task_type stringclasses 1
value | problem stringlengths 23 3.94k | answer stringlengths 1 231 | problem_tokens int64 8 1.39k | answer_tokens int64 1 200 |
|---|---|---|---|---|
math | Example 2 Color each vertex of a square pyramid with one color, and make the endpoints of the same edge have different colors. If only 5 colors are available, then the total number of different coloring methods is $\qquad$ | 420 | 47 | 3 |
math | 6. Let $0 \leqslant x_{i} \leqslant 1(i=1,2, \cdots, 5)$. Then
$$
\begin{array}{l}
\left|x_{1}-x_{2}\right|^{3}+\left|x_{2}-x_{3}\right|^{3}+\left|x_{3}-x_{4}\right|^{3}+ \\
\left|x_{4}-x_{5}\right|^{3}+\left|x_{5}-x_{1}\right|^{3}
\end{array}
$$
The maximum value is . $\qquad$ | 4 | 139 | 1 |
math | 3. Find the sum of the first 10 elements that are common to both the arithmetic progression $\{5,8,11,13, \ldots\}$ and the geometric progression $\{20,40,80,160, \ldots\}$. (10 points) | 6990500 | 67 | 7 |
math | The numbers 1 and 2 are written on the board. Each day, the scientific consultant Vybegallo replaces the two written numbers with their arithmetic mean and harmonic mean.
a) One day, one of the written numbers (it is unknown which one) turned out to be 941664/665857. What was the other number at that moment?
b) Will ... | No | 95 | 1 |
math | 10. (HUN 2) Three persons $A, B, C$, are playing the following game: A $k$ element subset of the set $\{1, \ldots, 1986\}$ is randomly chosen, with an equal probability of each choice, where $k$ is a fixed positive integer less than or equal to 1986. The winner is $A, B$ or $C$, respectively, if the sum of the chosen n... | 3 \nmid k | 147 | 5 |
math | ## Task 4 - 160524
Each student needs 15 notebooks per year. From 1 ton of paper, 25000 notebooks can be produced. How many students in total can be supplied with notebooks for one year from 3 tons of paper? | 5000 | 61 | 4 |
math | 5. In an isosceles trapezoid \(ABCD\) with bases \(AD\) and \(BC\), perpendiculars \(BH\) and \(DK\) are drawn from vertices \(B\) and \(D\) to the diagonal \(AC\). It is known that the feet of the perpendiculars lie on the segment \(AC\) and \(AC=20\), \(AK=19\), \(AH=3\). Find the area of trapezoid \(ABCD\).
(10 poi... | 120 | 109 | 3 |
math | Example 13. Find $\int \frac{d x}{x^{2} \sqrt{1+x^{2}}}$. | C-\frac{\sqrt{1+x^{2}}}{x} | 27 | 14 |
math | Let $P$, $A$, $B$, $C$, $D$ be points on a plane such that $PA = 9$, $PB = 19$, $PC = 9$, $PD = 5$, $\angle APB = 120^\circ$, $\angle BPC = 45^\circ$, $\angle CPD = 60^\circ$, and $\angle DPA = 135^\circ$. Let $G_1$, $G_2$, $G_3$, and $G_4$ be the centroids of triangles $PAB$, $PBC$, $PCD$, $PDA$. $[G_1G_2G_3G_4]$ can ... | 29 | 199 | 2 |
math | Problem 4. A rectangle and a square have equal areas. The numerical values of their dimensions are natural numbers from the first ten. If the width of the rectangle is $2 \mathrm{~cm}$, determine the dimensions of the rectangle and the square. | 4\, | 53 | 3 |
math | Pratyya and Payel have a number each, $n$ and $m$ respectively, where $n>m.$ Everyday, Pratyya multiplies his number by $2$ and then subtracts $2$ from it, and Payel multiplies his number by $2$ and then add $2$ to it. In other words, on the first day their numbers will be $(2n-2)$ and $(2m+2)$ respectively. Find minim... | 4 | 133 | 1 |
math | Example 11 Let the polynomial $h(x)$, when divided by $x-1, x-2, x-$ 3, have remainders $4,8,16$ respectively. Find the remainder when $h(x)$ is divided by $(x-1)(x-2)(x-3)$. | 2x^2-2x+4 | 67 | 9 |
math | 14. Find all prime numbers $x$ and $y$ that satisfy $x^{3}-x=y^{7}-y^{3}$.
