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 | Let $S_n$ be the sum of reciprocal values of non-zero digits of all positive integers up to (and including) $n$. For instance, $S_{13} = \frac{1}{1}+ \frac{1}{2}+ \frac{1}{3}+ \frac{1}{4}+ \frac{1}{5}+ \frac{1}{6}+ \frac{1}{7}+ \frac{1}{8}+ \frac{1}{9}+ \frac{1}{1}+ \frac{1}{1}+ \frac{1}{1}+ \frac{1}{1}+ \frac{1}{2}+ \... | 7 | 192 | 3 |
math | 4. Let $A=$ the set of red balls, $B=$ the set of yellow balls, $C=$ the set of green balls, where $A, B, C$ are pairwise disjoint.
We have: $\operatorname{card} A+\operatorname{cardB}=17, \operatorname{cardB}+\operatorname{card} C=29$ (2p)
Subtract the first relation from the second and obtain cardC - cardA $=12$. .... | \operatorname{card}A=12,\operatorname{cardB}=5,\operatorname{card}C=24 | 194 | 28 |
math | 13. Given variables $x, y$ satisfy the constraint conditions
$$
\left\{\begin{array}{l}
x-y+2 \leqslant 0, \\
x \geqslant 1, \\
x+y-7 \leqslant 0 .
\end{array}\right.
$$
Then the range of $\frac{y}{x}$ is $\qquad$ | \left[\frac{9}{5}, 6\right] | 87 | 14 |
math | ## Problem Statement
Calculate the limit of the function:
$\lim _{x \rightarrow 0}\left(2-e^{x^{2}}\right)^{\frac{1}{1-\cos \pi x}}$ | e^{-\frac{2}{\pi^{2}}} | 46 | 12 |
math | 36. Name a fraction greater than $\frac{11}{23}$ but less than $\frac{12}{23}$. | \frac{1}{2} | 30 | 7 |
math | 1. (5 points) $20132014 \times 20142013$ - $20132013 \times 20142014=$ $\qquad$ | 10000 | 53 | 5 |
math | (20) Given the ellipse $\frac{x^{2}}{5^{2}}+\frac{y^{2}}{4^{2}}=1$, a line through its left focus $F_{1}$ intersects the ellipse at points $A$ and $B$. Point $D(a, 0)$ is to the right of $F_{1}$. Lines $A D$ and $B D$ intersect the left directrix of the ellipse at points $M$ and $N$, respectively. If the circle with di... | 5 | 128 | 1 |
math | Problem 4. A farmer's field has the shape of a rectangle, with a length that is twice the width. It is fenced with three strands of wire, for which $360 \mathrm{~m}$ of wire was used. The field was sown with wheat. How many kilograms of wheat were harvested, if $35 \mathrm{~kg}$ of wheat is obtained from every $100 \ma... | 280\mathrm{~} | 96 | 8 |
math | 1.1.9 ** For a subset $S$ of the set $\{1,2, \cdots, 15\}$, if the positive integer $n$ and $n+|S|$ are both elements of $S$, then $n$ is called a "good number" of $S$. If a set $S$ has an element that is a "good number", then $S$ is called a "good set". Suppose 7 is a "good number" of some "good set" $X$. How many suc... | 2^{12} | 122 | 5 |
math | When a right triangle is rotated about one leg, the volume of the cone produced is $800 \pi$ $\text{cm}^3$. When the triangle is rotated about the other leg, the volume of the cone produced is $1920 \pi$ $\text{cm}^3$. What is the length (in cm) of the hypotenuse of the triangle? | 26 | 84 | 2 |
math | Calculate the sum of $n$ addends
$$7 + 77 + 777 +...+ 7... 7.$$ | 7 \left( \frac{10^{n+1} - 9n - 10}{81} \right) | 30 | 29 |
math | A set $S$ of positive integers is $\textit{sum-complete}$ if there are positive integers $m$ and $n$ such that an integer $a$ is the sum of the elements of some nonempty subset of $S$ if and only if $m \le a \le n$.
