Unnamed: 0 int64 0 40.3k | problem stringlengths 10 5.15k | ground_truth stringlengths 1 1.22k | solved_percentage float64 0 100 |
|---|---|---|---|
16,400 | Given an arithmetic sequence $\{a_n\}$ with a non-zero common difference, the sum of the first $n$ terms is $S_n$, satisfying $S_3 = a_4 + 4$, and $a_2$, $a_6$, $a_{18}$ form a geometric sequence.
1. Find the general formula for the sequence $\{a_n\}$.
2. Let $b_n = \frac{a_n}{2^n}$, find the sum of the first $n$ terms... | \frac{1}{4} | 14.84375 |
16,401 | Given a function f(n) defined on the set of positive integers, where f(1) = 2: For even n, f(n) = f(n-1) + 2; For odd n > 1, f(n) = f(n-2) + 2. Calculate the value of f(2017). | 2018 | 84.375 |
16,402 | A company allocates 5 employees to 3 different departments, with each department being allocated at least one employee. Among them, employees A and B must be allocated to the same department. Calculate the number of different allocation methods. | 36 | 79.6875 |
16,403 | Given vectors $m=(\sin x,-1)$ and $n=\left( \sqrt{3}\cos x,-\frac{1}{2}\right)$, and the function $f(x)=(m+n)\cdot m$.
1. Find the interval where the function $f(x)$ is monotonically decreasing.
2. Given $a$, $b$, and $c$ are the sides opposite to angles $A$, $B$, and $C$ respectively in $\triangle ABC$, with $A$ bein... | 2\sqrt{3} | 35.9375 |
16,404 | Given that $\sin2α + \sinα = 0, α ∈ (\frac{π}{2}, π)$, find the value of $\tan(α + \frac{π}{4})$. | \sqrt{3} - 2 | 38.28125 |
16,405 | A number $x$ is randomly chosen from the interval $[-1, 1]$. What is the probability that the value of $\cos \frac{\pi x}{2}$ lies between $0$ and $\frac{1}{2}$? | $\frac{1}{3}$ | 0 |
16,406 | Given the parabola $y^{2}=4x$, a line $l$ passing through its focus $F$ intersects the parabola at points $A$ and $B$ (with point $A$ in the first quadrant), such that $\overrightarrow{AF}=3\overrightarrow{FB}$. A line passing through the midpoint of $AB$ and perpendicular to $l$ intersects the $x$-axis at point $G$. C... | \frac{32\sqrt{3}}{9} | 17.1875 |
16,407 | A number of tourists want to take a cruise, and it is required that the number of people on each cruise ship is the same. If each cruise ship carries 12 people, there will be 1 person left who cannot board. If one cruise ship leaves empty, then all tourists can be evenly distributed among the remaining ships. It is kno... | 169 | 14.0625 |
16,408 | Determine the area of the smallest square that can contain a circle with a radius of 6. | 144 | 90.625 |
16,409 | Given that the area of $\triangle ABC$ is 360, and point $P$ is a point on the plane of the triangle, with $\overrightarrow {AP}= \frac {1}{4} \overrightarrow {AB}+ \frac {1}{4} \overrightarrow {AC}$, then the area of $\triangle PAB$ is \_\_\_\_\_\_. | 90 | 94.53125 |
16,410 | Let $q(x)$ be a monic quartic polynomial such that $q(1) = 3$, $q(2) = 6, $ $q(3) = 11$, and $q(4) = 18$. Find $q(5)$. | 51 | 0 |
16,411 | For how many ordered pairs of positive integers $(x, y)$ with $x < y$ is the harmonic mean of $x$ and $y$ equal to $4^{15}$? | 29 | 92.1875 |
16,412 | If the sum of the coefficients of each term in the expansion of $(x- \frac {4}{x})^{n}$ is $81$, then the constant term in the expansion is ________. | 96 | 88.28125 |
16,413 | Let $Q$ be a point outside of circle $C.$ A line from $Q$ is tangent to circle $C$ at point $R.$ A secant from $Q$ intersects $C$ at $X$ and $Y,$ such that $QX < QY.$ If $QX = 5$ and $QR = XY - QX,$ what is $QY$? | 20 | 89.84375 |
16,414 | The Rotokas alphabet, consisting of letters A, E, G, I, K, O, P, R, T, U, and V (note S is removed from the original alphabet), is used to design four-letter license plates. How many license plates are possible that start with E, end with O, cannot contain I, and must not have repeated letters? | 56 | 65.625 |
