Unnamed: 0
int64 0
40.3k
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stringlengths 10
5.15k
| ground_truth
stringlengths 1
1.22k
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float64 0
100
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|---|---|---|---|
31,100 | For the graph $y = mx + 3$, determine the maximum value of $a$ such that the line does not pass through any lattice points for $0 < x \leq 50$ when $\frac{1}{3} < m < a$. | \frac{17}{51} | 9.375 |
31,101 | $(1)$ Calculate: $\sqrt{3}\tan 45^{\circ}-\left(2023-\pi \right)^{0}+|2\sqrt{3}-2|+(\frac{1}{4})^{-1}-\sqrt{27}$; <br/>$(2)$ Simplify first, then evaluate: $\frac{{{x^2}-x}}{{{x^2}+2x+1}}\div (\frac{2}{{x+1}}-\frac{1}{x})$, simplify it first, then choose an integer you like as the value of $x$ within the range of $-2 \lt x \lt 3$ to substitute and evaluate. | \frac{4}{3} | 59.375 |
31,102 | If $A = 3009 \div 3$, $B = A \div 3$, and $Y = A - B$, then what is the value of $Y$? | 669 | 27.34375 |
31,103 | In Class 3 (1), consisting of 45 students, all students participate in the tug-of-war. For the other three events, each student participates in at least one event. It is known that 39 students participate in the shuttlecock kicking competition and 28 students participate in the basketball shooting competition. How many students participate in all three events? | 22 | 53.90625 |
31,104 | Given that $f(\alpha) = \left(\sqrt{\frac{1 - \sin{\alpha}}{1 + \sin{\alpha}}} + \sqrt{\frac{1 + \sin{\alpha}}{1 - \sin{\alpha}}}\right)\cos^3{\alpha} + 2\sin{\left(\frac{\pi}{2} + \alpha\right)}\cos{\left(\frac{3\pi}{2} + \alpha\right)}$ (where $\alpha$ is an angle in the third quadrant),
(I) find the value of $f(\alpha)$ when $\tan{\alpha} = 2$;
(II) find the value of $\tan{\alpha}$ when $f(\alpha) = \frac{2}{5}\cos{\alpha}$. | \frac{3}{4} | 13.28125 |
31,105 | Using the digits 0, 1, 2, 3, 4, and 5, form six-digit numbers without repeating any digit.
(1) How many such six-digit odd numbers are there?
(2) How many such six-digit numbers are there where the digit 5 is not in the unit place?
(3) How many such six-digit numbers are there where the digits 1 and 2 are not adjacent? | 408 | 26.5625 |
31,106 | Define the *hotel elevator cubic*as the unique cubic polynomial $P$ for which $P(11) = 11$ , $P(12) = 12$ , $P(13) = 14$ , $P(14) = 15$ . What is $P(15)$ ?
*Proposed by Evan Chen* | 13 | 70.3125 |
31,107 | a) In how many ways can 9 people arrange themselves I) on a bench II) around a circular table?
b) In how many ways can 5 men and 4 women arrange themselves on a bench such that I) no two people of the same gender sit next to each other? II) the men and women sit in separate groups (only 1 man and 1 woman sit next to each other)? | 5760 | 41.40625 |
31,108 | In the expansion of $(1+x)^3+(1+x)^4+\ldots+(1+x)^{19}$, the coefficient of the $x^2$ term is \_\_\_\_\_\_. | 1139 | 39.84375 |
31,109 | An octagon $ABCDEFGH$ is divided into eight smaller equilateral triangles, such as $\triangle ABJ$ (where $J$ is the center of the octagon), shown in boldface in the diagram. By connecting every third vertex, we obtain a larger equilateral triangle $\triangle ADE$, which is also shown in boldface. Compute the ratio $[\triangle ABJ]/[\triangle ADE]$.
