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|---|---|---|---|
29,800 | Positive integer $n$ cannot be divided by $2$ and $3$ , there are no nonnegative integers $a$ and $b$ such that $|2^a-3^b|=n$ . Find the minimum value of $n$ . | 35 | 94.53125 |
29,801 | Given that $\sqrt{51.11}\approx 7.149$ and $\sqrt{511.1}\approx 22.608$, determine the value of $\sqrt{511100}$. | 714.9 | 53.90625 |
29,802 | In the diagram, two circles touch at \( P \). Also, \( QP \) and \( SU \) are perpendicular diameters of the larger circle that intersect at \( O \). Point \( V \) is on \( QP \) and \( VP \) is a diameter of the smaller circle. The smaller circle intersects \( SU \) at \( T \), as shown. If \( QV = 9 \) and \( ST = 5 \), what is the sum of the lengths of the diameters of the two circles? | 91 | 0 |
29,803 | Given the hyperbola $C$: $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ ($a>0, b>0$) with the right focus $F$ and left vertex $A$, where $|FA|=2+\sqrt{5}$, the distance from $F$ to the asymptote of $C$ is $1$. A line $l$ passing through point $B(4,0)$ intersects the right branch of the hyperbola $C$ at points $P$ and $Q$. The lines $AP$ and $AQ$ intersect the $y$-axis at points $M$ and $N$, respectively. <br/>$(1)$ Find the standard equation of the hyperbola $C$;<br/>$(2)$ If the slopes of lines $MB$ and $NB$ are $k_{1}$ and $k_{2}$, respectively, determine whether $k_{1}k_{2}$ is a constant. If it is, find the value of this constant; if not, explain why. | -\frac{1}{48} | 11.71875 |
29,804 | Around the circle, a pentagon is described, the lengths of the sides of which are integers, and the first and third sides are equal to 1. Into which segments does the point of tangency divide the second side? | \frac{1}{2} | 3.90625 |
29,805 | \( p(x, y, z) \) is a polynomial with real coefficients such that:
1. \( p(tx, ty, tz) = t^2 f(y - x, z - x) \) for all real \( x, y, z, t \) (and some function \( f \));
2. \( p(1, 0, 0) = 4 \), \( p(0, 1, 0) = 5 \), and \( p(0, 0, 1) = 6 \);
3. \( p(\alpha, \beta, \gamma) = 0 \) for some complex numbers \( \alpha, \beta, \gamma \) such that \( |\beta - \alpha| = 10 \).
Find \( |\gamma - \alpha| \). | \frac{5 \sqrt{30}}{3} | 13.28125 |
29,806 | In right triangle $XYZ$ with $\angle Z = 90^\circ$, the length $XY = 15$ and the length $XZ = 8$. Find $\sin Y$. | \frac{\sqrt{161}}{15} | 1.5625 |
29,807 | [b]Problem Section #1
a) A set contains four numbers. The six pairwise sums of distinct elements of the set, in no particular order, are $189, 320, 287, 264, x$ , and y. Find the greatest possible value of: $x + y$ .
<span style="color:red">NOTE: There is a high chance that this problems was copied.</span> | 761 | 0.78125 |
29,808 | Given the quadratic function \( y = ax^2 + bx + c \) with its graph intersecting the \( x \)-axis at points \( A \) and \( B \), and its vertex at point \( C \):
(1) If \( \triangle ABC \) is a right-angled triangle, find the value of \( b^2 - 4ac \).
(2) Consider the quadratic function
\[ y = x^2 - (2m + 2)x + m^2 + 5m + 3 \]
with its graph intersecting the \( x \)-axis at points \( E \) and \( F \), and it intersects the linear function \( y = 3x - 1 \) at two points, with the point having the smaller \( y \)-coordinate denoted as point \( G \).
(i) Express the coordinates of point \( G \) in terms of \( m \).
(ii) If \( \triangle EFG \) is a right-angled triangle, find the value of \( m \). | -1 | 14.0625 |
29,809 | Let $f(x)$ be an odd function defined on $\mathbb{R}$. When $x > 0$, $f(x)=x^{2}+2x-1$.
