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25,900 | Given an ellipse with the equation \\(\\dfrac{x^{2}}{a^{2}}+\\dfrac{y^{2}}{b^{2}}=1(a > b > 0)\\) and an eccentricity of \\(\\dfrac{\\sqrt{3}}{2}\\). A line $l$ is drawn through one of the foci of the ellipse, perpendicular to the $x$-axis, and intersects the ellipse at points $M$ and $N$, with $|MN|=1$. Point $P$ is located at $(-b,0)$. Point $A$ is any point on the circle $O:x^{2}+y^{2}=b^{2}$ that is different from point $P$. A line is drawn through point $P$ perpendicular to $PA$ and intersects the circle $x^{2}+y^{2}=a^{2}$ at points $B$ and $C$.
(1) Find the standard equation of the ellipse;
(2) Determine whether $|BC|^{2}+|CA|^{2}+|AB|^{2}$ is a constant value. If it is, find that value; if not, explain why. | 26 | 3.125 |
25,901 | On a blackboard, the number 123456789 is written. Select two adjacent digits from this number, and if neither of them is 0, subtract 1 from each and swap their positions. For example: \( 123456789 \rightarrow 123436789 \rightarrow \cdots \). After performing this operation several times, what is the smallest possible number that can be obtained? The answer is __. | 101010101 | 0 |
25,902 | 4.3535… is a decimal, which can be abbreviated as , and the repeating cycle is . | 35 | 14.0625 |
25,903 | In triangle $ \triangle ABC $, the sides opposite angles A, B, C are respectively $ a, b, c $, with $ A = \frac{\pi}{4} $, $ \sin A + \sin(B - C) = 2\sqrt{2}\sin 2C $ and the area of $ \triangle ABC $ is 1. Find the length of side $ BC $. | \sqrt{5} | 6.25 |
25,904 | The scores (in points) of the 15 participants in the final round of a math competition are as follows: $56$, $70$, $91$, $98$, $79$, $80$, $81$, $83$, $84$, $86$, $88$, $90$, $72$, $94$, $78$. What is the $80$th percentile of these 15 scores? | 90.5 | 0.78125 |
25,905 | Let the sequence $\{x_n\}$ be defined by $x_1 \in \{5, 7\}$ and, for $k \ge 1, x_{k+1} \in \{5^{x_k} , 7^{x_k} \}$ . For example, the possible values of $x_3$ are $5^{5^5}, 5^{5^7}, 5^{7^5}, 5^{7^7}, 7^{5^5}, 7^{5^7}, 7^{7^5}$ , and $7^{7^7}$ . Determine the sum of all possible values for the last two digits of $x_{2012}$ . | 75 | 39.0625 |
25,906 | Consider a coordinate plane where at each lattice point, there is a circle with radius $\frac{1}{8}$ and a square with sides of length $\frac{1}{4}$, whose sides are parallel to the coordinate axes. A line segment runs from $(0,0)$ to $(729, 243)$. Determine how many of these squares and how many of these circles are intersected by the line segment, and find the total count of intersections, i.e., $m + n$. | 972 | 1.5625 |
25,907 | The number of integer solutions to the inequality $\log_{3}|x-2| < 2$. | 17 | 92.1875 |
25,908 | Given that \( x_{1}, x_{2}, \cdots, x_{n} \) are real numbers, find the minimum value of
\[
E(x_{1}, x_{2}, \cdots, x_{n}) = \sum_{i=1}^{n} x_{i}^{2} + \sum_{i=1}^{n-1} x_{i} x_{i+1} + \sum_{i=1}^{n} x_{i}
\] | -1/2 | 0 |
25,909 | Eight congruent copies of the parabola \( y = x^2 \) are arranged symmetrically around a circle such that each vertex is tangent to the circle, and each parabola is tangent to its two neighbors. Find the radius of the circle. Assume that one of the tangents to the parabolas corresponds to the line \( y = x \tan(45^\circ) \). | \frac{1}{4} | 0 |
25,910 | Jia and Yi are dividing 999 playing cards numbered 001, 002, 003, ..., 998, 999. All the cards whose numbers have all three digits not greater than 5 belong to Jia; cards whose numbers have one or more digits greater than 5 belong to Yi.
(1) How many cards does Jia get?