[asy]
size(150); defaultpen(linewidth(0.8)); dotfactor=5;
pair[] oct = new pair[8];
string[] octlabels = {"$H$", "$G$", "$F$", "$E$", "$D$", "$C$", "$B$", "$A$"};
octlabels.cyclic=true;
oct[0] = dir(0);
for(int i = 1; i <= 8; ++i){
oct[i] = dir(45*i);
draw(oct[i] -- oct[i-1]);
dot(octlabels[i],oct[i],oct[i]);
}
draw(oct[0]--oct[3]--oct[6]--cycle, linewidth(1.3));
draw(oct[0]--oct[1]--(0,0)--cycle, linewidth(1.3));
dot("$J$",(0,0),2*S);
[/asy] | \frac{1}{4} | 35.9375 |
31,110 | A solid rectangular block is created using $N$ congruent 1-cm cubes adhered face-to-face. When observing the block to maximize visibility of its surfaces, exactly $252$ of the 1-cm cubes remain hidden from view. Determine the smallest possible value of $N.$ | 392 | 12.5 |
31,111 | Given that $f(x+6) + f(x-6) = f(x)$ for all real $x$, determine the least positive period $p$ for these functions. | 36 | 70.3125 |
31,112 | Evaluate the expression $\frac {1}{3 - \frac {1}{3 - \frac {1}{3 - \frac{1}{4}}}}$. | \frac{11}{29} | 4.6875 |
31,113 | In a sequence, every (intermediate) term is half of the arithmetic mean of its neighboring terms. What relationship exists between any term, and the terms that are 2 positions before and 2 positions after it? The first term of the sequence is 1, and the 9th term is 40545. What is the 25th term? | 57424611447841 | 36.71875 |
31,114 | A merchant had 10 barrels of sugar, which he arranged into a pyramid as shown in the illustration. Each barrel, except one, was numbered. The merchant accidentally arranged the barrels such that the sum of the numbers along each row equaled 16. Could you rearrange the barrels such that the sum of the numbers along each row equals the smallest possible number? Note that the central barrel (which happened to be number 7 in the illustration) does not count in the sum. | 13 | 2.34375 |
31,115 | One hundred and one of the squares of an $n\times n$ table are colored blue. It is known that there exists a unique way to cut the table to rectangles along boundaries of its squares with the following property: every rectangle contains exactly one blue square. Find the smallest possible $n$ . | 101 | 60.9375 |
31,116 | In an arithmetic sequence $\{a_n\}$, if $a_3 - a_2 = -2$ and $a_7 = -2$, find the value of $a_9$. | -6 | 82.03125 |
31,117 | Two different natural numbers are selected from the set $\{1, 2, 3, \ldots, 8\}$. What is the probability that the greatest common factor of these two numbers is one? Express your answer as a common fraction. | \frac{3}{4} | 3.125 |
31,118 | The greatest common divisor of 15 and some number between 75 and 90 is 5. What is the number? | 85 | 3.90625 |
31,119 | Compute the length of the segment tangent from the origin to the circle that passes through the points $(4,5)$, $(8,10)$, and $(10,25)$. | \sqrt{82} | 0.78125 |
31,120 | A company sells a brand of cars in two places, A and B, with profits (in units of 10,000 yuan) of $L_{1}=-x^{2}+21x$ and $L_{2}=2x$ respectively, where $x$ is the sales volume (in units). If the company sells a total of 15 cars in both places, what is the maximum profit it can achieve? | 120 | 38.28125 |
31,121 | Given that points P and Q are on the curve $f(x) = x^2 - \ln x$ and the line $x-y-2=0$ respectively, find the minimum distance between points P and Q. | \sqrt{2} | 75.78125 |
31,122 | The instantaneous rate of change of carbon-14 content is $-\frac{\ln2}{20}$ (becquerel/year) given that at $t=5730$. Using the formula $M(t) = M_0 \cdot 2^{-\frac{t}{5730}}$, determine $M(2865)$. | 573\sqrt{2}/2 | 0 |
31,123 | Calculate the probability that the line $y=kx+k$ intersects with the circle ${{\left( x-1 \right)}^{2}}+{{y}^{2}}=1$. | \dfrac{1}{3} | 0 |
31,124 | Given the function $f(x)=\sqrt{2}\sin(2\omega x-\frac{\pi}{12})+1$ ($\omega > 0$) has exactly $3$ zeros in the interval $\left[0,\pi \right]$, determine the minimum value of $\omega$. | \frac{5}{3} | 26.5625 |
31,125 | A number is divided by \(7, 11, 13\). The sum of the quotients is 21, and the sum of the remainders is 21. What is the number? | 74 | 0 |
31,126 | In the arithmetic sequence $\left\{ a_n \right\}$, $a_{15}+a_{16}+a_{17}=-45$, $a_{9}=-36$, and $S_n$ is the sum of the first $n$ terms.
