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25,800 | 1. Given that $α$ and $β$ are acute angles, and $\cos α= \frac{4}{5}$, $\cos (α+β)=- \frac{16}{65}$, find the value of $\cos β$.
2. Given that $0 < β < \frac{π}{4} < α < \frac{3}{4}π$, $\cos ( \frac{π}{4}-α)= \frac{3}{5}$, $\sin ( \frac{3π}{4}+β)= \frac{5}{13}$, find the value of $\sin (α+β)$. | \frac{56}{65} | 13.28125 |
25,801 | Calculate the infinite sum:
\[
\sum_{n=1}^\infty \frac{n^3 - n}{(n+3)!}
\] | \frac{1}{6} | 13.28125 |
25,802 | For arithmetic sequences $\{a_{n}\}$ and $\{b_{n}\}$, the sums of the first $n$ terms are $S_{n}$ and $T_{n}$, respectively. If $\frac{{S}_{n}}{{T}_{n}}=\frac{2n+1}{3n+2}$, then $\frac{{a}_{2}+{a}_{5}+{a}_{17}+{a}_{20}}{{b}_{8}+{b}_{10}+{b}_{12}+{b}_{14}}=\_\_\_\_\_\_$. | \frac{43}{65} | 26.5625 |
25,803 | If $x$ and $y$ are positive integers such that $xy - 8x + 7y = 775$, what is the minimal possible value of $|x - y|$? | 703 | 27.34375 |
25,804 | A two-digit integer $AB$ equals $\frac{1}{9}$ of the three-digit integer $CCB$, where $C$ and $B$ represent distinct digits from 1 to 9. What is the smallest possible value of the three-digit integer $CCB$? | 225 | 1.5625 |
25,805 | In the decimal number system, the operation rule is "ten carries one". Analogous to this operation rule, perform the four arithmetic operations in the octal system and calculate $53_{(8)} \times 26_{(8)} =$ _______ (the operation result must be represented in octal numbers). | 1662_{(8)} | 64.84375 |
25,806 | Given $ \dfrac {3\pi}{4} < \alpha < \pi$, $\tan \alpha+ \dfrac {1}{\tan \alpha}=- \dfrac {10}{3}$.
$(1)$ Find the value of $\tan \alpha$;
$(2)$ Find the value of $ \dfrac {5\sin ^{2} \dfrac {\alpha}{2}+8\sin \dfrac {\alpha}{2}\cos \dfrac {\alpha}{2}+11\cos ^{2} \dfrac {\alpha}{2}-8}{ \sqrt {2}\sin (\alpha- \dfrac {\pi}{4})}$. | - \dfrac {5}{4} | 2.34375 |
25,807 | The TV station continuously plays 5 advertisements, consisting of 3 different commercial advertisements and 2 different Olympic promotional advertisements. The requirements are that the last advertisement must be an Olympic promotional advertisement, and the 2 Olympic promotional advertisements can be played consecutively. Determine the total number of different playback methods. | 36 | 23.4375 |
25,808 | In square $EFGH$, $EF$ is 8 centimeters, and $N$ is the midpoint of $\overline{GH}$. Let $P$ be the intersection of $\overline{EC}$ and $\overline{FN}$, where $C$ is a point on segment $GH$ such that $GC = 6$ cm. What is the area ratio of triangle $EFP$ to triangle $EPG$? | \frac{2}{3} | 0 |
25,809 | Let \( f(m, n) = 3m + n + (m + n)^2 \). Find \( \sum_{m=0}^{\infty} \sum_{n=0}^{\infty} 2^{-f(m, n)} \). | 4/3 | 64.0625 |
25,810 | The railway between Station A and Station B is 840 kilometers long. Two trains start simultaneously from the two stations towards each other, with Train A traveling at 68.5 kilometers per hour and Train B traveling at 71.5 kilometers per hour. After how many hours will the two trains be 210 kilometers apart? | 7.5 | 0 |
25,811 | Given that $\sin\alpha + \sin\beta = \frac{1}{3}$, find the maximum and minimum values of $y = \sin\beta - \cos^2\alpha$. | -\frac{11}{12} | 17.96875 |
25,812 | In trapezoid $PQRS$, the lengths of the bases $PQ$ and $RS$ are 10 and 21 respectively. The legs of the trapezoid are extended beyond $P$ and $Q$ to meet at point $T$. What is the ratio of the area of triangle $TPQ$ to the area of trapezoid $PQRS$? Express your answer as a common fraction. | \frac{100}{341} | 61.71875 |
25,813 | Three spheres with radii $11,$ $13,$ and $19$ are mutually externally tangent. A plane intersects the spheres in three congruent circles centered at $A,$ $B,$ and $C,$ respectively, and the centers of the spheres all lie on the same side of this plane. Suppose that $AB^2 = 560.$ Find $AC^2.$ | 756 | 0 |
25,814 | A natural number $k > 1$ is called *good* if there exist natural numbers $$ a_1 < a_2 < \cdots < a_k $$ such that $$ \dfrac{1}{\sqrt{a_1}} + \dfrac{1}{\sqrt{a_2}} + \cdots + \dfrac{1}{\sqrt{a_k}} = 1 $$ .