(1) Find $f(-2)$;
(2) Find the expression of $f(x)$. | -7 | 25.78125 |
29,810 | The sum of the dimensions of a rectangular prism is the sum of the number of edges, corners, and faces, where the dimensions are 2 units by 3 units by 4 units. Calculate the resulting sum. | 26 | 75 |
29,811 | What is the value of $\log_{10}{16} + 3\log_{5}{25} + 4\log_{10}{2} + \log_{10}{64} - \log_{10}{8}$? | 9.311 | 0 |
29,812 | In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. Given that $\cos A= \frac {c}{a}\cos C$, $b+c=2+ \sqrt {2}$, and $\cos B= \frac {3}{4}$, find the area of $\triangle ABC$. | \frac { \sqrt {7}}{2} | 0 |
29,813 | Alli rolls a standard $8$-sided die twice. What is the probability of rolling integers that differ by $3$ on her first two rolls? Express your answer as a common fraction. | \frac{1}{8} | 7.8125 |
29,814 | In the polar coordinate system, the curve $C\_1$: $ρ=2\cos θ$, and the curve $C\_2$: $ρ\sin ^{2}θ=4\cos θ$. Establish a rectangular coordinate system $(xOy)$ with the pole as the coordinate origin and the polar axis as the positive semi-axis $x$. The parametric equation of the curve $C$ is $\begin{cases} x=2+ \frac {1}{2}t \ y= \frac {\sqrt {3}}{2}t\end{cases}$ ($t$ is the parameter).
(I) Find the rectangular coordinate equations of $C\_1$ and $C\_2$;
(II) The curve $C$ intersects $C\_1$ and $C\_2$ at four distinct points, arranged in order along $C$ as $P$, $Q$, $R$, and $S$. Find the value of $||PQ|-|RS||$. | \frac {11}{3} | 1.5625 |
29,815 | There are a total of 2008 black triangles, marked as "▲" and "△", arranged in a certain pattern as follows: ▲▲△△▲△▲▲△△▲△▲▲…, then there are black triangles. | 1004 | 10.15625 |
29,816 | Given six senior students (including 4 boys and 2 girls) are arranged to intern at three schools, A, B, and C, with two students at each school, and the two girls cannot be at the same school or at school C, and boy A cannot go to school A, calculate the total number of different arrangements. | 18 | 0.78125 |
29,817 | The cells of a $5 \times 7$ table are filled with numbers so that in each $2 \times 3$ rectangle (either vertical or horizontal) the sum of the numbers is zero. By paying 100 rubles, you can choose any cell and find out what number is written in it. What is the smallest amount of rubles needed to determine the sum of all the numbers in the table with certainty? | 100 | 35.9375 |
29,818 | How many lattice points lie on the hyperbola $x^2 - y^2 = 3^4 \cdot 17^2$? | 30 | 18.75 |
29,819 | A computer program evaluates expressions without parentheses in the following way:
1) First, it performs multiplications and divisions from left to right one by one.
2) Then, it performs additions and subtractions from left to right.
For example, the value of the expression $1-2 / 3-4$ is $-3 \frac{2}{3}$. How many different results can we get if in the following expression each $*$ is independently replaced by one of the operators $+$, $-$, $/$, $\times$?
$1 * 1 * 1 * 1 * 1 * 1 * 1 * 1$ | 15 | 88.28125 |
29,820 | During a class's May Day gathering, the original program schedule included 5 events. Just before the performance, 2 additional events were added. If these 2 new events are to be inserted into the original schedule, how many different insertion methods are there? | 42 | 41.40625 |
29,821 | In the geometric sequence $\{a_n\}$, if $a_5 + a_6 + a_7 + a_8 = \frac{15}{8}$ and $a_6a_7 = -\frac{9}{8}$, then find the value of $\frac{1}{a_5} + \frac{1}{a_6} + \frac{1}{a_7} + \frac{1}{a_8}$. | -\frac{5}{3} | 28.90625 |
29,822 | Given vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ that satisfy: $|\overrightarrow{a}| = |\overrightarrow{b}| = 1$, and $|k\overrightarrow{a} + \overrightarrow{b}| = \sqrt{3}|\overrightarrow{a} - k\overrightarrow{b}| (k > 0)$. Find the maximum value of the angle between vectors $\overrightarrow{a}$ and $\overrightarrow{b}$. | \frac{\pi}{3} | 69.53125 |
29,823 | If set $A=\{-4, 2a-1, a^2\}$, $B=\{a-5, 1-a, 9\}$, and $A \cap B = \{9\}$, then the value of $a$ is. | -3 | 25 |
29,824 | Let $S$ be a subset of $\{1, 2, 3, \ldots, 100\}$ such that no pair of distinct elements in $S$ has a sum divisible by $5$. What is the maximum number of elements in $S$? | 40 | 6.25 |
29,825 | The room numbers of a hotel are all three-digit numbers. The first digit represents the floor and the last two digits represent the room number. The hotel has rooms on five floors, numbered 1 to 5. It has 35 rooms on each floor, numbered $\mathrm{n}01$ to $\mathrm{n}35$ where $\mathrm{n}$ is the number of the floor. In numbering all the rooms, how many times will the digit 2 be used?
