Unnamed: 0
int64 0
40.3k
| problem
stringlengths 10
5.15k
| ground_truth
stringlengths 1
1.22k
| solved_percentage
float64 0
100
|
|---|---|---|---|
29,000 | Determine the value of $b$, where $b$ is a positive number, such that the terms $10, b, \frac{10}{9}, \frac{10}{81}$ are the first four terms, respectively, of a geometric sequence. | 10 | 0 |
29,001 | When purchasing goods with a total amount of at least 1000 rubles, the store provides a 50% discount on subsequent purchases. With 1200 rubles in her pocket, Dasha wanted to buy 4 kg of strawberries and 6 kg of sugar. In the store, strawberries were sold at a price of 300 rubles per kg, and sugar at a price of 30 rubles per kg. Realizing that she did not have enough money for the purchase, Dasha still managed to buy everything she planned. How did she do that? | 1200 | 0 |
29,002 | Given a square initially painted black, with $\frac{1}{2}$ of the square black and the remaining part white, determine the fractional part of the original area of the black square that remains black after six changes where the middle fourth of each black area turns white. | \frac{729}{8192} | 15.625 |
29,003 | Find the number of different complex numbers $z$ such that $|z|=1$ and $z^{7!}-z^{6!}$ is a real number. | 44 | 0 |
29,004 | Given that $x = \frac{3}{4}$ is a solution to the equation $108x^2 + 61 = 145x - 7,$ what is the other value of $x$ that solves the equation? Express your answer as a common fraction. | \frac{68}{81} | 8.59375 |
29,005 | Triangles $ABC$ and $ADF$ have areas $4014$ and $14007,$ respectively, with $B=(0,0), C=(447,0), D=(1360,760),$ and $F=(1378,778).$ What is the sum of all possible $x$-coordinates of $A$? | 2400 | 7.03125 |
29,006 | Given the hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ ($a > 0$, $b > 0$) with its right focus at point $F$, and a point $P$ on the left branch of the hyperbola. Also given is that $PF$ is tangent to the circle $x^2 + y^2 = a^2$ at point $M$, where $M$ is precisely the midpoint of the line segment $PF$. Find the eccentricity of the hyperbola. | \sqrt{5} | 4.6875 |
29,007 | Given that Mr. A initially owns a home worth $\$15,000$, he sells it to Mr. B at a $20\%$ profit, then Mr. B sells it back to Mr. A at a $15\%$ loss, then Mr. A sells it again to Mr. B at a $10\%$ profit, and finally Mr. B sells it back to Mr. A at a $5\%$ loss, calculate the net effect of these transactions on Mr. A. | 3541.50 | 4.6875 |
29,008 | There is a five-digit number that, when divided by each of the 12 natural numbers 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, and 13, gives different remainders. What is this five-digit number? | 83159 | 21.875 |
29,009 | Two \(10 \times 24\) rectangles are inscribed in a circle as shown. Find the shaded area. | 169\pi - 380 | 0 |
29,010 |
An economist-cryptographer received a cryptogram from the ruler that contained a secret decree to introduce a per-unit tax in a certain market. The cryptogram specified the amount of tax revenue that needed to be collected, emphasizing that it was not possible to collect a higher amount of tax revenue in this market. Unfortunately, the economist-cryptographer deciphered the cryptogram incorrectly, rearranging the digits in the amount of tax revenue. Based on this incorrect data, a decision was made to introduce a per-unit tax on consumers of 30 monetary units per unit of the product. It is known that the market supply function is \( Q_s = 6P - 312 \), and the market demand is linear. In the situation with no taxes, the price elasticity of market supply at the equilibrium point is 1.5 times the absolute value of the price elasticity of market demand. After the tax was introduced, the consumer price increased to 118 monetary units.