Let $f(n)$ be the sum of the first $n$ *[good* numbers, $n \geq$ 1. Find the sum of all values of $n$ for which $f(n+5)/f(n)$ is an integer. | 18 | 10.15625 |
25,815 | In Mr. Smith's science class, there are 3 boys for every 4 girls. If there are 42 students in total in his class, what percent of them are boys? | 42.857\% | 0 |
25,816 | Given a sequence $\{x_n\}$ that satisfies $x_{n+2}=|x_{n+2}-x_n|$ (where $n \in \mathbb{N}^*$), if $x_1=1$, $x_2=a$ (where $a \leqslant 1$ and $a \neq 0$), and $x_{n+3}=x_n$ for any positive integer $n$, then the sum of the first 2017 terms of the sequence $\{x_n\}$ is ______. | 1345 | 15.625 |
25,817 | Petya approaches the entrance door with a combination lock, which has buttons numbered from 0 to 9. To open the door, three correct buttons need to be pressed simultaneously. Petya does not remember the code and tries combinations one by one. Each attempt takes Petya 2 seconds.
a) How much time will Petya need to definitely get inside?
b) On average, how much time will Petya need?
c) What is the probability that Petya will get inside in less than a minute? | \frac{29}{120} | 6.25 |
25,818 | In equilateral triangle $ABC$ a point $P$ lies such that $PA = 7$, $PB = 7$, and $PC = 14$. Determine the area of the triangle $ABC$.
**A)** $49\sqrt{3}$ \\
**B)** $98\sqrt{3}$ \\
**C)** $42\sqrt{3}$ \\
**D)** $21\sqrt{3}$ \\
**E)** $98$ | 49\sqrt{3} | 36.71875 |
25,819 | If the total sum of squared deviations of a set of data is 100, and the correlation coefficient is 0.818, then the sum of squared residuals is. | 33.0876 | 27.34375 |
25,820 | Polina custom makes jewelry for a jewelry store. Each piece of jewelry consists of a chain, a stone, and a pendant. The chains can be silver, gold, or iron. Polina has stones - cubic zirconia, emerald, quartz - and pendants in the shape of a star, sun, and moon. Polina is happy only when three pieces of jewelry are laid out in a row from left to right on the showcase according to the following rules:
- There must be a piece of jewelry with a sun pendant on an iron chain.
- Next to the jewelry with the sun pendant there must be gold and silver jewelry.
- The three pieces of jewelry in the row must have different stones, pendants, and chains.
How many ways are there to make Polina happy? | 24 | 11.71875 |
25,821 | Let \( p, q, r, s \) be distinct real numbers such that the roots of \( x^2 - 12px - 13q = 0 \) are \( r \) and \( s \), and the roots of \( x^2 - 12rx - 13s = 0 \) are \( p \) and \( q \). Find the value of \( p + q + r + s \). | 2028 | 46.09375 |
25,822 | Consider an octagonal lattice where each vertex is evenly spaced and one unit from its nearest neighbor. How many equilateral triangles have all three vertices in this lattice? Every side of the octagon is extended one unit outward with a single point placed at each extension, keeping the uniform distance of one unit between adjacent points. | 24 | 10.9375 |
25,823 | The number $0.324375$ can be written as a fraction $\frac{a}{b}$ for positive integers $a$ and $b$. When this fraction is in simplest terms, what is $a+b$? | 2119 | 62.5 |
25,824 | A two-digit number has its unit digit greater than the tens digit by 2. If this two-digit number is less than 30, find the number. | 24 | 34.375 |
25,825 | A *substring* of a number $n$ is a number formed by removing some digits from the beginning and end of $n$ (possibly a different number of digits is removed from each side). Find the sum of all prime numbers $p$ that have the property that any substring of $p$ is also prime.