(2) What is the sum of the numbers on all the cards Jia gets? | 59940 | 35.9375 |
25,911 | A 30 foot ladder is placed against a vertical wall of a building. The foot of the ladder is 11 feet from the base of the building. If the top of the ladder slips 6 feet, then the foot of the ladder will slide how many feet? | 9.49 | 2.34375 |
25,912 | Given the universal set $U=\{1, 2, 3, 4, 5, 6, 7, 8\}$, a set $A=\{a_1, a_2, a_3, a_4\}$ is formed by selecting any four elements from $U$, and the set of the remaining four elements is denoted as $\complement_U A=\{b_1, b_2, b_3, b_4\}$. If $a_1+a_2+a_3+a_4 < b_1+b_2+b_3+b_4$, then the number of ways to form set $A$ is \_\_\_\_\_\_. | 31 | 98.4375 |
25,913 | If a number is randomly selected from the set $\left\{ \frac{1}{3}, \frac{1}{4}, 3, 4 \right\}$ and denoted as $a$, and another number is randomly selected from the set $\left\{ -1, 1, -2, 2 \right\}$ and denoted as $b$, then the probability that the graph of the function $f(x) = a^{x} + b$ ($a > 0, a \neq 1$) passes through the third quadrant is ______. | \frac{3}{8} | 0.78125 |
25,914 | There are 10 numbers written in a circle, and their sum is 100. It is known that the sum of any three consecutive numbers is not less than 29.
Determine the smallest number \( A \) such that in any such set of numbers, each number does not exceed \( A \). | 13 | 63.28125 |
25,915 | Given that in triangle $\triangle ABC$, the opposite sides of angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively, and it is known that $\frac{{c \sin C}}{a} - \sin C = \frac{{b \sin B}}{a} - \sin A$, $b = 4$. Find:
$(1)$ The measure of angle $B$;
$(2)$ If $c = \frac{{4\sqrt{6}}}{3}$, find the area of $\triangle ABC$. | 4 + \frac{{4\sqrt{3}}}{3} | 0 |
25,916 | The diagram shows a square and a regular decagon that share an edge. One side of the square is extended to meet an extended edge of the decagon. What is the value of \( x \)? | 18 | 1.5625 |
25,917 | What is the smallest prime whose digits sum to 23? | 1993 | 0 |
25,918 | Rectangle ABCD has AB = 6 and BC = 3. Point M is chosen on side AB so that ∠AMD = ∠CMD. Calculate the degree measure of ∠AMD. | 67.5 | 0.78125 |
25,919 | If I have a $5\times 5$ chess board, in how many ways can I place five distinct pawns on the board such that no row and no column contains more than one pawn? | 14400 | 0 |
25,920 | Define $\displaystyle{f(x) = x + \sqrt{x + \sqrt{x + \sqrt{x + \sqrt{x + \ldots}}}}}$ . Find the smallest integer $x$ such that $f(x)\ge50\sqrt{x}$ .
(Edit: The official question asked for the "smallest integer"; the intended question was the "smallest positive integer".) | 2500 | 0 |
25,921 | Two players take turns placing Xs and Os in the cells of a $9 \times 9$ square (the first player places Xs, and their opponent places Os). At the end of the game, the number of rows and columns where there are more Xs than Os are counted as points for the first player. The number of rows and columns where there are more Os than Xs are counted as points for the second player. How can the first player win (score more points)? | 10 | 7.8125 |
25,922 | Find \(\sin \alpha\) if \(\cos \alpha = \operatorname{tg} \beta\), \(\cos \beta = \operatorname{tg} \gamma\), \(\cos \gamma = \operatorname{tg} \alpha\) \((0 < \alpha < \frac{\pi}{2}, 0 < \beta < \frac{\pi}{2}, 0 < \gamma < \frac{\pi}{2})\). | \frac{\sqrt{2}}{2} | 21.875 |
25,923 | Given $a$ and $b$ are the roots of the equation $x^2-2cx-5d = 0$, and $c$ and $d$ are the roots of the equation $x^2-2ax-5b=0$, where $a,b,c,d$ are distinct real numbers, find $a+b+c+d$. | 30 | 33.59375 |
25,924 | Star lists the whole numbers $1$ through $50$ once. Emilio copies Star's numbers, but he replaces each occurrence of the digit $2$ by the digit $1$ and each occurrence of the digit $3$ by the digit $2$. Calculate the difference between Star's sum and Emilio's sum. | 210 | 0 |
25,925 | If $\lceil{\sqrt{x}}\rceil=12$, how many possible integer values of $x$ are there? | 23 | 89.0625 |
25,926 | In a 7x7 geoboard, points A and B are positioned at (3,3) and (5,3) respectively. How many of the remaining 47 points will result in triangle ABC being isosceles? | 10 | 8.59375 |
25,927 | Two circles \(C_{1}\) and \(C_{2}\) touch each other externally and the line \(l\) is a common tangent. The line \(m\) is parallel to \(l\) and touches the two circles \(C_{1}\) and \(C_{3}\). The three circles are mutually tangent. If the radius of \(C_{2}\) is 9 and the radius of \(C_{3}\) is 4, what is the radius of \(C_{1}\)? | 12 | 1.5625 |
25,928 | A tourist city was surveyed, and it was found that the number of tourists per day $f(t)$ (in ten thousand people) and the time $t$ (in days) within the past month (calculated as $30$ days) approximately satisfy the function relationship $f(t)=4+ \frac {1}{t}$. The average consumption per person $g(t)$ (in yuan) and the time $t$ (in days) approximately satisfy the function relationship $g(t)=115-|t-15|$.