16,415 | Calculate: $(\sqrt{3}+\sqrt{2})(\sqrt{3}-\sqrt{2})^{2}=\_\_\_\_\_\_$. | \sqrt{3}-\sqrt{2} | 92.1875 |
16,416 | What is the value of $x$ in the equation $\sqrt{\frac{72}{25}} = \sqrt[4]{\frac{x}{25}}$? | 207.36 | 63.28125 |
16,417 | Given that Liz had no money initially, and her friends gave her one-sixth, one-fifth, and one-fourth of their respective amounts, find the fractional part of the group's total money that Liz has. | \frac{1}{5} | 2.34375 |
16,418 | Starting with an equilateral triangle as shown in diagram a, each side of the triangle is divided into three equal parts, and at the middle segment, new equilateral triangles are constructed outward, as shown in diagram b, forming a "snowflake hexagon." Next, each of the 12 sides of the "snowflake hexagon" is divided i... | 40/27 | 78.125 |
16,419 | When drawing 20 numbers from 2005 numbers using systematic sampling, calculate the interval of sampling. | 100 | 62.5 |
16,420 | Find how many integer values of \( x \) are there such that \( \lceil{\sqrt{x}}\rceil=18 \)? | 35 | 88.28125 |
16,421 | (Exploring with a counter) Write any multiple of 3, cube each of its digits and add them together, then cube each digit of the new number obtained and add them together to get another new number, and keep repeating this process...
(1) Write out the calculation formulas;
(2) Describe your findings. | 153 | 57.03125 |
16,422 | A bag contains four balls of the same shape and size, numbered $1$, $2$, $3$, $4$ respectively.
$(1)$ Randomly draw two balls from the bag, and calculate the probability that the sum of the numbers on the drawn balls is even.
$(2)$ First, randomly draw a ball from the bag, and record its number as $m$. Then, return the... | \frac{5}{8} | 36.71875 |
16,423 | For $-1<s<1$, let $T(s)$ denote the sum of the geometric series \[20 + 20s + 20s^2 + 20s^3+\cdots .\] Let $b$ between $-1$ and $1$ satisfy $T(b)T(-b) = 4800$. Find $T(b) + T(-b)$. | 480 | 84.375 |
16,424 | In the expansion of \((x+y+z)^{8}\), find the sum of the coefficients for all terms of the form \(x^{2} y^{a} z^{b}\) (where \(a, b \in \mathbf{N}\)). | 1792 | 82.03125 |
16,425 | Wang Hong's father deposited 20,000 yuan in the bank for a fixed term of three years, with an annual interest rate of 3.33%. How much money, including the principal and interest, can Wang Hong's father withdraw at the end of the term? | 21998 | 32.03125 |
16,426 | A function \(f(x)\) is defined for all real numbers \(x\). For all non-zero values \(x\), we have
\[3f\left(x\right) + f\left(\frac{1}{x}\right) = 15x + 8.\]
Let \(S\) denote the sum of all of the values of \(x\) for which \(f(x) = 2004\). Compute the integer nearest to \(S\). | 356 | 74.21875 |
16,427 | Find the largest positive integer $m$ such that an $m \times m$ square can be exactly divided into 7 rectangles with pairwise disjoint interiors, and the lengths of the 14 sides of these 7 rectangles are $1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14$. | 22 | 0 |
16,428 | For integers a, b, c, and d the polynomial $p(x) =$ $ax^3 + bx^2 + cx + d$ satisfies $p(5) + p(25) = 1906$ . Find the minimum possible value for $|p(15)|$ . | 47 | 75 |
16,429 | Calculate the value of $8\cos ^{2}25^{\circ}-\tan 40^{\circ}-4$. | \sqrt{3} | 14.0625 |
16,430 | Determine the volume in cubic inches of a right, rectangular prism where the areas of the side, front, and bottom faces are 20 square inches, 12 square inches, and 8 square inches, respectively. | 8\sqrt{30} | 39.84375 |
16,431 | A particle begins at a point P on the parabola y = x^2 - 2x - 8 where the y-coordinate is 8. It rolls along the parabola to the nearest point Q where the y-coordinate is -8. Calculate the horizontal distance traveled by the particle, defined as the absolute difference between the x-coordinates of P and Q. | \sqrt{17} - 1 | 51.5625 |
