Unnamed: 0
int64 0
40.3k
| problem
stringlengths 10
5.15k
| ground_truth
stringlengths 1
1.22k
| solved_percentage
float64 0
100
|
|---|---|---|---|
10,900 |
For each natural number from 1 to 999, Damir subtracted the last digit from the first digit and wrote all the resulting 1000 differences on the board. For example, for the number 7, Damir wrote 0; for the number 105, he wrote (-4); for the number 61, he wrote 5.
What is the sum of all the numbers on the board?
|
495
| 68.75 |
10,901 |
How many six-digit numbers exist that do not contain the digits zero and eight?
|
262144
| 55.46875 |
10,902 |
Let $p$ and $q$ be positive integers such that \[\frac{3}{5} < \frac{p}{q} < \frac{5}{8}\] and $q$ is as small as possible. What is $p+q$?
|
21
| 10.15625 |
10,903 |
How many of the integers \(1, 2, \ldots, 2004\) can be represented as \(\frac{mn+1}{m+n}\) for positive integers \(m\) and \(n\)?
|
2004
| 59.375 |
10,904 |
What positive integer can $n$ represent if it is known that by erasing the last three digits of the number $n^{3}$, we obtain the number $n$?
|
32
| 38.28125 |
10,905 |
Point \( O \) is located inside an isosceles right triangle \( ABC \). The distance from \( O \) to the vertex \( A \) of the right angle is 6, to the vertex \( B \) is 4, and to the vertex \( C \) is 8. Find the area of triangle \( ABC \).
|
20 + 6\sqrt{7}
| 2.34375 |
10,906 |
Given an arithmetic-geometric sequence $\{a\_n\}$, where $a\_1 + a\_3 = 10$ and $a\_4 + a\_6 = \frac{5}{4}$, find its fourth term and the sum of the first five terms.
|
\frac{31}{2}
| 10.15625 |
10,907 |
Mr. and Mrs. Smith have three children. They own a family van with a driver's seat, a passenger seat in the front, and two seats in the back. Either Mr. Smith or Mrs. Smith must sit in the driver's seat. How many seating arrangements are possible if one of the children insists on sitting in the front passenger seat?
|
12
| 48.4375 |
10,908 |
In $\triangle ABC$, point $D$ is the midpoint of side $BC$. Point $E$ is on $AC$ such that $AE:EC = 1:2$. Point $F$ is on $AD$ such that $AF:FD = 2:1$. If the area of $\triangle DEF$ is 24, determine the area of $\triangle ABC$.
|
432
| 62.5 |
10,909 |
Find the largest real number \(x\) such that
\[
\frac{x^{2} + x - 1 + \left|x^{2} - (x - 1)\right|}{2} = 35x - 250.
\]
|
25
| 100 |
10,910 |
Jill has 8 red marbles and 4 blue marbles in a bag. She removes a marble at random, records the color, puts it back, and then repeats this process until she has withdrawn 8 marbles. What is the probability that exactly four of the marbles that she removes are red? Express your answer as a decimal rounded to the nearest thousandth.
|
0.171
| 32.03125 |
10,911 |
Given a right triangle \( ABC \) with \(\angle A = 60^\circ\) and hypotenuse \( AB = 2 + 2\sqrt{3} \), a line \( p \) is drawn through vertex \( B \) parallel to \( AC \). Points \( D \) and \( E \) are placed on line \( p \) such that \( AB = BD \) and \( BC = BE \). Let \( F \) be the intersection point of lines \( AD \) and \( CE \). Find the possible value of the perimeter of triangle \( DEF \).
Options:
\[
\begin{gathered}
2 \sqrt{2} + \sqrt{6} + 9 + 5 \sqrt{3} \\
\sqrt{3} + 1 + \sqrt{2} \\
9 + 5 \sqrt{3} + 2 \sqrt{6} + 3 \sqrt{2}
\end{gathered}
\]
\[
1 + \sqrt{3} + \sqrt{6}
\]
|
1 + \sqrt{3} + \sqrt{6}
| 25 |
10,912 |
The coefficient of $x^{2}$ in the expansion of $\left( \frac {3}{x}+x\right)\left(2- \sqrt {x}\right)^{6}$ is ______.
|
243
| 60.9375 |
10,913 |
In $\triangle ABC$, $AC=8$, $BC=7$, $\cos B=-\frac{1}{7}$.
