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 | For all $n \geq 1$, simplify the expression
$$
\binom{n}{0}+\binom{n}{1}+\ldots+\binom{n}{n-1}+\binom{n}{n}
$$ | 2^n | 50 | 2 |
math | 1. Given the set of real numbers $\{1,2,3, x\}$, the maximum element of the set is equal to the sum of all elements in the set, then the value of $x$ is $\qquad$ . | -3 | 51 | 2 |
math | When drawing all diagonals in a regular pentagon, one gets an smaller pentagon in the middle. What's the ratio of the areas of those pentagons? | \frac{7 - 3\sqrt{5}}{2} | 34 | 15 |
math | 10.252. The sides of a triangle are in the ratio $5: 4: 3$. Find the ratio of the segments of the sides into which they are divided by the point of tangency of the inscribed circle. | 1:3;1:2;2:3 | 51 | 11 |
math | [ $\left[\begin{array}{l}\text { Generating functions } \\ {\left[\begin{array}{l}\text { Chebyshev Polynomials }\end{array}\right]} \\ \text { ] }\end{array}\right]$ [ Special polynomials (other).]
Find the generating functions of the sequences of Chebyshev polynomials of the first and second kind:
$$
F_{T}(x, z)=\s... | F_{T}(x,z)=(1-xz)(1-2xz+z^{2})^{-1},\quadF_{U}(x,z)=(1-2xz+z^{2})^{-1} | 164 | 42 |
math | 2. In an acute triangle $ABC$, $AD$, $DE$, $CF$ are altitudes, $H$ is the orthocenter of the triangle. If $EF$ bisects the area of triangle $ABC$, then $DE^2 + EF^2 + FD^2 =$ $\qquad$ (Given the side lengths of the triangle are $3$, $2\sqrt{2}$, $\sqrt{5}$) | 5 | 92 | 1 |
math | Problem 1. Let the natural numbers $x$ and $y$ be such that the fractions $\frac{7 x-5 y-3}{3 x+9 y-15}$ and $\frac{1}{5}$ are equivalent. Determine the value of the expression $(16 x-17 y)^{2015}$. | 0 | 73 | 1 |
math | ## Problem Statement
Write the decomposition of vector $x$ in terms of vectors $p, q, r$:
$x=\{-13 ; 2 ; 18\}$
$p=\{1 ; 1 ; 4\}$
$q=\{-3 ; 0 ; 2\}$
$r=\{1 ; 2 ;-1\}$ | 2p+5q | 76 | 5 |
math | 4. Find all non-negative real solutions $\left(x_{1}, x_{2}, \ldots, x_{n}\right)$ of the system of equations
$$
x_{i+1}=x_{i}^{2}-\left(x_{i-1}-1\right)^{2}, \quad i=1,2, \ldots, n
$$
(Indices are taken cyclically modulo $n$.) | x_{1}=x_{2}=\cdots=x_{n}=1 | 91 | 16 |
math | (a) Does there exist a polynomial $ P(x)$ with coefficients in integers, such that $ P(d) \equal{} \frac{2008}{d}$ holds for all positive divisors of $ 2008$?
(b) For which positive integers $ n$ does a polynomial $ P(x)$ with coefficients in integers exists, such that $ P(d) \equal{} \frac{n}{d}$ holds for all positi... | n = 1 | 98 | 5 |
math | 1. Convert the following fractions to decimals:
(i) $\frac{371}{6250}$,
(ii) $\frac{190}{37}$,
(iii) $\frac{13}{28}$,
(iv) $\frac{a}{875}, a=4,29,139,361$. | \begin{array}{l}
\text{(i) } 0.05936 \\
\text{(ii) } 5 . \dot{1} 3 \dot{5} \\
\text{(iii) } 0.46 \dot{4} 2857 \dot{1} \\
\text{(iv) } \begin{array}{l}
\frac{4}{875}=0.00 | 76 | 97 |
math | The $\textit{arithmetic derivative}$ $D(n)$ of a positive integer $n$ is defined via the following rules:
[list]
[*] $D(1) = 0$;
[*] $D(p)=1$ for all primes $p$;
[*] $D(ab)=D(a)b+aD(b)$ for all positive integers $a$ and $b$.
