winning Game B, which involves 5 coin tosses with the first two tosses and the ... | \frac{89}{256} | 3.125 |
16,732 | Solve for $x$ and $y$ given:
1. $\frac{4x - 2}{5x - 5} = \frac{3}{4}$
2. $x + y = 3$ | 10 | 0 |
16,733 | When $\sqrt[4]{5^9 \cdot 7^2}$ is fully simplified, the result is $a\sqrt[4]{b}$, where $a$ and $b$ are positive integers. What is $a+b$? | 270 | 73.4375 |
16,734 | Select 2 different numbers from 1, 3, 5, and 3 different numbers from 2, 4, 6, 8 to form a five-digit number, and determine the total number of even numbers among these five-digit numbers. | 864 | 72.65625 |
16,735 | In triangle $\triangle ABC$, $a$, $b$, $c$ are the opposite sides of the internal angles $A$, $B$, $C$, respectively, and $\sin ^{2}A+\sin A\sin C+\sin ^{2}C+\cos ^{2}B=1$.
$(1)$ Find the measure of angle $B$;
$(2)$ If $a=5$, $b=7$, find $\sin C$. | \frac{3\sqrt{3}}{14} | 69.53125 |
16,736 | Solve the equation \[\frac{x^2 + 3x + 4}{x + 5} = x + 6.\] | -\frac{13}{4} | 78.90625 |
16,737 | In the acute triangle \(KLM\), \(V\) is the intersection of its heights, and \(X\) is the foot of the height onto side \(KL\). The angle bisector of angle \(XVL\) is parallel to side \(LM\), and the angle \(MKL\) measures \(70^\circ\).
What are the measures of the angles \(KLM\) and \(KML\)? | 55 | 20.3125 |
16,738 | In the Cartesian coordinate system $xoy$, the parametric equation of line $l$ is $\begin{cases}x= \frac{ \sqrt{2}}{2}t \\ y=3+ \frac{ \sqrt{2}}{2}t\end{cases} (t$ is the parameter$)$, in the polar coordinate system with $O$ as the pole and the positive half-axis of $x$ as the polar axis, the polar equation of curve $C$... | \frac{2 \sqrt{5}}{3} | 50 |
16,739 | The digits of 2021 can be rearranged to form other four-digit whole numbers between 1000 and 3000. Find the largest possible difference between two such four-digit whole numbers. | 1188 | 39.84375 |
16,740 | Let $a_{n}(n\geqslant 2, n\in N^{*})$ be the coefficient of the linear term of $x$ in the expansion of ${({3-\sqrt{x}})^n}$. Find the value of $\frac{3^2}{a_2}+\frac{3^3}{a_3}+…+\frac{{{3^{18}}}}{{{a_{18}}}}$. | 17 | 92.1875 |
16,741 | At a community gathering there are only single women and married men with their wives. The probability that a randomly selected woman is single is $\frac{3}{7}$. Calculate the fraction of the people in the gathering who are married men. | \frac{4}{11} | 92.96875 |
16,742 | A heptagonal prism has ____ faces and ____ vertices. | 14 | 28.125 |
16,743 | What is the probability of drawing a number that is a multiple of 15 from a five-digit number formed without repeating digits using 1, 2, 3, 4, and 5? | \frac{1}{5} | 53.125 |
16,744 | Take a standard set of 28 dominoes and put back double 3, double 4, double 5, and double 6, as they will not be needed. Arrange the remaining dominoes to form 3 square frames, as shown in the image, so that the sum of the points along each side is equal. In the given example, these sums are equal to 15. If this is one ... | 15 | 9.375 |
16,745 | In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively, and $\cos A= \frac{4}{5}$.
(1) Find the value of $\sin ^{2} \frac{B+C}{2}+\cos 2A$;
(2) If $b=2$, the area of $\triangle ABC$ is $S=3$, find $a$. | \sqrt{13} | 71.875 |
16,746 | A series of lockers, numbered 1 through 100, are all initially closed. Student 1 goes through and opens every locker. Student 3 goes through and "flips" every 3rd locker ("flipping") a locker means changing its state: if the locker is open he closes it, and if the locker is closed he opens it). Thus, Student 3 will clo... | 10 | 83.59375 |
16,747 | A supermarket purchases two types of goods, $A$ and $B$. Buying 4 items of type $A$ costs $10$ yuan less than buying 5 items of type $B$. Buying 20 items of type $A$ and 10 items of type $B$ costs a total of $160$ yuan.
$(1)$ Find the cost price per item of goods $A$ and $B$ respectively.