| 576 | 78.90625 |
25,826 | Find the minimum value of
$$
\begin{aligned}
A & =\sqrt{\left(1264-z_{1}-\cdots-z_{n}\right)^{2}+x_{n}^{2}+y_{n}^{2}}+ \\
& \sqrt{z_{n}^{2}+x_{n-1}^{2}+y_{n-1}^{2}}+\cdots+\sqrt{z_{2}^{2}+x_{1}^{2}+y_{1}^{2}}+ \\
& \sqrt{z_{1}^{2}+\left(948-x_{1}-\cdots-x_{n}\right)^{2}+\left(1185-y_{1}-\cdots-y_{n}\right)^{2}}
\end{aligned}
$$
where \(x_{i}, y_{i}, z_{i}, i=1,2, \cdots, n\) are non-negative real numbers. | 1975 | 71.09375 |
25,827 | Let $p$, $q$, $r$, $s$, $t$, and $u$ be positive integers with $p+q+r+s+t+u = 2023$. Let $N$ be the largest of the sum $p+q$, $q+r$, $r+s$, $s+t$ and $t+u$. What is the smallest possible value of $N$? | 810 | 0.78125 |
25,828 | If $\{a_n\}$ is an arithmetic sequence, with the first term $a_1 > 0$, $a_{2012} + a_{2013} > 0$, and $a_{2012} \cdot a_{2013} < 0$, then the largest natural number $n$ for which the sum of the first $n$ terms $S_n > 0$ is. | 2012 | 2.34375 |
25,829 | Find $\tan A$ in the right triangle shown below.
[asy]
pair A,B,C;
A = (0,0);
B = (40,0);
C = (0,15);
draw(A--B--C--A);
draw(rightanglemark(B,A,C,20));
label("$A$",A,SW);
label("$B$",B,SE);
label("$C$",C,N);
label("$41$", (B+C)/2,NE);
label("$40$", B/2,S);
[/asy] | \frac{9}{40} | 59.375 |
25,830 | The function $f(n)$ defined on the set of natural numbers $\mathbf{N}$ is given by:
$$
f(n)=\left\{\begin{array}{ll}
n-3 & (n \geqslant 1000); \\
f[f(n+7)] & (n < 1000),
\end{array}\right.
$$
What is the value of $f(90)$? | 999 | 67.1875 |
25,831 | Given that Three people, A, B, and C, are applying to universities A, B, and C, respectively, where each person can only apply to one university, calculate the conditional probability $P\left(A|B\right)$. | \frac{1}{2} | 7.8125 |
25,832 | If I have a $5\times5$ chess board, in how many ways can I place five distinct pawns on the board such that each column and row of the board contains no more than one pawn? | 14400 | 1.5625 |
25,833 | Each triangle in a sequence is either a 30-60-90 triangle or a 45-45-90 triangle. The hypotenuse of each 30-60-90 triangle serves as the longer leg of the adjacent 30-60-90 triangle, except for the final triangle which is a 45-45-90 triangle. The hypotenuse of the largest triangle is 16 centimeters. What is the length of the leg of the last 45-45-90 triangle? Express your answer as a common fraction. | \frac{6\sqrt{6}}{2} | 0 |
25,834 | Let $ABCD$ be a square with each side of 4 units. Construct equilateral triangles $ABE$, $BCF$, $CDG$, and $DAH$ inscribed on each side of the square, inside the square. Let $E, F, G, H$ be the centers, respectively, of these equilateral triangles. What is the ratio of the area of square $EFGH$ to the area of square $ABCD$?
A) $\frac{2}{3}$
B) $\frac{1}{2}$
C) $\frac{1}{3}$
D) $\sqrt{3}$
E) $\sqrt{2}$ | \frac{2}{3} | 19.53125 |
25,835 | In $\triangle ABC$ , three lines are drawn parallel to side $BC$ dividing the altitude of the triangle into four equal parts. If the area of the second largest part is $35$ , what is the area of the whole $\triangle ABC$ ?