1. Restore the market demand function.
2. Determine the amount of tax revenue collected at the chosen tax rate.
3. Determine the per-unit tax rate that would meet the ruler’s decree.
4. What is the amount of tax revenue specified by the ruler? | 8640 | 3.125 |
29,011 | From the numbers $1, 2, \cdots, 1000$, choose $k$ numbers such that any three of the chosen numbers can form the side lengths of a triangle. Find the minimum value of $k$. | 16 | 24.21875 |
29,012 | A large chest contains 10 smaller chests. In each of the smaller chests, either 10 even smaller chests are placed or nothing is placed. In each of those smaller chests, either 10 smaller chests are placed or none, and so on. After this, there are exactly 2006 chests with contents. How many are empty? | 18054 | 4.6875 |
29,013 | What digits should replace the asterisks to make the number 454** divisible by 2, 7, and 9? | 45486 | 9.375 |
29,014 | Determine the numerical value of $k$ such that
\[\frac{12}{x + z} = \frac{k}{z - y} = \frac{5}{y - x}.\] | 17 | 1.5625 |
29,015 | Find the greatest common divisor of $8!$ and $(6!)^2.$ | 5760 | 75.78125 |
29,016 | A cylindrical glass vessel is heated to a temperature of $t = 1200^{\circ}$C, causing part of the air to be expelled. After covering the vessel with a well-sealing glass plate, it is left to cool to room temperature at $t^{\prime}=16^{\circ}$C. The remaining air inside becomes less dense, and the external air pressure presses the glass plate against the top edge of the vessel. What is the force in kilograms due to this pressure? What is the degree of rarefaction (the ratio of the internal and external air densities)? The cylinder has a radius of $r = 8.00 \text{ cm}$, a height of $h = 24.16 \text{ cm}$, and the prevailing atmospheric pressure is $p = 745.6 \text{ mmHg}$ (with mercury density at room temperature $\sigma = 13.56$). The expansion of the glass vessel is negligible, and the coefficient of air expansion is $\alpha = \frac{1}{273}$. | 19.6\% | 0 |
29,017 | A solid right prism $ABCDEF$ has a height of 20, as illustrated. The bases are equilateral triangles with a side length of 14. Points $X$, $Y$, and $Z$ are the midpoints of edges $AC$, $BC$, and $DC$, respectively. Determine the perimeter of triangle $XYZ$. | 7 + 2\sqrt{149} | 6.25 |
29,018 | Given a regular tetrahedron with vertices $A, B, C$, and $D$ and edge length 1, where point $P$ lies on edge $AC$ one third of the way from vertex $A$ to vertex $C$, and point $Q$ lies on edge $BD$ one third of the way from vertex $B$ to vertex $D$, calculate the least possible distance between points $P$ and $Q$. | \frac{2}{3} | 0 |
29,019 | A certain item has a cost price of $4$ yuan and is sold at a price of $5$ yuan. The merchant is preparing to offer a discount on the selling price, but the profit margin must not be less than $10\%$. Find the maximum discount rate that can be offered. | 12\% | 60.15625 |
29,020 |
23. Two friends, Marco and Ian, are talking about their ages. Ian says, "My age is a zero of a polynomial with integer coefficients."
Having seen the polynomial \( p(x) \) Ian was talking about, Marco exclaims, "You mean, you are seven years old? Oops, sorry I miscalculated! \( p(7) = 77 \) and not zero."
"Yes, I am older than that," Ian's agreeing reply.
Then Marco mentioned a certain number, but realizes after a while that he was wrong again because the value of the polynomial at that number is 85.
Ian sighs, "I am even older than that number."
Determine Ian's age. | 14 | 4.6875 |
29,021 | Given the general term of the sequence $\{a_n\}$ is $a_n=n^2\left(\cos^2 \frac{n\pi}{3}-\sin^2 \frac{n\pi}{3}\right)$, calculate the value of $S_{30}$. | 470 | 45.3125 |
29,022 | Let point $P$ be a moving point on the curve $C\_1$: $(x-2)^2 + y^2 = 4$. Establish a polar coordinate system with the coordinate origin $O$ as the pole and the positive semi-axis of the $x$-axis as the polar axis. Rotate point $P$ counterclockwise by $90^{{∘}}$ around the pole $O$ to obtain point $Q$. Denote the trajectory equation of point $Q$ as curve $C\_2$.