(1) Find the measure of angle $A$;
(2) Find the area of $\triangle ABC$.
|
6\sqrt{3}
| 5.46875 |
10,914 |
Given that Jessie moves from 0 to 24 in six steps, and travels four steps to reach point x, then one more step to reach point z, and finally one last step to point y, calculate the value of y.
|
24
| 68.75 |
10,915 |
Let $S_n$ be the sum of the first $n$ terms of an arithmetic sequence $\{a_n\}$. If $S_7=3(a_1+a_9)$, then the value of $\frac{a_5}{a_4}$ is \_\_\_\_\_.
|
\frac{7}{6}
| 100 |
10,916 |
In the rectangular coordinate system $(xOy)$, the parametric equations of the curve $C$ are given by: $\begin{cases} x=2\cos\theta \\ y=\sin\theta \end{cases}$. Establish a polar coordinate system with the coordinate origin $O$ as the pole and the positive half of the $x$-axis as the polar axis.
(1) If the horizontal coordinate of each point on the curve $C$ remains unchanged and the vertical coordinate is stretched to twice its original length, obtain the curve $C_{1}$. Find the polar equation of $C_{1}$;
(2) The polar equation of the straight line $l$ is $\rho\sin\left(\theta+\frac{\pi}{3}\right)=\sqrt{3}$, which intersects the curve $C_{1}$ at points $A$ and $B$. Calculate the area of triangle $AOB$.
|
\sqrt{3}
| 87.5 |
10,917 |
Given that \(a \leq b < c\) are the side lengths of a right triangle, find the maximum constant \(M\) such that
$$
\frac{1}{a}+\frac{1}{b}+\frac{1}{c} \geq \frac{M}{a+b+c} .
$$
|
5 + 3\sqrt{2}
| 50.78125 |
10,918 |
Real numbers \( x \) and \( y \) satisfy \( 4x^2 - 5xy + 4y^2 = 5 \). Let \( S = x^2 + y^2 \). Find the value of \( \frac{1}{S_{\max}} + \frac{1}{S_{\min}} \).
|
8/5
| 61.71875 |
10,919 |
Given that vectors $a$ and $b$ satisfy $(2a+3b) \perp b$, and $|b|=2\sqrt{2}$, find the projection of vector $a$ onto the direction of $b$.
|
-3\sqrt{2}
| 3.90625 |
10,920 |
One day while Tony plays in the back yard of the Kubik's home, he wonders about the width of the back yard, which is in the shape of a rectangle. A row of trees spans the width of the back of the yard by the fence, and Tony realizes that all the trees have almost exactly the same diameter, and the trees look equally spaced. Tony fetches a tape measure from the garage and measures a distance of almost exactly $12$ feet between a consecutive pair of trees. Tony realizes the need to include the width of the trees in his measurements. Unsure as to how to do this, he measures the distance between the centers of the trees, which comes out to be around $15$ feet. He then measures $2$ feet to either side of the first and last trees in the row before the ends of the yard. Tony uses these measurements to estimate the width of the yard. If there are six trees in the row of trees, what is Tony's estimate in feet?
[asy]
size(400);
defaultpen(linewidth(0.8));
draw((0,-3)--(0,3));
int d=8;
for(int i=0;i<=5;i=i+1)
{
draw(circle(7/2+d*i,3/2));
}
draw((5*d+7,-3)--(5*d+7,3));
draw((0,0)--(2,0),Arrows(size=7));
draw((5,0)--(2+d,0),Arrows(size=7));
draw((7/2+d,0)--(7/2+2*d,0),Arrows(size=7));
label(" $2$ ",(1,0),S);
label(" $12$ ",((7+d)/2,0),S);
label(" $15$ ",((7+3*d)/2,0),S);
[/asy]
|
82
| 12.5 |
10,921 |
A rectangle with dimensions $8 \times 2 \sqrt{2}$ and a circle with a radius of 2 have a common center. Find the area of their overlapping region.
|
2 \pi + 4
| 0.78125 |
10,922 |
The probability of an event occurring in each of 900 independent trials is 0.5. Find a positive number $\varepsilon$ such that with a probability of 0.77, the absolute deviation of the event frequency from its probability of 0.5 does not exceed $\varepsilon$.