Translate the above text into English, please retain the original text's line breaks and format, and output the translation result directly. | (x,y)=(5,2) | 56 | 7 |
math | Factorize the polynomial $P(X)=2 X^{3}+3 X^{2}-3 X-2$ in $\mathbb{R}$. | 2(X-1)(X+2)(X+\frac{1}{2}) | 32 | 17 |
math | An equilateral pentagon $AMNPQ$ is inscribed in triangle $ABC$ such that $M\in\overline{AB}$, $Q\in\overline{AC}$, and $N,P\in\overline{BC}$.
Suppose that $ABC$ is an equilateral triangle of side length $2$, and that $AMNPQ$ has a line of symmetry perpendicular to $BC$. Then the area of $AMNPQ$ is $n-p\sqrt{q}$, whe... | 5073 | 158 | 4 |
math | 1. Find the term that does not contain $x$ in the expansion (according to the binomial formula) of $\left(2 x+\frac{\sqrt[3]{x^{2}}}{x}\right)^{n}$ if the sum of all binomial coefficients in this expansion is 256. | 112 | 65 | 3 |
math | In the parallelepiped $A B C D A 1 B 1 C 1 D 1$, a segment is drawn connecting vertex $A$ to the midpoint of edge $C C 1$. In what ratio does this segment get divided by the plane $B D A 1$? | 2:3 | 62 | 3 |
math | 【Question 14】Select 5 digits from $1 \sim 9$, to form a five-digit number, such that this five-digit number can be evenly divided by any of the 5 selected digits, but cannot be evenly divided by any of the 4 unselected digits. Then, the smallest value of this five-digit number is $\qquad$.
---
Note: The blank at the ... | 14728 | 96 | 5 |
math | Three, (15 points) The three coefficients $a, b, c$ and the two roots $x_{1}, x_{2}$ of the quadratic equation $a x^{2}+b x+c$ are written in some order as five consecutive integers. Find all such quadratic equations, and briefly describe the reasoning process. | 2x^{2}-2,-2x^{2}+2 | 68 | 14 |
math | A line contains 40 characters: 20 crosses and 20 zeros. In one move, you can swap any two adjacent characters. What is the minimum number of moves required to ensure that 20 consecutive characters are crosses?
# | 200 | 50 | 3 |
math | ## Task B-4.2.
Determine $n \in \mathbb{N}$ such that the polynomial $P(x)=\left(2 x^{2}+1\right)^{n+1}-(3 x+15)^{2 n}$ is divisible by the polynomial $Q(x)=x+2$. In this case, determine all the non-real roots of the polynomial $P$. | x_{1,2}=\frac{-3\i\sqrt{119}}{4} | 86 | 22 |
math | Ondra, Matěj, and Kuba are returning from gathering nuts, and they have a total of 120. Matěj complains that Ondra, as always, has the most. Their father orders Ondra to give Matěj enough nuts to double his amount. Now Kuba complains that Matěj has the most. On their father's command, Matěj gives Kuba enough nuts to do... | 55,35,30 | 168 | 8 |
math | Task B-1.8. The perimeter of a right triangle is $14 \mathrm{~cm}$. A square is constructed outward on each side. The sum of the areas of all the squares is $72 \mathrm{~cm}^{2}$. What is the area of the given right triangle? | 7 | 66 | 1 |
math | The polynomial of seven variables
$$
Q(x_1,x_2,\ldots,x_7)=(x_1+x_2+\ldots+x_7)^2+2(x_1^2+x_2^2+\ldots+x_7^2)
$$
is represented as the sum of seven squares of the polynomials with nonnegative integer coefficients:
$$
Q(x_1,\ldots,x_7)=P_1(x_1,\ldots,x_7)^2+P_2(x_1,\ldots,x_7)^2+\ldots+P_7(x_1,\ldots,x_7)^2.
$$
Find all... | 3 | 170 | 1 |
math | Example 1 In $\triangle A B C$, $A B=A C, \angle A=$ $80^{\circ}, D$ is a point inside the shape, and $\angle D A B=\angle I B A=$ $10^{\circ}$. Find the degree measure of $\angle A C D$.