[/list]
Find the sum of all positive integers $n$ below $1000$ satisfying $D(n)=n$. | 31 | 108 | 2 |
math | 13. (10 points) There are two warehouses, A and B. Warehouse B originally had 1200 tons of inventory. When $\frac{7}{15}$ of the goods in Warehouse A and $\frac{1}{3}$ of the goods in Warehouse B are moved, and then 10% of the remaining goods in Warehouse A are moved to Warehouse B, the weights of the goods in Warehous... | 1875 | 107 | 4 |
math | Example 3 Given that $a, b, c, d$ are all prime numbers (allowing $a, b, c, d$ to be the same), and $abcd$ is the sum of 35 consecutive positive integers. Then the minimum value of $a+b+c+d$ is $\qquad$. ${ }^{[3]}$
(2011, Xin Zhi Cup Shanghai Junior High School Mathematics Competition) | 22 | 90 | 2 |
math | 39th Putnam 1978 Problem A3 Let p(x) = 2(x 6 + 1) + 4(x 5 + x) + 3(x 4 + x 2 ) + 5x 3 . Let a = ∫ 0 ∞ x/p(x) dx, b = ∫ 0 ∞ x 2 /p(x) dx, c = ∫ 0 ∞ x 3 /p(x) dx, d = ∫ 0 ∞ x 4 /p(x) dx. Which of a, b, c, d is the smallest? Solution | b | 134 | 1 |
math | ## Task 1
From $4 \mathrm{~kg}$ of wheat flour, 10 small white loaves are baked.
a) How many small white loaves can be made from 40 dt of flour?
b) And how many large white loaves, which are twice as heavy as the small ones, can be baked from it? | 10000 | 73 | 5 |
math | 3. The perimeter of a right-angled triangle is $2 p$ ( $p>0$ ), and its height dropped to the hypotenuse has a length of $h$. Determine its sides. | \begin{aligned}&=\frac{p}{+2p}(+p+\sqrt{(p-)^{2}-2^{2}}),\\&b=\frac{p}{+2p}(+p-\sqrt{(p-)^{2}-2^{2}}),\\&=\frac{2p^{2}}{+2p}\end{aligned} | 42 | 77 |
math | In a three-dimensional Euclidean space, by $\overrightarrow{u_1}$ , $\overrightarrow{u_2}$ , $\overrightarrow{u_3}$ are denoted the three orthogonal unit vectors on the $x, y$, and $z$ axes, respectively.
a) Prove that the point $P(t) = (1-t)\overrightarrow{u_1} +(2-3t)\overrightarrow{u_2} +(2t-1)\overrightarrow{u_3}$ ... | x + 3y - 2z + 47 = 0 | 291 | 16 |
math | 12. (10 points) Cut a pentagon along a straight line into two polygons, then cut one of the polygons along a straight line into two parts, resulting in three polygons, and then cut one of the polygons along a straight line into two parts, $\cdots$, and so on. To have 20 pentagons among the resulting polygons, what is t... | 38 | 84 | 2 |
math | [ Game Theory_(miscellaneous) ] [Evenness and Oddness $]
Under the Christmas tree, there are 2012 cones. Winnie-the-Pooh and donkey Eeyore are playing a game: they take cones for themselves in turns. On his turn, Winnie-the-Pooh takes one or four cones, and Eeyore takes one or three. Pooh goes first. The player who ca... | Winnie-the-Pooh | 110 | 6 |
math | Which is the six-digit number (abcdef) in the decimal system, whose 2, 3, 4, 5, 6 times multiples are also six-digit and their digits are formed by cyclic permutations of the digits of the above number and start with $c, b, e, f, d$ respectively? | 142857 | 66 | 6 |
math | Find the greatest value of the expression \[ \frac{1}{x^2-4x+9}+\frac{1}{y^2-4y+9}+\frac{1}{z^2-4z+9} \] where $x$, $y$, $z$ are nonnegative real numbers such that $x+y+z=1$. | \frac{7}{18} | 77 | 8 |
math | ## Task 6 - 140736
Claudia tells her friend Sabine that she has drawn a triangle $A B C$ in which the altitude from $A$ to $B C$ passes exactly through the intersection of the perpendicular bisector of $A B$ and the angle bisector of $\angle A B C$.