$(2)$ If the store purchas... | 100 | 16.40625 |
16,748 | For how many $n=2,3,4,\ldots,109,110$ is the base-$n$ number $432143_n$ a multiple of $11$? | 10 | 76.5625 |
16,749 | The mean of the set of numbers $\{91, 89, 85, 88, 90, 87, y\}$ is 88. What is the median of the set of seven numbers? | 88 | 92.1875 |
16,750 | Dr. Math's four-digit house number $WXYZ$ contains no zeroes and can be split into two different two-digit primes ``$WX$'' and ``$YZ$'' where the digits $W$, $X$, $Y$, and $Z$ are not necessarily distinct. If each of the two-digit primes is less than 50, how many such house numbers are possible? | 110 | 5.46875 |
16,751 | A triangular region is bounded by the two coordinate axes and the line given by the equation $3x + y = 9$. Check if the point (1,1) lies inside this triangular region and find the region's area in square units. | \frac{27}{2} | 28.125 |
16,752 | If a positive integer is a multiple of 3, and we sum the cubes of its digits to perform the first operation; then, we take the new number and sum the cubes of its digits to perform the second operation; if we repeat the above operations several times, you will find that eventually, this number will remain unchanged, an... | 153 | 45.3125 |
16,753 | Given that $| \overrightarrow{a}|=2$, $\overrightarrow{e}$ is a unit vector, and the angle between $\overrightarrow{a}$ and $\overrightarrow{e}$ is $\dfrac {\pi}{3}$, find the projection of $\overrightarrow{a}+ \overrightarrow{e}$ on $\overrightarrow{a}- \overrightarrow{e}$. | \sqrt {3} | 0 |
16,754 | The equation $x^2 - kx - 24 = 0$ has only integer solutions for certain positive integers $k$. What is the sum of all such values of $k$? | 40 | 82.03125 |
16,755 | Determine the ratio $\frac{s}{r}$, where $r$ is the total number of rectangles and $s$ is the number of squares formed by the grid of a $7\times7$ checkerboard. Express $\frac{s}{r}$ in its simplest form and find the sum of the numerator and denominator. | 33 | 67.96875 |
16,756 | It is now 3:15:20 PM, as read on a 12-hour digital clock. In 305 hours, 45 minutes, and 56 seconds, the time will be $X:Y:Z$. What is the value of $X + Y + Z$? | 26 | 5.46875 |
16,757 | Using the Horner's method (also known as Qin Jiushao's algorithm), calculate the value of the polynomial \\(f(x)=12+35x-8x^{2}+79x^{3}+6x^{4}+5x^{5}+3x^{6}\\) when \\(x=-4\\), and determine the value of \\(V_{3}\\). | -57 | 59.375 |
16,758 | Given that the sequence $\{a_n\}$ is a geometric sequence, and the sequence $\{b_n\}$ is an arithmetic sequence. If $a_1-a_6-a_{11}=-3\sqrt{3}$ and $b_1+b_6+b_{11}=7\pi$, then the value of $\tan \frac{b_3+b_9}{1-a_4-a_3}$ is ______. | -\sqrt{3} | 43.75 |
16,759 | In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are denoted as $a$, $b$, and $c$, respectively. It is given that $(2c-a)\cos B = b\cos A$.
1. Find angle $B$.
2. If $b=6$ and $c=2a$, find the area of $\triangle ABC$. | 6\sqrt{3} | 85.15625 |
16,760 | Let $a$ and $b$ be positive integers that satisfy $ab-7a-11b+13=0$ . What is the minimum possible value of $a+b$ ? | 34 | 81.25 |
16,761 | $(1)$ Given real numbers $x \gt 0$, $y \gt 0$, and $\frac{2}{x}+y=1$, find the minimum value of $2x+\frac{1}{3y}$;<br/>$(2)$ Given real numbers $x$ and $y$ satisfying $x+y=1$, $x \gt 0$, $y \gt 0$, find the minimum value of $\frac{1}{2x}+\frac{x}{y+1}$. | \frac{5}{4} | 48.4375 |
16,762 | If $\alpha$ , $\beta$ , and $\gamma$ are the roots of $x^3 - x - 1 = 0$ , compute $\frac{1+\alpha}{1-\alpha} + \frac{1+\beta}{1-\beta} + \frac{1+\gamma}{1-\gamma}$ . | -7 | 58.59375 |
16,763 | The price of a pair of shoes at Barry's Boutique was $50. Find the price of the shoes on Thursday after a 15% price increase. Then, calculate the price of the shoes on Friday after a 20% discount is applied to the new price. | 46 | 12.5 |
16,764 | In a triangular pyramid $P-ABC$, $PC \perp$ plane $ABC$, $\angle CAB=90^{\circ}$, $PC=3$, $AC=4$, $AB=5$, find the surface area of the circumscribed sphere of the triangular pyramid. | 50\pi | 46.09375 |
16,765 | What is the largest value of $n$ less than 50,000 for which the expression $3(n-3)^2 - 4n + 28$ is a multiple of 7? | 49999 | 25 |
16,766 | The ratio $AC:CB$ is $3:4$, in $\triangle ABC$. The external angle bisector of $\angle C$ intersects the extension of $BA$ at $P$, where $A$ is between $P$ and $B$. Find the ratio $PA:AB$. | 3:1 | 4.6875 |
16,767 | Given the function $f(x)=(\sin x+\cos x)^{2}+2\cos ^{2}x-2$.
