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 | 12.159. The sides of the parallelogram are in the ratio $p: q$, and the diagonals are in the ratio m:n. Find the angles of the parallelogram. | \arccos\frac{(p^{2}+q^{2})(n^{2}-^{2})}{2pq(^{2}+n^{2})};\pi-\arccos\frac{(p^{2}+q^{2})(n^{2}-^{2})}{2pq(^{2}+n^{2})} | 43 | 74 |
math | Let $ p \geq 3$ be a prime, and let $ p$ points $ A_{0}, \ldots, A_{p-1}$ lie on a circle in that order. Above the point $ A_{1+\cdots+k-1}$ we write the number $ k$ for $ k=1, \ldots, p$ (so $ 1$ is written above $ A_{0}$). How many points have at least one number written above them? | \frac{p+1}{2} | 102 | 9 |
math | Example 4. Find $\lim _{y \rightarrow 0} \frac{e^{y}+\sin y-1}{\ln (1+y)}$. | 2 | 35 | 1 |
math | Example 8 If $(x, y)$ satisfies $x^{2}+x y+y^{2} \leqslant 1$, find the maximum and minimum values of $x-y+2 x y$.
| \frac{25}{24} | 46 | 9 |
math | A disk with radius $1$ is externally tangent to a disk with radius $5$. Let $A$ be the point where the disks are tangent, $C$ be the center of the smaller disk, and $E$ be the center of the larger disk. While the larger disk remains fixed, the smaller disk is allowed to roll along the outside of the larger disk until t... | 58 | 186 | 2 |
math | Find all functions $f : \mathbb N \mapsto \mathbb N$ such that the following identity
$$f^{x+1}(y)+f^{y+1}(x)=2f(x+y)$$
holds for all $x,y \in \mathbb N$ | f(f(n)) = f(n+1) | 61 | 11 |
math | The number $123454321$ is written on a blackboard. Evan walks by and erases some (but not all) of the digits, and notices that the resulting number (when spaces are removed) is divisible by $9$. What is the fewest number of digits he could have erased?
[i]Ray Li[/i] | 2 | 74 | 1 |
math | Example 7 Let $a>0, b>0$, and $\sqrt{a}(\sqrt{a}+2 \sqrt[3]{b})$ $=\sqrt[3]{b}(\sqrt{a}+6 \sqrt[3]{b})$. Then the value of $\frac{2 a^{4}+a^{3} b-128 a b^{2}-64 b^{3}+b^{4}}{a^{4}+2 a^{3} b-64 a b^{2}-128 b^{3}+2 b^{4}}$ is | \frac{1}{2} | 130 | 7 |
math | 1. Let
$$
S=\frac{1}{2}, E=\frac{\frac{3}{7}+1}{\frac{3}{7}-1}, D=-2: \frac{5}{3}+1.1, A=3-0.2 \cdot 2, M=100 \cdot 0.03-5.25: \frac{1}{2}
$$
Calculate the value of the expression $S+E: D-A \cdot M$. | 45 | 107 | 2 |
math | \section*{Problem 7}
A tangent to the inscribed circle of a triangle drawn parallel to one of the sides meets the other two sides at \(\mathrm{X}\) and \(\mathrm{Y}\). What is the maximum length \(\mathrm{XY}\), if the triangle has perimeter \(\mathrm{p}\) ?
| \frac{p}{8} | 71 | 7 |
math | 32nd BMO 1996 Problem 4 Find all positive real solutions to w + x + y + z = 12, wxyz = wx + wy + wz + xy + xz + yz + 27. | w=x=y=z=3 | 52 | 6 |
math | Find all quadruples $(a, b, c, d)$ of positive integers which satisfy
$$
\begin{aligned}
a b+2 a-b & =58 \\
b c+4 b+2 c & =300 \\
c d-6 c+4 d & =101
\end{aligned}
$$ | (3,26,7,13),(15,2,73,7) | 71 | 21 |
math | 3. Given that $\left\{a_{n}\right\}$ is a geometric sequence, and $a_{1} a_{2017}=1$, if $f(x)=\frac{2}{1+x^{2}}$, then $f\left(a_{1}\right)+f\left(a_{2}\right)+f\left(a_{3}\right)+\cdots+f\left(a_{2017}\right)=$ | 2017 | 96 | 4 |
math | How many positive five-digit integers are there that have the product of their five digits equal to $900$?