|
0.02
| 10.9375 |
10,923 |
Four steel balls, each with a radius of 1, are completely packed into a container in the shape of a regular tetrahedron. Find the minimum height of this regular tetrahedron.
|
2+\frac{2 \sqrt{6}}{3}
| 46.875 |
10,924 |
Given a cube \(ABCD-A_1B_1C_1D_1\) with side length 1, and \(E\) as the midpoint of \(D_1C_1\), find the following:
1. The distance between skew lines \(D_1B\) and \(A_1E\).
2. The distance from \(B_1\) to plane \(A_1BE\).
3. The distance from \(D_1C\) to plane \(A_1BE\).
4. The distance between plane \(A_1DB\) and plane \(D_1CB_1\).
|
\frac{\sqrt{3}}{3}
| 33.59375 |
10,925 |
Given the sequence elements \( a_{n} \) such that \( a_{1}=1337 \) and \( a_{2n+1}=a_{2n}=n-a_{n} \) for all positive integers \( n \). Determine the value of \( a_{2004} \).
|
2004
| 46.09375 |
10,926 |
Given that the amount of cultural and tourism vouchers issued is $2.51 million yuan, express this amount in scientific notation.
|
2.51 \times 10^{6}
| 0 |
10,927 |
The fraction \(\frac{p}{q}\) is in its simplest form. If \(\frac{7}{10} < \frac{p}{q} < \frac{11}{15}\), where \(q\) is the smallest possible positive integer and \(c = pq\), find the value of \(c\).
|
35
| 60.15625 |
10,928 |
In the right triangular prism $ABC - A_1B_1C_1$, $\angle ACB = 90^\circ$, $AC = 2BC$, and $A_1B \perp B_1C$. Find the sine of the angle between $B_1C$ and the lateral face $A_1ABB_1$.
|
\frac{\sqrt{10}}{5}
| 17.1875 |
10,929 |
In a convex pentagon, all diagonals are drawn. For each pair of diagonals that intersect inside the pentagon, the smaller of the angles between them is found. What values can the sum of these five angles take?
|
180
| 54.6875 |
10,930 |
How many six-digit numbers exist in which each subsequent digit is less than the previous one?
|
210
| 36.71875 |
10,931 |
Given the function $f(x) = 2\sin^2x + \cos\left(\frac{\pi}{3} - 2x\right)$.
(1) Find the decreasing interval of $f(x)$ on $[0, \pi]$.
(2) Let $\triangle ABC$ have internal angles $A$, $B$, $C$ opposite sides $a$, $b$, $c$ respectively. If $f(A) = 2$, and the vector $\overrightarrow{m} = (1, 2)$ is collinear with the vector $\overrightarrow{n} = (\sin B, \sin C)$, find the value of $\frac{a}{b}$.
|
\sqrt{3}
| 26.5625 |
10,932 |
Professor Severus Snape brewed three potions, each in a volume of 600 ml. The first potion makes the drinker intelligent, the second makes them beautiful, and the third makes them strong. To have the effect of the potion, it is sufficient to drink at least 30 ml of each potion. Severus Snape intended to drink his potions, but he was called away by the headmaster and left the signed potions in large jars on his table. Taking advantage of his absence, Harry, Hermione, and Ron approached the table with the potions and began to taste them.
The first to try the potions was Hermione: she approached the first jar of the intelligence potion and drank half of it, then poured the remaining potion into the second jar of the beauty potion, mixed the contents thoroughly, and drank half of it. Then it was Harry's turn: he drank half of the third jar of the strength potion, then poured the remainder into the second jar, thoroughly mixed everything in the jar, and drank half of it. Now all the contents were left in the second jar, which Ron ended up with. What percentage of the contents of this jar does Ron need to drink to ensure that each of the three potions will have an effect on him?
|
40
| 1.5625 |
10,933 |
Calculate the sum of the following fractions: $\frac{1}{12} + \frac{2}{12} + \frac{3}{12} + \frac{4}{12} + \frac{5}{12} + \frac{6}{12} + \frac{7}{12} + \frac{8}{12} + \frac{9}{12} + \frac{65}{12} + \frac{3}{4}$.