Sabine claims that from this information alone, one can determine th... | 60 | 128 | 2 |
math | 1. Calculate: $2018 \cdot 146-[2018-18 \cdot(4+5 \cdot 20)] \cdot 18$. | 146\cdot2000 | 41 | 9 |
math | ## Problem Statement
Calculate the limit of the function:
$\lim _{x \rightarrow \frac{\pi}{2}} \frac{2+\cos x \cdot \sin \frac{2}{2 x-\pi}}{3+2 x \sin x}$ | \frac{2}{3+\pi} | 55 | 9 |
math | Let $P$ be a given quadratic polynomial. Find all functions $f : \mathbb{R}\to\mathbb{R}$ such that $$f(x+y)=f(x)+f(y)\text{ and } f(P(x))=f(x)\text{ for all }x,y\in\mathbb{R}.$$ | f(x) = 0 | 70 | 7 |
math | 31. In $\triangle A B C, D C=2 B D, \angle A B C=45^{\circ}$ and $\angle A D C=60^{\circ}$. Find $\angle A C B$ in degrees. | 75 | 53 | 2 |
math | Let $A$, $B$, $C$ and $D$ be the vertices of a regular tetrahedron, each of whose edges measures $1$ meter. A bug, starting from vertex $A$, observes the following rule: at each vertex it chooses one of the three edges meeting at that vertex, each edge being equally likely to be chosen, and crawls along that edge to th... | 182 | 128 | 3 |
math | 8.1. For all triples $(x, y, z)$ satisfying the system
$$
\left\{\begin{array}{l}
2 \sin x=\operatorname{tg} y \\
2 \cos y=\operatorname{ctg} z \\
\sin z=\operatorname{tg} x
\end{array}\right.
$$
find the smallest value of the expression $\cos x-\sin z$. | -\frac{5\sqrt{3}}{6} | 89 | 12 |
math | 3. A swimming pool is in the shape of a circle with diameter $60 \mathrm{ft}$. The depth varies linearly along the east-west direction from $3 \mathrm{ft}$ at the shallow end in the east to $15 \mathrm{ft}$ at the diving end in the west (this is so that divers look impressive against the sunset) but does not vary at al... | 8100\pi | 107 | 6 |
math | 3. In $\triangle A B C$, let $B C=a, C A=b, A B=c$. If $9 a^{2}+9 b^{2}-19 c^{2}=0$, then $\frac{\cot C}{\cot A+\cot B}=$
Translate the above text into English, please retain the original text's line breaks and format, and output the translation result directly. | \frac{5}{9} | 85 | 7 |
math | Let's find the binomial such that in its 7th power, the third term is $6 \frac{2}{9} a^{5}$, and the fifth term is $52 \frac{1}{2} a^{3}$. | (\frac{3}{2}+\frac{2}{3}) | 53 | 14 |
math | Let $(a,b)=(a_n,a_{n+1}),\forall n\in\mathbb{N}$ all be positive interger solutions that satisfies
$$1\leq a\leq b$$ and $$\dfrac{a^2+b^2+a+b+1}{ab}\in\mathbb{N}$$
And the value of $a_n$ is [b]only[/b] determined by the following recurrence relation:$ a_{n+2} = pa_{n+1} + qa_n + r$
Find $(p,q,r)$. | (p, q, r) = (5, -1, -1) | 123 | 18 |
math | ## Problem Statement
Calculate the limit of the numerical sequence:
$\lim _{n \rightarrow \infty} \frac{(2 n+1)^{3}+(3 n+2)^{3}}{(2 n+3)^{3}-(n-7)^{3}}$ | 5 | 61 | 1 |
math | B2. Let $A B C D$ be a square with side length 1 . Points $X$ and $Y$ are on sides $B C$ and $C D$ respectively such that the areas of triangles $A B X, X C Y$, and $Y D A$ are equal. Find the ratio of the area of $\triangle A X Y$ to the area of $\triangle X C Y$. | \sqrt{5} | 87 | 5 |
math | A courtyard has the shape of a parallelogram ABCD. At the corners of the courtyard there stand poles AA', BB', CC', and DD', each of which is perpendicular to the ground. The heights of these poles are AA' = 68 centimeters, BB' = 75 centimeters, CC' = 112 centimeters, and DD' = 133 centimeters. Find the distance in cen... | 14 | 103 | 2 |
math | 1389. Write the first five terms of the series $\sum_{n=1}^{\infty}(-1)^{n-1} \frac{n}{n+1}$. | \frac{1}{2}-\frac{2}{3}+\frac{3}{4}-\frac{4}{5}+\frac{5}{6}-\ldots | 42 | 38 |