Let $S$ be a sum-complete set such that $\{1, 3\} \subset S$ and $|S| = 8$. Find the greatest possible ... | 223 | 122 | 3 |
math | 53rd Putnam 1992 Problem A3 Find all positive integers a, b, m, n with m relatively prime to n such that (a 2 + b 2 ) m = (ab) n . Solution | ,b,,n=2^r,2^r,2r,2r+1foranypositiveintegerr | 49 | 25 |
math | Let $\mathbb{N}=\{1,2,3, \ldots\}$ be the set of positive integers. Find all functions $f$, defined on $\mathbb{N}$ and taking values in $\mathbb{N}$, such that $(n-1)^{2}<f(n) f(f(n))<n^{2}+n$ for every positive integer $n$. | f(n)=n | 84 | 4 |
math | 6. Let $\mathbb{R}^{+}$ be the set of positive real numbers. Find all functions $f: \mathbb{R}^{+} \rightarrow \mathbb{R}^{+}$ such that $f(x+f(y))=y f(x y+1)$, for all $x, y>0$. | f(y)=\frac{1}{y} | 72 | 10 |
math | Determine the smallest positive real number $r$ such that there exist differentiable functions $f: \mathbb{R} \rightarrow \mathbb{R}$ and $g: \mathbb{R} \rightarrow \mathbb{R}$ satisfying
(a) $f(0)>0$,
(b) $g(0)=0$,
(c) $\left|f^{\prime}(x)\right| \leq|g(x)|$ for all $x$,
(d) $\left|g^{\prime}(x)\right| \leq|f(x)|$... | \frac{\pi}{2} | 139 | 7 |
math | 352. Find a natural number that is equal to the sum of all its preceding natural numbers. Does more than one such number exist, or is there only one? | 3 | 35 | 1 |
math | 2. In the product $5^{-2} \cdot 5^{-4} \cdot 5^{-8} \cdot \ldots \cdot 5^{-x}$, where the exponents form a geometric sequence, determine $x$ such that $5^{-2} \cdot 5^{-4} \cdot 5^{-8} \cdot \ldots \cdot 5^{-x}=5^{-16382}$. | 8192 | 92 | 4 |
math | 13.439 If the desired two-digit number is increased by 46, the resulting number will have a product of its digits equal to 6. Find this number given that the sum of its digits is 14. | 77or86 | 49 | 5 |
math | $33 \quad m$ and $n$ are two positive integers satisfying $1 \leq m \leq n \leq 40$. Find the number of pairs of $(m, n)$ such that their product $m n$ is divisible by 33 . | 64 | 59 | 2 |
math | \section*{Problem 1 - 091221}
Given a sequence of real numbers \(a_{1}, a_{2}, \ldots, a_{n}, \ldots\) by the (independent) representation
\[
a_{n}=c_{2} n^{2}+c_{1} n+c_{0}
\]
where \(c_{0}, c_{1}, c_{2}\) are real numbers. The first difference sequence is denoted by \(D(1)_{n}=a_{n+1}-a_{n}\) and the second differ... | 2c_{2} | 262 | 5 |
math | 5. The solution set of the inequality
$$
\log _{\frac{1}{12}}\left(x^{2}+2 x-3\right)>x^{2}+2 x-16
$$
is | (-5,-3)\cup(1,3) | 50 | 11 |
math | Determine the prime factorization of $25^{2}+72^{2}$. | 157\times37 | 20 | 7 |
math | Let $ABCD$ be a rectangle with $AB = 6$ and $BC = 6 \sqrt 3$. We construct four semicircles $\omega_1$, $\omega_2$, $\omega_3$, $\omega_4$ whose diameters are the segments $AB$, $BC$, $CD$, $DA$. It is given that $\omega_i$ and $\omega_{i+1}$ intersect at some point $X_i$ in the interior of $ABCD$ for every $i=1,2,3,4... | 243 | 156 | 3 |
math | 92. The Meeting Problem. Two people agreed to meet at a certain place between 12 o'clock and 1 o'clock. According to the agreement, the one who arrives first waits for the other for 15 minutes, after which he leaves. What is the probability that these people will meet, if each of them chooses the moment of their arriva... | \frac{7}{16} | 94 | 8 |
math | 1. (Easy) Find the largest value of $x$ such that $\sqrt[3]{x}+\sqrt[3]{10-x}=1$. | 5+2\sqrt{13} | 33 | 9 |
math | Sally's salary in 2006 was $\$37,500$. For 2007 she got a salary increase of $x$ percent. For 2008 she got another salary increase of $x$ percent. For 2009 she got a salary decrease of $2x$ percent. Her 2009 salary is $\$34,825$. Suppose instead, Sally had gotten a $2x$ percent salary decrease for 2007, an $x$ percent ... | 34,825 | 151 | 6 |
math | A train 280 meters long travels at a speed of 18 meters per second across a bridge. It takes 20 seconds from the moment the tail of the train gets on the bridge until the head of the train leaves the bridge. Therefore, the length of the bridge is $\qquad$ meters. | 640 | 65 | 3 |
math | ## Task $32 / 70$