(I) Find the function relationship of the daily tourism income $w(t)$ (in ten thousand yuan) and time $t(1\leqslant t\leqslant 30,t\in N)$ of this city;
(II) Find the minimum value of the daily tourism income of this city (in ten thousand yuan). | 403 \frac {1}{3} | 0 |
25,929 | Given two geometric sequences $\{a_n\}$ and $\{b_n\}$, satisfying $a_1=a$ ($a>0$), $b_1-a_1=1$, $b_2-a_2=2$, and $b_3-a_3=3$.
(1) If $a=1$, find the general formula for the sequence $\{a_n\}$.
(2) If the sequence $\{a_n\}$ is unique, find the value of $a$. | \frac{1}{3} | 0.78125 |
25,930 | A truck delivered 4 bags of cement. They are stacked in the truck. A worker can carry one bag at a time either from the truck to the gate or from the gate to the shed. The worker can carry the bags in any order, each time taking the top bag, carrying it to the respective destination, and placing it on top of the existing stack (if there are already bags there). If given a choice to carry a bag from the truck or from the gate, the worker randomly chooses each option with a probability of 0.5. Eventually, all the bags end up in the shed.
a) (7th grade level, 1 point). What is the probability that the bags end up in the shed in the reverse order compared to how they were placed in the truck?
b) (7th grade level, 1 point). What is the probability that the bag that was second from the bottom in the truck ends up as the bottom bag in the shed? | \frac{1}{8} | 7.03125 |
25,931 | Five packages are delivered to five houses, one to each house. If the packages are randomly delivered, what is the probability that exactly three of them are delivered to their correct houses? | \frac{1}{6} | 3.125 |
25,932 | Solve the following quadratic equation: $x^2 + 5x - 4 = 0.$ | \frac{-5 - \sqrt{41}}{2} | 3.125 |
25,933 | Quadratic equations of the form $ax^2 + bx + c = 0$ have real roots with coefficients $a$, $b$, and $c$ selected from the set of integers $\{1, 2, 3, 4\}$, where $a \neq b$. Calculate the total number of such equations. | 17 | 0 |
25,934 | Five integers have an average of 69. The middle integer (the median) is 83. The most frequently occurring integer (the mode) is 85. The range of the five integers is 70. What is the second smallest of the five integers? | 77 | 79.6875 |
25,935 | The parametric equation of curve $C_{1}$ is $\begin{cases} x=2+2\cos \alpha \\ y=2\sin \alpha \end{cases}$ ($\alpha$ is the parameter), with the origin $O$ as the pole and the positive $x$-axis as the polar axis, a polar coordinate system is established. The curve $C_{2}$: $\rho=2\cos \theta$ intersects with the polar axis at points $O$ and $D$.
(I) Write the polar equation of curve $C_{1}$ and the polar coordinates of point $D$;
(II) The ray $l$: $\theta=\beta (\rho > 0, 0 < \beta < \pi)$ intersects with curves $C_{1}$ and $C_{2}$ at points $A$ and $B$, respectively. Given that the area of $\triangle ABD$ is $\frac{\sqrt{3}}{2}$, find $\beta$. | \frac{\pi}{3} | 53.125 |
25,936 | Given that the sequence $\{a_{n}\}$ is a geometric sequence with a common ratio $q\neq 1$, $a_{1}=3$, $3a_{1}$, $2a_{2}$, $a_{3}$ form an arithmetic sequence, and the terms of the sequence $\{a_{n}\}$ are arranged in a certain order as $a_{1}$, $a_{1}$, $a_{2}$, $a_{1}$, $a_{2}$, $a_{3}$, $a_{1}$, $a_{2}$, $a_{3}$, $a_{4}$, $\ldots$, determine the value of the sum $S_{23}$ of the first 23 terms of the new sequence $\{b_{n}\}$. | 1641 | 4.6875 |
25,937 | Calculate the sum of the repeating decimals $0.\overline{2}$, $0.\overline{02}$, and $0.\overline{0002}$ as a common fraction. | \frac{224422}{9999} | 0 |
25,938 | In triangle $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. It is known that $4a = \sqrt{5}c$ and $\cos C = \frac{3}{5}$.