math | $$
\begin{array}{l}
\text { 6. Given the quadratic equation in } x \text { is } \\
x^{2}+a x+(m+1)(m+2)=0
\end{array}
$$
For any real number $a$, the equation has real roots. Then the range of real number $m$ is . $\qquad$ | -2 \leqslant m \leqslant -1 | 81 | 15 |
math | Example 3.42. Investigate the function for extremum
$$
z=x^{2}+2 y^{2}-2 x y-x-2 y
$$ | (2,\frac{3}{2}) | 37 | 9 |
math | Example 2 Let any real numbers $x_{0}>x_{1}>x_{2}>x_{3}>0$. To make $\log _{\frac{x_{0}}{x_{1}}} 1993+\log _{\frac{x_{1}}{x_{2}}} 1993+\log _{\frac{x_{2}}{x_{3}}} 1993 \geqslant$ $k \log _{\frac{x_{0}}{x_{3}}} 1993$ always hold, then the maximum value of $k$ is $\qquad$
(1993, National Competition) | 9 | 138 | 1 |
math | 13.024. A tractor team can plow 5/6 of a plot of land in 4 hours and 15 minutes. Before the lunch break, the team worked for 4.5 hours, after which 8 hectares remained unplowed. How large was the plot? | 68 | 62 | 2 |
math | 1. Depending on the real parameter $a$, solve the equation
$$
\sqrt{2 x-a}-\sqrt{x-1}=2
$$
in the set of real numbers. | \begin{pmatrix}for-6&norealsolutions\\for-6&x=5\\for-6<\leq-2&x_{1/2}=11+\4\sqrt{6+}\\for-2<x=11+ | 40 | 56 |
math | Example 3 Given the family of curves $2(2 \sin \theta-\cos \theta+3) x^{2}-(8 \sin \theta+\cos \theta+1) y=0$, where $\theta$ is a parameter. Try to find the maximum value of the length of the chord intercepted by the line $y=2 x$ on this family of curves.
$(1995$, National High School Mathematics Competition) | 8 \sqrt{5} | 92 | 6 |
math | 88. A discrete random variable $X$ is given by the distribution series:
$$
X: \begin{array}{cccc}
x_{i} & 1 & 2 & 4 \\
p_{i} & 0.1 & 0.3 & 0.6^{\circ}
\end{array}
$$
Find its mathematical expectation, variance, and standard deviation. | 1.1358 | 83 | 6 |
math | 10. From $1,2,3, \cdots, 2003$, select $k$ numbers such that among the selected $k$ numbers, there are definitely three numbers that can form the side lengths of an acute triangle. Find the minimum value of $k$ that satisfies the above condition. | 29 | 66 | 2 |
math | [ Product of the lengths of the chord segments and the lengths of the secant segments ]
A line passing through point $A$, located outside the circle at a distance of 7 from its center, intersects the circle at points $B$ and $C$. Find the radius of the circle, given that $AB=3$, $BC=5$.
# | 5 | 73 | 1 |
math | Example 1. If in a Pythagorean triple, the difference between the hypotenuse and one of the legs is 1, then the form of the Pythagorean triple is
$$
2 a+1,2 a^{2}+2 a, 2 a^{2}+2 a+1 .
$$ | 2a+1, 2a^{2}+2a, 2a^{2}+2a+1 | 68 | 26 |
math | 5. Out of three hundred eleventh-grade students, excellent and good grades were received by $77 \%$ on the first exam, $71 \%$ on the second exam, and $61 \%$ on the third exam. What is the smallest number of participants who could have received excellent and good grades on all three exams?
Answer: 27. | 27 | 75 | 2 |
math | 3.2.1 * Given the sequence $\left\{a_{n}\right\}$ satisfies $a_{1}=1, a_{n+1}=\frac{1}{3} a_{n}+1, n=1,2, \cdots$. Find the general term of the sequence $\left\{a_{n}\right\}$. | a_{n}=\frac{3}{2}-\frac{1}{2}(\frac{1}{3})^{n-1} | 77 | 30 |
math | Memories all must have at least one out of five different possible colors, two of which are red and green. Furthermore, they each can have at most two distinct colors. If all possible colorings are equally likely, what is the probability that a memory is at least partly green given that it has no red?