Determine the smallest natural number $k$ that begins with the digit 7 (when represented in the decimal system) and satisfies the following additional property:
If the leading digit 7 is removed and appended to the end, the newly formed number $z=\frac{1}{3} k$. | 7241379310344827586206896551 | 70 | 28 |
math | 14. Suppose that $(21.4)^{e}=(0.00214)^{b}=100$. Find the value of $\frac{1}{a}-\frac{1}{b}$. | 2 | 49 | 1 |
math | A triangle $\triangle ABC$ satisfies $AB = 13$, $BC = 14$, and $AC = 15$. Inside $\triangle ABC$ are three points $X$, $Y$, and $Z$ such that:
[list]
[*] $Y$ is the centroid of $\triangle ABX$
[*] $Z$ is the centroid of $\triangle BCY$
[*] $X$ is the centroid of $\triangle CAZ$
[/list]
What is the area of $\trian... | \frac{84}{13} | 119 | 9 |
math | 7.240. $\lg ^{4}(x-1)^{2}+\lg ^{2}(x-1)^{3}=25$. | 1.1;11 | 35 | 6 |
math | 1. (10 points) The value of the expression $\frac{3}{2} \times\left[2 \frac{2}{3} \times\left(1.875-\frac{5}{6}\right)\right] \div\left[\left(0.875+1 \frac{5}{6}\right) \div 3 \frac{1}{4}\right]$ is | 5 | 90 | 1 |
math | Four men are each given a unique number from $1$ to $4$, and four women are each given a unique number from $1$ to $4$. How many ways are there to arrange the men and women in a circle such that no two men are next to each other, no two women are next to each other, and no two people with the same number are next to ea... | 12 | 102 | 2 |
math | 7. Let $A, B$ be two points on the curve $x y=1(x, y>0)$ in the Cartesian coordinate system $x O y$, and let the vector $\vec{m}=(1,|O A|)$. Then the minimum value of the dot product $\vec{m} \cdot \overrightarrow{O B}$ is $\qquad$ . | 2\sqrt[4]{2} | 81 | 8 |
math | Solve the equation
$$
x^{2}-8(x+3) \sqrt{x-1}+22 x-7=0
$$ | 5 | 32 | 1 |
math | 1. Let $x$ and $y$ be real numbers that satisfy the following system of equations:
$$
\left\{\begin{array}{l}
\frac{x}{x^{2} y^{2}-1}-\frac{1}{x}=4 \\
\frac{x^{2} y}{x^{2} y^{2}-1}+y=2
\end{array}\right.
$$
Find all possible values of the product $x y$. | xy=\\frac{1}{\sqrt{2}} | 99 | 12 |
math | (19) Let $\alpha, \beta$ be the two real roots of the quadratic equation $x^{2}-2 k x+k+20=0$. Find the minimum value of $(\alpha+1)^{2}+$ $(\beta+1)^{2}$, and indicate the value of $k$ when the minimum value is achieved. | 18 | 75 | 2 |
math | 2. A new math teacher came to the class. He conducted a survey among the students of this class, asking if they love math. It turned out that $50 \%$ love math, and $50 \%$ do not. The teacher conducted the same survey at the end of the school year. This time, $60 \%$ of the students answered "yes," and $40 \%$ answere... | 10;90 | 119 | 5 |
math | 7. Given $a, b>0$. Then the minimum value of $\frac{b^{2}+2}{a+b}+\frac{a^{2}}{a b+1}$ is . $\qquad$ | 2 | 47 | 1 |
math | 725. The number $11 \ldots 122 \ldots 2$ (written with 100 ones and 100 twos) can be expressed as the product of two consecutive natural numbers. | 33\ldots3\cdot33\ldots34 | 50 | 15 |
math | Task 2. By which smallest natural number should the number 63000 be multiplied so that the resulting product is a perfect square? | 70 | 30 | 2 |
math | 20th BMO 1984 Problem 3 Find the maximum and minimum values of cos x + cos y + cos z, where x, y, z are non-negative reals with sum 4π/3. | max\frac{3}{2}at0,0,\frac{4\pi}{3};0at0,\frac{2\pi}{3},\frac{2\pi}{3} | 48 | 42 |
math | ## [ Product of lengths of chord segments and lengths of secant segments ]
The radii of two concentric circles are in the ratio $1: 2$. A chord of the larger circle is divided into three equal parts by the smaller circle. Find the ratio of this chord to the diameter of the larger circle. | \frac{3\sqrt{6}}{8} | 64 | 12 |
math | Let $a$ be a positive real number. Then prove that the polynomial
\[ p(x)=a^3x^3+a^2x^2+ax+a \]
has integer roots if and only if $a=1$ and determine those roots. | a = 1 | 54 | 5 |
math | 4. Determine all triples $(x, y, z)$ of natural numbers such that
$$
x y z + x y + y z + z x + x + y + z = 243
$$ | (1,1,60),(1,60,1),(60,1,1) | 45 | 22 |
math | In the given operation, the letters $a, b$, and $c$ represent distinct digits and are different from 1. Determine the values of $a, b$, and $c$.