A 60
В 65
C 95
D 100
E 105 | 105 | 41.40625 |
29,826 | Inside a non-isosceles acute triangle \(ABC\) with \(\angle ABC = 60^\circ\), point \(T\) is marked such that \(\angle ATB = \angle BTC = \angle ATC = 120^\circ\). The medians of the triangle intersect at point \(M\). The line \(TM\) intersects the circumcircle of triangle \(ATC\) at point \(K\) for the second time. Find \( \frac{TM}{MK} \). | 1/2 | 50 |
29,827 | On the $x O y$ coordinate plane, there is a Chinese chess "knight" at the origin $(0,0)$. The "knight" needs to be moved to the point $P(1991,1991)$ using the movement rules of the chess piece. Calculate the minimum number of moves required. | 1328 | 17.1875 |
29,828 | If a four-digit natural number $\overline{abcd}$ has digits that are all different and not equal to $0$, and satisfies $\overline{ab}-\overline{bc}=\overline{cd}$, then this four-digit number is called a "decreasing number". For example, the four-digit number $4129$, since $41-12=29$, is a "decreasing number"; another example is the four-digit number $5324$, since $53-32=21\neq 24$, is not a "decreasing number". If a "decreasing number" is $\overline{a312}$, then this number is ______; if the sum of the three-digit number $\overline{abc}$ formed by the first three digits and the three-digit number $\overline{bcd}$ formed by the last three digits of a "decreasing number" is divisible by $9$, then the maximum value of the number that satisfies the condition is ______. | 8165 | 82.8125 |
29,829 | In trapezoid $ABCD$ with $\overline{BC}\parallel\overline{AD}$, let $BC = 700$ and $AD = 1400$. Let $\angle A = 45^\circ$, $\angle D = 45^\circ$, and $P$ and $Q$ be the midpoints of $\overline{BC}$ and $\overline{AD}$, respectively. Find the length $PQ$. | 350 | 32.8125 |
29,830 | The main structure of the Chinese space station includes the Tianhe core module, the Wentian experiment module, and the Mengtian experiment module. Assuming that the space station needs to arrange 6 astronauts, including astronauts A and B, to conduct experiments, with each module having at least one person and at most three people, find the number of different arrangements. | 450 | 2.34375 |
29,831 | Let $x, y, z$ be real numbers such that
\[
x + y + z = 5,
\]
\[
x^2 + y^2 + z^2 = 11.