(1) Find the minimum value of $S_n$ and the corresponding value of $n$;
(2) Calculate $T_n = \left| a_1 \right| + \left| a_2 \right| + \ldots + \left| a_n \right|$. | -630 | 17.1875 |
31,127 | What is the probability of rolling four standard, six-sided dice and getting at least three distinct numbers, with at least one die showing a '6'? Express your answer as a common fraction. | \frac{5}{18} | 3.125 |
31,128 | The area of the square is $s^2$ and the area of the rectangle is $3s \times \frac{9s}{2}$. | 7.41\% | 0 |
31,129 | Cindy leaves school at the same time every day. If she cycles at \(20 \ \text{km/h}\), she arrives home at 4:30 in the afternoon. If she cycles at \(10 \ \text{km/h}\), she arrives home at 5:15 in the afternoon. At what speed, in \(\text{km/h}\), must she cycle to arrive home at 5:00 in the afternoon? | 12 | 38.28125 |
31,130 | Determine the number of ways to arrange the letters of the word "PERCEPTION". | 453,600 | 0 |
31,131 | An investor put $\$12,\!000$ into a three-month savings certificate that paid a simple annual interest rate of $8\%$. After three months, she reinvested the resulting amount in another three-month certificate. The investment value after another three months was $\$12,\!435$. If the annual interest rate of the second certificate is $s\%,$ what is $s?$ | 6.38\% | 9.375 |
31,132 | In the rectangle PQRS, side PQ is of length 2 and side QR is of length 4. Points T and U lie inside the rectangle such that ∆PQT and ∆RSU are equilateral triangles. Calculate the area of quadrilateral QRUT. | 4 - \sqrt{3} | 34.375 |
31,133 | In triangle $\triangle ABC$, the sides opposite angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. Given that $\left(\sin A+\sin B\right)\left(a-b\right)=c\left(\sin C-\sin B\right)$, and $D$ is a point on side $BC$ such that $AD$ bisects angle $BAC$ and $AD=2$. Find:<br/>
$(1)$ The measure of angle $A$;<br/>
$(2)$ The minimum value of the area of triangle $\triangle ABC$. | \frac{4\sqrt{3}}{3} | 42.1875 |
31,134 | How many different routes can Samantha take by biking on streets to the southwest corner of City Park, then taking a diagonal path through the park to the northeast corner, and then biking on streets to school? | 400 | 9.375 |
31,135 | How many natural numbers between 200 and 400 are divisible by 8? | 25 | 54.6875 |
31,136 | In the geometric sequence $\{a_n\}$, it is given that $a_{13}=1$ and $a_{12} > a_{13}$. Find the largest integer $n$ for which $(a_1-\frac{1}{a_1})+(a_2-\frac{1}{a_2})+(a_3-\frac{1}{a_3})+\cdots+(a_n-\frac{1}{a_n}) > 0$. | 24 | 13.28125 |
31,137 | Suppose that \(x_1+1=x_2+2=x_3+3=\cdots=x_{2010}+2010=x_1+x_2+x_3+\cdots+x_{2010}+2011\). Find the value of \(\left\lfloor|T|\right\rfloor\), where \(T=\sum_{n=1}^{2010}x_n\). | 1005 | 30.46875 |
31,138 | In $\triangle ABC$, if $\frac {\tan A}{\tan B}+ \frac {\tan A}{\tan C}=3$, then the maximum value of $\sin A$ is ______. | \frac { \sqrt {21}}{5} | 0 |
31,139 | A regular octahedron is formed by joining the midpoints of the edges of a regular tetrahedron. Calculate the ratio of the volume of this octahedron to the volume of the original tetrahedron. | \frac{1}{2} | 50.78125 |
31,140 | At the beginning of the school year, Jinshi Middle School has 17 classrooms. Teacher Dong Helong takes 17 keys to open the doors, knowing that each key can only open one door, but not knowing which key matches which door. What is the maximum number of attempts he needs to make to open all 17 locked doors? | 136 | 24.21875 |
31,141 | In the Cartesian coordinate system, with the origin as the pole and the x-axis as the positive semi-axis, a polar coordinate system is established. The polar equation of circle C is $\rho=6\cos\theta$, and the parametric equation of line $l$ is
$$
\begin{cases}
x=3+ \frac {1}{2}t \\
y=-3+ \frac { \sqrt {3}}{2}t
\end{cases}
$$
($t$ is the parameter).