1. Find the polar coordinate equations of curves $C\_1$ and $C\_2$.
2. The ray $\theta = \frac{\pi}{3} (\rho > 0)$ intersects curves $C\_1$ and $C\_2$ at points $A$ and $B$, respectively. Let $M(2,0)$ be a fixed point. Calculate the area of $\triangle MAB$. | 3 - \sqrt{3} | 80.46875 |
29,023 | A cone is inscribed in a regular quadrilateral pyramid. Find the ratio of the total surface area of the cone to the lateral surface area of the cone, given that the side length of the pyramid's base is 4 and the angle between the pyramid's height and the plane of its lateral face is $30^{\circ}$. | \frac{3 + \sqrt{3}}{3} | 2.34375 |
29,024 | Given a sequence $ \{a_n\} $ with the first term $ \dfrac{3}{5} $, and the sequence $ \{a_n\} $ satisfies $ a_{n+1} = 2 - \dfrac{1}{a_n} $, calculate $ a_{2018} $. | \dfrac{4031}{4029} | 0 |
29,025 | Four balls numbered $1, 2, 3$, and $4$ are placed in an urn. One ball is drawn, its number noted, and then returned to the urn. This process is repeated three times. If the sum of the numbers noted is $9$, determine the probability that the ball numbered $3$ was drawn all three times. | \frac{1}{10} | 35.15625 |
29,026 |
\[\log_{10} x + \log_{\sqrt{10}} x + \log_{\sqrt[3]{10}} x + \ldots + \log_{\sqrt[1]{10}} x = 5.5\] | \sqrt[10]{10} | 4.6875 |
29,027 | Let $A_1A_2A_3A_4A_5A_6A_7A_8$ be convex 8-gon (no three diagonals concruent).
The intersection of arbitrary two diagonals will be called "button".Consider the convex quadrilaterals formed by four vertices of $A_1A_2A_3A_4A_5A_6A_7A_8$ and such convex quadrilaterals will be called "sub quadrilaterals".Find the smallest $n$ satisfying:
We can color n "button" such that for all $i,k \in\{1,2,3,4,5,6,7,8\},i\neq k,s(i,k)$ are the same where $s(i,k)$ denote the number of the "sub quadrilaterals" has $A_i,A_k$ be the vertices and the intersection of two its diagonals is "button". | 14 | 26.5625 |
29,028 | Given $y=y_{1}+y_{2}$, where $y_{1}$ is directly proportional to $(x+1)$, and $y_{2}$ is inversely proportional to $(x+1)$. When $x=0$, $y=-5$; when $x=2$, $y=-7$. Find the value of $x$ when $y=5$. | - \frac{5}{2} | 22.65625 |
29,029 | In a row with 120 seats, some of the seats are already occupied. If a new person arrives and must sit next to someone regardless of their choice of seat, what is the minimum number of people who were already seated? | 40 | 37.5 |
29,030 | Given non-zero vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ satisfy $|\overrightarrow{a}|=2|\overrightarrow{b}|$, and $(\overrightarrow{a}-\overrightarrow{b})\bot \overrightarrow{b}$, then the angle between $\overrightarrow{a}$ and $\overrightarrow{b}$ is ______. | \frac{\pi}{3} | 96.875 |
29,031 | A trapezoid $ABCD$ lies on the $xy$ -plane. The slopes of lines $BC$ and $AD$ are both $\frac 13$ , and the slope of line $AB$ is $-\frac 23$ . Given that $AB=CD$ and $BC< AD$ , the absolute value of the slope of line $CD$ can be expressed as $\frac mn$ , where $m,n$ are two relatively prime positive integers. Find $100m+n$ .
*Proposed by Yannick Yao* | 1706 | 8.59375 |
29,032 | Given three coplanar vectors $\overrightarrow{a}$, $\overrightarrow{b}$, and $\overrightarrow{c}$, where $\overrightarrow{a}=(\sqrt{2}, 2)$, $|\overrightarrow{b}|=2\sqrt{3}$, $|\overrightarrow{c}|=2\sqrt{6}$, and $\overrightarrow{a}$ is parallel to $\overrightarrow{c}$.