(Karl Czakler) | 210 | 31 | 3 |
math | 5. In acute triangle $ABC$, the lines tangent to the circumcircle of $ABC$ at $A$ and $B$ intersect at point $D$. Let $E$ and $F$ be points on $CA$ and $CB$ such that $DECF$ forms a parallelogram. Given that $AB = 20$, $CA=25$ and $\tan C = 4\sqrt{21}/17$, the value of $EF$ may be expressed as $m/n$ for relatively prim... | 267 | 144 | 3 |
math | What is the smallest positive integer with six positive odd integer divisors and twelve positive even integer divisors? | 180 | 21 | 3 |
math | 7.073. $9^{x^{2}-1}-36 \cdot 3^{x^{2}-3}+3=0$. | -\sqrt{2};-1;1;\sqrt{2} | 33 | 14 |
math | 1. Calculate $\sqrt{(31)(30)(29)(28)+1}$. | 869 | 21 | 3 |
math | 9. (14 points) Given the ellipse $C: \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1(a>b \geqslant 0)$, its eccentricity is $\frac{4}{5}$, and one of its directrix equations is $x=\frac{25}{4}$.
(1) Find the equation of the ellipse; (2) Let point $A$ have coordinates (6, $0), B$ be a moving point on the ellipse $C$, and cons... | \frac{(y-6)^{2}}{25}+\frac{(x-6)^{2}}{9}=1 | 167 | 28 |
math | Find all positive integer $n$ such that for all $i=1,2,\cdots,n$, $\frac{n!}{i!(n-i+1)!}$ is an integer.
[i]Proposed by ckliao914[/i] | n = p-1 | 53 | 6 |
math | Initially 247 Suppose there are $n$ square cards of the same size, each inscribed with a positive integer. These $n$ positive integers are consecutive two-digit numbers, and the sum of these two-digit numbers equals the "composite number" of the first and last terms (such as $1+2+\cdots+5=15,4+5+\cdots$ $+29=429$). If ... | 13,14, \cdots, 52,53 | 133 | 16 |
math | 2. For the arithmetic sequence $\left.\mid a_{n}\right\}$, the first term $a_{1}=8$, and there exists a unique $k$ such that the point $\left(k, a_{k}\right)$ lies on the circle $x^{2}+y^{2}=10^{2}$. Then the number of such arithmetic sequences is $\qquad$. | 17 | 82 | 2 |
math | Let $n$ be a positive integer and let $P$ be the set of monic polynomials of degree $n$ with complex coefficients. Find the value of
\[ \min_{p \in P} \left \{ \max_{|z| = 1} |p(z)| \right \} \] | 1 | 68 | 3 |
math | Each pair of vertices of a regular $67$-gon is joined by a line segment. Suppose $n$ of these segments are selected, and each of them is painted one of ten available colors. Find the minimum possible value of $n$ for which, regardless of which $n$ segments were selected and how they were painted, there will always be a... | n = 2011 | 88 | 8 |
math | Let's determine the pairs of positive integers $\left(a_{1} ; a_{2}\right)$ for which the sequence defined by the recursion $a_{n+2}=\frac{a_{n}+a_{n+1}}{\left(a_{n}, a_{n+1}\right)}(n \geq 1)$ is periodic. | a_{1}=a_{2}=2 | 75 | 9 |
math | 12. Let $a \in \mathbf{R}$, and for any real number $b$ we have $\max _{x \in[0,1]}\left|x^{2}+a x+b\right| \geq 1$, find the range of values for $a$. | \geq1or\leq-3 | 65 | 10 |
math | 2. The two positive integers $a, b$ with $a>b$ are such that $a \%$ of $b \%$ of $a$ and $b \%$ of $a \%$ of $b$ differ by 0.003 . Find all possible pairs $(a, b)$. | (5,2),(5,3),(6,1),(6,5) | 65 | 17 |
math | Let $a,b,c,d,e$ be positive real numbers. Find the largest possible value for the expression
$$\frac{ab+bc+cd+de}{2a^2+b^2+2c^2+d^2+2e^2}.$$ | \sqrt{\frac{3}{8}} | 56 | 9 |
math | Find the remainder when\[\binom{\binom{3}{2}}{2} + \binom{\binom{4}{2}}{2} + \dots + \binom{\binom{40}{2}}{2}\]is divided by $1000$. | 4 | 64 | 1 |
math | A merchant received four bags of potatoes and wants to measure the weight of each one. He knows that the weights of these bags in kilograms are whole and distinct numbers. Suppose the weights of the bags (in kilograms) are $a, b, c$, and $d$, with $a<b<c<d$.