$$
\begin{array}{r}
a b b \\
\times \quad c \\
\hline b c b 1
\end{array}
$$ | =5,b=3,=7 | 74 | 8 |
math | Task B-4.1. Determine the natural number $n \geq 2$ such that the equality
$$
\frac{(n-1)^{2} n(n+1)!}{(n+2)!}=\binom{n}{2}
$$
holds. | 4 | 59 | 1 |
math | 11. Billie has a die with the numbers $1,2,3,4,5$ and 6 on its six faces.
Niles has a die which has the numbers $4,4,4,5,5$ and 5 on its six faces.
When Billie and Niles roll their dice the one with the larger number wins. If the two numbers are equal it is a draw.
The probability that Niles wins, when written as a fra... | 181 | 126 | 3 |
math | The side lengths $a,b,c$ of a triangle $ABC$ are positive integers. Let:\\
\[T_{n}=(a+b+c)^{2n}-(a-b+c)^{2n}-(a+b-c)^{2n}+(a-b-c)^{2n}\]
for any positive integer $n$. If $\frac{T_{2}}{2T_{1}}=2023$ and $a>b>c$ , determine all possible perimeters of the triangle $ABC$. | 49 | 107 | 2 |
math | 10. (12 points) 1 kilogram of soybeans can be made into 3 kilograms of tofu, and 1 kilogram of soybean oil requires 6 kilograms of soybeans. Tofu sells for 3 yuan per kilogram, and soybean oil sells for 15 yuan per kilogram. A batch of soybeans weighs 460 kilograms, and after being made into tofu or soybean oil and sol... | 360 | 121 | 3 |
math |
2. Solve for integers $x, y, z$ :
$$
x+y=1-z, \quad x^{3}+y^{3}=1-z^{2}
$$
| (,-,1),(0,1,0),(-2,-3,6),(1,0,0),(0,-2,3),(-2,0,3),(-3,-2,6) | 40 | 44 |
math | *Ni, (25 minutes) In space, there are $n$ planes $(n \geqslant 4)$, no two of which are parallel, and no three of which are concurrent. Among their pairwise intersection lines, what is the maximum number of pairs of skew lines? | 3 C_{n}^{4} | 60 | 8 |
math | Find all positive integers $n$ which satisfy the following tow conditions:
(a) $n$ has at least four different positive divisors;
(b) for any divisors $a$ and $b$ of $n$ satisfying $1<a<b<n$, the number $b-a$ divides $n$.