(1) Find the Cartesian coordinate equation of circle C;
(2) Find the ratio of the lengths of the two arcs into which line $l$ divides circle C. | 1:2 | 81.25 |
31,142 | Given a sequence $\{a_n\}$ satisfying $a_1=1$, $a_{n+1}=2S_n+1$, where $S_n$ is the sum of the first $n$ terms of $\{a_n\}$, and $n\in\mathbb{N}^*$.
(1) Find $a_n$;
(2) If the sequence $\{b_n\}$ satisfies $b_n=\dfrac{1}{(1+\log_{3}a_n)(3+\log_{3}a_n)}$, and the sum of the first $n$ terms of $\{b_n\}$ is $T_n$, and for any positive integer $n$, $T_n < m$, find the minimum value of $m$. | \dfrac{3}{4} | 92.1875 |
31,143 | If the legs of a right triangle are in the ratio $3:4$, find the ratio of the areas of the two triangles created by dropping an altitude from the right-angle vertex to the hypotenuse. | \frac{9}{16} | 35.9375 |
31,144 | If \( \left\lfloor n^2/9 \right\rfloor - \lfloor n/3 \rfloor^2 = 5 \), then find all integer values of \( n \). | 14 | 4.6875 |
31,145 | Find the number of non-congruent scalene triangles whose sides all have integral length, and the longest side has length $11$ . | 20 | 98.4375 |
31,146 | Three cards are chosen at random from a standard 52-card deck. What is the probability that the first card is a spade, the second card is a 10, and the third card is a queen? | \frac{17}{11050} | 0 |
31,147 | Two trains, each composed of 15 identical cars, were moving towards each other at constant speeds. Exactly 28 seconds after their first cars met, a passenger named Sasha, sitting in the compartment of the third car, passed by a passenger named Valera in the opposite train. Moreover, 32 seconds later, the last cars of these trains completely passed each other. In which car was Valera traveling? | 12 | 2.34375 |
31,148 | A sample has a capacity of $80$. After grouping, the frequency of the second group is $0.15$. Then, the frequency of the second group is ______. | 12 | 53.125 |
31,149 | Given point P(-2,0) and line l: (1+3λ)x + (1+2λ)y = 2+5λ (λ ∈ ℝ), find the maximum value of the distance from point P to line l. | \sqrt{10} | 3.125 |
31,150 | What is the area of the region enclosed by $x^2 + y^2 = |x| - |y|$? | \frac{\pi}{2} | 42.96875 |
31,151 | Let $ABCD$ be a convex quadrilateral with $\angle DAB =\angle B DC = 90^o$ . Let the incircles of triangles $ABD$ and $BCD$ touch $BD$ at $P$ and $Q$ , respectively, with $P$ lying in between $B$ and $Q$ . If $AD = 999$ and $PQ = 200$ then what is the sum of the radii of the incircles of triangles $ABD$ and $BDC$ ? | 799 | 3.90625 |
31,152 | Given an ellipse $C$: $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1\left(a \gt b \gt 0\right)$ with the left and right foci $F_{1}$ and $F_{2}$ respectively. Let $F_{1}$ be the center of a circle with radius $|F_{1}F_{2}|$, intersecting the $y$-axis at points $A$ and $B$ (both points $A$ and $B$ are outside of $C$). Connect $F_{1}A$ with $C$ at point $P$. If $\overrightarrow{{F}_{2}P}\cdot \overrightarrow{{F}_{2}B}=0$, then $\angle AF_{1}B=$____; the eccentricity of the ellipse $C$ is ____. | \sqrt{3} - 1 | 0.78125 |
31,153 | In the diagram, the area of triangle $ABC$ is 36 square units. What is the area of triangle $BCD$ if the length of segment $CD$ is 39 units?
[asy]
draw((0,0)--(39,0)--(10,18)--(0,0)); // Adjusted for new problem length
dot((0,0));
label("$A$",(0,0),SW);
label("9",(4.5,0),S); // New base length of ABC
dot((9,0));
label("$C$",(9,0),S);
label("39",(24,0),S); // New length for CD
dot((39,0));
label("$D$",(39,0),SE);
dot((10,18)); // Adjusted location of B to maintain proportionality
label("$B$",(10,18),N);
draw((9,0)--(10,18));
[/asy] | 156 | 82.03125 |
31,154 | A quadrilateral $ABCD$ has a right angle at $\angle ABC$ and satisfies $AB = 12$ , $BC = 9$ , $CD = 20$ , and $DA = 25$ . Determine $BD^2$ .