$(Ⅰ)$ Find the value of $\sin A$.
$(Ⅱ)$ If $b = 11$, find the area of $\triangle ABC$. | 22 | 36.71875 |
25,939 | Let \( a, b, \) and \( c \) be positive real numbers. Find the minimum value of
\[
\frac{(a^2 + 4a + 2)(b^2 + 4b + 2)(c^2 + 4c + 2)}{abc}.
\] | 216 | 7.03125 |
25,940 | Fill in numbers in the boxes below so that the sum of the entries in each three consecutive boxes is $2005$ . What is the number that goes into the leftmost box?
[asy]
size(300);
label("999",(2.5,.5));
label("888",(7.5,.5));
draw((0,0)--(9,0));
draw((0,1)--(9,1));
for (int i=0; i<=9; ++i)
{
draw((i,0)--(i,1));
}
[/asy] | 118 | 45.3125 |
25,941 | There are $12$ small balls in a bag, which are red, black, and yellow respectively (these balls are the same in other aspects except for color). The probability of getting a red ball when randomly drawing one ball is $\frac{1}{3}$, and the probability of getting a black ball is $\frac{1}{6}$ more than getting a yellow ball. What are the probabilities of getting a black ball and a yellow ball respectively? | \frac{1}{4} | 10.15625 |
25,942 | In acute triangle $\triangle ABC$, if $\sin A = 3\sin B\sin C$, then the minimum value of $\tan A\tan B\tan C$ is \_\_\_\_\_\_. | 12 | 30.46875 |
25,943 | Rectangle $ABCD$ has an area of $32$, and side $\overline{AB}$ is parallel to the x-axis. Side $AB$ measures $8$ units. Vertices $A,$ $B$, and $C$ are located on the graphs of $y = \log_a x$, $y = 2\log_a x$, and $y = 4\log_a x$, respectively. Determine the value of $a$.
A) $\sqrt[3]{\frac{1 + \sqrt{33}}{2} + 8}$
B) $\sqrt[4]{\frac{1 + \sqrt{33}}{2} + 8}$
C) $\sqrt{\frac{1 + \sqrt{33}}{2} + 8}$
D) $\sqrt[6]{\frac{1 + \sqrt{33}}{2} + 8}$
E) $\sqrt[5]{\frac{1 + \sqrt{43}}{2} + 8}$ | \sqrt[4]{\frac{1 + \sqrt{33}}{2} + 8} | 24.21875 |
25,944 | Given a parallelogram \\(ABCD\\) where \\(AD=2\\), \\(∠BAD=120^{\\circ}\\), and point \\(E\\) is the midpoint of \\(CD\\), if \\( \overrightarrow{AE} \cdot \overrightarrow{BD}=1\\), then \\( \overrightarrow{BD} \cdot \overrightarrow{BE}=\\) \_\_\_\_\_\_. | 13 | 3.90625 |
25,945 | Compute the integral: \(\int_{0}^{\pi / 2}\left(\sin ^{2}(\sin x) + \cos ^{2}(\cos x)\right) \,dx\). | \frac{\pi}{4} | 0.78125 |
25,946 | Define a positive integer $n$ to be a factorial tail if there is some positive integer $m$ such that the decimal representation of $m!$ ends with exactly $n$ zeroes. How many positive integers less than $2500$ are not factorial tails? | 500 | 12.5 |
25,947 | Let the two foci of the conic section \\(\Gamma\\) be \\(F_1\\) and \\(F_2\\), respectively. If there exists a point \\(P\\) on the curve \\(\Gamma\\) such that \\(|PF_1|:|F_1F_2|:|PF_2|=4:3:2\\), then the eccentricity of the curve \\(\Gamma\\) is \_\_\_\_\_\_\_\_ | \dfrac{3}{2} | 29.6875 |
25,948 | Let $ABC$ be a triangle such that $AB = 7$ , and let the angle bisector of $\angle BAC$ intersect line $BC$ at $D$ . If there exist points $E$ and $F$ on sides $AC$ and $BC$ , respectively, such that lines $AD$ and $EF$ are parallel and divide triangle $ABC$ into three parts of equal area, determine the number of possible integral values for $BC$ . | 13 | 3.90625 |
25,949 | Mark had a box of chocolates. He consumed $\frac{1}{4}$ of them and then gave $\frac{1}{3}$ of what remained to his friend Lucy. Mark and his father then each ate 20 chocolates from what Mark had left. Finally, Mark's sister took between five and ten chocolates, leaving Mark with four chocolates. How many chocolates did Mark start with? | 104 | 0.78125 |
