task_type stringclasses 4
values | problem stringlengths 14 5.23k | solution stringlengths 1 8.29k | problem_tokens int64 9 1.02k | solution_tokens int64 1 1.98k |
|---|---|---|---|---|
math | A robot has 6 modules, each needing a base and a cap attached sequentially. In how many different orders can the robot attach its bases and caps, assuming that, on each module, the base must be attached before the cap?
A) $6! \cdot 2^6$
B) $(6!)^2$
C) $12!$
D) $\frac{12!}{2^6}$
E) $2^{12} \cdot 6!$ | \frac{12!}{2^6} | 102 | 11 |
math | During the "Christmas" sale at Jiuzhou Department Store, all goods are sold at a 40% discount. Therefore, an item with a marked price of $a$ yuan is sold for , and an item sold for $b$ yuan is originally priced at yuan. | \frac{5}{3}b | 64 | 8 |
math | Points $A$, $B$, and $C$ are on the same line. Given that $AB=5$ cm and $BC=4$ cm, find the distance between points $A$ and $C$. | 1\, \text{cm} \text{ or } 9\, \text{cm} | 45 | 22 |
math | Let $n$ be a positive integer. If $b \equiv (5^{2n} + 6)^{-1} \pmod{11}$, what is the remainder when $b$ is divided by 11? | b = 8 | 50 | 4 |
math | The standard equation of a parabola with the directrix \\(x=1\\) is $y^2 = -4x$. | y^{2}=-4x | 29 | 7 |
math | Given the function $f(x)=x^{3}+ax^{2}+bx+c$ where $a, b, c \in \mathbb{R}$. If the solution set of the inequality $f(x) < 0$ is $\{x | x < m+1$ and $x \neq m\}$, find the extreme value of the function $f(x)$. | -\frac{4}{27} | 84 | 8 |
math | Given an arithmetic sequence $\{a\_n\}$ with a common difference $d > 0$, and $a\_2$, $a\_5-1$, $a\_{10}$ form a geometric sequence. If $a\_1=5$, and $S\_n$ represents the sum of the first $n$ terms of the sequence, find the minimum value of $\frac{2S\_n+n+32}{a\_n+1}$. | \frac{20}{3} | 98 | 8 |
math | A certain natural disaster occurred in a region, causing the local tap water to be contaminated. A department decided to purify the water quality by adding a certain chemical into the water after testing. It is known that after adding a chemical with a mass of $m$ units, the concentration $y$ (mg/L) of the chemical rel... | [5,6] | 278 | 5 |
math | The sum of several consecutive natural numbers (more than 1) is 120. What is the largest possible number among these natural numbers? | 26 | 30 | 2 |
math | A cube of edge $3$ cm is cut into $N$ smaller cubes, not all the same size. If the edge of each of the smaller cubes is a whole number of centimeters, calculate the total number of smaller cubes. | 20 | 48 | 2 |
math | Given vectors $\overrightarrow{a}$, $\overrightarrow{b}$, and $\overrightarrow{c}$ lie in the same plane, and $\overrightarrow{a}=(3,2)$, $\overrightarrow{b}=(-2,1)$.
$(1)$ If $k\overrightarrow{a}-\overrightarrow{b}$ is perpendicular to $\overrightarrow{a}+2\overrightarrow{b}$, find the value of $k$;
$(2)$ If $\ove... | \frac{\pi}{2} | 167 | 7 |
math | Given the expression $3^5 \cdot 6^5 \cdot 3^6 \cdot 6^6$, evaluate the expression. | 18^{11} | 30 | 6 |
math | Find the solution of the system:
\[ x^{4} + y^{4} = 17 \]
\[ x + y = 3 \] | \{1, 2\} | 32 | 8 |
math | Given positive integers $a, b,$ and $c$ with $a + b + c = 20$ .