.
| 769 | 17.96875 |
31,155 | An equilateral triangle of side length $ n$ is divided into unit triangles. Let $ f(n)$ be the number of paths from the triangle in the top row to the middle triangle in the bottom row, such that adjacent triangles in a path share a common edge and the path never travels up (from a lower row to a higher row) or revisits a triangle. An example is shown on the picture for $ n \equal{} 5$ . Determine the value of $ f(2005)$ . | 2005! | 0 |
31,156 | Given the function $f(x)=x^{2-m}$ defined on the interval $[-3-m,m^{2}-m]$, which is an odd function, find $f(m)=$____. | -1 | 30.46875 |
31,157 | Given that the point $(1, \frac{1}{3})$ lies on the graph of the function $f(x)=a^{x}$ ($a > 0$ and $a \neq 1$), and the sum of the first $n$ terms of the geometric sequence $\{a_n\}$ is $f(n)-c$, the first term and the sum $S_n$ of the sequence $\{b_n\}$ ($b_n > 0$) satisfy $S_n-S_{n-1}= \sqrt{S_n}+ \sqrt{S_{n+1}}$ ($n \geqslant 2$).
(1) Find the general formula for the sequences $\{a_n\}$ and $\{b_n\}$.
(2) If the sum of the first $n$ terms of the sequence $\left\{ \frac{1}{b_n b_{n+1}} \right\}$ is $T_n$, what is the smallest positive integer $n$ for which $T_n > \frac{1000}{2009}$? | 112 | 25.78125 |
31,158 | Simplify $(2^8 + 4^5)(2^3 - (-2)^3)^7$. | 1280 \cdot 16^7 | 0 |
31,159 | If $f(x) = \frac{1 + x}{1 - 3x}, f_1(x) = f(f(x)), f_2(x) = f(f_1(x)),$ and in general $f_n(x) = f(f_{n-1}(x)),$ then $f_{2023}(2)=$ | \frac{1}{7} | 74.21875 |
31,160 | In triangle $\triangle ABC$, $a=7$, $b=8$, $A=\frac{\pi}{3}$.
1. Find the value of $\sin B$.
2. If $\triangle ABC$ is an obtuse triangle, find the height on side $BC$. | \frac{12\sqrt{3}}{7} | 8.59375 |
31,161 | Let the function be $$f(x)= \sqrt {3}\sin 2x+2\cos^{2}x+2$$.
(I) Find the smallest positive period and the range of $f(x)$;
(II) In $\triangle ABC$, $a$, $b$, and $c$ are the sides opposite to angles $A$, $B$, and $C$, respectively. If $$A= \frac {\pi }{3}$$ and the area of $\triangle ABC$ is $$\frac { \sqrt {3}}{2}$$, find $f(A)$ and the value of $a$. | \sqrt {3} | 0 |
31,162 | The sum of seven test scores has a mean of 84, a median of 85, and a mode of 88. Calculate the sum of the three highest test scores. | 264 | 14.84375 |
31,163 | Given that each of the smaller rectangles has a shorter side of 7 feet and a longer side with a length three times that of the shorter side, and that two smaller rectangles are placed adjacent to each other by their longer sides and the third rectangle is placed adjacent by its long side to one of the other two rectangles, calculate the area of the larger rectangle EFGH in square feet. | 588 | 11.71875 |
31,164 | For real numbers $s,$ the intersection points of the lines $2x + 3y = 8s + 5$ and $x + 2y = 3s + 2$ are plotted. Determine the slope of the line on which all these points lie. | -\frac{7}{2} | 0 |
31,165 | Consider a regular octagon with side length 3, inside of which eight semicircles lie such that their diameters coincide with the sides of the octagon. Determine the area of the shaded region, which is the area inside the octagon but outside all of the semicircles.