A) $\frac{119}{12}$
B) $9$
C) $\frac{113}{12}$
D) $10$
|
\frac{119}{12}
| 80.46875 |
10,934 |
Using the digits $2, 4, 6$ to construct six-digit numbers, how many such numbers are there if no two consecutive digits in the number can both be 2 (for example, 626442 is allowed, but 226426 is not allowed)?
|
448
| 82.8125 |
10,935 |
We have created a convex polyhedron using pentagons and hexagons where three faces meet at each vertex. Each pentagon shares its edges with 5 hexagons, and each hexagon shares its edges with 3 pentagons. How many faces does the polyhedron have?
|
32
| 83.59375 |
10,936 |
You have a \(2 \times 3\) grid filled with integers between 1 and 9. The numbers in each row and column are distinct. The first row sums to 23, and the columns sum to 14, 16, and 17 respectively.
Given the following grid:
\[
\begin{array}{c|c|c|c|}
& 14 & 16 & 17 \\
\hline
23 & a & b & c \\
\hline
& x & y & z \\
\hline
\end{array}
\]
What is \(x + 2y + 3z\)?
|
49
| 10.15625 |
10,937 |
Given \( A \cup B = \left\{a_{1}, a_{2}, a_{3}\right\} \) and \( A \neq B \), where \((A, B)\) and \((B, A)\) are considered different pairs, find the number of such pairs \((A, B)\).
|
26
| 70.3125 |
10,938 |
For each pair of distinct natural numbers \(a\) and \(b\), not exceeding 20, Petya drew the line \( y = ax + b \) on the board. That is, he drew the lines \( y = x + 2, y = x + 3, \ldots, y = x + 20, y = 2x + 1, y = 2x + 3, \ldots, y = 2x + 20, \ldots, y = 3x + 1, y = 3x + 2, y = 3x + 4, \ldots, y = 3x + 20, \ldots, y = 20x + 1, \ldots, y = 20x + 19 \). Vasia drew a circle of radius 1 with center at the origin on the same board. How many of Petya’s lines intersect Vasia’s circle?
|
190
| 20.3125 |
10,939 |
Add $5.467$ and $3.92$ as a decimal.
|
9.387
| 100 |
10,940 |
Among 6 courses, if person A and person B each choose 3 courses, the number of ways in which exactly 1 course is chosen by both A and B is \_\_\_\_\_\_.
|
180
| 59.375 |
10,941 |
If the first digit of a four-digit number, which is a perfect square, is decreased by 3, and the last digit is increased by 3, it also results in a perfect square. Find this number.
|
4761
| 44.53125 |
10,942 |
Given an ellipse $\frac {x^{2}}{a^{2}} + \frac {y^{2}}{b^{2}} = 1 (a > b > 0)$ with vertex $B$ at the top, vertex $A$ on the right, and right focus $F$. Let $E$ be a point on the lower half of the ellipse such that the tangent at $E$ is parallel to $AB$. If the eccentricity of the ellipse is $\frac {\sqrt{2}}{2}$, then the slope of line $EF$ is __________.
|
\frac {\sqrt {2}}{4}
| 0 |
10,943 |
A square is initially divided into sixteen equal smaller squares. The center four squares are then each divided into sixteen smaller squares of equal area, and this pattern continues indefinitely. In each set of sixteen smaller squares, the four corner squares are shaded. What fractional part of the entire square is shaded?
|
\frac{1}{3}
| 51.5625 |
10,944 |
Two adjacent faces of a tetrahedron, which are equilateral triangles with side length 1, form a dihedral angle of 60 degrees. The tetrahedron is rotated around the common edge of these faces. Find the maximum area of the projection of the rotating tetrahedron onto a plane containing the given edge. (12 points)
|
\frac{\sqrt{3}}{4}
| 8.59375 |
10,945 |
Given a hyperbola $C$ with an eccentricity of $\sqrt {3}$, foci $F\_1$ and $F\_2$, and a point $A$ on the curve $C$. If $|F\_1A|=3|F\_2A|$, then $\cos \angle AF\_2F\_1=$ \_\_\_\_\_\_.