25,950 | Given that the leftmost digit is odd and less than 5, the second digit is an even number less than 6, all four digits are different, and the number is divisible by 5, find the number of four-digit numbers that satisfy these conditions. | 48 | 0.78125 |
25,951 | Triangle $DEF$ has side lengths $DE = 15$, $EF = 36$, and $FD = 39$. Rectangle $WXYZ$ has vertex $W$ on $\overline{DE}$, vertex $X$ on $\overline{DF}$, and vertices $Y$ and $Z$ on $\overline{EF}$. In terms of the side length $WX = \omega$, the area of $WXYZ$ can be expressed as the quadratic polynomial
\[Area(WXYZ) = \alpha \omega - \beta \omega^2.\]
Then the coefficient $\beta = \frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$. | 17 | 36.71875 |
25,952 | The value of \( 2 \frac{1}{10} + 3 \frac{11}{100} \) can be calculated. | 5.21 | 3.125 |
25,953 | Find the integer $x$ that satisfies the equation $10x + 3 \equiv 7 \pmod{18}$. | 13 | 0 |
25,954 | In $\triangle ABC$, let $a$, $b$, and $c$ be the lengths of the sides opposite angles $A$, $B$, and $C$ respectively, and the three interior angles $A$, $B$, $C$ satisfy $A+C=2B$.
$\text{(1)}$ If $b=2$, find the maximum value of the area of $\triangle ABC$ and determine the shape of the triangle when the maximum area is achieved;
$\text{(2)}$ If $\dfrac {1}{\cos A} + \dfrac {1}{\cos C} = -\dfrac {\sqrt {2}}{\cos B}$, find the value of $\cos \dfrac {A-C}{2}$. | \dfrac{\sqrt{2}}{2} | 32.8125 |
25,955 | Given the function $f(x)=2\sqrt{2}\cos x\sin\left(x+\frac{\pi}{4}\right)-1$.
(I) Find the value of $f\left(\frac{\pi}{4}\right)$;
(II) Find the maximum and minimum values of $f(x)$ in the interval $\left[0, \frac{\pi}{2}\right]$. | -1 | 66.40625 |
25,956 | Marissa constructs a large spherical snow sculpture by placing smaller snowballs inside it with radii of 4 inches, 6 inches, and 8 inches. Assuming all snowballs are perfectly spherical and fit exactly inside the sculpture, and the sculpture itself is a sphere whose radius is such that the sum of the volumes of the smaller snowballs equals the volume of the sculpture, find the radius of the sculpture. Express your answer in terms of \( \pi \). | \sqrt[3]{792} | 47.65625 |
25,957 | In three-dimensional space, the volume of the geometric body formed by points whose distance to line segment $A B$ is no greater than three units is $216 \pi$. What is the length of the line segment $A B$? | 20 | 81.25 |
25,958 | In the arithmetic sequence $\{a_{n}\}$, $\frac{a_{1010}}{a_{1009}} < -1$. If its first $n$ terms' sum $S_{n}$ has a maximum value, then the maximum positive integer value of $n$ that makes $S_{n} > 0$ is $\_\_\_\_\_\_\_\_\_\_.$ | 2018 | 1.5625 |
25,959 | Given the function $f(x) = \begin{cases} x-5, & x\geq 2000 \\ f[f(x+8)], & x<2000 \end{cases}$, calculate $f(1996)$. | 2002 | 0 |
25,960 | Given the sequence ${a_n}$ that satisfies the equation $a_{n+1}+(-1)^{n}a_{n}=3n-1,(n∈N^{*})$, determine the sum of the first 40 terms of the sequence ${a_n}$. | 1240 | 3.90625 |
25,961 | Sara used $\frac{5}{8}$ of a roll of wrapping paper to wrap four presents. She used an additional $\frac{1}{24}$ of a roll on one of the presents for decorative purposes. How much wrapping paper did she use on each of the other three presents? | \frac{7}{36} | 16.40625 |
25,962 | In triangle $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively, and $(\sin A + \sin B)(a-b) = c(\sin C - \sqrt{3}\sin B)$.