---
Note: The symbol $\mathrm{I}$ in the original text seems to be a typo or a misprint, and it should likely be $D$ for consisten... | 30^{\circ} | 182 | 6 |
math | 14. Given real numbers $a, b, c$, satisfying $a+b+c=0, a^{2}+b^{2}+c^{2}=6$. Then the maximum value of $a$ is | 2 | 46 | 1 |
math | Example 1 Given the equation about $x$: $\lg (x-1)+\lg (3-x)=\lg (a-x)$ has two distinct real roots, find the range of values for $a$. | 3<<\frac{13}{4} | 44 | 10 |
math | 3. From Nizhny Novgorod to Astrakhan, a steamboat takes 5 days, and on the return trip - 7 days. How long will it take for rafts to float from Nizhny Novgorod to Astrakhan? | 35 | 56 | 2 |
math | 2. Let $x, y$ be positive numbers, and $S$ be the smallest number among $x, y+\frac{1}{x}, \frac{1}{y}$. Find the maximum possible value of $S$. For what values of $x, y$ is this maximum value achieved?
(6th All-Soviet Union Olympiad) | \sqrt{2} | 75 | 5 |
math | 10.184. Calculate the area of a trapezoid, the parallel sides of which are 16 and 44 cm, and the non-parallel sides are 17 and 25 cm. | 450\mathrm{~}^{2} | 48 | 11 |
math | ## problem statement
Find the point $M^{\prime}$ symmetric to the point $M$ with respect to the plane.
$M(3 ;-3 ;-1)$
$2 x-4 y-4 z-13=0$ | M^{\}(2;-1;1) | 51 | 10 |
math | 4・114 Try to solve the system of equations
$$\left\{\begin{array}{l}
\lg x+\lg y=1 \\
x^{2}+y^{2}-3 x-3 y=8
\end{array}\right.$$ | x_1=2, y_1=5; x_2=5, y_2=2 | 57 | 23 |
math | Given the circle $O$ and the line $e$, on which the point $A$ moves. The polar of point $A$ with respect to the circle intersects the perpendicular line to $e$ at point $A$ at point $M$. What is the geometric locus of point $M$? | y^{2}=r^{2}- | 62 | 8 |
math | 12. In the Cartesian coordinate system $x O y$, it is known that points $A_{1}(-2,0), A_{2}(2,0)$, and a moving point $P(x, y)$ satisfies the product of the slopes of lines $A_{1} P$ and $A_{2} P$ is $-\frac{3}{4}$. Let the trajectory of point $P$ be curve $C$.
(1) Find the equation of $C$;
(2) Suppose point $M$ is on ... | 0 | 188 | 1 |
math | Find the real number $k$ such that $a$, $b$, $c$, and $d$ are real numbers that satisfy the system of equations
\begin{align*}
abcd &= 2007,\\
a &= \sqrt{55 + \sqrt{k+a}},\\
b &= \sqrt{55 - \sqrt{k+b}},\\
c &= \sqrt{55 + \sqrt{k-c}},\\
d &= \sqrt{55 - \sqrt{k-d}}.
\end{align*} | 1018 | 113 | 4 |
math | 46. Solve the equation in integers
$$
\sqrt{\sqrt{x+\sqrt{x+\sqrt{x+\ldots+\sqrt{x}}}}}=y
$$ | 0 | 34 | 1 |
math | 331. Find the last two digits of the number $137^{42}$. | 69 | 21 | 2 |
math | Let the functions $f(\alpha,x)$ and $g(\alpha)$ be defined as \[f(\alpha,x)=\dfrac{(\frac{x}{2})^\alpha}{x-1}\qquad\qquad\qquad g(\alpha)=\,\dfrac{d^4f}{dx^4}|_{x=2}\] Then $g(\alpha)$ is a polynomial is $\alpha$. Find the leading coefficient of $g(\alpha)$. | \frac{1}{16} | 99 | 8 |
math | 1. Let $Q$ be a polynomial
$$
Q(x)=a_{0}+a_{1} x+\cdots+a_{n} x^{n},
$$
where $a_{0}, \ldots, a_{n}$ are nonnegative integers. Given that $Q(1)=4$ and $Q(5)=152$, find $Q(6)$. | 254 | 83 | 3 |
math | 3. A basketball championship has been played in a double round-robin system (each pair of teams play each other twice) and without ties (if the game ends in a tie, there are overtimes until one team wins). The winner of the game gets 2 points and the loser gets 1 point. At the end of the championship, the sum of the po... | 39 | 99 | 2 |