16,432 | At a bus station, there are three buses departing to a school between 6:30 AM and 7:30 AM each day. The ticket prices for the buses are the same, but the comfort levels vary. Xiao Jie, a student, observes before boarding. When the first bus arrives, he does not get on but carefully observes its comfort level. If the co... | \frac{1}{2} | 18.75 |
16,433 | In a finite sequence of real numbers, the sum of any 7 consecutive terms is negative while the sum of any 11 consecutive terms is positive. What is the maximum number of terms in such a sequence? | 16 | 46.875 |
16,434 | Let $p$, $q$, and $r$ be the distinct roots of the polynomial $x^3 - 15x^2 + 50x - 60$. It is given that there exist real numbers $A$, $B$, and $C$ such that \[\dfrac{1}{s^3 - 15s^2 + 50s - 60} = \dfrac{A}{s-p} + \dfrac{B}{s-q} + \frac{C}{s-r}\]for all $s\not\in\{p,q,r\}$. What is $\tfrac1A+\tfrac1B+\tfrac1C$?
A) 133
B... | 135 | 49.21875 |
16,435 | In $\triangle PQR,$ $PQ=PR=30$ and $QR=28.$ Points $M, N,$ and $O$ are located on sides $\overline{PQ},$ $\overline{QR},$ and $\overline{PR},$ respectively, such that $\overline{MN}$ and $\overline{NO}$ are parallel to $\overline{PR}$ and $\overline{PQ},$ respectively. What is the perimeter of parallelogram $PMNO$? | 60 | 23.4375 |
16,436 | A particular integer is the smallest multiple of 72, each of whose digits is either 0 or 1. How many digits does this integer have? | 12 | 30.46875 |
16,437 | Two distinct non-consecutive positive integers $x$ and $y$ are factors of 48. If $x\cdot y$ is not a factor of 48, what is the smallest possible value of $x\cdot y$? | 18 | 0.78125 |
16,438 | Given that $| \overrightarrow{a}|=3 \sqrt {2}$, $| \overrightarrow{b}|=4$, $\overrightarrow{m}= \overrightarrow{a}+ \overrightarrow{b}$, $\overrightarrow{n}= \overrightarrow{a}+λ \overrightarrow{b}$, $ < \overrightarrow{a}, \overrightarrow{b} > =135^{\circ}$, if $\overrightarrow{m} \perp \overrightarrow{n}$, find the v... | -\frac{3}{2} | 92.96875 |
16,439 | Calculate the integrals:
1) \(\int_{0}^{\frac{\pi}{2}} \sin ^{3} x \, dx\);
2) \(\int_{0}^{\ln 2} \sqrt{e^{x}-1} \, dx\);
3) \(\int_{-a}^{a} x^{2} \sqrt{a^{2}-x^{2}} \, dx\);
4) \(\int_{1}^{2} \frac{\sqrt{x^{2}-1}}{x} \, dx\). | \sqrt{3} - \frac{\pi}{3} | 55.46875 |
16,440 | Given that point P is a point on the graph of the function f(x)=e^(2x), find the minimum distance from point P to the line l:y=2x. | \frac{\sqrt{5}}{5} | 85.9375 |
16,441 | A cylindrical can has a circumference of 24 inches and a height of 7 inches. A spiral strip is painted on the can such that it winds around the can precisely once, reaching from the bottom to the top. However, instead of reaching directly above where it started, it ends 3 inches horizontally to the right. What is the l... | \sqrt{778} | 28.125 |
16,442 | The minimum number of digits to the right of the decimal point needed to express the fraction $\frac{987654321}{2^{30}\cdot 5^3}$ as a decimal. | 30 | 56.25 |
16,443 | Determine the smallest natural number $n$ for which there exist distinct nonzero naturals $a, b, c$ , such that $n=a+b+c$ and $(a + b)(b + c)(c + a)$ is a perfect cube. | 10 | 95.3125 |
16,444 | Point $P$ moves on the ellipse $\dfrac{x^2}{9} + \dfrac{y^2}{25} = 1$, and points $A$ and $B$ move respectively on the circles $x^2 + (y-4)^2 = 16$ and $x^2 + (y+4)^2 = 4$. The maximum value of $PA + PB$ is \_\_\_\_\_\_. | 16 | 35.9375 |
16,445 | A space probe travels $5,555,555,555,555$ kilometers in its journey towards a distant star. After reaching its destination, it has traveled $3,333,333,333,333$ kilometers from its last refueling point. How many kilometers did the probe travel before the last refueling point?