$(1)$ Find the measure of angle $A$;
$(2)$ If $\cos \angle ABC = -\frac{1}{7}$, $D$ is a point on segment $AC$, $\angle ABD = \angle CBD$, $BD = \frac{7\sqrt{7}}{3}$, find $c$. | 7\sqrt{3} | 0 |
25,963 | Given an ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ ($a > b > 0$) with a focal distance of $2\sqrt{3}$, the line $l_1: y = kx$ ($k \neq 0$) intersects the ellipse at points A and B. A line $l_2$ passing through point B with a slope of $\frac{1}{4}k$ intersects the ellipse at another point D, and $AD \perp AB$.
1. Find the equation of the ellipse.
2. Suppose the line $l_2$ intersects the x-axis and y-axis at points M and N, respectively. Find the maximum value of the area of $\triangle OMN$. | \frac{9}{8} | 14.84375 |
25,964 | A bird discovered $543_{8}$ different ways to build a nest in each of its eight tree homes. How many ways are there in base 10? | 355 | 94.53125 |
25,965 | A group of $6$ friends are to be seated in the back row of an otherwise empty movie theater with $8$ seats in a row. Euler and Gauss are best friends and must sit next to each other with no empty seat between them, while Lagrange cannot sit in an adjacent seat to either Euler or Gauss. Calculate the number of different ways the $6$ friends can be seated in the back row. | 3360 | 3.125 |
25,966 | Given that
$$
\begin{array}{l}
a + b + c = 5, \\
a^2 + b^2 + c^2 = 15, \\
a^3 + b^3 + c^3 = 47.
\end{array}
$$
Find the value of \((a^2 + ab + b^2)(b^2 + bc + c^2)(c^2 + ca + a^2)\). | 625 | 1.5625 |
25,967 | Given $\overrightarrow{a}=(\tan (\theta+ \frac {\pi}{12}),1)$ and $\overrightarrow{b}=(1,-2)$, where $\overrightarrow{a} \perp \overrightarrow{b}$, find the value of $\tan (2\theta+ \frac {5\pi}{12})$. | - \frac{1}{7} | 54.6875 |
25,968 | Given two lines $l_{1}$: $mx+2y-2=0$ and $l_{2}$: $5x+(m+3)y-5=0$, if $l_{1}$ is parallel to $l_{2}$, find the value of $m$. | -5 | 16.40625 |
25,969 | Given the line $l: x-y+4=0$ and the circle $C: \begin{cases}x=1+2\cos \theta \\ y=1+2\sin \theta\end{cases} (\theta$ is a parameter), find the distance from each point on $C$ to $l$. | 2 \sqrt{2}-2 | 53.90625 |
25,970 | Triangle $ABC$ has $AB = 2$ , $BC = 3$ , $CA = 4$ , and circumcenter $O$ . If the sum of the areas of triangles $AOB$ , $BOC$ , and $COA$ is $\tfrac{a\sqrt{b}}{c}$ for positive integers $a$ , $b$ , $c$ , where $\gcd(a, c) = 1$ and $b$ is not divisible by the square of any prime, find $a+b+c$ .