[i](4th Middle European Mathematical Olympiad, Individual Competition, Problem 4)[/i] | 6, 8, 12 | 83 | 8 |
math | Point, point $E, F$ are the centroids of $\triangle A B D$ and $\triangle A C D$ respectively, connecting $E, F$ intersects $A D$ at point $G$. What is the value of $\frac{D G}{G A}$? (1991-1992 Guangzhou, Luoyang, Fuzhou, Wuhan, Chongqing Junior High School League) | \frac{1}{2} | 90 | 7 |
math | 8. The sequence $\left\{a_{n}\right\}$ with all terms being positive integers is defined as follows: $a_{0}=m, a_{n+1}=a_{n}^{5}+487(n \in \mathbf{N})$, then the value of $m$ that makes the sequence $\left\{a_{n}\right\}$ contain the most perfect squares is $\qquad$. | 9 | 92 | 1 |
math | Of a rhombus $ABCD$ we know the circumradius $R$ of $\Delta ABC$ and $r$ of $\Delta BCD$. Construct the rhombus. | ABCD | 38 | 3 |
math | 7. In the tetrahedron $S-ABC$,
$$
SA=SB=SC=\sqrt{21}, BC=6 \text{. }
$$
If the projection of point $A$ onto the plane of the side $SBC$ is exactly the orthocenter of $\triangle SBC$, then the volume of the inscribed sphere of the tetrahedron $S-ABC$ is . $\qquad$ | \frac{4\pi}{3} | 92 | 9 |
math | 4. Starting from 1, alternately add 4 and 3, to get the following sequence of numbers $1,5,8,12,15,19,22 \ldots \ldots$ The number in this sequence that is closest to 2013 is $\qquad$ . | 2014 | 68 | 4 |
math | In English class, you have discovered a mysterious phenomenon -- if you spend $n$ hours on an essay, your score on the essay will be $100\left( 1-4^{-n} \right)$ points if $2n$ is an integer, and $0$ otherwise. For example, if you spend $30$ minutes on an essay you will get a score of $50$, but if you spend $35$ minute... | 75 | 172 | 2 |
math | [Text problems $]$
Liza is 8 years older than Nastya. Two years ago, she was three times as old as Nastya. How old is Liza?
# | 14 | 40 | 2 |
math | 18. Numbers $a, b$ and $c$ are such that $\frac{a}{b+c}=\frac{b}{c+a}=\frac{c}{a+b}=k$. How many possible values of $k$ are there? | 2 | 53 | 1 |
math | 1. Find the number of points in the $x O y$ plane having natural coordinates $(x, y)$ and lying on the parabola $y=-\frac{x^{2}}{3}+7 x+54$. | 8 | 49 | 1 |
math | 7. Given that the complex number $z$ satisfies $|z|=1$, then the maximum value of $\left|z^{3}-3 z-2\right|$ is $\qquad$ . | 3\sqrt{3} | 42 | 6 |
math | 69. A reconnaissance aircraft is flying in a circle centered at point $A$. The radius of the circle is $10 \mathrm{km}$, and the speed of the aircraft is 1000 km/h. At some moment, a missile is launched from point $A$ with the same speed as the aircraft, and it is controlled such that it always remains on the line conn... | \frac{\pi}{200} | 102 | 9 |
math | 8. Given the sequences $\left\{a_{n}\right\}$ and $\left\{b_{n}\right\}$ with the general terms $a_{n}=2^{n}, b_{n}=5 n-2$. Then the sum of all elements in the set
$$
\left\{a_{1}, a_{2}, \cdots, a_{2019}\right\} \cap\left\{b_{1}, b_{2}, \cdots, b_{2019}\right\}
$$
is $\qquad$ | 2184 | 123 | 4 |
math | Problem 6.1. Find any solution to the puzzle
$$
\overline{A B}+A \cdot \overline{C C C}=247
$$
where $A, B, C$ are three different non-zero digits; the notation $\overline{A B}$ represents a two-digit number composed of the digits $A$ and $B$; the notation $\overline{C C C}$ represents a three-digit number consisting... | 251 | 121 | 3 |
math | For a real number $a$ and an integer $n(\geq 2)$, define
$$S_n (a) = n^a \sum_{k=1}^{n-1} \frac{1}{k^{2019} (n-k)^{2019}}$$
Find every value of $a$ s.t. sequence $\{S_n(a)\}_{n\geq 2}$ converges to a positive real. | 2019 | 99 | 4 |
math | 1. In the cube $A B C D-A_{1} B_{1} C_{1} D_{1}$, $E$, $F$, and $G$ are the midpoints of edges $B C$, $C C_{1}$, and $C D$ respectively. Then the angle formed by line $A_{1} G$ and plane $D E F$ is $\qquad$ | 90^{\circ} | 86 | 6 |
math | 52. Find the remainder when the number $50^{13}$ is divided by 7. | 1 | 22 | 1 |
math | 14. $[8]$ Evaluate the infinite sum $\sum_{n=1}^{\infty} \frac{n}{n^{4}+4}$. | \frac{3}{8} | 35 | 7 |
math | 48. How old is each of us, if I am now twice as old as you were when I was as old as you are now, and we are both 63 years old in total. | I36old,You27old | 42 | 9 |
math | 2.18. Find the divergence of the vector field:
a) $\vec{A}=x y^{2} \vec{i}+x^{2} y \vec{j}+z^{3} \vec{k}$ at the point $M(1,-1,3)$;
b) the gradient of the function $\varphi=x y^{2} z^{2}$;
c) $\dot{A}=y z(4 x \vec{i}-y \vec{j}-z \vec{k})$. | 29,2x(y^{2}+z^{2}),0 | 108 | 15 |
math | \section*{Problem \(4-341034=340934\)}
A square \(A B C D\) is divided into 25 congruent smaller squares.