A) $2,111,111,111,111$
B) $2,222,222,222,222... | 2,222,222,222,222 | 58.59375 |
16,446 | Find the minimum value of the expression:
\[(15 - x)(8 - x)(15 + x)(8 + x).\] | -6480.25 | 68.75 |
16,447 | An equilateral triangle of side length $12$ is completely filled by non-overlapping equilateral triangles of side length $2$. | 36 | 95.3125 |
16,448 | In the Cartesian coordinate system $xOy$, the curve $C_{1}$ is defined by $\begin{cases} x=-2+\cos \alpha \\ y=-1+\sin \alpha \end{cases}$ (where $\alpha$ is a parameter). In the polar coordinate system with the origin $O$ as the pole and the positive half-axis of $x$ as the polar axis, the curve $C_{2}$ is defined by ... | \frac {1}{2} | 53.90625 |
16,449 | If $\theta \in \left[\dfrac{\pi}{4}, \dfrac{\pi}{2}\right]$ and $\sin 2\theta = \dfrac{3\sqrt{7}}{8}$, find the value of $\sin \theta$. | \dfrac{3}{4} | 65.625 |
16,450 | Find the greatest common divisor of 4,004 and 10,010. | 2002 | 100 |
16,451 | Given that complex numbers $a,$ $b,$ and $c$ are zeros of a polynomial $P(z) = z^3 + qz + r,$ and $|a|^2 + |b|^2 + |c|^2 = 300$. The points corresponding to $a,$ $b,$ and $c$ on the complex plane are the vertices of a right triangle. Find the square of the length of the hypotenuse, $h^2$, given that the triangle's cent... | 450 | 2.34375 |
16,452 | Let \[P(x) = (3x^4 - 39x^3 + ax^2 + bx + c)(4x^4 - 96x^3 + dx^2 + ex + f),\] where $a, b, c, d, e, f$ are real numbers. Suppose that the set of all complex roots of $P(x)$ is $\{1, 2, 2, 3, 3, 4, 6\}.$ Find $P(7).$ | 86400 | 29.6875 |
16,453 | How many distinct arrangements of the letters in the word "balloon" are there? | 1260 | 42.1875 |
16,454 | Given a right-angled triangle, one of whose acute angles is $\alpha$. Find the ratio of the radii of the circumscribed and inscribed circles and determine for which value of $\alpha$ this ratio will be the smallest. | \sqrt{2} + 1 | 57.8125 |
16,455 | The five books "Poetry," "Documents," "Rites," "Changes," and "Spring and Autumn" all have different numbers of pages. The differences in the number of pages between the books are as follows:
1. "Poetry" and "Documents" differ by 24 pages.
2. "Documents" and "Rites" differ by 17 pages.