*Proposed by Michael Tang* | 152 | 0 |
25,971 | What is the smallest positive integer $n$ for which $11n-8$ and $5n + 9$ share a common factor greater than $1$? | 165 | 21.875 |
25,972 | Find all natural numbers \( x \) that satisfy the following conditions: the product of the digits of \( x \) is equal to \( 44x - 86868 \), and the sum of the digits is a cube of a natural number. | 1989 | 98.4375 |
25,973 | In $\vartriangle ABC$ points $D, E$ , and $F$ lie on side $\overline{BC}$ such that $\overline{AD}$ is an angle bisector of $\angle BAC$ , $\overline{AE}$ is a median, and $\overline{AF}$ is an altitude. Given that $AB = 154$ and $AC = 128$ , and $9 \times DE = EF,$ fi nd the side length $BC$ . | 94 | 1.5625 |
25,974 | A car is braking to a complete stop. It is known that its speed at the midpoint of the distance was 100 km/h. Determine its initial speed. | 141.4 | 0 |
25,975 | Given the parabola $C_1$: $y^{2}=4x$ with focus $F$ that coincides with the right focus of the ellipse $C_2$: $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1 (a > b > 0)$, and the line connecting the intersection points of the curves $C_1$ and $C_2$ passes through point $F$, determine the length of the major axis of the ellipse $C_2$. | 2\sqrt{2}+2 | 1.5625 |
25,976 | Given a random variable $0.4987X \sim N\left( 9, \sigma^2 \right)$, and $P(X < 6) = 0.2$, determine the probability that $9 < X < 12$. | 0.3 | 11.71875 |
25,977 | Evaluate the expression $\sqrt{25\sqrt{15\sqrt{9}}}$. | 5\sqrt{15} | 0 |
25,978 | There are 3 females and 3 males to be arranged in a sequence of 6 contestants, with the restriction that no two males can perform consecutively and the first contestant cannot be female contestant A. Calculate the number of different sequences of contestants. | 132 | 34.375 |
25,979 | Brown lives on a street with more than 20 but fewer than 500 houses (all houses are numbered sequentially: $1, 2, 3$, etc.). Brown discovered that the sum of all the numbers from the first house to his own house (inclusive) equals half the sum of all the numbers from the first to the last house (inclusive).
What is the number of his house? | 84 | 35.9375 |
25,980 | It is known that ship $A$ is located at $80^{\circ}$ north by east from lighthouse $C$, and the distance from $A$ to $C$ is $2km$. Ship $B$ is located at $40^{\circ}$ north by west from lighthouse $C$, and the distance between ships $A$ and $B$ is $3km$. Find the distance from $B$ to $C$ in $km$. | \sqrt {6}-1 | 0 |
25,981 | Given that the complex number $z\_1$ satisfies $((z\_1-2)(1+i)=1-i)$, the imaginary part of the complex number $z\_2$ is $2$, and $z\_1z\_2$ is a real number, find $z\_2$ and $|z\_2|$. | 2 \sqrt {5} | 0 |
25,982 | At a conference, the 2016 participants were registered from P1 to P2016. Each participant from P1 to P2015 shook hands with exactly the same number of participants as the number on their registration form. How many hands did the 2016th participant shake? | 1008 | 28.125 |
25,983 | Given that a rancher bought 600 sheep at an unknown cost price, then sold 550 sheep for the cost price of 600 sheep, and the remaining 50 sheep at the same price per head as the first 550, calculate the percent gain on the entire transaction. | 9.09\% | 71.09375 |
25,984 | Determine the number of functions $f: \{1, 2, 3\} \rightarrow \{1, 2, 3\}$ satisfying the property $f(f(x)) = f(x)$. | 10 | 68.75 |
25,985 | A king summoned two wise men. He gave the first one 100 blank cards and instructed him to write a positive number on each (the numbers do not have to be different), without showing them to the second wise man. Then, the first wise man can communicate several distinct numbers to the second wise man, each of which is either written on one of the cards or is a sum of the numbers on some cards (without specifying exactly how each number is derived). The second wise man must determine which 100 numbers are written on the cards. If he cannot do this, both will be executed; otherwise, a number of hairs will be plucked from each of their beards equal to the amount of numbers the first wise man communicated. How can the wise men, without colluding, stay alive and lose the minimum number of hairs? | 101 | 2.34375 |
25,986 | (a) The natural number \( n \) is less than 150. What is the largest remainder that the number 269 can give when divided by \( n \)?
(b) The natural number \( n \) is less than 110. What is the largest remainder that the number 269 can give when divided by \( n \)? | 109 | 5.46875 |
25,987 | Four identical isosceles triangles $A W B, B X C, C Y D$, and $D Z E$ are arranged with points $A, B, C, D$, and $E$ lying on the same straight line. A new triangle is formed with sides the same lengths as $A X, A Y,$ and $A Z$. If $A Z = A E$, what is the largest integer value of $x$ such that the area of this new triangle is less than 2004? | 22 | 1.5625 |
25,988 | In an opaque bag, there are three balls, each labeled with the numbers $-1$, $0$, and $\frac{1}{3}$, respectively. These balls are identical except for the numbers on them. Now, a ball is randomly drawn from the bag, and the number on it is denoted as $m$. After putting the ball back and mixing them, another ball is drawn, and the number on it is denoted as $n$. The probability that the quadratic function $y=x^{2}+mx+n$ does not pass through the fourth quadrant is ______. | \frac{5}{9} | 24.21875 |
25,989 | In the Cartesian coordinate system $xoy$, the parametric equation of curve $C_1$ is $$\begin{cases} x=a\cos t+ \sqrt {3} \\ y=a\sin t\end{cases}$$ (where $t$ is the parameter, $a>0$). In the polar coordinate system with the origin as the pole and the positive $x$-axis as the polar axis, the equation of curve $C_2$ is $$\rho^{2}=2\rho\sin\theta+6$$.