Determine the number of possible integer values for $\frac{a + b}{c}.$ | 6 | 50 | 1 |
math | The domain of the function $f(x)$ is $C$. If it satisfies: ① $f(x)$ is a monotonic function within $C$; ② there exists $[m, n] \subseteq D$ such that the range of $f(x)$ on $[m, n]$ is $[\frac{m}{2}, \frac{n}{2}]$, then $y=f(x)$ is called a "hope function". If the function $f(x) = \log_a(a^x + t)$ ($a>0$, $a \neq 1$) i... | (0, \frac{1}{4}) | 145 | 10 |
math | The value of $\sin 600^\circ$ can be calculated. | -\frac{\sqrt{3}}{2} | 16 | 10 |
math | The United States Postal Service has updated its postage rules. An extra charge of $\$0.11$ is added if the length of an envelope, in inches, divided by its height, in inches, is less than $1.4$ or greater than $2.4$ or if the area of the envelope (length times height) is less than $18$ square inches. Determine the num... | 3 | 157 | 1 |
math | Given non-zero real numbers \(a, b, c\) satisfy:
\[
a + b + c = 0, \quad a^{4} + b^{4} + c^{4} = 128.
\]
Find all possible values of \(ab + bc + ca\). | -8 | 62 | 2 |
math | The GDP of a certain city last year was 84,300,000 ten thousand yuan. Represent this number in scientific notation. | 8.43 \times 10^7 | 31 | 11 |
math | For the lines $x+ay+6=0$ and $(a-2)x+3y+2a=0$ to be parallel, determine the value of $a$. | -1 \text{ or } 3 | 38 | 9 |
math | Given proposition $p$: The domain of the function $f(x) = \lg (ax^2 - x + \frac{1}{16}a)$ is $\mathbb{R}$; and proposition $q$: The inequality $\sqrt{2x + 1} < 1 + ax$ holds for all positive real numbers. If the proposition "$p$ or $q$" is true and the proposition "$p$ and $q$" is false, then the range of values for th... | [1, 2] | 117 | 6 |
math | If $a$ men can build $d$ walls in $b$ days, how many days will it take $b$ men working at the same rate to build $a$ walls?
**A)** $\frac{d^2}{a}$
**B)** $\frac{a^2}{d}$
**C)** $\frac{b^2}{d}$
**D)** $\frac{d}{a^2}$ | \frac{a^2}{d} | 90 | 9 |
math | Points $K$, $L$, $M$, and $N$ lie in the plane of the square $ABCD$ so that $AKB$, $BLC$, $CMD$, and $DNA$ are equilateral triangles. If $ABCD$ has an area of 16, find the area of $KLMN$. Express your answer in simplest radical form.
[asy]
pair K,L,M,I,A,B,C,D;
D=(0,0);
C=(10,0);
B=(10,10);
A=(0,10);
I=(-8.7,5);
L=(18... | 32 + 16\sqrt{3} | 309 | 11 |
math | When $x \neq 1$ and $x \neq 0$, we can find the sum of the first $n$ terms, $S_n = 1 + 2x + 3x^2 + \ldots nx^{n-1}$ ($n \in \mathbb{N}^*$) of the sequence $\{nx^{n-1}\}$ by using the "staggered subtraction method" of sequence summation. It can also be derived by summing the geometric series $x + x^2 + x^3 + \ldots + x^... | n(n+3) \cdot 2^{n-2} | 409 | 14 |
math | Calculate: $-3x \cdot (2x^2 - x + 4) = \quad ; \quad (2a - b) \quad = 4a^2 - b^2.$ | 2a + b | 44 | 4 |
math | What are the last $2$ digits of the number $2018^{2018}$ when written in base $7$ ? | 44 | 35 | 2 |
math | During a rainstorm, 15 liters of water fell per square meter. By how much did the water level in Michael's outdoor pool rise? | 1.5 \, \text{cm} | 30 | 10 |
math | You have $5$ red shirts, $7$ green shirts, $8$ pairs of blue pants, $6$ pairs of green pants, $8$ green hats, and $8$ red hats, all of which are distinct. How many outfits can you make consisting of one shirt, one pair of pants, and one hat where:
1. The shirt and hat colors do not match, and
2. The pants color must ma... | 688 | 96 | 3 |
math | Find the last non-zero digit of the product of all natural numbers from 1 to 40. | 6 | 21 | 1 |
math | Given that $e$ is the base of the natural logarithm, if the function $g(t)=e^{t}⋅(t^{3}-6t^{2}+3t+m)$ satisfies $∀t∈[1,a]$, $∃m∈[0,5]$, such that $g(t)≤t$ holds true, then the maximum value of the positive integer $a$ is _______. | 5 | 87 | 1 |
math | Cici and Shanshan each have a certain number of points cards.