$(1)$ Find the smallest positive period and the intervals of monotonic increase for the function $f(x)$;
$(2)$ When $x\in\left[ \frac {\pi}{4}, \frac {3\pi}{4}\right]$, find the maximum and minimum values of the function $f(x)$. | - \sqrt {2} | 0 |
16,768 | If \( x^{4} + ax^{2} + bx + c = 0 \) has roots 1, 2, and 3 (one root is repeated), find \( a + c \).
(17th Annual American High School Mathematics Examination, 1966) | -61 | 10.15625 |
16,769 | Consider the set of all triangles $OPQ$ where $O$ is the origin and $P$ and $Q$ are distinct points in the plane with nonnegative integer coordinates $(x,y)$ such that $39x + y = 1953$. Find the number of such distinct triangles whose area is a positive integer and where $x_1 \neq x_2$. | 625 | 42.96875 |
16,770 | Given a set of paired data $(18,24)$, $(13,34)$, $(10,38)$, $(-1,m)$, the regression equation for these data is $y=-2x+59.5$. Find the correlation coefficient $r=$______(rounded to $0.001$). | -0.998 | 14.0625 |
16,771 | Triangle $PQR$ has vertices $P = (4,0)$, $Q = (0,4)$, and $R$, where $R$ is on the line $x + y = 8$ and also on the line $y = 2x$. Find the area of $\triangle PQR$.
A) $\frac{4}{3}$
B) $\frac{6}{3}$
C) $\frac{8}{3}$
D) $\frac{10}{3}$
E) $\frac{12}{3}$ | \frac{8}{3} | 46.875 |
16,772 | Two students, A and B, each select two courses from four elective courses. The probability that they share exactly one course in common is ______. | \frac{2}{3} | 71.875 |
16,773 | Simplify and evaluate:
(1) Calculate the value of $\frac {1}{\log_{4}6}+6^{\log_{6} \sqrt {3}-1}-2\log_{6} \frac {1}{3}$;
(2) Given $\tan\alpha=2$ and $\sin\alpha+\cos\alpha < 0$, find the value of $\frac {\tan(\pi-\alpha)\cdot \sin(-\alpha+ \frac {3\pi}{2})}{\cos(\pi +\alpha )\cdot \sin(-\pi -\alpha )}$. | \sqrt{5} | 49.21875 |
16,774 | A cylinder has a radius of 2 inches and a height of 3 inches. What is the radius of a sphere that has the same volume as this cylinder? | \sqrt[3]{9} | 49.21875 |
16,775 | Find $x$, such that $3^7 \cdot 3^x = 81$. | -3 | 45.3125 |
16,776 | Let the function $f\left( x \right)=\sin \left( wx-\frac{\pi }{6} \right)+\sin \left( wx-\frac{\pi }{2} \right)$, where $0 < w < 3$, and it is known that $f\left( \frac{\pi }{6} \right)=0$,
$(I)$ Find $w$
$(II)$ Stretch the x-coordinates of the points on the graph of $y=f\left( x \right)$ by a factor of $2$ (the y-co... | -\frac{3}{2} | 18.75 |
16,777 | Two cards are dealt from a standard deck of 52 cards. What is the probability that the first card dealt is a $\heartsuit$ and the second card dealt is a $\clubsuit$? | \frac{13}{204} | 100 |
16,778 | Given $\tan \alpha = -\frac{1}{2}$, find the value of $\frac{1+2\sin \alpha \cos \alpha}{\sin^2 \alpha - \cos^2 \alpha}$. | -\frac{1}{3} | 96.09375 |
16,779 | The constant term in the expansion of $(x^2-2)\left(x-\frac{2}{\sqrt{x}}\right)^{6}$ is ______. | -480 | 85.15625 |
16,780 | Given that $x$, $y$, and $z$ are all non-negative numbers and $x + y + z = 2$, find the minimum value of $\frac{1}{3}x^{3} + y^{2} + z$. | \frac{13}{12} | 50 |
16,781 | Let $a,$ $b,$ $c,$ $d$ be real numbers such that $a + b + c + d = 10$ and
\[ab + ac + ad + bc + bd + cd = 20.\] Find the largest possible value of $d$. | \frac{5 + 5\sqrt{21}}{2} | 0 |
16,782 | Given the hyperbola $x^{2}-4y^{2}=4$ with left and right foci $F_{1}$ and $F_{2}$. A line passing through $F_{1}$ intersects the left branch at points $A$ and $B$. If $|AB|=3$, find the perimeter of $\triangle AF_{2}B$. | 14 | 59.375 |
16,783 | Given the binomial $(ax+b)^{n}=a_{0}+a_{1}x+a_{2}x^{2}+a_{3}x^{3}+…+a_{n}x^{n}(a，b∈R，n∈N^{*})$.