|
\frac{\sqrt{3}}{3}
| 68.75 |
10,946 |
Given that there are 5 balls of each of the three colors: black, white, and red, each marked with the numbers 1, 2, 3, 4, 5, calculate the number of different ways to draw 5 balls such that their numbers are all different and all three colors are present.
|
150
| 28.90625 |
10,947 |
Given the function $f(x)= \dfrac {2-\cos \left( \dfrac {\pi}{4}(1-x)\right)+\sin \left( \dfrac {\pi}{4}(1-x)\right)}{x^{2}+4x+5}(-4\leqslant x\leqslant 0)$, find the maximum value of $f(x)$.
|
2+ \sqrt {2}
| 0 |
10,948 |
When \( n \) is a positive integer, the function \( f \) satisfies \( f(n+3)=\frac{f(n)-1}{f(n)+1} \), with \( f(1) \neq 0 \) and \( f(1) \neq \pm 1 \). Find the value of \( f(8) \cdot f(2018) \).
|
-1
| 42.96875 |
10,949 |
Find the largest possible value of \( n \) such that there exist \( n \) consecutive positive integers whose sum is equal to 2010.
|
60
| 80.46875 |
10,950 |
In how many ways can a committee of three people be formed if the members are to be chosen from four married couples?
|
32
| 82.03125 |
10,951 |
A standard six-sided die is rolled 3 times. If the sum of the numbers rolled on the first two rolls equals the number rolled on the third roll, what is the probability that at least one of the numbers rolled is a 2?
|
$\frac{8}{15}$
| 0 |
10,952 |
Given that $-\frac{\pi}{2}<\alpha<\frac{\pi}{2}, 2 \tan \beta=\tan 2\alpha, \tan (\beta-\alpha)=-2 \sqrt{2}$, find the value of $\cos \alpha$.
|
\frac{\sqrt{3}}{3}
| 82.8125 |
10,953 |
For a given positive integer \( k \), let \( f_{1}(k) \) represent the square of the sum of the digits of \( k \), and define \( f_{n+1}(k) = f_{1}\left(f_{n}(k)\right) \) for \( n \geq 1 \). Find the value of \( f_{2005}\left(2^{2006}\right) \).
|
169
| 57.8125 |
10,954 |
Let $A = 3^7 + \binom{7}{2}3^5 + \binom{7}{4}3^3 + \binom{7}{6}3$, and $B = \binom{7}{1}3^6 + \binom{7}{3}3^4 + \binom{7}{5}3^2 + 1$. Find the value of $A - B$.
|
128
| 83.59375 |
10,955 |
In a square \(ABCD\), let \(P\) be a point on the side \(BC\) such that \(BP = 3PC\) and \(Q\) be the midpoint of \(CD\). If the area of the triangle \(PCQ\) is 5, what is the area of triangle \(QDA\)?
|
20
| 55.46875 |
10,956 |
A frustum of a right circular cone is formed by cutting a smaller cone from a larger cone. Suppose the frustum has a lower base radius of 8 inches, an upper base radius of 2 inches, and a height of 5 inches. Calculate the total surface area of the frustum.
|
10\pi \sqrt{61} + 68\pi
| 22.65625 |
10,957 |
Given the function $f(x)=2\ln(3x)+8x$, calculate the value of $\lim_{\triangle x \to 0} \frac{f(1-2\triangle x)-f(1)}{\triangle x}$.
|
-20
| 91.40625 |
10,958 |
The sum of the interior numbers in the sixth row of Pascal's Triangle is 30. What is the sum of the interior numbers of the eighth row?
|
126
| 21.09375 |
10,959 |
A circle with a radius of 15 is tangent to two adjacent sides \( AB \) and \( AD \) of square \( ABCD \). On the other two sides, the circle intercepts segments of 6 and 3 cm from the vertices, respectively. Find the length of the segment that the circle intercepts from vertex \( B \) to the point of tangency.
|
12
| 11.71875 |
10,960 |
Calculate the volume of the solid bounded by the surfaces \(x + z = 6\), \(y = \sqrt{x}\), \(y = 2\sqrt{x}\), and \(z = 0\) using a triple integral.