[i]Proposed by M... | \frac{2}{5} | 70 | 7 |
math | Find all pairs of positive integers $(a, b)$ such that $a-b$ is a prime and $ab$ is a perfect square. Answer: Pairs $(a, b)=\left(\left(\frac{p+1}{2}\right)^{2},\left(\frac{p-1}{2}\right)^{2}\right)$, where $p$ is a prime greater than 2. | (a, b)=\left(\left(\frac{p+1}{2}\right)^{2},\left(\frac{p-1}{2}\right)^{2}\right) | 87 | 40 |
math | Bootin D. ..
Several guard brigades of the same size slept for the same number of nights. Each guard slept more nights than there are guards in a brigade, but fewer than the number of brigades. How many guards are in a brigade if all the guards together slept for 1001 person-nights? | 7 | 66 | 1 |
math | 1. Given
$$
\begin{array}{l}
(\sqrt{2017} x-\sqrt{2027})^{2017} \\
=a_{1} x^{2017}+a_{2} x^{2016}+\cdots+a_{2017} x+a_{2018} .
\end{array}
$$
Then $\left(a_{1}+a_{3}+\cdots+a_{2017}\right)^{2}-$
$$
\begin{aligned}
& \left(a_{2}+a_{4}+\cdots+a_{2018}\right)^{2} \\
= &
\end{aligned}
$$ | -10^{2017} | 157 | 9 |
math | (15) From any point \( P \) on the line \( l: \frac{x}{12}+\frac{y}{8}=1 \), draw tangents \( PA \) and \( PB \) to the ellipse \( C: \frac{x^{2}}{24}+\frac{y^{2}}{16}=1 \), with points of tangency \( A \) and \( B \), respectively. Find the locus of the midpoint \( M \) of segment \( AB \). | \frac{(x-1)^{2}}{\frac{5}{2}}+\frac{(y-1)^{2}}{\frac{5}{3}}=1 | 108 | 36 |
math | A Christmas tree seller sold spruces for 220 Kč, pines for 250 Kč, and firs for 330 Kč. In the morning, he had the same number of spruces, firs, and pines. In the evening, he had sold all the trees and earned a total of 36000 Kč for them.
How many trees did the seller sell that day?
Hint. Count in threes. | 135 | 99 | 3 |
math | 1. A bridge over a river connects two different regions of the country. Once, one of the regions repainted the part of the bridge that belonged to it. If the freshly painted part of the bridge were 1.2 times larger, it would make up exactly half of the entire bridge. What part of the bridge needs to be repainted to mak... | \frac{1}{12} | 77 | 8 |
math | 10. For the non-empty subsets of the set $A=\{1,2, \cdots, 10\}$, subsets whose elements' sum is a multiple of 10 are called good subsets. Therefore, the number of good subsets of $A$ is $\qquad$. | 103 | 62 | 3 |
math | 9. (16 points) Let the ellipse $C: \frac{(x-1)^{2}}{16}+\frac{(y-1)^{2}}{9}=1$, and the line $l: y=a x+b$. If two tangents are drawn from any two points $M, N$ on the line $l$ to the ellipse $C$, and the lines connecting the points of tangency are $m, n$, then $m \parallel n$. Try to find the locus of the point $(a, b)... | a+b=1 | 141 | 4 |
math | 214. A dog is chasing a rabbit that is 150 feet ahead of it. It makes a leap of 9 feet every time the rabbit jumps 7 feet. How many jumps must the dog make to catch the rabbit?
## Herbert's Problem. | 75 | 55 | 2 |
math | A package of 8 greeting cards comes with 10 envelopes. Kirra has 7 cards but no envelopes. What is the smallest number of packages that Kirra needs to buy to have more envelopes than cards?
(A) 3
(B) 4
(C) 5
(D) 6
(E) 7 | 4 | 68 | 1 |
math | Determine the smallest positive integer $n$ with the following property: For all positive integers $x, y$, and $z$ with $x \mid y^{3}$ and $y \mid z^{3}$ and $z \mid x^{3}$, it always holds that $x y z \mid (x+y+z)^{n}$.