(1) When $a=b=1$, $n=6$, find:
① The value of $a_{1}+a_{2}+a_{3}+…+a_{n}$.
② The value of $a_{1}+2a_{2}+3a_{3}+…+na_{n}$.
(2) When $a=1$, $b=- \sqrt {3}$, $n=8$, find the value of $(a_{0}+a_{2}+a_{4}+a_{6}+a_{8})^{2}-(... | 256 | 75 |
16,784 | Given the function $f(x)=\sin(2x+ \frac{\pi}{6})+\sin(2x- \frac{\pi}{6})+\cos 2x+a$ ($a\in\mathbb{R}$, $a$ is a constant),
(1) Find the smallest positive period of the function;
(2) Find the intervals of monotonic increase of the function;
(3) If $x\in\left[0, \frac{\pi}{2}\right]$ and the minimum value of $f(x)$ is... | -1 | 20.3125 |
16,785 | If $\sqrt{3\sqrt{s-3}} = \sqrt[4]{9 - s}$, then find $s$. | 3.6 | 40.625 |
16,786 | Convert \(531_8\) to base 7. | 1002_7 | 57.03125 |
16,787 | Given the function $f(x)=x^{3}+ax^{2}+bx+a^{2}$ where $a,b \in \mathbb{R}$. If the function $f(x)$ has an extremum of $10$ at $x=1$, then the value of $b$ is \_\_\_\_\_\_. | -11 | 46.875 |
16,788 | Consider the graph of \( y = g(x) \), with \( 1 \) unit between grid lines, where \( g(x) = \frac{(x-4)(x-2)(x)(x+2)(x+4)(x+6)}{720} - 2.5 \), defined only on the shown domain.
Determine the sum of all integers \( c \) for which the equation \( g(x) = c \) has exactly \( 4 \) solutions.
[asy]
size(150);
real f(real ... | -5 | 39.0625 |
16,789 | The fraction of the area of rectangle P Q R S that is shaded must be calculated. | \frac{1}{2} | 24.21875 |
16,790 | Given the function $f(x)=\begin{cases} (\frac{1}{2})^{x} & x\geqslant 4 \\ f(x+1) & x < 4 \end{cases}$, find the value of $f(2+\log_{2}3)$. | \frac{1}{24} | 65.625 |
16,791 | Given the real numbers $a$, $b$, $c$, $d$ that satisfy $$\frac {a-2e^{a}}{b}= \frac {2-c}{d-1}=1$$, where $e$ is the base of the natural logarithm, find the minimum value of $(a-c)^2+(b-d)^2$. | \frac{25}{2} | 22.65625 |
16,792 | A 9 by 9 checkerboard has alternating black and white squares. How many distinct squares, with sides on the grid lines of the checkerboard (horizontal and vertical) and containing at least 6 black squares, can be drawn on the checkerboard? | 91 | 12.5 |
16,793 | Given sets $A=\{x|x^2-2x-3>0\}$, $B=\{x|x^2+ax+b\leq0\}$, if $A\cup B=\mathbb{R}$ and $A\cap B=\{x|3<x\leq4\}$, then the value of $a+b$ equals to. | -7 | 75.78125 |
16,794 | What is the smallest natural number that can be added to 40,317 to make it a palindrome? | 87 | 0.78125 |
16,795 | Simplify: $$\sqrt[3]{5488000}$$ | 176.4 | 0 |
16,796 | Laura typically performs her routine using two 10-pound dumbbells for 30 repetitions. If she decides to use two 8-pound dumbbells instead, how many repetitions must she complete to lift the same total weight as her usual routine? | 37.5 | 4.6875 |
16,797 | In triangle $XYZ$, medians $XM$ and $YN$ intersect at $Q$, $QN=3$, $QM=4$, and $MN=5$. What is the area of $XMYN$? | 54 | 0 |
16,798 | Given that $\tan \alpha = \frac{1}{3}$ and $\tan (\alpha + \beta) = \frac{1}{2}$, find the value of $\tan \beta$. | \frac{1}{7} | 98.4375 |
16,799 | In $\triangle ABC$ , point $D$ lies on side $AC$ such that $\angle ABD=\angle C$ . Point $E$ lies on side $AB$ such that $BE=DE$ . $M$ is the midpoint of segment $CD$ . Point $H$ is the foot of the perpendicular from $A$ to $DE$ . Given $AH=2-\sqrt{3}$ and $AB=1$ , find the size of $\angle AME$ . | 15 | 3.125 |
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