math | 4. There is an unlimited supply of square glasses in 10 colors. In how many ways can 4 glasses be inserted into a $2 \times 2$ window frame so that some color appears in both the upper and lower halves of the window. | 3430 | 53 | 4 |
math | Let the major axis of an ellipse be $AB$, let $O$ be its center, and let $F$ be one of its foci. $P$ is a point on the ellipse, and $CD$ a chord through $O$, such that $CD$ is parallel to the tangent of the ellipse at $P$. $PF$ and $CD$ intersect at $Q$. Compare the lengths of $PQ$ and $OA$. | PQ = OA | 92 | 4 |
math | 15. (3 points) There are 300 chess pieces in black and white. The black crow divides the black and white pieces into 100 piles, with 3 pieces in each pile. Among them, there are 27 piles with only $l$ white piece, 42 piles with 2 or 3 black pieces, and the number of piles with 3 white pieces is equal to the number of p... | 158 | 113 | 3 |
math | 2. There is a class of four-digit numbers that, when divided by 5, leave a remainder of 1; when divided by 7, leave a remainder of 4; and when divided by 11, leave a remainder of 9. What is the smallest four-digit number in this class? | 1131 | 64 | 4 |
math | Since $(36,83)=1$, by Lemma $1-$ there must exist two integers $x$, $y$ such that
$$36 x+83 y=1$$
holds, find $x, y$. | x=30, y=-13 | 49 | 9 |
math | There are $n$ white and $n$ black balls in an urn. We take out $n$ balls. What is the probability that among the $n$ balls taken out, $k$ will be white? | \frac{\binom{n}{k}^{2}}{\binom{2n}{n}} | 45 | 21 |
math | 4. Find all real solutions to the system of equations
$$
\left\{\begin{array}{l}
\sqrt{x-997}+\sqrt{y-932}+\sqrt{z-796}=100 \\
\sqrt{x-1237}+\sqrt{y-1121}+\sqrt{3045-z}=90 \\
\sqrt{x-1621}+\sqrt{2805-y}+\sqrt{z-997}=80 \\
\sqrt{2102-x}+\sqrt{y-1237}+\sqrt{z-932}=70
\end{array}\right.
$$
(L. S. Korechkova, A. A. Tessl... | 2021 | 170 | 4 |
math | ## Task B-1.2.
Let $n$ be a natural number and let
$$
A=0.1^{n} \cdot 0.01^{n} \cdot 0.001^{n} \cdot 0.0001^{n} \cdot \ldots \cdot 0 . \underbrace{00 \ldots 0}_{199 \text { zeros }} 1^{n} .
$$
The number $A^{-1}$ has 140701 digits. Determine the number $n$. | 7 | 123 | 1 |
math | 3. Six points A, B, C, D, E, F are connected with segments length of $1$. Each segment is painted red or black probability of $\frac{1}{2}$ independence. When point A to Point E exist through segments painted red, let $X$ be. Let $X=0$ be non-exist it. Then, for $n=0,2,4$, find the probability of $X=n$. | P(X=0) = \frac{69}{128}, P(X=2) = \frac{7}{16}, P(X=4) = \frac{3}{128} | 97 | 46 |
math | Example $\mathbf{1}$ Let $f(x)$ be monotonic on $\mathbf{R}$, and $f\left(\frac{x+y}{2}\right)=\frac{1}{2}(f(x)+f(y))$. Find $f(x)$. | f(x)=+b | 57 | 5 |
math | [ $\left.\begin{array}{c}\text { Theorem of Bezout. Factorization }\end{array}\right]$
Find the solution to the system
$$
x^{4}+y^{4}=17
$$
$$
x+y=3 \text {. } \quad .
$$
# | {1,2} | 67 | 5 |
math | ## Task Condition
Are the vectors $a, b$ and $c$ coplanar?
$a=\{1 ; 5 ; 2\}$
$b=\{-1 ; 1 ;-1\}$
$c=\{1 ; 1 ; 1\}$ | -2 | 57 | 2 |
math | 6. (6 points) Fill in a natural number in each of the $\square$ and $\triangle$ to make the equation true.