Shanshan said to Cici: "If you give me 2 cards, my number of cards will be twice yours."
Cici said to Shanshan: "If you give me 3 cards, my number of cards will be three times yours."
Shanshan said to Cici: "If you give me 4 cards, my number of cards will b... | 15 | 180 | 2 |
math | For each ordered pair of real numbers $(x,y)$ satisfying \[\log_2(2x+y) = \log_4(x^2+xy+7y^2)\]there is a real number $K$ such that \[\log_3(3x+y) = \log_9(3x^2+4xy+Ky^2).\]Find the product of all possible values of $K$. | 189 | 89 | 3 |
math | If
\[\frac{x}{a} + \frac{y}{b} + \frac{z}{c} = 4 \quad \text{and} \quad \frac{a}{x} + \frac{b}{y} + \frac{c}{z} = 2,\]find \(\frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2}.\) | 12 | 103 | 2 |
math | For a positive real number $x > 1,$ the Riemann zeta function $\zeta(x)$ is defined by
\[\zeta(x) = \sum_{n = 1}^\infty \frac{1}{n^x}.\]Compute
\[\sum_{k = 2}^\infty \{\zeta(2k - 1)\}.\]Note: For a real number $x,$ $\{x\}$ denotes the fractional part of $x.$ | \frac{1}{4} | 106 | 7 |
math | Find the equation of the line that satisfies the following conditions:
(1) Passes through point P(3, 2) and has equal intercepts on both coordinate axes;
(2) Passes through point A(-1, -3), and its angle of inclination is twice that of the line $y=3x$. | 3x+4y+15=0 | 69 | 10 |
math | Out of two hundred ninth-grade students, $80\%$ received excellent grades on the first exam, $70\%$ on the second exam, and $59\%$ on the third exam. What is the minimum number of students who could have received excellent grades on all three exams? | 18 | 63 | 2 |
math | Evaluate the expression $3^{3^{3^3}}$ with all possible arrangements of parentheses. How many distinct values can be obtained? | 4 | 28 | 1 |
math | If b² = c² + ac in an acute triangle ABC, where a, b, and c are the sides opposite to angles A, B, and C respectively, determine the range of $\frac {b}{c}$. | (1, 2) | 47 | 6 |
math | The price of a product was increased by $r\%$ and later decreased by $s\%$. If the final price was one dollar, what was the initial price of the product?
A) $\frac{10000}{10000 - rs}$
B) $\frac{10000}{10000 + 100(r-s)}$
C) $\frac{10000}{10000 + 100(r-s) - rs}$
D) $\frac{1}{1 - \frac{rs}{10000}}$
E) $\frac{10000 + rs}{1... | \frac{10000}{10000 + 100(r-s) - rs} | 152 | 25 |
math | Consider two lines $l$ and $m$ with equations $y = -2x + 8$, and $y = -3x + 8$. What is the probability that a point randomly selected in the 1st quadrant and below $l$ will fall between $l$ and $m$? Express your answer as a decimal to the nearest hundredth. | 0.33 | 76 | 4 |
math | Given a sequence $\{a_n\}$ with the sum of its first $n$ terms denoted as $S_n$, and the point $(n,\frac{S_n}{n})$ lies on the line $y=\frac{1}{2}x+\frac{11}{2}$. Another sequence $\{b_n\}$ satisfies $b_{n+2}-2b_{n+1}+b_n=0$ $(n\in \mathbb{N}^*)$, and $b_3=11$, with the sum of its first $9$ terms being $153$.
$(1)$ Fi... | m=11 | 304 | 4 |
math | Solve for $n$: $3^n \cdot 9^n = 256^{n-50}$. | n = \frac{-400 \log_3 2}{3 - 8 \log_3 2} | 26 | 26 |
math | In the Cartesian coordinate system, determine the quadrant in which the point $P(-3,2)$ is located. | 2 | 23 | 1 |
math | Let $n$ be a positive integer and $a,b,c,d$ be integers such that $a\equiv c^{-1} \pmod{n}$ and $b\equiv d^{-1} \pmod{n}$. What is the remainder when $(ab + cd)$ is divided by $n$? | 2 | 64 | 1 |
math | How many ordered triples of positive integers $(a, b, c)$ are there for which $a^{4} b^{2} c=54000$ ? | 16 | 36 | 2 |
math | A certain company produces a type of electronic chip, and the annual production of this chip does not exceed 350,000 units. The planned selling price of this electronic chip is 160,000 yuan per 10,000 units. It is known that the cost of producing this type of electronic chip is divided into fixed costs and variable cos... | x = 9 | 312 | 4 |
math | An infinite sequence of digits $1$ and $2$ is determined by the following two properties:
i) The sequence is built by writing, in some order, blocks $12$ and blocks $112.$ ii) If each block $12$ is replaced by $1$ and each block $112$ by $2$ , the same sequence is again obtained.