3. "Rites" and "Changes" differ b... | 34 | 32.03125 |
16,456 | When arranging the performance order of 6 singers, requiring that both singers B and C are either before or after singer A, calculate the total number of different arrangements. | 480 | 3.90625 |
16,457 | The robotics club has 30 members: 12 boys and 18 girls. A 6-person committee is chosen at random. What is the probability that the committee has at least 1 boy and at least 1 girl? | \frac{574,287}{593,775} | 0 |
16,458 | In triangle $\triangle ABC$, the sides opposite to the internal angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. It is known that $b=2\left(a\cos B-c\right)$. Find:<br/>
$(1)$ The value of angle $A$;<br/>
$(2)$ If $a\cos C=\sqrt{3}$ and $b=1$, find the value of $c$. | 2\sqrt{3} - 2 | 72.65625 |
16,459 | In a regular octagon, find the ratio of the length of the shortest diagonal to the longest diagonal. Express your answer as a common fraction in simplest radical form. | \frac{\sqrt{2}}{2} | 38.28125 |
16,460 | Given that square $ABCE$ has side lengths $AF = 3FE$ and $CD = 3DE$, calculate the ratio of the area of $\triangle AFD$ to the area of square $ABCE$. | \frac{3}{8} | 30.46875 |
16,461 | Given that there are 4 qualified and 2 defective products, determine the probability of finding the last defective product exactly on the fourth inspection when selectins products one at a time and not returning them after each selection. | \frac{1}{5} | 23.4375 |
16,462 | Choose one of the following conditions from (1) $a\sin \left(B+C\right)+c\sin C-b\sin B=2a\sin C\sin B$, (2) $\frac{cosB}{cosC}+\frac{b}{c-\sqrt{2}a}=0$, (3) $2a^{2}=(a^{2}+b^{2}-c^{2})(1+\tan C)$, and fill in the blank in the question below, and answer the corresponding questions.
Given $\triangle ABC$ with sides $a... | \frac{3\sqrt{2}}{5} | 18.75 |
16,463 | For $n$ measured in degrees, let $T(n) = \cos^2(30^\circ -n) - \cos(30^\circ -n)\cos(30^\circ +n) +\cos^2(30^\circ +n)$ . Evaluate $$ 4\sum^{30}_{n=1} n \cdot T(n). $$ | 1395 | 63.28125 |
16,464 | Simplify \[\frac{1}{\dfrac{3}{\sqrt{5}+2} + \dfrac{4}{\sqrt{7}-2}}.\] | \frac{9\sqrt{5} + 4\sqrt{7} + 10}{(9\sqrt{5} + 4\sqrt{7})^2 - 100} | 0 |
16,465 | Given a quadratic function $f(x) = ax^2 + bx + c$ (where $a$, $b$, and $c$ are constants). If the solution set of the inequality $f(x) \geq 2ax + b$ is $\mathbb{R}$ (the set of all real numbers), then the maximum value of $\frac{b^2}{a^2 + c^2}$ is __________. | 2\sqrt{2} - 2 | 12.5 |
16,466 | In triangle $ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively, with $c=2$ and $b=\sqrt{2}a$. Find the maximum value of the area of $\triangle ABC$. | 2\sqrt{2} | 16.40625 |
16,467 | Two right triangles, $ABC$ and $ACD$, are joined as shown. Squares are drawn on four of the sides. The areas of three of the squares are 25, 49, and 64 square units. What is the number of square units in the area of the fourth square?