1. Find $|\overrightarrow{c}-\overrightarrow{a}|$;
2. If $\overrightarrow{a}-\overrightarrow{b}$ is perpendicular to $3\overrightarrow{a}+2\overrightarrow{b}$, find the value of $\overrightarrow{a}\cdot(\overrightarrow{a}+\overrightarrow{b}+\overrightarrow{c})$. | -12 | 12.5 |
29,033 | Let positive integers \( a \) and \( b \) be such that \( 15a + 16b \) and \( 16a - 15b \) are both perfect squares. Find the smallest possible value of the smaller of these two squares.
| 481^2 | 0 |
29,034 | Let $n$ be a positive integer and let $d_{1},d_{2},,\ldots ,d_{k}$ be its divisors, such that $1=d_{1}<d_{2}<\ldots <d_{k}=n$ . Find all values of $n$ for which $k\geq 4$ and $n=d_{1}^{2}+d_{2}^{2}+d_{3}^{2}+d_{4}^{2}$ . | 130 | 100 |
29,035 | John now has 15 marbles of different colors, including two red, two green, and one blue marble. In how many ways can he choose 5 marbles, if exactly two of the chosen marbles are red and one is green? | 110 | 51.5625 |
29,036 | In the interval [1, 6], three different integers are randomly selected. The probability that these three numbers are the side lengths of an obtuse triangle is ___. | \frac{1}{4} | 1.5625 |
29,037 | Given a regular tetrahedron $P-ABC$, where points $P$, $A$, $B$, and $C$ are all on the surface of a sphere with radius $\sqrt{3}$. If $PA$, $PB$, and $PC$ are mutually perpendicular, calculate the distance from the center of the sphere to the plane $ABC$. | \dfrac{\sqrt{3}}{3} | 18.75 |
29,038 | Please write down an irrational number whose absolute value is less than $3: \_\_\_\_\_\_.$ | \sqrt{3} | 0 |
29,039 | Two families of three, with a total of $4$ adults and $2$ children, agree to go on a countryside trip together on Sunday using two cars, an "Audi" and a "Jetta". Each car can accommodate a maximum of $4$ people, and the two children cannot be left alone in one car. Determine the number of different seating arrangements possible. | 48 | 4.6875 |
29,040 |
Let ellipse $\Gamma: \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1 (a > b > 0)$ have an eccentricity of $\frac{\sqrt{3}}{2}$. A line with slope $k (k > 0)$ passes through the left focus $F$ and intersects the ellipse $\Gamma$ at points $A$ and $B$. If $\overrightarrow{A F}=3 \overrightarrow{F B}$, find $k$. | \sqrt{2} | 20.3125 |
29,041 | The quadrilateral \(ABCD\) is inscribed in a circle. \(I\) is the incenter of triangle \(ABD\). Find the minimum value of \(BD\) given that \(AI = BC = CD = 2\). | 2\sqrt{3} | 3.125 |
29,042 | How many triangles with positive area can be formed with vertices at points $(i,j)$ in the coordinate plane, where $i$ and $j$ are integers between $1$ and $6$, inclusive? | 6788 | 0.78125 |
29,043 | A \(3 \times 3 \times 3\) cube composed of 27 unit cubes rests on a horizontal plane. Determine the number of ways of selecting two distinct unit cubes (order is irrelevant) from a \(3 \times 3 \times 1\) block with the property that the line joining the centers of the two cubes makes a \(45^{\circ}\) angle with the horizontal plane. | 18 | 10.15625 |
29,044 | Find the least positive integer such that when its leftmost digit is deleted, the resulting integer is 1/19 of the original integer. | 950 | 5.46875 |
29,045 | In triangle $\triangle ABC$, it is known that $\overrightarrow{CD}=2\overrightarrow{DB}$, $P$ is a point on segment $AD$, and satisfies $\overrightarrow{CP}=\frac{1}{2}\overrightarrow{CA}+m\overrightarrow{CB}$. If the area of $\triangle ABC$ is $\sqrt{3}$ and $∠ACB=\frac{π}{3}$, then the minimum value of the length of segment $CP$ is ______. | \sqrt{2} | 2.34375 |
29,046 | Calculate the total surface area of two hemispheres of radius 8 cm each, joined at their bases to form a complete sphere. Assume that one hemisphere is made of a reflective material that doubles the effective surface area for purposes of calculation. Express your answer in terms of $\pi$. | 384\pi | 42.96875 |
29,047 | Given that the weights (in kilograms) of 4 athletes are all integers, and they weighed themselves in pairs for a total of 5 times, obtaining weights of 99, 113, 125, 130, 144 kilograms respectively, and there are two athletes who did not weigh together, determine the weight of the heavier one among these two athletes. | 66 | 1.5625 |
29,048 | Let $[x]$ represent the greatest integer less than or equal to $x$. Given that the natural number $n$ satisfies $\left[\frac{1}{15}\right] + \left[\frac{2}{15}\right] + \left[\frac{3}{15}\right] + \cdots + \left[\frac{n-1}{15}\right] + \left[\frac{n}{15}\right] > 2011$, what is the smallest value of $n$? | 253 | 23.4375 |
29,049 | A lemming starts at a corner of a rectangular area measuring 8 meters by 15 meters. It dashes diagonally across the rectangle towards the opposite corner for 11.3 meters. Then the lemming makes a $90^{\circ}$ right turn and sprints upwards for 3 meters. Calculate the average of the shortest distances to each side of the rectangle. | 5.75 | 18.75 |
29,050 | Four fair dice are tossed at random. What is the probability that the four numbers on the dice can be arranged to form an arithmetic progression with a common difference of one? | \frac{1}{432} | 0.78125 |
29,051 | The graphs of five functions are labelled from **(1) through (5)**. Provided below are descriptions of three:
1. The domain of function (2) is from $$\{-6,-5,-4,-3,-2,-1,0,1,2,3\}.$$ It is graphed as a set of discrete points.
2. Function (4) is defined by the equation $$y = x^3$$ and is graphed from $$x = -3$$ to $$x = 3$$.
3. Function (5) is a rational function defined by $$y = \frac{5}{x}$$, excluding the origin from its domain.
Determine the product of the labels of the functions that are invertible. | 20 | 24.21875 |
29,052 | Given the function $f(x)=\ln x-mx (m\in R)$.
(I) Discuss the monotonic intervals of the function $f(x)$;
(II) When $m\geqslant \frac{ 3 \sqrt {2}}{2}$, let $g(x)=2f(x)+x^{2}$ and its two extreme points $x\_1$, $x\_2 (x\_1 < x\_2)$ are exactly the zeros of $h(x)=\ln x-cx^{2}-bx$. Find the minimum value of $y=(x\_1-x\_2)h′( \frac {x\_1+x\_2}{2})$. | -\frac{2}{3} + \ln 2 | 1.5625 |
29,053 | We call any eight squares in a diagonal of a chessboard as a fence. The rook is moved on the chessboard in such way that he stands neither on each square over one time nor on the squares of the fences (the squares which the rook passes is not considered ones it has stood on). Then what is the maximum number of times which the rook jumped over the fence? | 47 | 0.78125 |
29,054 | Alvin rides on a flat road at $25$ kilometers per hour, downhill at $35$ kph, and uphill at $10$ kph. Benny rides on a flat road at $35$ kph, downhill at $45$ kph, and uphill at $15$ kph. Alvin goes from town $X$ to town $Y$, a distance of $15$ km all uphill, then from town $Y$ to town $Z$, a distance of $25$ km all downhill, and then back to town $X$, a distance of $30$ km on the flat. Benny goes the other way around using the same route. What is the time in minutes that it takes Alvin to complete the $70$-km ride, minus the time in minutes it takes Benny to complete the same ride? | 33 | 10.9375 |
29,055 | Two circles touch in $M$ , and lie inside a rectangle $ABCD$ . One of them touches the sides $AB$ and $AD$ , and the other one touches $AD,BC,CD$ . The radius of the second circle is four times that of the first circle. Find the ratio in which the common tangent of the circles in $M$ divides $AB$ and $CD$ . | 1:1 | 10.15625 |
29,056 | On Tony's map, the distance from Saint John, NB to St. John's, NL is $21 \mathrm{~cm}$. The actual distance between these two cities is $1050 \mathrm{~km}$. What is the scale of Tony's map? | 1:5 000 000 | 49.21875 |
29,057 | What is the smallest positive integer that has exactly eight distinct positive divisors, where all divisors are powers of prime numbers? | 24 | 93.75 |
29,058 | For certain real values of $a, b, c,$ and $d,$ the equation $x^4+ax^3+bx^2+cx+d=0$ has four non-real roots. The product of two of these roots is $7-3i$ and the sum of the other two roots is $5-2i,$ where $i^2 = -1.$ Find $b.$ | 43 | 7.03125 |
29,059 | How many ways are there to arrange the letters of the word $\text{ZOO}_1\text{M}_1\text{O}_2\text{M}_2\text{O}_3$, in which the three O's and the two M's are considered distinct? | 5040 | 32.03125 |
29,060 | A recipe calls for $5 \frac{3}{4}$ cups of flour and $2 \frac{1}{2}$ cups of sugar. If you make one-third of the recipe, how many cups of flour and sugar do you need? Express your answers as mixed numbers. | \frac{5}{6} | 1.5625 |