[asy]
defaultpen(linewidth(0.7)); size(120);
pair B = (0,0), C = (1,0), A = (0.7,1); pair[] AB, AC;
draw(A--B--C--cycle);
for(int i = 1; i < 4; ++i) {
AB.push((i*A + (4-i)*B)/4); AC.push((i*A + (4-i)*C)/4);
draw(AB[i-1] -- AC[i-1]);
}
filldraw(AB[1]--AB[0]--AC[0]--AC[1]--cycle, gray(0.7));
label(" $A$ ",A,N); label(" $B$ ",B,S); label(" $C$ ",C,S);[/asy] | 560/3 | 0.78125 |
25,836 | Find all $t$ such that $x-t$ is a factor of $4x^2 + 11x - 3$. | -3 | 3.90625 |
25,837 | What is the area of a hexagon where the sides alternate between lengths of 2 and 4 units, and the triangles cut from each corner have base 2 units and altitude 3 units? | 36 | 4.6875 |
25,838 | Given the number 2550, calculate the sum of its prime factors. | 27 | 12.5 |
25,839 | Given Daphne's four friends visit her every 4, 6, 8, and 10 days respectively, all four friends visited her yesterday, calculate the number of days in the next 365-day period when exactly two friends will visit her. | 129 | 0 |
25,840 | 40 pikes were released into a pond. A pike is considered well-fed if it has eaten three other pikes (whether well-fed or hungry). What is the maximum number of pikes that can be well-fed? | 13 | 16.40625 |
25,841 | In triangle $XYZ$, $XY=153$, $XZ=147$, and $YZ=140$. The angle bisector of angle $X$ intersects $\overline{YZ}$ at point $D$, and the angle bisector of angle $Y$ intersects $\overline{XZ}$ at point $E$. Let $P$ and $Q$ be the feet of the perpendiculars from $Z$ to $\overline{YE}$ and $\overline{XD}$, respectively. Find $PQ$. | 67 | 0 |
25,842 | The pressure \( P \) exerted by wind on a sail varies jointly as the area \( A \) of the sail and the cube of the wind's velocity \( V \). When the velocity is \( 8 \) miles per hour, the pressure on a sail of \( 2 \) square feet is \( 4 \) pounds. Find the wind velocity when the pressure on \( 4 \) square feet of sail is \( 32 \) pounds. | 12.8 | 2.34375 |
25,843 | From the $8$ vertices of a cube, select any $4$ vertices. The probability that these $4$ points lie in the same plane is ______. | \frac{6}{35} | 28.90625 |
25,844 | Through vertex $A$ of parallelogram $ABCD$, a line is drawn that intersects diagonal $BD$, side $CD$, and line $BC$ at points $E$, $F$, and $G$, respectively. Find the ratio $BE:ED$ if $FG:FE=4$. Round your answer to the nearest hundredth if needed. | 2.24 | 0 |
25,845 | The minimum and maximum values of the function $f(x)=\cos 2x+\sin x+1$ are $\_\_\_\_\_\_$ and $\_\_\_\_\_\_$, respectively. | \frac{17}{8} | 49.21875 |
25,846 | Ellen wants to color some of the cells of a $4 \times 4$ grid. She wants to do this so that each colored cell shares at least one side with an uncolored cell and each uncolored cell shares at least one side with a colored cell. What is the largest number of cells she can color? | 12 | 3.90625 |
25,847 | Let $ABC$ be a triangle where $\angle$ **B=55** and $\angle$ **C = 65**. **D** is the mid-point of **BC**. Circumcircle of **ACD** and**ABD** cuts **AB** and**AC** at point **F** and **E** respectively. Center of circumcircle of **AEF** is**O**. $\angle$ **FDO** = ? | 30 | 78.125 |
25,848 | The distance between A and C is the absolute value of (k-7) plus the distance between B and C is the square root of ((k-4)^2 + (-1)^2). Find the value of k that minimizes the sum of these two distances. | \frac{11}{2} | 0.78125 |