a) Show that, when weighing the bags two at a time, the larg... | =41,b=45,=56,=71 | 225 | 15 |
math | Find all prime numbers $p$ such that $p^3$ divides the determinant
\[\begin{vmatrix} 2^2 & 1 & 1 & \dots & 1\\1 & 3^2 & 1 & \dots & 1\\ 1 & 1 & 4^2 & & 1\\ \vdots & \vdots & & \ddots & \\1 & 1 & 1 & & (p+7)^2 \end{vmatrix}.\] | p \in \{2, 3, 5, 181\} | 110 | 19 |
math | 7.194. $\log _{x} 2-\log _{4} x+\frac{7}{6}=0$. | \frac{1}{\sqrt[3]{4}};8 | 30 | 14 |
math | 10. From any point $P$ on the parabola $y^{2}=2 x$, draw a perpendicular to its directrix $l$, with the foot of the perpendicular being $Q$. The line connecting the vertex $O$ and $P$ and the line connecting the focus $F$ and $Q$ intersect at point $R$. Then the equation of the locus of point $R$ is $\qquad$. | y^{2}=-2x^{2}+x | 89 | 12 |
math | Let $P(x)=x^2-3x-9$. A real number $x$ is chosen at random from the interval $5\leq x \leq 15$. The probability that $\lfloor \sqrt{P(x)} \rfloor = \sqrt{P(\lfloor x \rfloor )}$ is equal to $\dfrac{\sqrt{a}+\sqrt{b}+\sqrt{c}-d}{e}$, where $a,b,c,d$ and $e$ are positive integers and none of $a,b,$ or $c$ is divisible by... | 850 | 138 | 3 |
math | Example 2 Given that the complex number satisfies $x^{2}+x+1=0$, find the value of $x^{14}+\frac{1}{x^{14}}$. | -1 | 42 | 2 |
math | Example 17 A worker takes care of three machine tools. In one hour, the probabilities that machine tools A, B, and C need attention are $0.9$, $0.8$, and $0.85$, respectively. In one hour, find:
(1) the probability that no machine tool needs attention;
(2) the probability that at least one machine tool needs attention. | 0.997 | 82 | 5 |
math | 11. Let $0 \leqslant x \leqslant \pi$, and
$$
3 \sin \frac{x}{2}=\sqrt{1+\sin x}-\sqrt{1-\sin x} \text {. }
$$
Then $\tan x=$ . $\qquad$ | 0 | 65 | 1 |
math | 7.1. The sides of a rectangle are in the ratio $3: 4$, and its area is numerically equal to its perimeter. Find the sides of the rectangle. | \frac{7}{2},\frac{14}{3} | 37 | 15 |
math | Each term of a sequence of natural numbers is obtained from the previous term by adding to it its largest digit. What is the maximal number of successive odd terms in such a sequence? | 5 | 36 | 1 |
math | $12 \cdot 88$ Find all positive integers $m, n$, such that $(m+n)^{m}=n^{m}+1413$.
(2nd Northeast China Three Provinces Mathematics Invitational Competition, 1987) | =3,\quadn=11 | 57 | 8 |
math | 3. Two fields are planted with roses and lavender. $65 \%$ of the area of the first field is roses, $45 \%$ of the area of the second field is roses, and $53 \%$ of the area of both fields together is roses. What percentage of the total area (of both fields together) is the area of the first field? | 40 | 77 | 2 |
math | Example 1. Find the rotor of the vector $\mathbf{v}=(x+z) \mathbf{i}+(y+z) \mathbf{j}+\left(x^{2}+z\right) \mathbf{k}$. | rot{}=-{i}-(2x-1){j} | 51 | 14 |
math | 5. In the Cartesian coordinate system, $F_{1}, F_{2}$ are the left and right foci of the hyperbola $\Omega: x^{2}-\frac{y^{2}}{3}=1$, respectively. A line $l$ passing through $F_{1}$ intersects $\Omega$ at two points $P, Q$. If $\overrightarrow{F_{1} F_{2}} \cdot \overrightarrow{F_{1} P}=16$, then the value of $\overri... | \frac{27}{13} | 133 | 9 |
math | 4. Including A, six people pass the ball to each other. The ball is passed from one person to another each time, starting with A. The total number of ways for the ball to return to A after six passes is $\qquad$ . | 2605 | 51 | 4 |
math | 17. The difference of the squares of two different real numbers is param 1 times greater than the difference of these numbers, and the difference of the cubes of these numbers is param 2 times greater than the difference of these numbers. By how many times is the difference of the fourth powers of these numbers greater... | 769 | 317 | 3 |
math | 18.2. (Jury, Sweden, 79). Find the maximum value of the product $x^{2} y^{2} z^{2} u$ given that $x, y, z, u \geqslant 0$ and
$$
2 x+x y+z+y z u=1
$$ | \frac{1}{512} | 71 | 9 |
math | Gapochkin A.i.