|
\frac{48}{5} \sqrt{6}
| 0 |
10,961 |
The four faces of a tetrahedral die are labelled $0, 1, 2,$ and $3,$ and the die has the property that, when it is rolled, the die promptly vanishes, and a number of copies of itself appear equal to the number on the face the die landed on. For example, if it lands on the face labelled $0,$ it disappears. If it lands on the face labelled $1,$ nothing happens. If it lands on the face labelled $2$ or $3,$ there will then be $2$ or $3$ copies of the die, respectively (including the original). Suppose the die and all its copies are continually rolled, and let $p$ be the probability that they will all eventually disappear. Find $\left\lfloor \frac{10}{p} \right\rfloor$ .
|
24
| 43.75 |
10,962 |
It is known that the numbers $\frac{x}{2}, 2x - 3, \frac{18}{x} + 1$, taken in the specified order, form a geometric progression. Find the common ratio of this progression. Round your answer to two decimal places.
|
2.08
| 85.9375 |
10,963 |
Five marbles are distributed at a random among seven urns. What is the expected number of urns with exactly one marble?
|
6480/2401
| 56.25 |
10,964 |
Evaluate or simplify:
1. $\log_2 \sqrt{2}+\log_9 27+3^{\log_3 16}$;
2. $0.25^{-2}+\left( \frac{8}{27} \right)^{-\frac{1}{3}}-\frac{1}{2}\lg 16-2\lg 5+\left( \frac{1}{2} \right)^{0}$.
|
\frac{33}{2}
| 75.78125 |
10,965 |
A fair coin is flipped 9 times. What is the probability that at least 6 of the flips result in heads?
|
\frac{65}{256}
| 0.78125 |
10,966 |
The set of positive even numbers $\{2, 4, 6, \cdots\}$ is grouped in increasing order such that the $n$-th group has $3n-2$ numbers:
\[
\{2\}, \{4, 6, 8, 10\}, \{12, 14, 16, 18, 20, 22, 24\}, \cdots
\]
Determine which group contains the number 2018.
|
27
| 43.75 |
10,967 |
A circle with a radius of 2 passes through the midpoints of three sides of triangle \(ABC\), where the angles at vertices \(A\) and \(B\) are \(30^{\circ}\) and \(45^{\circ}\), respectively.
Find the height drawn from vertex \(A\).
|
2 + 2\sqrt{3}
| 17.1875 |
10,968 |
In a checkers tournament, students from 10th and 11th grades participated. Each player played against every other player exactly once. A win earned a player 2 points, a draw earned 1 point, and a loss earned 0 points. The number of 11th graders was 10 times the number of 10th graders, and together they scored 4.5 times more points than all the 10th graders combined. How many points did the most successful 10th grader score?
|
20
| 57.03125 |
10,969 |
Jindra collects dice, all of the same size. Yesterday he found a box in which he started stacking the dice. He managed to fully cover the square base with one layer of dice. He similarly stacked five more layers, but he ran out of dice halfway through the next layer. Today, Jindra received 18 more dice from his grandmother, which were exactly the amount he needed to complete this layer.
How many dice did Jindra have yesterday?
|
234
| 30.46875 |
10,970 |
Given that the circumference of a sector is $20\,cm$ and its area is $9\,cm^2$, find the radian measure of the central angle of the sector.
|
\frac{2}{9}
| 69.53125 |
10,971 |
Given that $O$ is the origin of coordinates, and $M$ is a point on the ellipse $\frac{x^2}{2} + y^2 = 1$. Let the moving point $P$ satisfy $\overrightarrow{OP} = 2\overrightarrow{OM}$.
- (I) Find the equation of the trajectory $C$ of the moving point $P$;
- (II) If the line $l: y = x + m (m \neq 0)$ intersects the curve $C$ at two distinct points $A$ and $B$, find the maximum value of the area of $\triangle OAB$.
|
2\sqrt{2}
| 55.46875 |
10,972 |
Given that \( n! \), in decimal notation, has exactly 57 ending zeros, find the sum of all possible values of \( n \).
|
1185
| 39.84375 |
10,973 |
Given a triangle \(A B C\) with \(A B = A C\) and \(\angle A = 110^{\circ}\). Inside the triangle, a point \(M\) is chosen such that \(\angle M B C = 30^{\circ}\) and \(\angle M C B = 25^{\circ}\). Find \(\angle A M C\).
|
85
| 32.8125 |
10,974 |
New definition: Given that $y$ is a function of $x$, if there exists a point $P(a, a+2)$ on the graph of the function, then point $P$ is called a "real point" on the graph of the function. For example, the "real point" on the line $y=2x+1$ is $P(1,3)$.