25,849 | A high school with 2000 students held a "May Fourth" running and mountain climbing competition in response to the call for "Sunshine Sports". Each student participated in only one of the competitions. The number of students from the first, second, and third grades participating in the running competition were \(a\), \(b\), and \(c\) respectively, with \(a:b:c=2:3:5\). The number of students participating in mountain climbing accounted for \(\frac{2}{5}\) of the total number of students. To understand the students' satisfaction with this event, a sample of 200 students was surveyed. The number of second-grade students participating in the running competition that should be sampled is \_\_\_\_\_. | 36 | 38.28125 |
25,850 | 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 $A \triangle B = \_\_\_\_\_$;
(2) Let $\text{Card}(M)$ represent the number of elements in a finite set $M$. When $\text{Card}(X \triangle A) + \text{Card}(X \triangle B)$ takes the minimum value, the number of possible sets $X$ is $\_\_\_\_\_$. | 16 | 60.9375 |
25,851 | Given an isosceles triangle DEF with DE = DF = 5√3, a circle with radius 6 is tangent to DE at E and to DF at F. If the altitude from D to EF intersects the circle at its center, find the area of the circle that passes through vertices D, E, and F. | 36\pi | 3.90625 |
25,852 | Given that $b$ is a multiple of $570$, find the greatest common divisor of $4b^3 + 2b^2 + 5b + 171$ and $b$. | 171 | 4.6875 |
25,853 | In $\triangle ABC$, $\angle A = 60^\circ$, $AB > AC$, point $O$ is the circumcenter, and the altitudes $BE$ and $CF$ intersect at point $H$. Points $M$ and $N$ are on segments $BH$ and $HF$ respectively, such that $BM = CN$. Find the value of $\frac{MH + NH}{OH}$. | \sqrt{3} | 0 |
25,854 | A rectangular piece of paper $A B C D$ is folded and flattened as shown in the diagram, so that triangle $D C F$ falls onto triangle $D E F$, with vertex $E$ precisely landing on side $A B$. Given $\angle 1 = 22^\circ$, find the measure of $\angle 2$. | 44 | 2.34375 |
25,855 | Given the function \[f(x) = \left\{ \begin{aligned} x+3 & \quad \text{ if } x < 2 \\ x^2 & \quad \text{ if } x \ge 2 \end{aligned} \right.\] determine the value of \(f^{-1}(-5) + f^{-1}(-4) + \dots + f^{-1}(2) + f^{-1}(3). \) | -35 + \sqrt{2} + \sqrt{3} | 1.5625 |
25,856 | What is the smallest positive integer with exactly 12 positive integer divisors? | 96 | 0 |
25,857 | Given a square piece of paper with side length $s$ folded in half diagonally, then cut along a line perpendicular to the fold from the midpoint of the hypotenuse to the opposite side, forming a large rectangle and two smaller, identical triangles, find the ratio of the perimeter of one of the small triangles to the perimeter of the large rectangle. | \frac{2}{3} | 3.125 |
25,858 | A bus ticket is numbered with six digits: from 000000 to 999999. You buy one ticket. What is the probability that you will get a ticket whose digits are in ascending (or descending) order? | 0.00042 | 0.78125 |
25,859 | There is a unique two-digit positive integer $u$ for which the last two digits of $15\cdot u$ are $45$, and $u$ leaves a remainder of $7$ when divided by $17$. | 43 | 3.90625 |
25,860 | Find the greatest real $k$ such that, for every tetrahedron $ABCD$ of volume $V$ , the product of areas of faces $ABC,ABD$ and $ACD$ is at least $kV^2$ . | 9/2 | 0 |
25,861 | How many non-empty subsets $S$ of $\{1, 2, 3, \ldots, 10\}$ satisfy the following two properties?
1. No two consecutive integers belong to $S$.
2. If $S$ contains $k$ elements, then $S$ contains no number less than $k$. | 143 | 62.5 |
25,862 | Find the smallest \( n \) such that whenever the elements of the set \(\{1, 2, \ldots, n\}\) are colored red or blue, there always exist \( x, y, z, w \) (not necessarily distinct) of the same color such that \( x + y + z = w \). | 11 | 68.75 |