In which position is th... | 2 | 117 | 1 |
math | For each positive integer $k$ , let $S(k)$ be the sum of its digits. For example, $S(21) = 3$ and $S(105) = 6$ . Let $n$ be the smallest integer for which $S(n) - S(5n) = 2013$ . Determine the number of digits in $n$ . | 224 | 94 | 3 |
math | Given vectors $\overrightarrow{a}, \overrightarrow{b}$ that satisfy $|\overrightarrow{a}| = 1, |\overrightarrow{b}| = 2, \overrightarrow{a}\cdot \overrightarrow{b} = -\sqrt{3}$, find the angle between $\overrightarrow{a}$ and $\overrightarrow{b}$. | \frac{5\pi}{6} | 76 | 9 |
math | Solve the equations:<br/>
$(1) (x-5)^{2}=16$;<br/>
$(2) 2x^{2}-1=-4x$;<br/>
$(3) 5x(x+1)=2(x+1)$;<br/>
$(4) 2x^{2}-x-1=0$. | x_1 = -\frac{1}{2}, x_2 = 1 | 74 | 18 |
math | Music is composed of sounds of different frequencies. If the frequency of the first note $1(do)$ is $f$, then the frequencies of the seven notes $1(do)$, $2(re)$, $3(mi)$, $4(fa)$, $5(so)$, $6(la)$, $7(si)$ in the solfège are $f$, $\frac{9}{8}f$, $\frac{81}{64}f$, $\frac{4}{3}f$, $\frac{3}{2}f$, $\frac{27}{16}f$, $\fra... | 0 | 237 | 1 |
math | Let $p$, $q$, $r$ be the roots of the cubic polynomial $x^3 - 2x - 2 = 0$. Find
\[
p(q - r)^2 + q(r - p)^2 + r(p - q)^2.
\] | -6 | 59 | 2 |
math | Let $P$ be a point on the ellipse $C: \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (a > b > 0)$, and $F_{1}$, $F_{2}$ be the foci of $C$. It is given that $PF_{1} \perp PF_{2}$.
$(1)$ If $|PF_{1}| = 2|PF_{2}|$, find the eccentricity of the ellipse $C$.
$(2)$ If the coordinates of point $P$ are $(3,4)$, find the standard e... | \frac{x^2}{45} + \frac{y^2}{20} = 1 | 143 | 23 |
math | Given that $A=\{x|ax^{2}-ax+1\leqslant 0, x\in \mathbb{R}\}=\varnothing$, the range of values for $a$ is \_\_\_\_\_\_. | [0,4) | 53 | 5 |
math | Given an arithmetic sequence $\{a_n\}$ where the common difference $d > 0$, and the sum of the first $n$ terms is $S_n$, it is known that $a_2 \cdot a_3 = 45$ and $a_1 + a_5 = 18$.
(1) Find the general formula for the $n$th term of the sequence $\{a_n\}$.
(2) Let $b_n = \frac{S_n}{n + c}$ ($n \in \mathbb{N}^*$), is... | c = -\frac{1}{2} | 166 | 10 |
math | For the set $M$, define the function $f_M(x) = \begin{cases} -1, & x \in M \\ 1, & x \notin M \end{cases}$. For two sets $M$ and $N$, define the set $M \triangle N = \{x | f_M(x) \cdot f_N(x) = -1\}$. Given $A = \{2, 4, 6, 8, 10\}$ and $B = \{1, 2, 4, 8, 16\}$.