$(1)$ Determine whether there is a "real point" on the line $y=\frac{1}{3}x+4$. If yes, write down its coordinates directly; if not, explain the reason.
$(2)$ If there are two "real points" on the parabola $y=x^{2}+3x+2-k$, and the distance between the two "real points" is $2\sqrt{2}$, find the value of $k$.
$(3)$ If there exists a unique "real point" on the graph of the quadratic function $y=\frac{1}{8}x^{2}+\left(m-t+1\right)x+2n+2t-2$, and when $-2\leqslant m\leqslant 3$, the minimum value of $n$ is $t+4$, find the value of $t$.
|
t=-1
| 61.71875 |
10,975 |
The sum of three different numbers is 75. The two larger numbers differ by 5 and the two smaller numbers differ by 4. What is the value of the largest number?
|
\frac{89}{3}
| 0.78125 |
10,976 |
On a line, 6 points are given, and on a parallel line, 8 points are given. How many triangles can be formed with the vertices at the given points?
|
288
| 65.625 |
10,977 |
In a rectangular parallelepiped $A B C D A_{1} B_{1} C_{1} D_{1}$, the lengths of the edges are known: $A B=14$, $A D=60$, $A A_{1}=40$. A point $E$ is marked at the midpoint of edge $A_{1} B_{1}$, and point $F$ is marked at the midpoint of edge $B_{1} C_{1}$. Find the distance between the lines $A E$ and $B F$.
|
13.44
| 5.46875 |
10,978 |
$18 \cdot 92$ A square $ABCD$ has side lengths of 1. Points $E$ and $F$ lie on sides $AB$ and $AD$ respectively, such that $AE = AF$. If the quadrilateral $CDFE$ has the maximum area, what is the maximum area?
|
$\frac{5}{8}$
| 0 |
10,979 |
In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are denoted as $a$, $b$, and $c$ respectively, and it is given that $\cos \frac{A}{2}= \frac{2 \sqrt{5}}{5}, \overrightarrow{AB} \cdot \overrightarrow{AC}=15$.
$(1)$ Find the area of $\triangle ABC$;
$(2)$ If $\tan B=2$, find the value of $a$.
|
2 \sqrt{5}
| 71.875 |
10,980 |
Given that the hyperbola C passes through the point (4, 3) and shares the same asymptotes with $$\frac {x^{2}}{4}$$ - $$\frac {y^{2}}{9}$$ = 1, determine the equation of C and its eccentricity.
|
\frac {\sqrt {13}} {2}
| 0 |
10,981 |
The distance between cities $A$ and $B$ is 435 km. A train left city $A$ at a speed of 45 km/h. After 40 minutes, another train left city $B$ heading towards the first train at a speed of 55 km/h. What distance will be between them one hour before they meet?
|
100
| 48.4375 |
10,982 |
In the expression \((x+y+z)^{2024} + (x-y-z)^{2024}\), the parentheses are expanded and like terms are combined. How many monomials \(x^{a} y^{b} z^{c}\) have a non-zero coefficient?
|
1026169
| 25.78125 |
10,983 |
Consider a set of 150 cards numbered from 1 to 150. Each card is randomly placed in a box. A card is selected at random from this box. What is the probability that the number on the card is a multiple of 4, 5, or 6? Express your answer as a common fraction.
|
\frac{7}{15}
| 62.5 |
10,984 |
Given points $A(-2, 0)$ and $B(0, 2)$, $k$ is a constant, and $M$, $N$ are two distinct points on the circle ${x^2} + {y^2} + kx = 0$. $P$ is a moving point on the circle ${x^2} + {y^2} + kx = 0$. If $M$ and $N$ are symmetric about the line $x - y - 1 = 0$, find the maximum area of $\triangle PAB$.
|
3 + \sqrt{2}
| 35.15625 |
10,985 |
Given \(1 \leq x^{2}+y^{2} \leq 4\), find the sum of the maximum and minimum values of \(x^{2}-xy+y^{2}\).