Note that the diagram is not provided, but imagine it similarly to the reference whe... | 10 | 14.84375 |
16,468 | Simplify $(9 \times 10^{12}) \div (3 \times 10^4) + (2 \times 10^8) \div (4 \times 10^2)$. | 300,500,000 | 0 |
16,469 | A circle with center \( Q \) and radius 2 rolls around the inside of a right triangle \( DEF \) with side lengths 9, 12, and 15, always remaining tangent to at least one side of the triangle. When \( Q \) first returns to its original position, through what distance has \( Q \) traveled? | 24 | 66.40625 |
16,470 | Given the power function $f(x) = kx^a$ whose graph passes through the point $\left( \frac{1}{3}, 81 \right)$, find the value of $k + a$. | -3 | 60.9375 |
16,471 | If $z=1+i$, then $|{iz+3\overline{z}}|=\_\_\_\_\_\_$. | 2\sqrt{2} | 39.84375 |
16,472 | In the sequence ${a_{n}}$, $a_{1}=1$, $a_{n+2}+(-1)^{n}a_{n}=1$. Let $s_{n}$ be the sum of the first $n$ terms of the sequence ${a_{n}}$. Find $s_{100}$ = \_\_\_\_\_\_. | 1300 | 50 |
16,473 | Suppose that for a positive integer \( n \), \( 2^n + 1 \) is a prime number. What remainder can this prime have when divided by 240? | 17 | 47.65625 |
16,474 | In the Cartesian coordinate system $(xOy)$, the parametric equations of line $l$ are given by $\begin{cases}x=1+\frac{\sqrt{2}}{2}t\\y=\frac{\sqrt{2}}{2}t\end{cases}$ (where $t$ is the parameter), and in the polar coordinate system with the origin $O$ as the pole and the $x$-axis as the polar axis, the polar equation o... | \sqrt{2}-1 | 67.96875 |
16,475 | Given that $\operatorname{log}_{8}(p) = \operatorname{log}_{12}(q) = \operatorname{log}_{18}(p-q)$, calculate the value of $\frac{q}{p}$. | \frac{\sqrt{5} - 1}{2} | 15.625 |
16,476 | Let $ABC$ be an acute-angled triangle and $P$ be a point in its interior. Let $P_A,P_B$ and $P_c$ be the images of $P$ under reflection in the sides $BC,CA$ , and $AB$ , respectively. If $P$ is the orthocentre of the triangle $P_AP_BP_C$ and if the largest angle of the triangle that can be formed by the... | 120 | 90.625 |
16,477 | Express as a fraction in lowest terms: $0.\overline{23} + 0.\overline{056} + 0.\overline{004}$. | \frac{28917}{98901} | 0 |
16,478 | Given that all edges of a tetrahedron have a length of $\sqrt{2}$, and the four vertices are on the same sphere, calculate the surface area of this sphere. | 3 \pi | 99.21875 |
16,479 | How many sequences of 8 digits $x_1, x_2, \ldots, x_8$ can we form, given the condition that no two adjacent $x_i$ have the same parity and the sequence starts with an odd digit? Leading zeroes are allowed. | 390625 | 95.3125 |
16,480 | Given the function $f(x)=\cos^4x-2\sin x\cos x-\sin^4x.$
$(1)$ Find the smallest positive period of $f(x)$.
$(2)$ When $x\in\left[0, \frac{\pi}{2}\right]$, find the minimum value of $f(x)$ and the set of $x$ values for which this minimum is achieved. | \frac{3\pi}{8} | 48.4375 |
16,481 | Given the increasing sequence of positive integers \(a_{1}, a_{2}, a_{3}, \cdots\) satisfies the recurrence relation \(a_{n+2} = a_{n} + a_{n+1}\) for \(n \geq 1\), and \(a_{7} = 120\), find the value of \(a_{8}\). | 194 | 85.9375 |
16,482 | Let $ABCD$ be a rectangle with sides $AB,BC,CD$ and $DA$ . Let $K,L$ be the midpoints of the sides $BC,DA$ respectivily. The perpendicular from $B$ to $AK$ hits $CL$ at $M$ . Find $$ \frac{[ABKM]}{[ABCL]} $$ | 2/3 | 11.71875 |
16,483 | Let the function be $$f(x)=1-2\sin^{2}x-\cos(2x+ \frac {\pi}{3})$$
(1) Find the smallest positive period of the function $f(x)$.
(2) For triangle ABC, the sides $a$, $b$, $c$ are opposite to the angles $A$, $B$, $C$, respectively. Given $b=5$, and $$f\left( \frac {B}{2}\right)=1$$, find the maximum area of triangle... | \frac {25 \sqrt {3}}{4} | 0 |
16,484 | Given $α∈(\frac{\pi}{2},π)$, $\sin α =\frac{\sqrt{5}}{5}$.