(1) List the elements of the set $A \triangle B = \_\_\_\... | 16 | 216 | 2 |
math | Let \( n \) (\( n \geqslant 2 \)) be a given positive integer. Find the value of \(\prod_{k=1}^{n-1} \sin \frac{n(k-1)+k}{n(n+1)} \pi \). | \frac{1}{2^{n-1}} | 60 | 11 |
math | In triangle $ABC$, $AB=15$ and $AC=8$. The angle bisector of $\angle A$ intersects $BC$ at point $D$, and point $M$ is the midpoint of $AD$. Let $P$ be the point of the intersection of $AB$ and $CM$. The ratio $BP$ to $PA$ can be expressed in the form $\dfrac{m}{n}$, where $m$ and $n$ are relatively prime positive inte... | 31 | 108 | 2 |
math | Determine which of the following statements are true:
1. $y=1$ is a power function;
2. An odd function $y=f(x)$ defined on $\mathbb{R}$ satisfies $f(0)=0$;
3. The function $f(x)=\lg(x+ \sqrt{x^2+1})$ is an odd function;
4. For $a<0$, $(a^2)^{\frac{3}{2}}=a^3$;
5. The function $y=1$ has two zeros;
List all the correct s... | 1, 2, 3 | 121 | 7 |
math | The vertices of a regular $2012$ -gon are labeled $A_1,A_2,\ldots, A_{2012}$ in some order. It is known that if $k+\ell$ and $m+n$ leave the same remainder when divided by $2012$ , then the chords $A_kA_{\ell}$ and $A_mA_n$ have no common points. Vasya walks around the polygon and sees that the first two ve... | A_{28} | 148 | 5 |
math | Let $T$ be the set of all points $(x, y, z)$ where $x, y, z$ each belong to the set $\{-1, 0, 1\}$. Determine the number of equilateral triangles whose vertices all belong to $T$. | 80 | 57 | 2 |
math | A coin has a probability of $3/4$ of turning up heads. Given that this coin is tossed 60 times, calculate the probability that the number of heads obtained is divisible by 3. | \frac{1}{3} | 42 | 7 |
math | Evaluate the expression \( 3000(3000^{2999})^2 \). | 3000^{5999} | 24 | 10 |
math | The notebook lists all the irreducible fractions with numerator 15 that are greater than $\frac{1}{16}$ and less than $\frac{1}{15}$. How many such fractions are listed in the notebook? | 8 | 47 | 1 |
math | Let $f : \mathbb{R} \to \mathbb{R}$ be a function such that
\[ f(f(x) + y) = f(x - y) + 2f(x)y\]
for all real numbers $x$ and $y$. Determine the possible values for $f(4)$ if $f(x) = cx$ for some constant $c$. | 0 \text{ and } 4 | 82 | 8 |
math | Given that $(x-a)(x+2)^5=a_0+a_1(x+1)+a_2(x+1)^2+...+a_6(x+1)^6$, if $a_0+a_1+a_2+...+a_6=-96$, find the value of $a_4$. | a_4=-10 | 71 | 6 |
math | The domain of the function $f(x)= \sqrt {4-|x|}+ \lg \dfrac {x^{2}-5x+6}{x-3}$ is what interval. | (2,3) \cup (3,4] | 42 | 12 |
math | Calculate the indefinite integral:
$$
\int \ln \left(4 x^{2}+1\right) d x
$$ | x \ln (4 x^2 + 1) - 8x + 4 \arctan (2x) + C | 28 | 29 |
math | Let
\[
\mathbf{B} = \renewcommand{\arraystretch}{1.5} \begin{pmatrix} \cos\left(\frac{\pi}{4}\right) & 0 & -\sin\left(\frac{\pi}{4}\right) \\ 0 & 1 & 0 \\ \sin\left(\frac{\pi}{4}\right) & 0 & \cos\left(\frac{\pi}{4}\right) \end{pmatrix} \renewcommand{\arraystretch}{1}.