29,061 | Let the common ratio of the geometric sequence $\{a_n\}$ be $q$, and the sum of the first $n$ terms be $S_n$. If $S_{n+1}$, $S_n$, and $S_{n+2}$ form an arithmetic sequence, find the value of $q$. | -2 | 92.96875 |
29,062 | Fix a sequence $a_1,a_2,a_3\ldots$ of integers satisfying the following condition:for all prime numbers $p$ and all positive integers $k$ ,we have $a_{pk+1}=pa_k-3a_p+13$ .Determine all possible values of $a_{2013}$ . | 13 | 8.59375 |
29,063 | Find the remainder when $5^{2021}$ is divided by $17$. | 14 | 35.9375 |
29,064 | Convert the binary number $110110100_2$ to base 4. | 31220_4 | 56.25 |
29,065 | In the Cartesian coordinate system $xOy$, let $D$ be the region represented by the inequality $|x| + |y| \leq 1$, and $E$ be the region consisting of points whose distance from the origin is no greater than 1. If a point is randomly thrown into $E$, the probability that this point falls into $D$ is. | \frac{4}{\pi} | 0.78125 |
29,066 | Given a circle $C$: $(x+1)^{2}+y^{2}=r^{2}$ and a parabola $D$: $y^{2}=16x$ intersect at points $A$ and $B$, and $|AB|=8$, calculate the area of circle $C$. | 25\pi | 1.5625 |
29,067 | A rectangular sheet of paper is 20 cm long and 12 cm wide. It is folded along its diagonal. What is the perimeter of the shaded region? | 64 | 25 |
29,068 | Inside the circle \(\omega\), there is a circle \(\omega_{1}\) that touches it at point \(K\). Circle \(\omega_{2}\) touches circle \(\omega_{1}\) at point \(L\) and intersects circle \(\omega\) at points \(M\) and \(N\). It turns out that points \(K\), \(L\), and \(M\) are collinear. Find the radius of circle \(\omega\) if the radii of circles \(\omega_{1}\) and \(\omega_{2}\) are 4 and 7, respectively. | 11 | 8.59375 |
29,069 | Determine the number of 6-digit even numbers that can be formed using the digits 1, 2, 3, 4, 5, 6 without repetition and ensuring that 1, 3, 5 are not adjacent to one another. | 36 | 0 |
29,070 | Record the outcome of hitting or missing for 6 consecutive shots in order.
① How many possible outcomes are there?
② How many outcomes are there where exactly 3 shots hit the target?
③ How many outcomes are there where 3 shots hit the target, and exactly two of those hits are consecutive? | 12 | 28.90625 |
29,071 | Compute \(\arccos(\cos 8.5)\). All functions are in radians. | 2.217 | 1.5625 |
29,072 | Compute the number of ordered pairs of integers $(x,y)$ with $1\le x<y\le 200$ such that $i^x+i^y$ is a real number. | 4950 | 3.90625 |
29,073 | How many sequences of $0$s and $1$s of length $20$ are there that begin with a $0$, end with a $0$, contain no two consecutive $0$s, and contain no four consecutive $1$s?
A) 65
B) 75
C) 85
D) 86
E) 95 | 86 | 18.75 |
29,074 | Around a circle, an isosceles trapezoid \(ABCD\) is described. The side \(AB\) touches the circle at point \(M\), and the base \(AD\) touches the circle at point \(N\). The segments \(MN\) and \(AC\) intersect at point \(P\), such that \(NP: PM=2\). Find the ratio \(AD: BC\). | 3:1 | 2.34375 |
29,075 | Compute $e^{\pi}+\pi^e$ . If your answer is $A$ and the correct answer is $C$ , then your score on this problem will be $\frac{4}{\pi}\arctan\left(\frac{1}{\left|C-A\right|}\right)$ (note that the closer you are to the right answer, the higher your score is).