\]
Find the smallest and largest possible values of $x$, and compute their sum. | \frac{10}{3} | 90.625 |
29,832 | How many positive three-digit integers are there in which each of the three digits is either prime or a perfect square? | 216 | 5.46875 |
29,833 | In the diagram, \( PQR \) is a straight line segment and \( QS = QT \). Also, \( \angle PQS = x^\circ \) and \( \angle TQR = 3x^\circ \). If \( \angle QTS = 76^\circ \), find the value of \( x \). | 38 | 61.71875 |
29,834 | Consider a function \( g \) that maps nonnegative integers to real numbers, with \( g(1) = 1 \), and for all nonnegative integers \( m \ge n \),
\[ g(m + n) + g(m - n) = \frac{g(3m) + g(3n)}{3} \]
Find the sum of all possible values of \( g(10) \). | 100 | 10.15625 |
29,835 | Jenna bakes a $24$-inch by $15$-inch pan of chocolate cake. The cake pieces are made to measure $3$ inches by $2$ inches each. Calculate the number of pieces of cake the pan contains. | 60 | 23.4375 |
29,836 | A spinner is divided into 8 equal sectors numbered from 1 to 8. Jane and her sister each spin the spinner once. If the non-negative difference of their numbers is less than 4, Jane wins. Otherwise, her sister wins. What is the probability that Jane wins? | \frac{11}{16} | 82.8125 |
29,837 | Compute the value of the expression:
\[ 2(1 + 4(1 + 4(1 + 4(1 + 4(1 + 4(1 + 4(1 + 4(1 + 4(1 + 4))))))))) \] | 699050 | 45.3125 |
29,838 | Given non-zero vectors $\overrightarrow{a}, \overrightarrow{b}$, if $(\overrightarrow{a} - 2\overrightarrow{b}) \perp \overrightarrow{a}$ and $(\overrightarrow{b} - 2\overrightarrow{a}) \perp \overrightarrow{b}$, find the angle between $\overrightarrow{a}$ and $\overrightarrow{b}$. | \frac{\pi}{3} | 100 |
29,839 | Given the parabola $y=x^2$ and the moving line $y=(2t-1)x-c$ have common points $(x_1, y_1)$, $(x_2, y_2)$, and $x_1^2+x_2^2=t^2+2t-3$.
(1) Find the range of the real number $t$;
(2) When does $t$ take the minimum value of $c$, and what is the minimum value of $c$? | \frac{11-6\sqrt{2}}{4} | 7.03125 |
29,840 | Find the largest possible value of $k$ for which $3^{13}$ is expressible as the sum of $k$ consecutive positive integers. | 1458 | 7.03125 |
29,841 | In triangle \( \triangle ABC \), \( \angle BAC = 90^\circ \), \( AC = AB = 4 \), and point \( D \) is inside \( \triangle ABC \) such that \( AD = \sqrt{2} \). Find the minimum value of \( BD + CD \). | 2\sqrt{10} | 18.75 |
29,842 | Given the hyperbola $\frac {x^{2}}{4}-y^{2}$=1, a line $l$ with a slope angle of $\frac {π}{4}$ passes through the right focus $F\_2$ and intersects the right branch of the hyperbola at points $M$ and $N$. The midpoint of the line segment $MN$ is $P$. Determine the vertical coordinate of point $P$. | \frac{\sqrt {5}}{3} | 0 |
29,843 | Evaluate
\[
\left(c^c - c (c - 1)^{c-1}\right)^c
\]
when \( c = 4 \). | 148^4 | 0 |
29,844 | On the eve of the Spring Festival, a convenience store sold a batch of goods that were purchased at 12 yuan per piece at a price of 20 yuan per piece, selling 240 pieces per day. After a period of sales, it was found that if the price per piece was increased by 0.5 yuan, then 10 fewer pieces would be sold per day; if the price per piece was decreased by 0.5 yuan, then 20 more pieces would be sold per day. In order to achieve a daily sales profit of 1980 yuan for the goods, how much should each piece be priced? | 23 | 14.0625 |
29,845 | Given a tetrahedron \( P-ABCD \) where the edges \( AB \) and \( BC \) each have a length of \(\sqrt{2}\), and all other edges have a length of 1, find the volume of the tetrahedron. | \frac{\sqrt{2}}{6} | 6.25 |
29,846 | Given that a meeting is convened with 2 representatives from one company and 1 representative from each of the other 3 companies, find the probability that 3 randomly selected individuals to give speeches come from different companies. | 0.6 | 0.78125 |
29,847 | Consider a rectangle $ABCD$ containing three squares. Two smaller squares each occupy a part of rectangle $ABCD$, and each smaller square has an area of 1 square inch. A larger square, also inside rectangle $ABCD$ and not overlapping with the smaller squares, has a side length three times that of one of the smaller squares. What is the area of rectangle $ABCD$, in square inches? | 11 | 7.8125 |
29,848 | Given a hyperbola $x^{2}- \frac {y^{2}}{3}=1$ and two points $M$, $N$ on it are symmetric about the line $y=x+m$, and the midpoint of $MN$ lies on the parabola $y^{2}=18x$. Find the value of the real number $m$. | -8 | 10.15625 |
29,849 | In triangle $\triangle ABC$, let $a$, $b$, and $c$ be the lengths of the sides opposite to the internal angles $A$, $B$, and $C$, respectively, and $\sin \left(A-B\right)\cos C=\cos B\sin \left(A-C\right)$. <br/>$(1)$ Determine the shape of triangle $\triangle ABC$; <br/>$(2)$ If triangle $\triangle ABC$ is an acute triangle and $a=\frac{1}{\sin B}$, find the maximum value of $\frac{{b}^{2}+{a}^{2}}{(ab)^{2}}$. | \frac{25}{16} | 15.625 |
29,850 | Given the function $f(x)=\cos x\bullet \sin (x+\frac{\pi }{3})-\sqrt{3}\cos^{2}x+\frac{\sqrt{3}}{4}$, where $x\in R$.