\]
Compute $\mathbf{B}^{2023}$. | \begin{pmatrix} \frac{\sqrt{2}}{2} & 0 & -\frac{\sqrt{2}}{2} \\ 0 & 1 & 0 \\ \frac{\sqrt{2}}{2} & 0 & \frac{\sqrt{2}}{2} \end{pmatrix} | 132 | 71 |
math | The domain of the function $f(x) = \ln(x - 1)$ is $\{x | x > 1\}$. | \{x | x > 1\} | 29 | 10 |
math | What is the tenth number in the row of Pascal's triangle that has 100 numbers? | \binom{99}{9} | 20 | 9 |
math | Two perpendicular lines intersect at the point $A(4,10)$. The product of their $y$-intercepts is 100. What is the area of triangle $\triangle APQ$, where $P$ and $Q$ are the $y$-intercepts of these lines? | 40 | 63 | 2 |
math | In a box, there are 6 light bulbs, among which 2 are defective and 4 are good. Two bulbs are randomly selected from the box. Calculate the probability of the following events:
(Ⅰ) Both selected bulbs are defective;
(Ⅱ) Exactly one of the selected bulbs is defective. | P= \frac{8}{15} | 65 | 10 |
math | What is the units digit of the product of all odd positive integers between 100 and 200 that do not end in 5? | 9 | 31 | 1 |
math | Consider a paper triangle whose vertices are \((0,0), (10,0),\) and \((4,6)\). The vertices of its midpoint triangle are the midpoints of its sides. A triangular pyramid is formed by folding the triangle along the sides of its midpoint triangle. What is the volume of this pyramid? | 5 | 67 | 1 |
math | Determine the number of significant digits in the measurement of the side of a square whose computed area is $2.3104$ square meters to the nearest hundred-thousandth of a square meter. | 5 | 42 | 1 |
math | In triangle $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively, and $\frac{1}{2}c\sin B=(c-a\cos B)\sin C$.
$(1)$ Find angle $A$;
$(2)$ If $D$ is a point on side $AB$ such that $AD=2DB$, $AC=2$, and $BC=\sqrt{7}$, find the area of triangle $\triangle ACD$. | \sqrt{3} | 114 | 5 |
math | Calculate the probability of randomly selecting an element from the set $\{x \mid x=\frac{n\pi}{5}, n=1,2,3,4,5,6,7,8\}$ that satisfies the inequality $\tan{x} > 0$. | \frac{1}{2} | 56 | 7 |
math | (1) The negation of the proposition "For all $x \in \mathbb{R}$, $x^2 + 2x + m \leqslant 0$" is ______.
(2) Given $A = \{ x | x^2 - x \leqslant 0 \}$, $B = \{ x | 2^{1-x} + a \leqslant 0 \}$. If $A \subseteq B$, then the range of the real number $a$ is ______.
(3) Let the function $f(x) = \begin{cases} x + 1, & x \le... | 4 | 242 | 1 |
math | Find all positive solutions (\(x_{1}>0, x_{2}>0, x_{3}>0, x_{4}>0, x_{5}>0\)) of the system of equations
$$
\left\{\begin{array}{l}
x_{1}+x_{2}=x_{3}^{2} \\
x_{2}+x_{3}=x_{4}^{2} \\
x_{3}+x_{4}=x_{5}^{2} \\
x_{4}+x_{5}=x_{1}^{2} \\
x_{5}+x_{1}=x_{2}^{2}
\end{array}\right.
$$ | x_1 = x_2 = x_3 = x_4 = x_5 = 2 | 147 | 22 |
math | Find the largest prime divisor of $2102012_7$. | 79 | 17 | 2 |
math | Defined on $\mathbf{R}$, the function $f$ satisfies
$$
f(1+x)=f(9-x)=f(9+x).
$$
Given $f(0)=0$, and $f(x)=0$ has $n$ roots in the interval $[-4020, 4020]$, find the minimum value of $n$. | 2010 | 80 | 4 |
math | Solve the pair of simultaneous equations
$$
\begin{aligned}
x^{2}-2 x y & =1, \\
5 x^{2}-2 x y+2 y^{2} & =5 .