A) $54 + 18\sqrt{2} - 9\pi$
B) $54 + 36\sqrt{2} - 9\pi$
C) $18 + 36\sqrt{2} - 9\pi$
D) $27 + 36\sqrt{2} - 4.5\pi$ | 54 + 36\sqrt{2} - 9\pi | 2.34375 |
31,166 | Determine the number of decreasing sequences of positive integers \(b_1 \geq b_2 \geq b_3 \geq \cdots \geq b_7 \leq 1500\) such that \(b_i - i\) is divisible by 3 for \(1 \leq i \le 7\). Express the number of such sequences as \({m \choose n}\) for some integers \(m\) and \(n\), and compute the remainder when \(m\) is divided by 1000. | 506 | 58.59375 |
31,167 | Out of 1500 people surveyed, 25% do not like television, and 15% of those who do not like television also do not like games. How many people surveyed do not like both television and games? | 56 | 92.96875 |
31,168 | In a box, there are 2 one-cent coins, 4 five-cent coins, and 6 ten-cent coins. Six coins are drawn in sequence without replacement, and each coin has an equal probability of being selected. What is the probability that the total value of the drawn coins is at least 50 cents? | $\frac{127}{924}$ | 0 |
31,169 | Let $O$ be the origin. $y = c$ intersects the curve $y = 2x - 3x^3$ at $P$ and $Q$ in the first quadrant and cuts the y-axis at $R$ . Find $c$ so that the region $OPR$ bounded by the y-axis, the line $y = c$ and the curve has the same area as the region between $P$ and $Q$ under the curve and above the line $y = c$ . | 4/9 | 10.9375 |
31,170 | In a sequence of positive integers starting from 1, some numbers are colored red according to the following rules: first color 1 red, then color the next 2 even numbers 2 and 4 red; next, color the 4 closest consecutive odd numbers after 4, which are 5, 7, 9, red; then color the 4 closest consecutive even numbers after 9, which are 10, 12, 14, 16, red; and so on, coloring the closest 5 consecutive odd numbers after 16, which are 17, 19, 21, 23, 25, red. Continue this pattern to get a red subsequence: 1, 2, 4, 5, 7, 9, 12, 14, 16, 17,…. The 57th number in this red subsequence, starting from 1, is. | 103 | 6.25 |
31,171 | Find \( g(2022) \) if for any real numbers \( x, y \) the following equality holds:
\[ g(x-y) = g(x) + g(y) - 2021(x+y) \] | 4086462 | 95.3125 |
31,172 | Let $b_n$ be the number obtained by writing the integers $1$ to $n$ from left to right, and then reversing the sequence. For example, $b_4 = 43211234$ and $b_{12} = 121110987654321123456789101112$. For $1 \le k \le 100$, how many $b_k$ are divisible by 9? | 22 | 96.875 |
31,173 | What is the smallest integer whose square is 78 more than three times the integer? | -6 | 21.875 |
31,174 | A standard deck of cards has 52 cards divided into 4 suits, each of which has 13 cards. Two of the suits ($\heartsuit$ and $\diamondsuit$) are red, and the other two ($\spadesuit$ and $\clubsuit$) are black. The cards in the deck are shuffled. What is the probability that the first card drawn is a heart and the second card drawn is a diamond? | \frac{169}{2652} | 0 |
31,175 | Four contestants \\(A\\), \\(B\\), \\(C\\), and \\(D\\) participate in three competitions: shooting, ball throwing, and walking on a balance beam. Each contestant has an equal chance of passing or failing each competition. At the end of the competitions, the judges will evaluate the performance of each contestant and award the top two contestants.