25,863 | A school library purchased 17 identical books. How much do they cost if they paid more than 11 rubles 30 kopecks, but less than 11 rubles 40 kopecks for 9 of these books? | 2142 | 0.78125 |
25,864 | Given points \(A=(8,15)\) and \(B=(16,9)\) are on a circle \(\omega\), and the tangent lines to \(\omega\) at \(A\) and \(B\) meet at a point \(P\) on the x-axis, calculate the area of the circle \(\omega\). | 250\pi | 1.5625 |
25,865 | How many ways can the integers from $-7$ to $7$ inclusive be arranged in a sequence such that the absolute value of the numbers in the sequence does not decrease? | 128 | 21.09375 |
25,866 | In triangle $ABC$, let vector $\vec{a} = (1, \cos B)$ and vector $\vec{b} = (\sin B, 1)$, and suppose $\vec{a}$ is perpendicular to $\vec{b}$. Find the magnitude of angle $B$. | \frac{3\pi}{4} | 32.8125 |
25,867 | The circular region of the sign now has an area of 50 square inches. To decorate the edge with a ribbon, Vanessa plans to purchase 5 inches more than the circle’s circumference. How many inches of ribbon should she buy if she estimates \(\pi = \frac{22}{7}\)? | 30 | 10.9375 |
25,868 | A man chooses two positive integers \( m \) and \( n \). He defines a positive integer \( k \) to be good if a triangle with side lengths \( \log m \), \( \log n \), and \( \log k \) exists. He finds that there are exactly 100 good numbers. Find the maximum possible value of \( mn \). | 134 | 0 |
25,869 | In the convex pentagon $ABCDE$, $\angle A = \angle B = 120^{\circ}$, $EA = AB = BC = 2$, and $CD = DE = 4$. Calculate the area of $ABCDE$. | 7\sqrt{3} | 10.15625 |
25,870 | Find $(100110_2 + 1001_2) \times 110_2 \div 11_2$. Express your answer in base 2. | 1011110_2 | 86.71875 |
25,871 | A dealer plans to sell a new type of air purifier. After market research, the following pattern was discovered: When the profit per purifier is $x$ (unit: Yuan, $x > 0$), the sales volume $q(x)$ (unit: hundred units) and $x$ satisfy the following relationship: If $x$ does not exceed $20$, then $q(x)=\dfrac{1260}{x+1}$; If $x$ is greater than or equal to $180$, then the sales volume is zero; When $20\leqslant x\leqslant 180$, $q(x)=a-b\sqrt{x}$ ($a,b$ are real constants).
$(1)$ Find the expression for the function $q(x)$;
$(2)$ At what value of $x$ does the total profit (unit: Yuan) reach its maximum value, and what is this maximum value? | 240000 | 0 |
25,872 | Calculate the number of five-digit numbers formed from the digits 1, 2, 3, 4, 5 without repeating any digits, and where the digits 1 and 2 are not adjacent to the digit 5. | 36 | 34.375 |
25,873 | In the "five rows by six columns" examination room, if two students A and B from the same school are to be seated with no two adjacent seats in any direction (front, back, left, right), calculate the number of different seating arrangements for students A and B. | 772 | 18.75 |
25,874 | Find the value of $(8x - 5)^2$ given that the number $x$ satisfies the equation $7x^2 + 6 = 5x + 11$. | \frac{2865 - 120\sqrt{165}}{49} | 9.375 |
25,875 | Determine the sum of all integer values $n$ for which $\binom{25}{n} + \binom{25}{12} = \binom{26}{13}$. | 13 | 0 |
25,876 | Given that the angle between the generating line and the axis of a cone is $\frac{\pi}{3}$, and the length of the generating line is $3$, find the maximum value of the cross-sectional area through the vertex. | \frac{9}{2} | 0.78125 |
25,877 | Let the function $f(x) = \frac{bx}{\ln x} - ax$, where $e$ is the base of the natural logarithm.
(I) If the tangent line to the graph of the function $f(x)$ at the point $(e^2, f(e^2))$ is $3x + 4y - e^2 = 0$, find the values of the real numbers $a$ and $b$.