(Ⅰ) Find the value of $\tan\left( \frac{\pi}{4}+2α \right)$;
(Ⅱ) Find the value of $\cos\left( \frac{5\pi}{6}-2α \right)$. | -\frac{3\sqrt{3}+4}{10} | 60.9375 |
16,485 | Evaluate the determinant of the given matrix:
\[
\begin{vmatrix} \sin \theta \sin \phi & \sin \theta \cos \phi & \cos \theta \\ \cos \phi & -\sin \phi & 0 \\ -\cos \theta \sin \phi & -\cos \theta \cos \phi & \sin \theta \end{vmatrix}
\] | -1 | 75.78125 |
16,486 | For each integer $n$ greater than 1, let $G(n)$ be the number of solutions of the equation $\sin x = \sin (n+1)x$ on the interval $[0, 2\pi]$. Calculate $\sum_{n=2}^{100} G(n)$. | 10296 | 6.25 |
16,487 | Rectangle \(WXYZ\) is divided into four smaller rectangles. The perimeters of three of these smaller rectangles are 11, 16, and 19. The perimeter of the fourth rectangle lies between 11 and 19. What is the length of the perimeter of \(WXYZ\)?
Options:
A) 28
B) 30
C) 32
D) 38
E) 40 | 30 | 46.875 |
16,488 | What percent of the palindromes between 1000 and 2000 contain at least one 3 or 5, except in the first digit? | 36\% | 12.5 |
16,489 | The third and twentieth terms of an arithmetic sequence are 10 and 65, respectively. What is the thirty-second term? | 103.8235294118 | 0 |
16,490 | The sequence $\{a_n\}$ satisfies $a_{n+1}=(2|\sin \frac{n\pi}{2}|-1)a_{n}+n$, then the sum of the first $100$ terms of the sequence $\{a_n\}$ is __________. | 2550 | 1.5625 |
16,491 | Calculate $1,000,000,000,000 - 888,777,888,777$. | 111,222,111,223 | 0 |
16,492 | Find \( n > m \geq 1 \) such that the last two digits of the decimal representation of \( 42^{n} \) are the same as those of \( 42^{m} \), and such that the sum \( m+n \) is minimal. | 24 | 14.0625 |
16,493 | Given that $a \in \mathbb{R}$, if the real part and the imaginary part of the complex number $\frac{a + i}{1 + i}$ (where $i$ is the imaginary unit) are equal, then $\_\_\_\_\_\_$, $| \overline{z}| = \_\_\_\_\_\_$. | \frac{\sqrt{2}}{2} | 67.1875 |
16,494 | How many whole numbers between $200$ and $500$ contain the digit $3$? | 138 | 78.125 |
16,495 | Determine the number of all positive integers which cannot be written in the form $80k + 3m$ , where $k,m \in N = \{0,1,2,...,\}$ | 79 | 22.65625 |
16,496 | Given sets \( A = \{1, 2, 3\} \) and \( B = \{4, 5, 6, 7\} \), with a mapping \( f: A \rightarrow B \), if for \( x \in A \), the expression \( x + f(x) + x \cdot f(x) \) is odd, then the number of such mappings \( f \) is: | 32 | 21.875 |
16,497 | Find the biggest real number $ k$ such that for each right-angled triangle with sides $ a$ , $ b$ , $ c$ , we have
\[ a^{3}\plus{}b^{3}\plus{}c^{3}\geq k\left(a\plus{}b\plus{}c\right)^{3}.\] | \frac{3\sqrt{2} - 4}{2} | 50 |
16,498 | It is known that the equation $ |x - 1| + |x - 2| +... + |x - 2001| = a$ has exactly one solution. Find $a$ . | 1001000 | 99.21875 |
16,499 | A circle with a radius of 3 units has its center at $(0, 0)$. A circle with a radius of 8 units has its center at $(18, 0)$. A line tangent to both circles intersects the $x$-axis at point $(x, 0)$ to the right of the origin. Find the value of $x$. Express your answer as a common fraction. | \frac{54}{11} | 60.9375 |
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