\end{aligned}
$$ | (x, y) = (1,0), (-1,0), \left(\frac{1}{3}, -\frac{4}{3}\right), \left(-\frac{1}{3}, \frac{4}{3}\right) | 51 | 53 |
math | How many non-congruent triangles with only integer side lengths have a perimeter of 15 units? | 7 | 21 | 1 |
math | Given $p$ is a prime number, such that both roots of the quadratic equation $x^2 - 2px + p^2 - 5p - 1 = 0$ are integers, find all possible values of $p$. | 3 \text{ or } 7 | 51 | 8 |
math | Find all real numbers $x$ such that \[4 \le \frac{x}{3x-7} < 9.\] (Give your answer in interval notation.) | \left(\frac{63}{26}, \frac{28}{11}\right] | 36 | 22 |
math | Let \( P(x) = a_0 + a_1 x + a_2 x^2 + \cdots + a_n x^n \) be a polynomial in \( x \) where the coefficients \( a_0, a_1, a_2, \ldots, a_n \) are non-negative integers. If \( P(1) = 25 \) and \( P(27) = 1771769 \), find the value of \( a_0 + 2a_1 + 3a_2 + \cdots + (n+1)a_n \). | 75 | 132 | 2 |
math | Consider the line $l: x + y - 4 = 0$ and the circle $x^2 + y^2 = 4$. Let point $A$ lie on line $l$. If there exist two points $B$ and $C$ on the circle such that $\angle BAC = 60^\circ$, find the maximum possible value of the $x$-coordinate of point $A$. | x = 4 | 87 | 4 |
math | Two particles move along the edges of an equilateral $\triangle ABC$, starting simultaneously from $A$ and $B$, respectively, and moving at the same speed. Each particle moves in the direction \[A\Rightarrow B\Rightarrow C\Rightarrow A.\] Determine the ratio of the area of the region $R$ to the area of $\triangle ABC$. | \frac{1}{4} | 73 | 7 |
math | Point $D$ lies on side $AC$ of an equilateral triangle $ABC$ such that the measure of angle $DBC$ is $30$ degrees. What is the ratio of the area of triangle $ADB$ to the area of triangle $CDB$? Express your answer as a common fraction in simplest radical form. | 1 | 68 | 1 |
math | The sum of all the digits used to write the whole numbers 10 through 13 is $1 + 0 + 1 + 1 + 1 + 2 + 1 + 3 = 10$. What is the sum of all the digits used to write the whole numbers 1 through 110, inclusive? | 957 | 72 | 3 |
math | The value of $y$ varies inversely as $x^2$ and when $x=3$, $y=2$. What is $x$ when $y=8$? | \frac{3}{2} | 39 | 7 |
math | Find
\[\sum_{n = 1}^\infty \frac{n^2 + n - 1}{(n + 2)!}.\] | \frac{1}{2} | 34 | 7 |
math | In an exhibition there are $100$ paintings each of which is made with exactly $k$ colors. Find the minimum possible value of $k$ if any $20$ paintings have a common color but there is no color that is used in all paintings. | 21 | 63 | 2 |
math | An ellipse C is defined by the equation $$\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1$$ ($a > b > 0$) with left and right foci $F_1$ and $F_2$, respectively, and an eccentricity of $$\frac{\sqrt{3}}{2}$$. The line passing through $F_1$ and perpendicular to the x-axis intersects the ellipse C at a segment of length 1.... | 1 | 182 | 1 |
math | Suppose $f(x)$ and $g(x)$ are functions which satisfy $f(g(x)) = x^3$ and $g(f(x)) = x^2$ for all $x \ge 1$. If $g(27) = 27$, compute $[g(3)]^2$. | 27 | 67 | 2 |
math | On a plate, there are various pancakes with three different fillings: 2 with meat, 3 with cottage cheese, and 5 with strawberries. Sveta consecutively ate all of them, choosing each next pancake at random. Find the probability that the first and the last pancakes she ate had the same filling. | \frac{14}{45} | 66 | 9 |
math | In trapezoid $JKLM$, sides $\overline{JK}$ and $\overline{LM}$ are parallel, $\angle J = 3\angle M$, and $\angle L = 2\angle K$. Find $\angle K$. | 72^\circ | 51 | 4 |
math | Pentagon $ANDD'Y$ has $AN \parallel DY$ and $AY \parallel D'N$ with $AN = D'Y$ and $AY = DN$ . If the area of $ANDY$ is 20, the area of $AND'Y$ is 24, and the area of $ADD'$ is 26, the area of $ANDD'Y$ can be expressed in the form $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$ . ... | 71 | 149 | 2 |
math | If the height on the base of an isosceles triangle is $18 \mathrm{~cm}$ and the median on the leg is $15 \mathrm{~cm}$, what is the area of this isosceles triangle? | 144 | 52 | 3 |
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