Let \(n\) be a positive integer with \(n \leq 25\). Then \(n\) different colors are chosen, and for each of these colors, 25 small tiles of the size of the smaller squares are provided.
From these \(n \cdot 25\)... | 1,2,3,4,5,6 | 254 | 11 |
math | The increasing sequence $1; 3; 4; 9; 10; 12; 13; 27; 28; 30; 31, \ldots$ is formed with positive integers which are powers of $3$ or sums of different powers of $3$. Which number is in the $100^{th}$ position? | 981 | 82 | 3 |
math | Test $\mathbf{E}$ Let $a, b, c, d, m, n$ be positive integers,
$$
a^{2}+b^{2}+c^{2}+d^{2}=1989, a+b+c+d=m^{2} \text {, }
$$
and the largest of $a, b, c, d$ is $n^{2}$, determine (and prove) the values of $m, n$. | =9,n=6 | 99 | 5 |
math | 16th USAMO 1987 Problem 2 The feet of the angle bisectors of the triangle ABC form a right-angled triangle. If the right-angle is at X, where AX is the bisector of angle A, find all possible values for angle A. Solution | 120 | 59 | 3 |
math | 11. Two concentric circles have radii 2006 and 2007. $A B C$ is an equilateral triangle inscribed in the smaller circle and $P$ is a point on the circumference of the larger circle. Given that a triangle with side lengths equal to $P A, P B$ and $P C$ has area $\frac{a \sqrt{b}}{c}$, where $a, b, c$ are positive intege... | 4020 | 287 | 4 |
math | 4. The numbers $a_{1}, a_{2}, \ldots, a_{20}$ satisfy the conditions:
$$
\begin{aligned}
& a_{1} \geq a_{2} \geq \ldots \geq a_{20} \geq 0 \\
& a_{1}+a_{2}=20 \\
& a_{3}+a_{4}+\ldots+a_{20} \leq 20
\end{aligned}
$$
What is the maximum value of the expression:
$$
a_{1}^{2}+a_{2}^{2}+\ldots+a_{20}^{2}
$$
For which va... | 400 | 177 | 3 |
math | ## Problem Statement
Find the cosine of the angle between vectors $\overrightarrow{A B}$ and $\overrightarrow{A C}$.
$A(-1, -2, 1), B(-4, -2, 5), C(-8, -2, 2)$ | \frac{1}{\sqrt{2}} | 59 | 10 |
math | 5. Given that the average of 8 numbers is 8, if one of the numbers is changed to 8, the average of these 8 numbers becomes 7, then the original number that was changed is $\qquad$ . | 16 | 49 | 2 |
math | ## Task 1
Calculate the results. Then order the results, starting with the largest.
$$
\begin{array}{rlll}
24-7 & 46-46 & 53+26 & 34+37 \\
45-29 & 52-25 & 55+37 & 68+0
\end{array}
$$ | 92,79,71,68,27,17,16,0 | 88 | 22 |
math | 12. (12 points) Nine cards are labeled with the numbers $2,3,4,5,6,7,8,9,10$ (they cannot be read upside down). Four people, A, B, C, and D, each draw two of these cards.