(1) Find the minimum positive period and symmetry center of $f(x)$.
(2) If the graph of the function $y=g(x)$ is obtained by shifting the graph of $y=f(x)$ to the left by $\frac{\pi }{4}$ units, find the maximum and minimum values of $g(x)$ on the interval $\left[-\frac{\pi }{6},\frac{\pi }{3}\right]$. | -\frac{1}{4} | 63.28125 |
29,851 | Given a sequence $\{a_n\}$ that satisfies: $a_{n+2}=\begin{cases} 2a_{n}+1 & (n=2k-1, k\in \mathbb{N}^*) \\ (-1)^{\frac{n}{2}} \cdot n & (n=2k, k\in \mathbb{N}^*) \end{cases}$, with $a_{1}=1$ and $a_{2}=2$, determine the maximum value of $n$ for which $S_n \leqslant 2046$. | 19 | 0.78125 |
29,852 | A right triangular pyramid has a base edge length of $2$, and its three side edges are pairwise perpendicular. Calculate the volume of this pyramid. | \frac{\sqrt{6}}{3} | 0 |
29,853 | An eight-sided die, with faces numbered from 1 to 8, is tossed three times. Given that the sum of the first two tosses equals the third, calculate the probability that at least one "2" is tossed. | \frac{11}{28} | 0 |
29,854 | What three-digit number with units digit 4 and hundreds digit 5 is divisible by 8 and has an even tens digit? | 544 | 5.46875 |
29,855 | Xiaoming's father departs from home to go shopping at the supermarket. If he first rides a bicycle for 12 minutes and then walks for 20 minutes, he can reach the supermarket; if he first rides a bicycle for 8 minutes and then walks for 36 minutes, he can also reach the supermarket. How many minutes will it take to reach the supermarket if he first rides a bicycle for 2 minutes and then walks? | 60 | 14.0625 |
29,856 | **Compute the sum of all four-digit numbers where every digit is distinct and then find the remainder when this sum is divided by 1000.** | 720 | 5.46875 |
29,857 | $\alpha$ and $\beta$ are two parallel planes. Four points are taken within plane $\alpha$, and five points are taken within plane $\beta$.
(1) What is the maximum number of lines and planes that can be determined by these points?