$$
\square^{2}+12=\triangle^{2} \text {, then: } \square+\triangle=6 \text {. }
$$ | 6 | 61 | 1 |
math | 74. The probability that the total length of flax plants of variety $A$ is $75-84$ cm is 0.6. What is the probability that among 300 flax plants of this variety, the relative frequency of plants of such length will deviate in absolute value from the probability of the appearance of plants of such length by no more than... | 0.9232 | 85 | 6 |
math | In the triangle $A B C$ the angle $B$ is not a right angle, and $A B: B C=k$. Let $M$ be the midpoint of $A C$. The lines symmetric to $B M$ with respect to $A B$ and $B C$ intersect $A C$ at $D$ and $E$. Find $B D: B E$. | k^2 | 81 | 3 |
math | Find all natural numbers $n$ such that the sum of the three largest divisors of $n$ is $1457$. | 987, 1023, 1085, 1175 | 28 | 21 |
math | 7. An ant starts from vertex $A$ of the rectangular prism $A B C D-A_{1} B_{1} C_{1} D_{1}$, and travels along the surface to reach vertex $C_{1}$. The shortest distance is 6. Then the maximum volume of the rectangular prism is $\qquad$ . | 12 \sqrt{3} | 71 | 7 |
math | \section*{Problem 4 - 330944}
Someone finds the statement
\[
22!=11240007277 * * 607680000
\]
In this, the two digits indicated by \(*\) are illegible. He wants to determine these digits without performing the multiplications that correspond to the definition of 22!.
Conduct such a determination and justify it! It ... | 77 | 139 | 2 |
math |
98.1. Determine all functions $f$ defined in the set of rational numbers and taking their values in the same set such that the equation $f(x+y)+f(x-y)=2 f(x)+2 f(y)$ holds for all rational numbers $x$ and $y$.
| f(x)=^{2} | 61 | 6 |
math | $7.30 \lg (\lg x)+\lg \left(\lg x^{3}-2\right)=0$.
Translate the above text into English, please retain the original text's line breaks and format, and output the translation result directly.
$7.30 \lg (\lg x)+\lg \left(\lg x^{3}-2\right)=0$. | 10 | 80 | 2 |
math | 1. Find the largest 65-digit number, the product of whose digits equals the sum of its digits. | 991111\ldots1 | 23 | 10 |
math | F5 (20-3, UK) Let $f, g: \mathbf{N}^{*} \rightarrow \mathbf{N}^{*}$ be strictly increasing functions, and
$$
f\left(\mathbf{N}^{*}\right) \cup g\left(\mathbf{N}^{*}\right)=\mathbf{N}^{*}, f\left(\mathbf{N}^{*}\right) \cap g\left(\mathbf{N}^{*}\right)=\varnothing, g(n)=f(f(n))+1 .
$$
Find $f(240)$. | 388 | 138 | 3 |
math | Let $S$ be a list of positive integers--not necessarily distinct--in which the number $68$ appears. The average (arithmetic mean) of the numbers in $S$ is $56$. However, if $68$ is removed, the average of the remaining numbers drops to $55$. What is the largest number that can appear in $S$? | 649 | 79 | 3 |
math | ## Task 4 - 170724
Determine all four-digit natural numbers that are divisible by 24 and whose digit representation has the form $9 x 7 y$!
Here, $x$ and $y$ are to be replaced by one of the ten digits $(0, \ldots, 9)$ each. | 9072,9672,9576 | 73 | 14 |
math | ## Task 32/74
For which natural numbers $n$ is $z=4^{n}+4^{4}+4^{8}$ a perfect square? | 11 | 38 | 2 |
math | ## Problem Statement
Calculate the area of the parallelogram constructed on vectors $a$ and $b$.