(Gerhard J. Woeginger)
Answer. The smallest such number is $n=13$. | 13 | 96 | 2 |
math | 3A. Solve the equation
$$
2^{3 x}-\frac{8}{2^{3 x}}-6\left(2^{x}-\frac{1}{2^{x-1}}\right)=1
$$ | 1 | 50 | 1 |
math | Example 5. Find the sum of the series $f(x)=\sum_{n=0}^{\infty}\left(n^{2}+3 n-5\right) \cdot x^{n}$. | f(x)=\frac{7x^{2}-14x+5}{(x-1)^{3}} | 46 | 25 |
math | The 79th question: Let $\mathrm{p}$ be an odd prime, and consider the set $\{1,2, \ldots, 2 \mathrm{p}\}$ of subsets $A$ that satisfy the following conditions: (1) $A$ contains exactly $\mathrm{p}$ elements; (2) the sum of all elements in $A$ is divisible by $\mathrm{p}$. Try to find the number of all such subsets. | \frac{1}{p}[C_{2p}^p+2(p-1)] | 97 | 20 |
math | Problem 2. For what least $n$ do there exist $n$ numbers from the interval $(-1 ; 1)$ such that their sum is 0 and the sum of their squares is 36?
# | 38 | 46 | 2 |
math | Let $ f:\mathbb{Z}_{>0}\rightarrow\mathbb{R} $ be a function such that for all $n > 1$ there is a prime divisor $p$ of $n$ such that \[ f(n)=f\left(\frac{n}{p}\right)-f(p). \]
Furthermore, it is given that $ f(2^{2014})+f(3^{2015})+f(5^{2016})=2013 $. Determine $ f(2014^2)+f(2015^3)+f(2016^5) $. | \frac{49}{3} | 141 | 8 |
math | Example 4-24 Use two colors to color the 6 faces and 8 vertices of a cube. How many different schemes are there? | 776 | 30 | 3 |
math | 11. At a railway station, there are $k$ traffic lights, each of which can transmit three signals: red, yellow, and green. How many different signals can be transmitted using all the traffic lights? | 3^k | 44 | 3 |
math | 5. Find the set of values of the expression $\frac{a \cos x + b \sin x + c}{\sqrt{a^{2} + b^{2} + c^{2}}}$, where $x, a, b, c$ are arbitrary numbers such that $a^{2} + b^{2} + c^{2} \neq 0$. (20 points) | [-\sqrt{2};\sqrt{2}] | 85 | 11 |
math | 3. In the Cartesian coordinate system $x O y$, two circles both pass through the point $(1,1)$, and are tangent to the line $y=\frac{4}{3} x$ and the $x$-axis. Then the sum of the radii of the two circles is . $\qquad$ | \frac{3}{2} | 67 | 7 |
math | Where does the following function take its extreme values?
$$
\sin \left(\cos ^{2} x\right)+\sin \left(\sin ^{2} x\right)
$$ | k\pi/4(k=0,\1,\2,\ldots) | 41 | 16 |
math | Problem 2. Solve the equation:
$$
\frac{x^{2}+1}{2}+\frac{2 x^{2}+1}{3}+\frac{3 x^{2}+1}{4}+\cdots+\frac{2015 x^{2}+1}{2016}=2015
$$ | {1,-1} | 75 | 5 |
math | 4. Extreme set (from 6th grade, $\mathbf{1}$ point). From the digits 1 to 9, three single-digit and three two-digit numbers are formed, with no digits repeating. Find the smallest possible arithmetic mean of the resulting set of numbers. | 16.5 | 57 | 4 |
math | ## Task A-4.5. (4 points)
For which $n \in \mathbb{N}$ do there exist an angle $\alpha$ and a convex $n$-gon with angles $\alpha, 2 \alpha, \ldots, n \alpha$? | n=3n=4 | 59 | 6 |
math | Example 6: There are three types of goods, A, B, and C. If you buy 3 pieces of A, 7 pieces of B, and 1 piece of C, it costs 3.15 yuan; if you buy 4 pieces of A, 10 pieces of B, and 1 piece of C, it costs 4.20 yuan. Now, if you buy 1 piece each of A, B, and C, it will cost $\qquad$ yuan.