\\((1)\\) The probability that contestant \\(D\\) gets at least two passing grades;
\\((2)\\) The probability that only one of contestants \\(C\\) and \\(D\\) receives an award. | \dfrac {2}{3} | 2.34375 |
31,176 | Place 6 balls, labeled from 1 to 6, into 3 different boxes with each box containing 2 balls. If the balls labeled 1 and 2 cannot be placed in the same box, the total number of different ways to do this is \_\_\_\_\_\_. | 72 | 9.375 |
31,177 | Given two non-zero vectors $\overrightarrow {a}$ and $\overrightarrow {b}$ that satisfy $| \overrightarrow {a} + \overrightarrow {b} | = | \overrightarrow {a} - \overrightarrow {b} | = \sqrt {2} | \overrightarrow {a} |$, find the angle between vector $\overrightarrow {b}$ and $\overrightarrow {a} + \overrightarrow {b}$. | \frac{\pi}{4} | 81.25 |
31,178 | Two circles of radius 3 are centered at $(3,0)$ and at $(0,3)$. What is the area of the intersection of the interiors of the two circles? Express your answer in fully expanded form in terms of $\pi$. | \frac{9\pi - 18}{2} | 14.84375 |
31,179 | A certain high school is planning to hold a coming-of-age ceremony for senior students on the "May Fourth" Youth Day to motivate the seniors who are preparing for the college entrance examination. The Student Affairs Office has prepared five inspirational songs, a video speech by an outstanding former student, a speech by a teacher representative, and a speech by a current student. Based on different requirements, find the arrangements for this event.<br/>$(1)$ If the three speeches cannot be adjacent, how many ways are there to arrange them?<br/>$(2)$ If song A cannot be the first one and song B cannot be the last one, how many ways are there to arrange them?<br/>$(3)$ If the video speech by the outstanding former student must be before the speech by the current student, how many ways are there to arrange them? (Provide the answer as a number) | 20160 | 29.6875 |
31,180 | For every positive integer $k$ , let $\mathbf{T}_k = (k(k+1), 0)$ , and define $\mathcal{H}_k$ as the homothety centered at $\mathbf{T}_k$ with ratio $\tfrac{1}{2}$ if $k$ is odd and $\tfrac{2}{3}$ is $k$ is even. Suppose $P = (x,y)$ is a point such that $$ (\mathcal{H}_{4} \circ \mathcal{H}_{3} \circ \mathcal{H}_2 \circ \mathcal{H}_1)(P) = (20, 20). $$ What is $x+y$ ?
(A *homothety* $\mathcal{H}$ with nonzero ratio $r$ centered at a point $P$ maps each point $X$ to the point $Y$ on ray $\overrightarrow{PX}$ such that $PY = rPX$ .) | 256 | 14.0625 |
31,181 | With the popularity of cars, the "driver's license" has become one of the essential documents for modern people. If someone signs up for a driver's license exam, they need to pass four subjects to successfully obtain the license, with subject two being the field test. In each registration, each student has 5 chances to take the subject two exam (if they pass any of the 5 exams, they can proceed to the next subject; if they fail all 5 times, they need to re-register). The first 2 attempts for the subject two exam are free, and if the first 2 attempts are unsuccessful, a re-examination fee of $200 is required for each subsequent attempt. Based on several years of data, a driving school has concluded that the probability of passing the subject two exam for male students is $\frac{3}{4}$ each time, and for female students is $\frac{2}{3}$ each time. Now, a married couple from this driving school has simultaneously signed up for the subject two exam. If each person's chances of passing the subject two exam are independent, their principle for taking the subject two exam is to pass the exam or exhaust all chances.
$(Ⅰ)$ Find the probability that this couple will pass the subject two exam in this registration and neither of them will need to pay the re-examination fee.
$(Ⅱ)$ Find the probability that this couple will pass the subject two exam in this registration and the total re-examination fees they incur will be $200. | \frac{1}{9} | 20.3125 |
31,182 | A box contains a collection of triangular, square, and rectangular tiles. There are 32 tiles in the box, consisting of 114 edges in total. Each rectangle has 5 edges due to a small notch cut on one side. Determine the number of square tiles in the box. | 10 | 16.40625 |
31,183 | Selected Exercise $(4-5)$: Inequality Lecture
Given the function $f(x)=|2x-a|+|x-1|$, where $a\in R$
(1) Find the range of values for the real number $a$ if the inequality $f(x)\leqslant 2-|x-1|$ has a solution;
(2) When $a < 2$, the minimum value of the function $f(x)$ is $3$, find the value of the real number $a$. | -4 | 29.6875 |
31,184 | Let $x_1$ satisfy $2x+2^x=5$, and $x_2$ satisfy $2x+2\log_2(x-1)=5$. Calculate the value of $x_1+x_2$. | \frac {7}{2} | 0.78125 |
31,185 | Twelve standard 6-sided dice are rolled. What is the probability that exactly two of the dice show a 1? Express your answer as a decimal rounded to the nearest thousandth. | 0.296 | 93.75 |
31,186 | Automobile license plates for a state consist of four letters followed by a dash and two single digits. How many different license plate combinations are possible if exactly two letters are repeated once each (meaning two pairs of identical letters), and digits can be repeated? | 390000 | 14.0625 |
31,187 | Two numbers are independently selected from the set of positive integers less than or equal to 7. What is the probability that the sum of the two numbers is less than their product? Express your answer as a common fraction. | \frac{36}{49} | 5.46875 |
31,188 | Each pair of vertices of a regular $67$ -gon is joined by a line segment. Suppose $n$ of these segments are selected, and each of them is painted one of ten available colors. Find the minimum possible value of $n$ for which, regardless of which $n$ segments were selected and how they were painted, there will always be a vertex of the polygon that belongs to seven segments of the same color. | 2011 | 7.03125 |
31,189 | A painting $20$" X $30$" is to be framed where the frame border at the top and bottom is three times as wide as the frame on the sides. If the total area of the frame (not including the painting) equals twice the area of the painting itself, find the ratio of the smaller dimension to the larger dimension of the entire framed painting.