A says: "The two numbers I got are coprime, because they are consecutive"
B says: "The two numbers I got are not coprime, and they are not multiples ... | 7 | 169 | 1 |
math | Problem 5.6. Vanya received three sets of candies as a New Year's gift. In the sets, there are three types of candies: lollipops, chocolate, and jelly. The total number of lollipops in all three sets is equal to the total number of chocolate candies in all three sets, as well as the total number of jelly candies in all... | 29 | 156 | 2 |
math | Let $S=\{1,2, \ldots ,10^{10}\}$. Find all functions $f:S \rightarrow S$, such that $$f(x+1)=f(f(x))+1 \pmod {10^{10}}$$ for each $x \in S$ (assume $f(10^{10}+1)=f(1)$). | f(x) = x \pmod{10^{10}} | 82 | 16 |
math | Find, with explanation, the maximum value of $f(x)=x^3-3x$ on the set of all real numbers $x$ satisfying $x^4+36\leq 13x^2$. | 18 | 48 | 2 |
math | Consider all 1000-element subsets of the set $\{1, 2, 3, ... , 2015\}$. From each such subset choose the least element. The arithmetic mean of all of these least elements is $\frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Find $p + q$. | 431 | 82 | 3 |
math | ## Problem I - 5
A positive number $\mathrm{x}$ satisfies the relation
$$
x^{2}+\frac{1}{x^{2}}=7
$$
Prove that
$$
x^{5}+\frac{1}{x^{5}}
$$
is an integer and find its value. | 123 | 67 | 3 |
math | 13.062. An apprentice turner is machining pawns for a certain number of chess sets. He wants to learn to produce 2 more pawns per day than he does now; then he would complete the same task 10 days faster. If he could learn to produce 4 more pawns per day than he does now, the time required to complete the same task wou... | 15 | 113 | 2 |
math | 5. (7 points) The king decided to test his hundred sages and announced that the next day he would line them up with their eyes blindfolded and put a black or white hat on each of them. After their eyes are uncovered, each, starting from the last in line, will name the supposed color of their hat. If he fails to guess c... | 99 | 104 | 2 |
math | 11. (20 points) For any real numbers $a_{1}, a_{2}, \cdots, a_{5}$ $\in[0,1]$, find the maximum value of $\prod_{1 \leq i<j \leq 5}\left|a_{i}-a_{j}\right|$. | \frac{3\sqrt{21}}{38416} | 70 | 17 |
math | 8.067. $\sin 3 x+\sin 5 x=2\left(\cos ^{2} 2 x-\sin ^{2} 3 x\right)$. | x_{1}=\frac{\pi}{2}(2n+1),x_{2}=\frac{\pi}{18}(4k+1),n,k\inZ | 42 | 38 |
math | Two circles, both with the same radius $r$, are placed in the plane without intersecting each other. A line in the plane intersects the first circle at the points $A,B$ and the other at points $C,D$, so that $|AB|=|BC|=|CD|=14\text{cm}$. Another line intersects the circles at $E,F$, respectively $G,H$ so that $|EF|=|FG... | 13 | 107 | 2 |
math | 15. (13 points) Given the ellipse $E: \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1(a>$ $b>0)$, the length of the minor axis is $2 \sqrt{3}$, and the eccentricity is $e_{1}$; the hyperbola $\frac{x^{2}}{m}-\frac{y^{2}}{n}=1(m 、 n>0)$ has asymptotes $y= \pm \sqrt{3} x$, and the eccentricity is $e_{2} ; e_{1} e_{2}=1$.
(1) F... | \\frac{\sqrt{6}}{2}x+1 | 285 | 13 |
math | 3. Let $a, b \in \mathbf{N}^{*},(a, b)=1$. Find the smallest positive integer $c_{0}$, such that for any $c \in \mathbf{N}^{*}, c \geqslant c_{0}$, the indeterminate equation $a x+b y=c$ has non-negative integer solutions. | --b+1 | 81 | 4 |
math | (7) Nine positive real numbers $a_{1}, a_{2}, \cdots, a_{9}$ form a geometric sequence, and $a_{1}+a_{2}=\frac{3}{4}$, $a_{3}+a_{4}+a_{5}+a_{6}=15$. Then $a_{7}+a_{8}+a_{9}=$ $\qquad$ . | 112 | 93 | 3 |
math | 2. 10 people go to the bookstore to buy books, it is known that: (1) each person bought three types of books; (2) any two people have at least one book in common.
How many people at most could have bought the book that was purchased by the fewest people? | 5 | 63 | 1 |
math | 7.1. Write the number 2013 several times in a row so that the resulting number is divisible by 9. Explain your answer. | 201320132013 | 32 | 12 |
math | 2. Given $a=\sqrt{3}-1$. Then $a^{2012}+2 a^{2011}-2 a^{2010}=$ | 0 | 39 | 1 |
math | 79. $\int\left(\sin \frac{x}{2}+\cos \frac{x}{2}\right)^{2} d x$.
Translate the above text into English, please keep the original text's line breaks and format, and output the translation result directly.
79. $\int\left(\sin \frac{x}{2}+\cos \frac{x}{2}\right)^{2} d x$. | x-\cosx+C | 88 | 5 |
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