|
6.5
| 0 |
10,986 |
Let \(ABCD\) be an isosceles trapezoid such that \(AD = BC\), \(AB = 3\), and \(CD = 8\). Let \(E\) be a point in the plane such that \(BC = EC\) and \(AE \perp EC\). Compute \(AE\).
|
2\sqrt{6}
| 2.34375 |
10,987 |
If $θ∈[0,π]$, then the probability of $\sin (θ+ \frac {π}{3}) > \frac {1}{2}$ being true is ______.
|
\frac {1}{2}
| 85.15625 |
10,988 |
In triangle $ABC$, $AB=3$, $AC=4$, and $\angle BAC=60^{\circ}$. If $P$ is a point in the plane of $\triangle ABC$ and $AP=2$, calculate the maximum value of $\vec{PB} \cdot \vec{PC}$.
|
10 + 2 \sqrt{37}
| 35.15625 |
10,989 |
Pasha wrote down an equation consisting of integers and arithmetic operation signs in his notebook. He then encoded each digit and operation sign in the expression on the left side of the equation with a letter, replacing identical digits or signs with identical letters, and different ones with different letters. He ended up with the equation:
$$
\text { VUZAKADEM = } 2023 .
$$
Create at least one version of the expression that Pasha could have encoded. Allowed arithmetic operations are addition, subtraction, multiplication, and division. Parentheses cannot be used.
|
2065 + 5 - 47
| 0 |
10,990 |
In quadrilateral $ABCD$ , $AB \parallel CD$ and $BC \perp AB$ . Lines $AC$ and $BD$ intersect at $E$ . If $AB = 20$ , $BC = 2016$ , and $CD = 16$ , find the area of $\triangle BCE$ .
*Proposed by Harrison Wang*
|
8960
| 49.21875 |
10,991 |
Given the equation of the circle $x^{2}+y^{2}-8x+15=0$, find the minimum value of $k$ such that there exists at least one point on the line $y=kx+2$ that can serve as the center of a circle with a radius of $1$ that intersects with circle $C$.
|
- \frac{4}{3}
| 86.71875 |
10,992 |
Let's divide a sequence of natural numbers into groups:
\((1), (2,3), (4,5,6), (7,8,9,10), \ldots\)
Let \( S_{n} \) denote the sum of the \( n \)-th group of numbers. Find \( S_{16} - S_{4} - S_{1} \).
|
2021
| 88.28125 |
10,993 |
The coefficient of $$\frac {1}{x}$$ in the expansion of $$(1-x^2)^4\left(\frac {x+1}{x}\right)^5$$ is __________.
|
-29
| 8.59375 |
10,994 |
Let $a_1 = \sqrt 7$ and $b_i = \lfloor a_i \rfloor$ , $a_{i+1} = \dfrac{1}{b_i - \lfloor b_i \rfloor}$ for each $i\geq i$ . What is the smallest integer $n$ greater than $2004$ such that $b_n$ is divisible by $4$ ? ( $\lfloor x \rfloor$ denotes the largest integer less than or equal to $x$ )
|
2005
| 36.71875 |
10,995 |
Given the function $f(x) = e^{x} \cos x - x$.
(I) Find the equation of the tangent line to the curve $y = f(x)$ at the point $(0, f(0))$;
(II) Find the maximum and minimum values of the function $f(x)$ on the interval $[0, \frac{\pi}{2}]$.
|
-\frac{\pi}{2}
| 77.34375 |
10,996 |
Find a positive integer $n$ with five non-zero different digits, which satisfies to be equal to the sum of all the three-digit numbers that can be formed using the digits of $n$ .
|
35964
| 98.4375 |
10,997 |
Find the number of 8-digit numbers where the product of the digits equals 9261. Present the answer as an integer.
|
1680
| 0 |
10,998 |
What is the last two digits of base- $3$ representation of $2005^{2003^{2004}+3}$ ?
|
11
| 3.125 |
10,999 |
Given that the sine and cosine values of angle $α$ are both negative, and $\cos(75^{\circ}+α)=\frac{1}{3}$, find the value of $\cos(105^{\circ}-α)+\sin(α-105^{\circ})$ = \_\_\_\_\_\_.
|
\frac{2\sqrt{2}-1}{3}
| 80.46875 |
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