(2) What is the maximum number of tetrahedrons that can be formed with these points as vertices? | 120 | 35.15625 |
29,858 | Given that the internal angles $A$ and $B$ of $\triangle ABC$ satisfy $\frac{\sin B}{\sin A} = \cos(A+B)$, find the maximum value of $\tan B$. | \frac{\sqrt{2}}{4} | 34.375 |
29,859 | Six people are arranged in a row. In how many ways can the three people A, B, and C be arranged such that they are not adjacent to each other? | 144 | 2.34375 |
29,860 | Let $S(n)$ denote the sum of digits of a natural number $n$ . Find all $n$ for which $n+S(n)=2004$ . | 2001 | 11.71875 |
29,861 | Given that the base area of a cone is $\pi$, and the lateral area is twice the base area, determine the surface area of the circumscribed sphere of the cone. | \frac{16\pi}{3} | 67.96875 |
29,862 | Given a set with three elements, it can be represented as $\{a, \frac{b}{a}, 1\}$ and also as $\{a^2, a+b, 0\}$. Find the value of $a^{2013} + b^{2013}$ \_\_\_\_\_\_. | -1 | 79.6875 |
29,863 | In a square, points \(P\) and \(Q\) are placed such that \(P\) is the midpoint of the bottom side and \(Q\) is the midpoint of the right side of the square. The line segment \(PQ\) divides the square into two regions. Calculate the fraction of the square's area that is not in the triangle formed by the points \(P\), \(Q\), and the top-left corner of the square. | \frac{7}{8} | 58.59375 |
29,864 | What three-digit integer is equal to the sum of the factorials of its digits, where one of the digits is `3`, contributing `3! = 6` to the sum? | 145 | 53.125 |
29,865 | Four students participate in a knowledge contest, each student must choose one of the two questions, A or B, to answer. Correctly answering question A earns 60 points, while an incorrect answer results in -60 points. Correctly answering question B earns 180 points, while an incorrect answer results in -180 points. The total score of these four students is 0 points. How many different scoring situations are there in total? | 44 | 1.5625 |
29,866 | $(1)$ Given the function $f(x) = |x+1| + |2x-4|$, find the solution to $f(x) \geq 6$;<br/>$(2)$ Given positive real numbers $a$, $b$, $c$ satisfying $a+2b+4c=8$, find the minimum value of $\frac{1}{a} + \frac{1}{b} + \frac{1}{c}$. | \frac{11+6\sqrt{2}}{8} | 12.5 |
29,867 | Three cards are dealt at random from a standard deck of 52 cards. What is the probability that the first card is an Ace, the second card is a $\spadesuit$, and the third card is a 3? | \frac{17}{11050} | 0 |
29,868 | Xiao Ming invests in selling a type of eye-protecting desk lamp with a cost price of $20$ yuan per unit. During the sales process, he found that the relationship between the monthly sales volume $y$ (units) and the selling price $x$ (yuan) can be approximated as a linear function: $y=-10x+500$. In the sales process, the selling price is not lower than the cost price, and the profit per unit is not higher than $60\%$ of the cost price. <br/>$(1)$ If Xiao Ming's monthly profit is denoted as $w$ (yuan), find the functional relationship between the monthly profit $w$ (yuan) and the selling price $x$ (yuan), and determine the range of values for the independent variable $x$. <br/>$(2)$ At what price should the selling price be set to maximize the monthly profit? What is the maximum monthly profit? | 2160 | 52.34375 |
29,869 | The pan containing 24-inch by 15-inch brownies is cut into pieces that measure 3 inches by 2 inches. Calculate the total number of pieces of brownies the pan contains. | 60 | 33.59375 |
29,870 | The vertices of an equilateral triangle lie on the hyperbola \( xy = 3 \). The centroid of this triangle is at the origin, \( (0,0) \). What is the square of the area of the triangle? | 108 | 16.40625 |
29,871 | On a balance scale, three different masses were put at random on each pan and the result is shown in the picture. The masses are 101, 102, 103, 104, 105, and 106 grams. What is the probability that the 106 gram mass stands on the heavier pan?