How many integers from 1 to 1997 have a sum of digits that is divisible by 5? | 399 | 30 | 3 |
math | Example 1.6. Find: $\int \frac{x_{3}+2 x^{2}-4 x-1}{(x-2)^{2}\left(x^{2}+x+1\right)} d x$. | -\frac{1}{(x-2)}+\frac{11}{7}|\lnx-2|-\frac{2}{7}\ln(x^{2}+x+1)+\frac{8}{7\sqrt{3}}\operatorname{arctg}\frac{2x+1}{\sqrt{3}}+C | 50 | 74 |
math | Eddy draws $6$ cards from a standard $52$-card deck. What is the probability that four of the cards that he draws have the same value? | \frac{3}{4165} | 35 | 10 |
math | ## Task 3
Uwe says: "My father is 42 years old. My father is two years older than my mother. My mother is twice as old as my brother and I. I am two years younger than my brother." How old are Uwe, his brother, and his mother? | Uweis9old,hisbrotheris11old,hismotheris40old | 63 | 19 |
math | Find the functions $f: \mathbb{R} \mapsto \mathbb{R}$ such that, for all real numbers $x$ and $y$, we have:
$$
f\left(x^{2}+x+f(y)\right)=y+f(x)+f(x)^{2}
$$ | f(x) = x | 65 | 5 |
math | 1. From point A to point B, which are 12 km apart, a pedestrian and a bus set out simultaneously. Arriving at point B in less than one hour, the bus, without stopping, turned around and started moving back towards point A at a speed twice its initial speed. After 12 minutes from its departure from point B, the bus met ... | 21 | 141 | 2 |
math | 12. Given the ellipse $C: \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1(a>b>0)$ with an eccentricity of $\frac{1}{2}$, $A_{1}, A_{2}$ are the left and right vertices of the ellipse $C$, $F_{1}, F_{2}$ are the left and right foci of the ellipse $C$, $B$ is the upper vertex of the ellipse $C$, and the circumradius of $\trian... | (0,\frac{9\sqrt{5}}{2}) | 295 | 14 |
math | Bethany is told to create an expression from $2 \square 0 \square 1 \square 7$ by putting a + in one box, a in another, and $a x$ in the remaining box. There are 6 ways in which she can do this. She calculates the value of each expression and obtains a maximum value of $M$ and a minimum value of $m$. What is $M-m$ ? | 15 | 89 | 2 |
math | $7.80 \quad\left\{\begin{array}{l}x^{x-2 y}=36, \\ 4(x-2 y)+\log _{6} x=9\end{array}\right.$
(find only integer solutions). | (6;2) | 57 | 5 |
math | 1. Let set $A=\{2,0,1,8\}, B=\{2 a \mid a \in A\}$, then the sum of all elements in $A \cup B$ is | 31 | 45 | 2 |
math | 9.2. What digit can the number $f(x)=[x]+[3 x]+[6 x]$ end with, where $x$ is an arbitrary positive real number? Here $[x]$ denotes the integer part of the number $x$, that is, the greatest integer not exceeding $x$.
| 0,1,3,4,6,7 | 65 | 11 |
math | Let $n$ be positive integer. Define a sequence $\{a_k\}$ by
\[a_1=\frac{1}{n(n+1)},\ a_{k+1}=-\frac{1}{k+n+1}+\frac{n}{k}\sum_{i=1}^k a_i\ \ (k=1,\ 2,\ 3,\ \cdots).\]
(1) Find $a_2$ and $a_3$.