*2017 CCA Math Bonanza Lightning Round #5.2* | 45.5999 | 0 |
29,076 | There are eight different symbols designed on $n\geq 2$ different T-shirts. Each shirt contains at least one symbol, and no two shirts contain all the same symbols. Suppose that for any $k$ symbols $(1\leq k\leq 7)$ the number of shirts containing at least one of the $k$ symbols is even. Determine the value of $n$ . | 255 | 6.25 |
29,077 | Calculate the product of $1101_2 \cdot 111_2$. Express your answer in base 2. | 1000111_2 | 0 |
29,078 | Given that \(15^{-1} \equiv 31 \pmod{53}\), find \(38^{-1} \pmod{53}\), as a residue modulo 53. | 22 | 59.375 |
29,079 | The sequence $(a_n)$ is defined recursively by $a_0=1$, $a_1=\sqrt[23]{3}$, and $a_n=a_{n-1}a_{n-2}^3$ for $n\geq 2$. Determine the smallest positive integer $k$ such that the product $a_1a_2\cdots a_k$ is an integer. | 22 | 10.15625 |
29,080 | Given the function $f(x)=a\frac{x+1}{x}+\ln{x}$, the equation of the tangent line at the point (1, f(1)) is y=bx+5.
1. Find the real values of a and b.
2. Find the maximum and minimum values of the function $f(x)$ on the interval $[\frac{1}{e}, e]$, where $e$ is the base of the natural logarithm. | 3+\ln{2} | 19.53125 |
29,081 | Given an arithmetic sequence $\{a_n\}$ with the first term $a$ ($a \in \mathbb{R}, a \neq 0$). Let the sum of the first $n$ terms of the sequence be $S_n$, and for any positive integer $n$, it holds that $$\frac {a_{2n}}{a_{n}}= \frac {4n-1}{2n-1}$$.
(1) Find the general formula for $\{a_n\}$ and $S_n$;
(2) Does there exist positive integers $n$ and $k$ such that $S_n$, $S_{n+1}$, and $S_{n+k}$ form a geometric sequence? If so, find the values of $n$ and $k$; if not, explain why. | k=3 | 1.5625 |
29,082 | Calculate the sum $E(1)+E(2)+E(3)+\cdots+E(200)$ where $E(n)$ denotes the sum of the even digits of $n$, and $5$ is added to the sum if $n$ is a multiple of $10$. | 902 | 0 |
29,083 | Given that one air conditioner sells for a 10% profit and the other for a 10% loss, and the two air conditioners have the same selling price, determine the percentage change in the shopping mall's overall revenue. | 1\% | 2.34375 |
29,084 | So, Xiao Ming's elder brother was born in a year that is a multiple of 19. In 2013, determine the possible ages of Xiao Ming's elder brother. | 18 | 0.78125 |
29,085 | In the polar coordinate system, the equation of curve C is $\rho^2= \frac{3}{1+2\sin^2\theta}$. Point R is at $(2\sqrt{2}, \frac{\pi}{4})$.
P is a moving point on curve C, and side PQ of rectangle PQRS, with PR as its diagonal, is perpendicular to the polar axis. Find the maximum and minimum values of the perimeter of rectangle PQRS and the polar angle of point P when these values occur. | \frac{\pi}{6} | 3.125 |
29,086 | Given 60 feet of fencing, what is the greatest possible number of square feet in the area of a pen, if the pen is designed as a rectangle subdivided evenly into two square areas? | 450 | 0.78125 |
29,087 | For $a>0$ , let $f(a)=\lim_{t\to\+0} \int_{t}^1 |ax+x\ln x|\ dx.$ Let $a$ vary in the range $0 <a< +\infty$ , find the minimum value of $f(a)$ . | \frac{\ln 2}{2} | 0 |
29,088 | Where is Tony's cheese? The bottom right corner of a 3x3 grid :0 We’re doing weekly posts of problems until the start of the in-person tournament on April 13th (Sign up **[here](https://www.stanfordmathtournament.com/competitions/smt-2024)**!) Registrations for our synchronous Online tournament are also open **[here](https://www.stanfordmathtournament.com/competitions/smt-2024-online)**. Also if you’re coming in-person and want to get some pre-competition action, submit your solutions to PoTW [here](https://forms.gle/2FpDoHFeVHg2aqWS6), and the top 3 people with the most correct solutions will receive surprise prize at the tournament :)**Easy:** Consider the curves $x^2 + y^2 = 1$ and $2x^2 + 2xy + y^2 - 2x -2y = 0$ . These curves intersect at two points, one of which is $(1, 0)$ . Find the other one. **Hard:** Tony the mouse starts in the top left corner of a $3x3$ grid. After each second, he randomly moves to an adjacent square with equal probability. What is the probability he reaches the cheese in the bottom right corner before he reaches the mousetrap in the center.