A) 75%
B) 80%
C) 90%
D) 95%
E) 100% | 80\% | 23.4375 |
29,872 | The positive integers $a$ and $b$ are such that the numbers $15a+16b$ and $16a-15b$ are both squares of positive integers. What is the least possible value that can be taken on by the smaller of these two squares? | 231361 | 22.65625 |
29,873 | If the square roots of a positive number are $2a+6$ and $3-a$, then the value of $a$ is ____. | -9 | 13.28125 |
29,874 | How many distinct digits can appear as the second to last digit (penultimate digit) of an integral perfect square number? | 10 | 75.78125 |
29,875 | For non-negative integers $n$, the function $f(n)$ is defined by $f(0) = 0$, $f(1) = 1$, and $f(n) = f\left(\left\lfloor \frac{1}{2} n \right\rfloor \right) + n - 2\left\lfloor \frac{1}{2} n \right\rfloor$. Find the maximum value of $f(n)$ for $0 \leq n \leq 1997$. | 10 | 96.09375 |
29,876 | In the equation
$$
\frac{x^{2}+p}{x}=-\frac{1}{4},
$$
with roots \(x_{1}\) and \(x_{2}\), determine \(p\) such that:
a) \(\frac{x_{1}}{x_{2}}+\frac{x_{2}}{x_{1}}=-\frac{9}{4}\),
b) one root is 1 less than the square of the other root. | -\frac{15}{8} | 40.625 |
29,877 | Two fair 8-sided dice, with sides numbered from 1 to 8, are rolled once. The sum of the numbers rolled determines the radius of a circle. Calculate the probability that the numerical value of the area of the circle is less than the numerical value of the circle's circumference. | \frac{1}{64} | 43.75 |
29,878 | The circle C is tangent to the y-axis and the line l: y = (√3/3)x, and the circle C passes through the point P(2, √3). Determine the diameter of circle C. | \frac{14}{3} | 23.4375 |
29,879 | There are 20 rooms, some with lights on and some with lights off. The occupants of these rooms prefer to match the majority of the rooms. Starting from room one, if the majority of the remaining 19 rooms have their lights on, the occupant will turn the light on; otherwise, they will turn the light off. Initially, there are 10 rooms with lights on and 10 rooms with lights off, and the light in the first room is on. After everyone in these 20 rooms has had a turn, how many rooms will have their lights off? | 20 | 0.78125 |
29,880 | Let \( a \), \( b \), and \( c \) be the roots of the polynomial equation \( x^3 - 2x^2 + x - 1 = 0 \). Calculate \( \frac{1}{a-2} + \frac{1}{b-2} + \frac{1}{c-2} \). | -5 | 60.9375 |
29,881 | Given Ricardo has $3000$ coins comprised of pennies ($1$-cent coins), nickels ($5$-cent coins), and dimes ($10$-cent coins), with at least one of each type of coin, calculate the difference in cents between the highest possible and lowest total value that Ricardo can have. | 26973 | 64.84375 |
29,882 | On a redesigned dartboard, the outer circle radius is increased to $8$ units and the inner circle has a radius of $4$ units. Additionally, two radii divide the board into four congruent sections, each labeled inconsistently with point values as follows: inner sections have values of $3$ and $4$, and outer sections have $2$ and $5$. The probability of a dart hitting a particular region is still proportional to its area. Calculate the probability that when three darts hit this board, the total score is exactly $12$.
A) $\frac{9}{2048}$
B) $\frac{9}{1024}$
C) $\frac{18}{2048}$
D) $\frac{15}{1024}$ | \frac{9}{1024} | 40.625 |
29,883 | What is the smallest square of an integer that ends with the longest sequence of the same digits?
For example, if the longest sequence of the same digits were five, then a suitable number would be 24677777 (of course, if it were the smallest square, but it is not). Zero is not considered an acceptable digit. | 1444 | 0 |
29,884 | Let $a$ , $b$ , $c$ , and $d$ be positive real numbers such that
\[a^2 + b^2 - c^2 - d^2 = 0 \quad \text{and} \quad a^2 - b^2 - c^2 + d^2 = \frac{56}{53}(bc + ad).\]
Let $M$ be the maximum possible value of $\tfrac{ab+cd}{bc+ad}$ . If $M$ can be expressed as $\tfrac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers, find $100m + n$ .