(2) Find the general term $a_k$.
(3) Let $b_n=\sum_{k=1}^n \sqrt{a_k}$. Prove that $\lim_{n\to\infty} b_n=\ln 2$.
50 point... | \ln 2 | 160 | 5 |
math | 1. In triangle $A B C$, there is a point $D$ on side $B C$ such that $B A=A D=D C$. Suppose $\angle B A D=80^{\circ}$. Determine the size of $\angle A C B$. | 25 | 56 | 2 |
math | 5. Given $f(x)$ is a function defined on the set of real numbers, and $f(x+2)[1-f(x)]=1+f(x)$. If $f(1)=$ $2+\sqrt{3}$, then the value of $f(1949)$ is $\qquad$ | \sqrt{3}-2 | 67 | 6 |
math | Find all positive integers $ n$ such that $ 20^n \minus{} 13^n \minus{} 7^n$ is divisible by $ 309$. | n = 1 + 6k | 37 | 9 |
math | Determine all real numbers $a, b, c, d$ that satisfy the following system of equations.
$$
\left\{\begin{array}{r}
a b c+a b+b c+c a+a+b+c=1 \\
b c d+b c+c d+d b+b+c+d=9 \\
c d a+c d+d a+a c+c+d+a=9 \\
d a b+d a+a b+b d+d+a+b=9
\end{array}\right.
$$ | =b==\sqrt[3]{2}-1,=5\sqrt[3]{2}-1 | 102 | 21 |
math | 9. There are 1000 lamps and 1000 switches, each switch controls all lamps whose numbers are multiples of its own, initially all lamps are on. Now pull the $2, 3, 5$ switches, then the number of lamps that are still on is $\qquad$.
| 499 | 66 | 3 |
math | 9. Let $A C$ be a diameter of a circle $\omega$ of radius 1 , and let $D$ be the point on $A C$ such that $C D=1 / 5$. Let $B$ be the point on $\omega$ such that $D B$ is perpendicular to $A C$, and let $E$ be the midpoint of $D B$. The line tangent to $\omega$ at $B$ intersects line $C E$ at the point $X$. Compute $A ... | 3 | 110 | 1 |
math | Problem 11.5. At the call of the voivode, 55 soldiers arrived: archers and swordsmen. All of them were dressed either in golden or black armor. It is known that swordsmen tell the truth when wearing black armor and lie when wearing golden armor, while archers do the opposite.
- In response to the question "Are you wea... | 22 | 147 | 2 |
math | 11.48 Which is greater: $3^{400}$ or $4^{300} ?$ | 3^{400}>4^{300} | 26 | 12 |
math | Find all functions $f: \mathbb{R} \longrightarrow \mathbb{R}$ such that $f(0)=0$, and for all $x, y \in \mathbb{R}$,
$$
(x-y)\left(f\left(f(x)^{2}\right)-f\left(f(y)^{2}\right)\right)=(f(x)+f(y))(f(x)-f(y))^{2}
$$ | f(x)=cx | 91 | 4 |
math | Find all the integer solutions of the equation
$$
9 x^{2} y^{2}+9 x y^{2}+6 x^{2} y+18 x y+x^{2}+2 y^{2}+5 x+7 y+6=0
$$ | (-2,0),(-3,0),(0,-2),(-1,2) | 61 | 19 |
math | Example 2 Calculate $[\sqrt{2008+\sqrt{2008+\cdots+\sqrt{2008}}}]$ (2008 appears 2008 times).