$a=p-3q$
$b=p+2q$
$|p|=\frac{1}{5}$
$|q|=1$
$(\widehat{p, q})=\frac{\pi}{2}$ | 1 | 69 | 1 |
math | Example 9 Find the greatest common divisor of $198,252,924$, and express it as an integer linear combination of 198,252 and 924. | 6 = 924 - 204 \cdot 252 + 255 \cdot 198 | 45 | 28 |
math | 7. The monotonic increasing interval of the function $f(x)=\log _{\frac{1}{3}}\left(x^{2}-5 x+6\right)$ is $\qquad$ . | (-\infty, 2) | 43 | 8 |
math | 7.4. In triangle $A B C$, the angles $A$ and $C$ at the base are $20^{\circ}$ and $40^{\circ}$, respectively. It is known that $A C - A B = 5$ (cm). Find the length of the angle bisector of angle $B$. | 5 | 72 | 1 |
math | I2.1 Suppose $P$ is an integer and $5<P<20$. If the roots of the equation $x^{2}-2(2 P-3) x+4 P^{2}-14 P+8=0$ are integers, find the value of $P$. | 12 | 62 | 2 |
math | \section*{Exercise 1 - 101041}
Form all sets of five one- or two-digit prime numbers such that in each of these sets, each of the digits 1 through 9 appears exactly once! | 8 | 49 | 1 |
math | 2.1. (14 points) A team consisting of juniors and masters from the "Vimpel" sports society went to a shooting tournament. The average number of points scored by the juniors turned out to be 22, by the masters - 47, and the average number of points for the entire team - 41. What is the share (in percent) of masters in t... | 76 | 87 | 2 |
math | 4- 195 Two drops of water fall successively from a steep cliff that is 300 meters high. When the first drop has fallen 0.001 millimeters, the second drop begins to fall. What is the distance between the two drops when the first drop reaches the bottom of the cliff (the answer should be accurate to 0.1 millimeters; air ... | 34.6 | 85 | 4 |
math | Consider the set $A=\{1,2,3, \ldots, 2011\}$. How many subsets of $A$ exist such that the sum of their elements is 2023060? | 4 | 50 | 1 |
math | Find all values of $n \in \mathbb{N}$ such that $3^{2 n}-2^{n}$ is prime. | 1 | 29 | 1 |
math | ## Problem Statement
Based on the definition of the derivative, find $f^{\prime}(0)$:
$$
f(x)=\left\{\begin{array}{c}
\sqrt{1+\ln \left(1+3 x^{2} \cos \frac{2}{x}\right)}-1, x \neq 0 \\
0, x=0
\end{array}\right.
$$ | 0 | 88 | 1 |
math | Task 1. Represent in the form of an irreducible fraction:
$$
\frac{4+8}{12}+\frac{16+20}{24}+\ldots+\frac{64+68}{72}
$$ | \frac{191}{20} | 54 | 10 |
math | 18 Given a regular triangular prism $A B C-A_{1} B_{1} C_{1}$, where $A A_{1}=2 A C=4$, extend $C B$ to $D$ such that $C B=B D$.
(1) Prove that line $C_{1} B$ // plane $A B_{1} D$;
(2) Find the sine value of the angle formed by plane $A B_{1} D$ and plane $A C B$. | \frac{4\sqrt{17}}{17} | 108 | 14 |
math | Example 6 A sports meet lasted for $n(n>1)$ days, and a total of $m$ medals were awarded. On the first day, one medal was awarded, and then $\frac{1}{7}$ of the remaining $m-1$ medals were awarded. On the second day, two medals were awarded, and then $\frac{1}{7}$ of the remaining medals were awarded, and so on. Finall... | 6 | 131 | 1 |
math | 450. A sample of size $n=50$ has been drawn from the population:
| variant | $x_{i}$ | 2 | 5 | 7 | 10 |
| :--- | :--- | ---: | ---: | ---: | ---: |
| frequency | $n_{i}$ | 16 | 12 | 8 | 14 |
Find the unbiased estimate of the population mean. | 5.76 | 95 | 4 |
math | # Problem 7. (4 points)
Natural numbers $a, b, c$ are such that $\operatorname{GCD}(\operatorname{LCM}(a, b), c) \cdot \operatorname{LCM}(\operatorname{GCD}(a, b), c)=200$.
What is the greatest value that $\operatorname{GCD}(\operatorname{LCM}(a, b), c)$ can take? | 10 | 97 | 2 |
math | 6. Discuss the factorization of the following polynomials over the integers using the prime factors of integers:
(1) $f(x)=x^{4}-3 x^{2}+9$;
(2) $f(x)=x^{4}-6 x^{3}+7 x^{2}+6 x-80$. | f(x)=(x-5)(x+2)(x^{2}-3x+8) | 70 | 20 |
math | 【Example 5】Find the sum of all four-digit numbers composed of $1,2,3,4,5$ without any repeated digits.
Translate the above text into English, please retain the original text's line breaks and format, and output the translation result directly. | 399960 | 56 | 6 |
math | Suggestion: Let $a=\sqrt{2 n+1}$ and $b=$ $\sqrt{2 n-1}$.
Facts that Help: Use the identity
$\left(a^{2}+a b+b^{2}\right)(a-b)=a^{3}-b^{3}$.