A) $\frac{1}{4}$
B) $\frac{1}{3}$
C) $\frac{1}{2}$
D) $\frac{2}{3}$
E) $\frac{3}{4}$ | \frac{1}{2} | 79.6875 |
31,190 | The maximum and minimum values of the function $y=2x^3-3x^2-12x+5$ in the interval $[0,3]$ are respectively what are the values? | -15 | 81.25 |
31,191 | Given that the general term of the sequence $\{a_{n}\}$ is ${a}_{n}=97-3n(n∈{N}^{*})$, find the value of $n$ for which the sum of the first $n$ terms of the sequence $\{{a}_{n}{a}_{n+1}{a}_{n+2}\}(n∈{N}^{*})$ reaches its maximum value. | 32 | 37.5 |
31,192 | How many solutions does the equation $\tan x = \tan (\tan x + \frac{\pi}{4})$ have in the interval $0 \leq x \leq \tan^{-1} 1884$? | 600 | 0.78125 |
31,193 | Let the even function $f(x)$ satisfy $f(x+6) = f(x) + f(3)$ for any $x \in \mathbb{R}$, and when $x \in (-3, -2)$, $f(x) = 5x$. Find the value of $f(201.2)$. | -14 | 18.75 |
31,194 | Source: 2018 Canadian Open Math Challenge Part B Problem 2
-----
Let ABCD be a square with side length 1. Points $X$ and $Y$ are on sides $BC$ and $CD$ respectively such that the areas of triangels $ABX$ , $XCY$ , and $YDA$ are equal. Find the ratio of the area of $\triangle AXY$ to the area of $\triangle XCY$ .
[center][/center] | \sqrt{5} | 3.90625 |
31,195 | Given numbers \( x_{1}, \cdots, x_{1991} \) satisfy the condition
$$
\left|x_{1}-x_{2}\right|+\cdots+\left|x_{1990}-x_{1991}\right|=1991 ,
$$
where \( y_{k}=\frac{1}{k}\left(x_{1}+\cdots+x_{k}\right) \) for \( k = 1, \cdots, 1991 \). Find the maximum possible value of the following expression:
$$
\left|y_{1}-y_{2}\right|+\cdots+\left|y_{1990}-y_{1991}\right| .
$$ | 1990 | 62.5 |
31,196 | Parallelogram $ABCD$ has $AB=CD=6$ and $BC=AD=10$ , where $\angle ABC$ is obtuse. The circumcircle of $\triangle ABD$ intersects $BC$ at $E$ such that $CE=4$ . Compute $BD$ . | 4\sqrt{6} | 3.90625 |
31,197 | The sides and vertices of a pentagon are labelled with the numbers $1$ through $10$ so that the sum of the numbers on every side is the same. What is the smallest possible value of this sum? | 14 | 61.71875 |
31,198 | In acute triangle $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. Given that $\frac{\sin B\sin C}{\sin A}=\frac{3\sqrt{7}}{2}$, $b=4a$, and $a+c=5$, find the area of $\triangle ABC$. | \frac{3\sqrt{7}}{4} | 16.40625 |
31,199 | Given the function $f(x)=\sin (\omega x+ \frac {\pi}{3})$ ($\omega > 0$), if $f( \frac {\pi}{6})=f( \frac {\pi}{3})$ and $f(x)$ has a minimum value but no maximum value in the interval $( \frac {\pi}{6}, \frac {\pi}{3})$, determine the value of $\omega$. | \frac {14}{3} | 10.9375 |
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