(2008, International Youth Math Invitational Competition)
【Analysis】Although there are 2008 square root operations, as long as you patiently estimate from the inside out, the pattern will naturally become apparent. | 45 | 91 | 2 |
math | 13.174. In four boxes, there is tea. When 9 kg were taken out of each box, the total that remained in all of them together was as much as there was in each one. How much tea was in each box? | 12\mathrm{} | 53 | 5 |
math | 3. If a square pyramid with a base edge length of 2 is inscribed with a sphere of radius $\frac{1}{2}$, then the volume of this square pyramid is .. $\qquad$ | \frac{16}{9} | 43 | 8 |
math | A sequence $\{a_n\}$ is defined by $a_n=\int_0^1 x^3(1-x)^n dx\ (n=1,2,3.\cdots)$
Find the constant number $c$ such that $\sum_{n=1}^{\infty} (n+c)(a_n-a_{n+1})=\frac{1}{3}$ | c = 5 | 83 | 5 |
math | 5. It is known that the polynomial $f(x)=8+32 x-12 x^{2}-4 x^{3}+x^{4}$ has 4 distinct real roots $\left\{x_{1}, x_{2}, x_{3}, x_{4}\right\}$. The polynomial $\quad$ of the form $g(x)=b_{0}+b_{1} x+b_{2} x^{2}+b_{3} x^{3}+x^{4}$ has $\quad$ roots $\left\{x_{1}^{2}, x_{2}^{2}, x_{3}^{2}, x_{4}^{2}\right\}$. Find the coe... | -1216 | 165 | 5 |
math | 1. A warehouse has coffee packed in bags of 15 kg and 8 kg. How many bags of coffee in total does the warehouseman need to prepare to weigh out 1998 kg of coffee, with the number of 8 kg bags being the smallest possible? | 136 | 58 | 3 |
math | 1. In the set $M=\{1,2, \cdots, 2018\}$, the sum of elements whose last digit is 8 is $\qquad$ | 204626 | 40 | 6 |
math | 2. All real number pairs $(x, y)$ that satisfy the equation
$$
(x+3)^{2}+y^{2}+(x-y)^{2}=3
$$
are $\qquad$ . | (-2, -1) | 47 | 6 |
math | 36. a) $(a+2 b)^{3}$;
б) $(5 a-b)^{3}$
в) $(2 a+3 b)^{3}$;
г) $\left(m^{3}-n^{2}\right)^{3}$. | )^{3}+6^{2}b+12^{2}+8b^{3};b)125^{3}-75^{2}b+15^{2}-b^{3};)8^{3}+36^{2}b+54^{2}+27b^{3};)^{9}-3^{6} | 57 | 79 |
math | 6. Find all values of the parameter $b$ such that the system
$$
\left\{\begin{array}{l}
x \cos a + y \sin a + 3 \leqslant 0 \\
x^{2} + y^{2} + 8x - 4y - b^{2} + 6b + 11 = 0
\end{array}\right.
$$
has at least one solution for any value of the parameter $a$. | b\in(-\infty;-2\sqrt{5}]\cup[6+2\sqrt{5};+\infty) | 105 | 29 |
math | For problem 11 , i couldn’t find the correct translation , so i just posted the hungarian version . If anyone could translate it ,i would be very thankful .
[tip=see hungarian]Az $X$ ́es$ Y$ valo ́s ́ert ́eku ̋ v ́eletlen v ́altoz ́ok maxim ́alkorrel ́acio ́ja az $f(X)$ ́es $g(Y )$ v ́altoz ́ok korrela ́cio ́j ́anak... | 0 | 563 | 1 |
math | 1. Choose 5 numbers (repetition allowed) from $\{1,2, \cdots, 100\}$. Then the expected number of composite numbers taken is | \frac{37}{10} | 38 | 9 |
math | $\frac{9}{10}$ is to be expressed as the sum of fractions, where the denominators of these fractions are powers of 6. | \frac{9}{10}=\frac{5}{6}+\frac{2}{6^{2}}+\frac{2}{6^{3}}+\ldots | 31 | 36 |
math | 4. If positive real numbers $x, y$ satisfy $y>2 x$, then the minimum value of $\frac{y^{2}-2 x y+x^{2}}{x y-2 x^{2}}$ is
Translate the above text into English, please retain the original text's line breaks and format, and output the translation result directly. | 4 | 74 | 1 |
math | ## Problem Statement
Find the derivative.