(II) When $b = 1$, if there exist $x_1, x_2 \in [e, e^2]$ such that $f(x_1) \leq f'(x_2) + a$ holds, find the minimum value of the real number $a$. | \frac{1}{2} - \frac{1}{4e^2} | 4.6875 |
25,878 | What is half of the absolute value of the difference of the squares of 21 and 15 added to the absolute value of the difference of their cubes? | 3051 | 2.34375 |
25,879 | What is the maximum value of $\frac{(3^t - 5t)t}{9^t}$ for real values of $t$? | \frac{1}{20} | 0 |
25,880 | Complex numbers $p,$ $q,$ $r$ form an equilateral triangle with side length 24 in the complex plane. If $|p + q + r| = 48,$ find $|pq + pr + qr|.$ These complex numbers have been translated by the same complex number $z$ compared to their original positions on the origin. | 768 | 32.03125 |
25,881 | Using the 0.618 method to select a trial point, if the experimental interval is $[2, 4]$, with $x_1$ being the first trial point and the result at $x_1$ being better than that at $x_2$, then the value of $x_3$ is ____. | 3.236 | 1.5625 |
25,882 | Two railway tracks intersect at a right angle. Two trains are simultaneously speeding towards the intersection point from different tracks: one from a station located 40 km from the intersection point, and the other from a station located 50 km away. The first train travels at 800 meters per minute, while the second train travels at 600 meters per minute. After how many minutes from departure will the distance between the trains be minimized? What is this minimum distance? | 16 | 14.84375 |
25,883 | Fnargs are either red or blue and have 2, 3, or 4 heads. A group of six Fnargs consisting of one of each possible form (one red and one blue for each number of heads) is made to line up such that no immediate neighbors are the same color nor have the same number of heads. How many ways are there of lining them up from left to right? | 12 | 0 |
25,884 | Let $m \ge 2$ be an integer and let $T = \{2,3,4,\ldots,m\}$. Find the smallest value of $m$ such that for every partition of $T$ into two subsets, at least one of the subsets contains integers $a$, $b$, and $c$ (not necessarily distinct) such that $a + b = c$. | 15 | 2.34375 |
25,885 | Let $(b_1, b_2, ... b_{12})$ be a list of the 12 integers from 4 to 15 inclusive such that for each $2 \le i \le 12$, either $b_i + 1$ or $b_i - 1$ or both appear somewhere before $b_i$ in the list. How many such lists are there? | 2048 | 7.03125 |
25,886 | In a triangle $ABC$ , the median $AD$ divides $\angle{BAC}$ in the ratio $1:2$ . Extend $AD$ to $E$ such that $EB$ is perpendicular $AB$ . Given that $BE=3,BA=4$ , find the integer nearest to $BC^2$ . | 29 | 0.78125 |
25,887 | Find the length of a line segment from a vertex to the center of a regular hexagon with a side length of 12. Express your answer in simplest radical form. | 4\sqrt{3} | 0 |
25,888 | A group of students during a sports meeting lined up for a team photo. When they lined up in rows of 5, there were two students left over. When they formed rows of 6 students, there were three extra students, and when they lined up in rows of 8, there were four students left over. What is the fewest number of students possible in this group? | 59 | 0 |
25,889 | Before the soccer match between the "North" and "South" teams, five predictions were made:
a) There will be no draw;
b) "South" will concede goals;
c) "North" will win;
d) "North" will not lose;
e) Exactly 3 goals will be scored in the match.
After the match, it was found that exactly three predictions were correct. What was the final score of the match? | 2-1 | 10.9375 |
25,890 | Let $p$ and $q$ be real numbers, and suppose that the roots of the equation \[x^3 - 9x^2 + px - q = 0\] are three distinct positive integers. Compute $p + q.$ | 38 | 0 |
25,891 | I have a bag containing red and green marbles. Initially, the ratio of red to green marbles is 3:2. If I remove 18 red marbles and add 15 green marbles, the new ratio becomes 2:5. How many red marbles were there in the bag initially? | 33 | 1.5625 |
25,892 | If the system of equations
\[
\begin{align*}
4x + y &= a, \\
3x + 4y^2 &= 3a,
\end{align*}
\]
has a solution $(x,y)$ when $x=3$, compute $a$. | 9.75 | 3.90625 |
25,893 | In a right triangle \(ABC\) with \(\angle C = 90^{\circ}\), a segment \(BD\) equal to the leg \(BC\) is laid out on the extension of the hypotenuse \(AB\), and point \(D\) is connected to \(C\). Find \(CD\) if \(BC = 7\) and \(AC = 24\). | 8 \sqrt{7} | 22.65625 |
25,894 | For all positive integers $n$, let $g(n)=\log_{3003} n^3$. Find $g(7)+g(11)+g(13)$. | \frac{9}{4} | 0 |
25,895 | When three standard dice are tossed, the numbers $x, y, z$ are obtained. Find the probability that $xyz = 8$. | \frac{1}{36} | 34.375 |
25,896 | Find the coefficient of \(x^8\) in the polynomial expansion of \((1-x+2x^2)^5\). | 80 | 32.8125 |
25,897 | How many of the natural numbers from 1 to 800, inclusive, contain the digit 7 at least once? | 233 | 0 |
25,898 | What is $1010101_2 + 111000_2$? Write your answer in base $10$. | 141 | 57.03125 |
25,899 | Consider the infinite series \(S\) represented by \(2 - 1 - \frac{1}{3} + \frac{1}{9} - \frac{1}{27} - \frac{1}{81} + \frac{1}{243} - \cdots\). Find the sum \(S\). | \frac{3}{4} | 10.9375 |
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