*Proposed by Robin Park* | 4553 | 2.34375 |
29,885 | Given that positive real numbers $x$ and $y$ satisfy $4x+3y=4$, find the minimum value of $\frac{1}{2x+1}+\frac{1}{3y+2}$. | \frac{3}{8}+\frac{\sqrt{2}}{4} | 0 |
29,886 | For any set \( S \), let \( |S| \) represent the number of elements in set \( S \) and let \( n(S) \) represent the number of subsets of set \( S \). If \( A \), \( B \), and \( C \) are three finite sets such that:
(1) \( |A|=|B|=2016 \);
(2) \( n(A) + n(B) + n(C) = n(A \cup B \cup C) \),
then the maximum value of \( |A \cap B \cap C| \) is ________. | 2015 | 41.40625 |
29,887 | Call a positive integer strictly monotonous if it is a one-digit number or its digits, read from left to right, form a strictly increasing or a strictly decreasing sequence, and no digits are repeated. Determine the total number of strictly monotonous positive integers. | 1013 | 36.71875 |
29,888 | If $S$, $H$, and $E$ are all distinct non-zero digits less than $6$ and the following is true, find the sum of the three values $S$, $H$, and $E$, expressing your answer in base $6$. $$\begin{array}{c@{}c@{}c@{}c} &S&H&E_6\\ &+&H&E_6\\ \cline{2-4} &H&E&S_6\\ \end{array}$$ | 15_6 | 2.34375 |
29,889 | Given a circle with center \(O\) and radius \(OD\) perpendicular to chord \(AB\), intersecting \(AB\) at point \(C\). Line segment \(AO\) is extended to intersect the circle at point \(E\). If \(AB = 8\) and \(CD = 2\), what is the area of \(\triangle BCE\)? | 12 | 12.5 |
29,890 | The surface area of a sphere with edge lengths 3, 4, and 5 on the rectangular solid is what? | 50\pi | 14.84375 |
29,891 |
In triangle \(ABC\), two identical rectangles \(PQRS\) and \(P_1Q_1R_1S_1\) are inscribed (with points \(P\) and \(P_1\) lying on side \(AB\), points \(Q\) and \(Q_1\) lying on side \(BC\), and points \(R, S, R_1,\) and \(S_1\) lying on side \(AC\)). It is known that \(PS = 3\) and \(P_1S_1 = 9\). Find the area of triangle \(ABC\). | 72 | 9.375 |
29,892 | Given the function $f(x)=\sin (2x+\varphi)$, if the graph is shifted to the left by $\dfrac {\pi}{6}$ units and the resulting graph is symmetric about the $y$-axis, determine the possible value of $\varphi$. | \dfrac {\pi}{6} | 41.40625 |
29,893 | Given a regular triangular pyramid $P-ABC$ (the base triangle is an equilateral triangle, and the vertex $P$ is the center of the base) with all vertices lying on the same sphere, and $PA$, $PB$, $PC$ are mutually perpendicular, and the side length of the base equilateral triangle is $\sqrt{2}$, calculate the volume of this sphere. | \frac{\sqrt{3}\pi}{2} | 0.78125 |
29,894 | Circles of radius $3$ and $6$ are externally tangent to each other and are internally tangent to a circle of radius $9$ . The circle of radius $9$ has a chord that is a common external tangent of the other two circles. Find the square of the length of this chord. | 72 | 13.28125 |
29,895 | Consider an alphabet of 2 letters. A word is any finite combination of letters. We will call a word unpronounceable if it contains more than two identical letters in a row. How many unpronounceable words of 7 letters exist? Points for the problem: 8. | 86 | 7.03125 |
29,896 | Given that Isabella's fort has dimensions $15$ feet in length, $12$ feet in width, and $6$ feet in height, with one-foot thick floor and walls, determine the number of one-foot cubical blocks required to construct this fort. | 430 | 25.78125 |
29,897 | The chess club has 20 members: 12 boys and 8 girls. A 4-person team is chosen at random. What is the probability that the team has at least 2 boys and at least 1 girl? | \frac{4103}{4845} | 0.78125 |
29,898 | Calculate the numerical value by listing.<br/>Select $5$ people from $8$ people including $A$, $B$, and $C$ to line up.<br/>$(1)$ If $A$ must be included, how many ways are there to line up?<br/>$(2)$ If $A$, $B$, and $C$ are not all included, how many ways are there to line up?<br/>$(3)$ If $A$, $B$, and $C$ are all included, $A$ and $B$ must be next to each other, and $C$ must not be next to $A$ or $B$, how many ways are there to line up?<br/>$(4)$ If $A$ is not allowed to be at the beginning or end, and $B$ is not allowed to be in the middle (third position), how many ways are there to line up? | 4440 | 7.03125 |
29,899 | Suppose we flip five coins simultaneously: a penny (1 cent), a nickel (5 cents), a dime (10 cents), a quarter (25 cents), and a half-dollar (50 cents). What is the probability that at least 40 cents worth of coins come up heads? | \frac{9}{16} | 0 |
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