$$
y=\ln \frac{\sqrt{5}+\tan \frac{x}{2}}{\sqrt{5}-\tan \frac{x}{2}}
$$ | \frac{\sqrt{5}}{6\cos^{2}\frac{x}{2}-1} | 42 | 21 |
math | 4. Given real numbers $x, y$ satisfy $\frac{x^{2}}{3}+y^{2}=1$. Then
$$
P=|2 x+y-4|+|4-x-2 y|
$$
the range of values for $P$ is $\qquad$. | [2,14] | 63 | 6 |
math | 7. (6 points) Xiao Ming is 12 years old this year, and his father is 40 years old. When Xiao Ming is $\qquad$ _ years old, his father's age will be 5 times Xiao Ming's age. | 7 | 53 | 1 |
math | Example 2.34. $I=$ V.p. $\int_{-\infty}^{\infty} \frac{(1+x) d x}{1+x^{2}}$. | \pi | 40 | 2 |
math | ## Task $31 / 73$
We are looking for all prime numbers $p$ for which $p^{4}-1$ is not divisible by 15. | p_{1}=3p_{2}=5 | 37 | 10 |
math | Example 4 Let $a=\frac{1994}{1995}, b=\frac{1993}{1994}$. Try to compare the sizes of $a$ and $b$. | a > b | 47 | 3 |
math | 7. Given that $x, y, z$ are positive real numbers, satisfying $x+y+z=1$, if $\frac{a}{x y z}=\frac{1}{x}+\frac{1}{y}+\frac{1}{z}-2$, then the range of the real number $a$ is | (0,\frac{7}{27}] | 68 | 10 |
math | Find all the integer solutions $(x, y, z)$ of the equation
\[
(x+y+z)^{5}=80 x y z\left(x^{2}+y^{2}+z^{2}\right)
\] | (x, y, z) \in\{(0, t,-t),(t, 0,-t),(t,-t, 0): t \in \mathbb{Z}\} | 50 | 40 |
math | [ Special cases of parallelepipeds (other).] Area of the section
The base of a right parallelepiped is a rhombus, the area of which is equal to $Q$. The areas of the diagonal sections are $S 1$ and $S 2$. Find the volume of the parallelepiped. | \sqrt{\frac{QS_{1}S_{2}}{2}} | 68 | 16 |
math | Solve the following system of equations:
$$
\begin{aligned}
& 1-\frac{12}{3 x+y}=\frac{2}{\sqrt{x}} \\
& 1+\frac{12}{3 x+y}=\frac{6}{\sqrt{y}}
\end{aligned}
$$ | 4+2\sqrt{3},12+6\sqrt{3} | 67 | 17 |
math | ## Problem Statement
Find the derivative.
$$
y=x+\frac{1}{\sqrt{2}} \ln \frac{x-\sqrt{2}}{x+\sqrt{2}}+a^{\pi^{\sqrt{2}}}
$$ | \frac{x^{2}}{x^{2}-2} | 52 | 13 |
math | 7.187. $x^{2} \cdot \log _{x} 27 \cdot \log _{9} x=x+4$. | 2 | 35 | 1 |
math | 2. A TV station is going to broadcast a 30-episode TV series. If it is required that the number of episodes aired each day must be different, what is the maximum number of days the TV series can be broadcast?
---
The translation maintains the original format and line breaks as requested. | 7 | 61 | 1 |
math | 3. $n$ people $(n \geqslant 3)$ stand in a circle, where two specified individuals $A$ and $B$ definitely do not stand next to each other. How many such arrangements are there?
Arrange the above text in English, preserving the original text's line breaks and format, and output the translation result directly. | (n-3)(n-2)! | 72 | 8 |
math | 5. Given real numbers $x, y$ satisfy
$$
\frac{4}{x^{4}}-\frac{2}{x^{2}}=3, y^{4}+y^{2}=3 \text {. }
$$
Then the value of $\frac{4}{x^{4}}+y^{4}$ is $\qquad$
(2008, "Mathematics Weekly Cup" National Junior High School Mathematics Competition) | 7 | 93 | 1 |
math | 3. For which values of $a$, for any $b$ there exists at least one $c$, such that the system
$$
\left\{\begin{array}{l}
2 x+b y=a c^{2}+c \\
b x+2 y=c-1
\end{array}\right.
$$
has at least one solution? | \in[-1,0] | 75 | 7 |
math | 5. For any two points $P, Q$ on the ellipse $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1(a>b>0)$, if $O P \perp O Q$, then the minimum value of $|O P| \cdot|O Q|$ is $\qquad$. | \frac{2 a^{2} b^{2}}{a^{2}+b^{2}} | 76 | 22 |
math | 5. Given the sequence $\left\{a_{n}\right\}$ satisfies
$$
a_{1}=2, a_{n+1}=\frac{1+a_{n}}{1-a_{n}}\left(n \in \mathbf{N}_{+}\right) \text {. }
$$
Let $T_{n}=a_{1} a_{2} \cdots a_{n}$. Then $T_{2010}=$ $\qquad$ | -6 | 103 | 2 |
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