Again feel free to reach out to us at [[email protected]](mailto:[email protected]) with any questions, or ask them in this thread!
Best,
[The SMT Team](https://www.stanfordmathtournament.com/our-team)
| \frac{1}{7} | 0.78125 |
29,089 | The regular octagon $ABCDEFGH$ has its center at $J$. Each of the vertices and the center are to be associated with one of the digits $1$ through $9$, with each digit used once, in such a way that the sums of the numbers on the lines $AJE$, $BJF$, $CJG$, and $DJH$ are equal. In how many ways can this be done? [asy]
size(175);
defaultpen(linewidth(0.8));
path octagon;
string labels[]={"A","B","C","D","E","F","G","H","I"};
for(int i=0;i<=7;i=i+1)
{
pair vertex=dir(135-45/2-45*i);
octagon=octagon--vertex;
label(" $"+labels[i]+"$ ",vertex,dir(origin--vertex));
}
draw(octagon--cycle);
dot(origin);
label(" $J$ ",origin,dir(0));
[/asy] | 1152 | 74.21875 |
29,090 | A triangle has sides of length $7$ and $23$. What is the smallest whole number greater than the perimeter of any triangle with these side lengths? | 60 | 10.9375 |
29,091 | \( S \) is the set of all ordered tuples \((a, b, c, d, e, f)\) where \(a, b, c, d, e, f\) are integers and \(a^2 + b^2 + c^2 + d^2 + e^2 = f^2\). Find the largest \( k \) such that \( k \) divides \( a b c d e f \) for all elements in \( S \). | 24 | 3.125 |
29,092 | From the 6 finalists, 1 first prize, 2 second prizes, and 3 third prizes are to be awarded. Calculate the total number of possible outcomes. | 60 | 72.65625 |
29,093 | Let $ABC$ be the triangle with vertices located at the center of masses of Vincent Huang's house, Tristan Shin's house, and Edward Wan's house; here, assume the three are not collinear. Let $N = 2017$ , and define the $A$ -*ntipodes* to be the points $A_1,\dots, A_N$ to be the points on segment $BC$ such that $BA_1 = A_1A_2 = \cdots = A_{N-1}A_N = A_NC$ , and similarly define the $B$ , $C$ -ntipodes. A line $\ell_A$ through $A$ is called a *qevian* if it passes through an $A$ -ntipode, and similarly we define qevians through $B$ and $C$ . Compute the number of ordered triples $(\ell_A, \ell_B, \ell_C)$ of concurrent qevians through $A$ , $B$ , $C$ , respectively.
*Proposed by Brandon Wang* | 2017^3 - 2 | 0 |
29,094 | Find the greatest root of the polynomial $f(x) = 16x^4 - 8x^3 + 9x^2 - 3x + 1$. | 0.5 | 14.84375 |
29,095 | Given that Bill's age in two years will be three times his current age, and the digits of both Jack's and Bill's ages are reversed, find the current age difference between Jack and Bill. | 18 | 0.78125 |
29,096 | Each of 8 balls is randomly and independently painted either black or white with equal probability. Calculate the probability that every ball is different in color from at least half of the other 7 balls. | \frac{35}{128} | 35.9375 |
29,097 | Pentagon $ABCDE$ is inscribed in a circle such that $ACDE$ is a square with area $12$. Determine the largest possible area of pentagon $ABCDE$. | 9 + 3\sqrt{2} | 21.09375 |
29,098 | Tam created the mosaic shown using a regular hexagon, squares, and equilateral triangles. If the side length of the hexagon is \( 20 \text{ cm} \), what is the outside perimeter of the mosaic? | 240 | 10.9375 |
29,099 | Given a square ABCD with a side length of 2, points M and N are the midpoints of sides BC and CD, respectively. If vector $\overrightarrow {MN}$ = x $\overrightarrow {AB}$ + y $\overrightarrow {AD}$, find the values of xy and $\overrightarrow {AM}$ • $\overrightarrow {MN}$. | -1 | 95.3125 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.