For a positive integer $n$ consider the function
$$
f(n)=\frac{4 n+\sqrt{4 n^{2}-1}}{\sqrt{2 n+1}+\sqrt{2 n-1}}
$$
Calculate the value of
$$
f(1)+f(2)+f(3)+\cdots+f(40)
$$ | 364 | 140 | 3 |
math | 368*. Solve the equation:
$$
x^{2}+\frac{x^{2}}{(x+1)^{2}}=1
$$ | \frac{(\sqrt{2}-1\\sqrt{2\sqrt{2}-1})}{2} | 32 | 23 |
math | 1. From the sequence of natural numbers, all numbers that are squares or cubes of integers have been erased. Which of the remaining numbers is in the hundredth place? | 112 | 34 | 3 |
math | 56. If the four-digit number $\overline{2 a 17}$ is a multiple of 19, then $a=$ | 7 | 30 | 1 |
math | ## 213. Math Puzzle $2 / 83$
Michael wishes his uncle a happy birthday, hoping that he may live to be at least 3 Gs old. How many years does this wish correspond to?
(1 Gs $=1$ Gigasecond $=10^{9}$ ) | 95.13 | 66 | 5 |
math | Calculate:
1. $\sum_{k=0}^{n}\binom{n}{k}$
2. $\sum_{k=0}^{n} k\binom{n}{k}$ | n2^{n-1} | 41 | 7 |
math | 1. DeAndre Jordan shoots free throws that are worth 1 point each. He makes $40 \%$ of his shots. If he takes two shots find the probability that he scores at least 1 point. | \frac{16}{25} | 44 | 9 |
math | Example 3 Solve the equation
$$
\frac{1}{6 x-5}-\frac{1}{5 x+4}=\frac{1}{5 x-4}-\frac{1}{4 x+5} .
$$ | x_{1}=9, x_{2}=0, x_{3}=1 | 51 | 17 |
math | 31. Find the particular solution of the differential equation $d y=\left(x^{2}-1\right) d x$, if $y=4$ when $x=1$. | \frac{x^{3}}{3}-x+\frac{14}{3} | 39 | 18 |
math | 1.a) Find 11 consecutive natural numbers whose sum is 99.
b) A four-digit natural number has its first two digits identical, and the units digit is 5. This number is divided by a two-digit number, and the remainder is 98. Find the dividend, divisor, and quotient.
Prof. Ana Marcela Popa | 4,5,6,\ldots,14 | 73 | 11 |
math | Example 5. Solve the integral equation
$$
\varphi(x)=x+\int_{x}^{\infty} \mathrm{e}^{2(x-t)} \varphi(t) d t
$$ | \varphi(x)=2x+1+Ce^{x} | 46 | 14 |
math | 10. A bag contains two red balls, three white balls, and four yellow balls, from which four balls are drawn at random. The probability that all three colors are included is $\qquad$ | \frac{4}{7} | 41 | 7 |
math | 28 Find the smallest positive integer $n$, such that in decimal notation $n^{3}$ ends with the digits 888. | 192 | 29 | 3 |
math | Given four positive numbers: $a, b, c, d$. Among the products $a b, a c, a d, b c, b d, c d$, we know the values of five of them, which are 2, 3, 4, 5, and 6. What is the value of the sixth product? | \frac{12}{5} | 72 | 8 |
math | 15. For natural numbers $n$ greater than 0, define an operation “ $G$ " as follows:
(1) When $n$ is odd, $G(n)=3 n+1$;
(2) When $n$ is even, $G(n)$ equals $n$ continuously divided by 2 until the quotient is odd;
The $k$-th “ $G$ ” operation is denoted as $G^{k}$, for example, $G^{1}(5)=3 \times 5+1=16, G^{2}(5)=G^{1}(1... | 63,34,4 | 238 | 7 |
math | Problem 3. Martin is 3 times younger than his father, and together they are 44 years old. In how many years will Martin be 2 times younger than his father. | 11 | 39 | 2 |
math | ## Problem B1
Let $a_{n}, b_{n}$ be two sequences of integers such that: (1) $a_{0}=0, b_{0}=8$; (2) $a_{n+2}=2 a_{n+1}-a_{n}+2$, $b_{n+2}=2 b_{n+1}-b_{n}$, (3) $a_{n}{ }^{2}+b_{n}{ }^{2}$ is a square for $n>0$. Find at least two possible values for $\left(a_{1992}\right.$, $\mathrm{b}_{1992}$ ).
## Answer
(1992. 19... | (1992\cdot1996,4\cdot1992+8),(1992\cdot1988,-4\cdot1992+8) | 192 | 43 |
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