(1985. National Junior High Scho... | 1.05 | 120 | 4 |
math | [ Algebraic equations and systems of equations (miscellaneous).]
Solve the equation $a^{2}+b^{2}+c^{2}+d^{2}-a b-b c-c d-d+2 / 5=0$.
# | =1/5,b=2/5,=3/5,=4/5 | 54 | 19 |
math | Let $f(n)=\displaystyle\sum_{k=1}^n \dfrac{1}{k}$. Then there exists constants $\gamma$, $c$, and $d$ such that \[f(n)=\ln(x)+\gamma+\dfrac{c}{n}+\dfrac{d}{n^2}+O\left(\dfrac{1}{n^3}\right),\] where the $O\left(\dfrac{1}{n^3}\right)$ means terms of order $\dfrac{1}{n^3}$ or lower. Compute the ordered pair $(c,d)$. | \left( \frac{1}{2}, -\frac{1}{12} \right) | 135 | 23 |
math | ## Aufgabe 3
a)
| $x$ | $y$ | $x-y$ |
| :---: | :---: | :---: |
| 100 | 63 | |
| 96 | 49 | |
| 83 | 26 | |
b) $27: a=9 ; \quad 18: a=9 ; \quad 45: a=9$
| 37,47,57,3,2,5 | 99 | 14 |
math | 3. For real numbers $a$ and $b$, it holds that $a^{3}=3 a b^{2}+11$ and $b^{3}=3 a^{2} b+2$. Calculate the value of the expression $a^{2}+b^{2}$. | 5 | 62 | 1 |
math | Each of the first $150$ positive integers is painted on a different marble, and the $150$ marbles are placed in a bag. If $n$ marbles are chosen (without replacement) from the bag, what is the smallest value of $n$ such that we are guaranteed to choose three marbles with consecutive numbers? | 101 | 72 | 3 |
math | ## Problem Statement
Calculate the limit of the function:
$$
\lim _{x \rightarrow 0} \frac{\sqrt{1+\tan x}-\sqrt{1+\sin x}}{x^{3}}
$$ | \frac{1}{4} | 47 | 7 |
math | ## Task 4.
Determine all triples $(a, b, c)$ of natural numbers for which
$$
a \mid (b+1), \quad b \mid (c+1) \quad \text{and} \quad c \mid (a+1)
$$ | (1,1,1),(1,1,2),(1,2,1),(2,1,1),(1,3,2),(2,1,3),(3,2,1),(3,5,4),(4,3,5),(5,4,3) | 60 | 61 |
math | Let $x, y, z$ be real numbers not equal to zero with
$$
\frac{x+y}{z}=\frac{y+z}{x}=\frac{z+x}{y} .
$$
Determine all possible values of
$$
\frac{(x+y)(y+z)(z+x)}{x y z}
$$
(Walther Janous)
Answer. The only possible values are 8 and -1. | 8-1 | 92 | 3 |
math | ## Task $2 / 84$
Given is a circle with diameter $d=2 r=A B$. A line perpendicular to $A B$ intersects the diameter at $P$ and the circle at $C$ and $D$. The perimeters of triangles $A P C$ and $B P D$ are in the ratio $\sqrt{3}: 1$.
What is the ratio $A P: P B$? | AP:PB=3:1 | 90 | 7 |
math | ### 6.244. Solve the equation
$$
\begin{aligned}
& x(x+1)+(x+1)(x+2)+(x+2)(x+3)+(x+3)(x+4)+\ldots \\
& \ldots+(x+8)(x+9)+(x+9)(x+10)=1 \cdot 2+2 \cdot 3+\ldots+8 \cdot 9+9 \cdot 10
\end{aligned}
$$ | x_{1}=0,x_{2}=-10 | 109 | 12 |
math | consider a $2008 \times 2008$ chess board. let $M$ be the smallest no of rectangles that can be drawn on the chess board so that sides of every cell of the board is contained in the sides of one of the rectangles. find the value of $M$. (eg for $2\times 3$ chessboard, the value of $M$ is 3.)