(1) Identify the type of curve $C_1$ and convert its equation into polar coordinates;
(2) Given that $C_1$ and $C_2$ intersect at points $A$ and $B$, and line segment $AB$ passes through the pole, find the length of segment $AB$. | 3 \sqrt {3} | 0 |
25,990 |
A man and his faithful dog simultaneously started moving along the perimeter of a block from point \( A \) at time \( t_{0} = 0 \) minutes. The man moved at a constant speed clockwise, while the dog ran at a constant speed counterclockwise. It is known that the first time they met was \( t_{1} = 2 \) minutes after starting, and this meeting occurred at point \( B \). Given that they continued moving after this, each in their original direction and at their original speed, determine the next time they will both be at point \( B \) simultaneously. Note that \( A B = C D = 100 \) meters and \( B C = A D = 200 \) meters. | 14 | 7.8125 |
25,991 | Let $ABC$ be a triangle such that $AB=2$ , $CA=3$ , and $BC=4$ . A semicircle with its diameter on $BC$ is tangent to $AB$ and $AC$ . Compute the area of the semicircle. | \frac{27\pi}{40} | 2.34375 |
25,992 | On a table, there are 20 cards numbered from 1 to 20. Xiaoming picks 2 cards each time, such that the number on one card is 2 times the number on the other card plus 2. What is the maximum number of cards Xiaoming can pick? | 12 | 28.125 |
25,993 | Two congruent squares, $ABCD$ and $JKLM$, each have side lengths of 12 units. Square $JKLM$ is placed such that its center coincides with vertex $C$ of square $ABCD$. Determine the area of the region covered by these two squares in the plane. | 216 | 10.9375 |
25,994 | Find the largest real $ T$ such that for each non-negative real numbers $ a,b,c,d,e$ such that $ a\plus{}b\equal{}c\plus{}d\plus{}e$ : \[ \sqrt{a^{2}\plus{}b^{2}\plus{}c^{2}\plus{}d^{2}\plus{}e^{2}}\geq T(\sqrt a\plus{}\sqrt b\plus{}\sqrt c\plus{}\sqrt d\plus{}\sqrt e)^{2}\] | \frac{\sqrt{30}}{30 + 12\sqrt{6}} | 0 |
25,995 | Find the smallest positive integer \( n \) such that the mean of the squares of the first \( n \) natural numbers (\( n > 1 \)) is an integer.
(Note: The mean of the squares of \( n \) numbers \( a_1, a_2, \cdots, a_n \) is given by \( \sqrt{\frac{a_{1}^2 + a_{2}^2 + \cdots + a_{n}^2}{n}} \).)
(Note: Fifteenth American Mathematical Olympiad, 1986) | 337 | 82.03125 |
25,996 | Given the curve $f(x)=x^2-2x$, find the slope angle of the tangent line at the point $(\frac{3}{2},f(\frac{3}{2}))$. | \frac{\pi}{4} | 76.5625 |
25,997 | Xiao Jun is playing a dice game. He starts at the starting square. If he rolls a 1 to 5, he moves forward by the number of spaces shown on the dice. If he rolls a 6 or moves beyond the final square at any time, he must immediately return to the starting square. How many possible ways are there for Xiao Jun to roll the dice three times and exactly reach the ending square? | 19 | 2.34375 |
25,998 | Four circles are inscribed such that each circle touches the midpoint of each side of a square. The side of the square is 10 cm, and the radius of each circle is 5 cm. Determine the area of the square not covered by any circle. | 100 - 50\pi | 0 |
25,999 | A function \( f(n) \) defined for positive integers satisfies:
\[ f(n) = \begin{cases}
n - 3 & \text{if } n \geq 1000 \\
f[f(n + 7)] & \text{if } n < 1000
\end{cases} \]
Determine \( f(90) \). | 999 | 52.34375 |
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