| 2009 | 86 | 4 |
math | 1. The Year of the Tiger 2022 has the property that it is a multiple of 6 and the sum of its digits is 6, such positive integers are called "White Tiger Numbers". Therefore, among the first 2022 positive integers, the number of "White Tiger Numbers" $n=$ $\qquad$ . | 30 | 72 | 2 |
math | 13. Given the sequence $\left\{a_{n}\right\}$ satisfies:
$$
a_{1}=a, a_{n+1}=\frac{5 a_{n}-8}{a_{n}-1}\left(n \in \mathbf{Z}_{+}\right) \text {. }
$$
(1) If $a=3$, prove that $\left\{\frac{a_{n}-2}{a_{n}-4}\right\}$ is a geometric sequence, and find the general term formula of the sequence $\left\{a_{n}\right\}$;
(2) I... | \in(3,+\infty) | 155 | 9 |
math | 1. When purchasing goods for an amount of no less than 900 rubles, the store provides a discount on subsequent purchases of $25 \%$. Having 1200 rubles in his pocket, Petya wanted to buy 3 kg of meat and 1 kg of onions. In the store, meat was sold at 400 rubles per kg, and onions at 50 rubles per kg. Realizing that he ... | 1200\mathrm{p} | 121 | 9 |
math | 1. Solve the equation: $2^{x^{5}} 4^{x^{4}}+256^{4}=3 \cdot 16^{x^{3}}$. | x=2 | 39 | 3 |
math | 3.081. $\sin ^{2}\left(\alpha-\frac{3 \pi}{2}\right)\left(1-\operatorname{tg}^{2} \alpha\right) \operatorname{tg}\left(\frac{\pi}{4}+\alpha\right) \cos ^{-2}\left(\frac{\pi}{4}-\alpha\right)$. | 2 | 82 | 1 |
math | Example 2 There are three types of goods, A, B, and C. If you buy 3 pieces of A, 7 pieces of B, and 1 piece of C, it costs a total of 315 yuan; if you buy 4 pieces of A, 10 pieces of B, and 1 piece of C, it costs a total of 420 yuan. Question: How much would it cost to buy one piece each of A, B, and C? | 105 | 104 | 3 |
math | 24. Find the number of permutations $a_{1} a_{2} a_{3} a_{4} a_{5} a_{6}$ of the six integers from 1 to 6 such that for all $i$ from 1 to $5, a_{i+1}$ does not exceed $a_{i}$ by 1 . | 309 | 75 | 3 |
math | 1. Milla and Zhena came up with a number each and wrote down all the natural divisors of their numbers on the board. Milla wrote down 10 numbers, Zhena wrote down 9 numbers, and the largest number written on the board twice is 50. How many different numbers are written on the board? | 13 | 69 | 2 |
math | 8. $[7]$ Two circles with radius one are drawn in the coordinate plane, one with center $(0,1)$ and the other with center $(2, y)$, for some real number $y$ between 0 and 1 . A third circle is drawn so as to be tangent to both of the other two circles as well as the $x$ axis. What is the smallest possible radius for th... | 3-2\sqrt{2} | 88 | 8 |
math | Example 1. Find $\frac{1}{1-\frac{1}{1-\cdots \frac{1}{1-\frac{355}{113}}}}$. | \frac{355}{113} | 39 | 11 |
math | Five. (15 points) Find the largest positive integer $n$ that satisfies the inequality
$$
\left[\frac{n}{2}\right]+\left[\frac{n}{3}\right]+\left[\frac{n}{11}\right]+\left[\frac{n}{13}\right]<n
$$
where $[x]$ denotes the greatest integer not exceeding the real number $x$. | 1715 | 82 | 4 |
math | 2. Given that the function $f(x)$ is an odd function on $\mathbf{R}$, and when $x \geqslant 0$, $f(x)=x^{2}$, then the solution set of the inequality $f(f(x))+$ $f(x-1)<0$ is $\qquad$ | (-\infty,\frac{\sqrt{5}-1}{2}) | 69 | 15 |
math | 31. How many natural numbers not exceeding 500 and not divisible by 2, 3, or 5 are there? | 134 | 29 | 3 |
math | In a tournament with 10 teams, each of them plays against each other exactly once. Additionally, there are no ties, and each team has a $50 \%$ chance of winning any match. What is the probability that, after tallying the scores from $\frac{10 \cdot 9}{2}=45$ games, no two players have the same number of wins? | \frac{10!}{2^{45}} | 81 | 12 |
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