task_type stringclasses 4
values | problem stringlengths 21 5.23k | answer stringlengths 1 8.29k | problem_tokens int64 11 1.16k | answer_tokens int64 1 2.04k |
|---|---|---|---|---|
math | 4B. Let $P(x)=(2+3x)^{1994}$. Calculate the sum of all coefficients of the polynomial $P(x)$ that stand before the odd powers of $x$.
Note. The final results should be given in the shortest form, i.e., sums of a large number of addends should not appear. | \frac{5^{1994}-1}{2} | 72 | 14 |
math | 52. One fountain fills the pool in 2.5 hours, the other in 3.75 hours. How long will it take for both fountains to fill the pool? | 1.5 | 39 | 3 |
math | Solve the following system of equations:
$$
\frac{x}{a}=\frac{y}{b}=\frac{z}{c}, x+y+z=d
$$ | \frac{}{+b+},\quadb\frac{}{+b+},\quad\frac{}{+b+} | 36 | 29 |
math | 13. If the expansion of $(a+b)^{n}$ has three consecutive terms whose binomial coefficients form an arithmetic sequence, then the largest three-digit positive integer $n=$ | 959 | 37 | 3 |
math | How many positive integers $k$ are there such that \[\dfrac k{2013}(a+b)=lcm(a,b)\] has a solution in positive integers $(a,b)$? | 1006 | 42 | 4 |
math | Three lines are drawn parallel to each of the three sides of $\triangle ABC$ so that the three lines intersect in the interior of $ABC$. The resulting three smaller triangles have areas $1$, $4$, and $9$. Find the area of $\triangle ABC$.
[asy]
defaultpen(linewidth(0.7)); size(120);
pair relpt(pair P, pair Q, real a, ... | 36 | 279 | 2 |
math | Example 4 Let $f(x)=a x^{2}+b x+c$ have all coefficients positive, and $a+b+c=1$. For all positive arrays $x_{1}, x_{2}, \cdots, x_{n}$ satisfying: $x_{1} x_{2} \cdots x_{n}=1$, find the minimum value of $f\left(x_{1}\right) f\left(x_{2}\right) \cdots f\left(x_{n}\right)$. (All-Russian Mathematical Olympiad problem) | 1 | 117 | 1 |
math | 1. Given the sequence $\left\{a_{n}\right\}$ satisfies:
$$
a_{1}=1, a_{n}=5 a_{n-1}+3^{n-1}(n \geqslant 2) \text {. }
$$
Find $a_{n}$ and the highest power of 2 that divides $a_{k}\left(k=2^{2019}\right)$ | 2021 | 91 | 4 |
math | 3. How many nine-digit numbers written using nine different digits 1, 2, 3, 4, 5, 6, 7, 8, 9 are there such that each three-digit number formed by three consecutive digits of that number is divisible by 3? | 1296 | 60 | 4 |
math | Task 3. Find all pairs of prime numbers $(p, q)$ for which
$$
p\left(p^{2}-p-1\right)=q(2 q+3)
$$ | (13,31) | 41 | 7 |
math | 76(1010). A collective farm team must harvest a potato crop by a certain deadline. After 60% of the potatoes were harvested, a combine was sent to assist the team, which reduced the harvesting time by 5 days. How many days would it have taken to harvest the potatoes without the combine's help, if it is known that the c... | 20 | 90 | 2 |
math | 6. The brakes of a car allow it to stand on an inclined asphalt surface with an angle at the base of no more than $15^{\circ}$. Determine the minimum braking distance of this car when moving at a speed of 20 m/s on a flat horizontal road with the same surface. The acceleration due to gravity $g=10 m / s^{2}, \cos 15^{\... | 74.6 | 120 | 4 |
math | 11.5. Find the number of different arrangements in a row of all natural numbers from 1 to 10 such that the sum of any three consecutive numbers is divisible by 3. | 1728 | 40 | 4 |
math | Let $n$ be a positive integer. For any $k$, denote by $a_k$ the number of permutations of $\{1,2,\dots,n\}$ with exactly $k$ disjoint cycles. (For example, if $n=3$ then $a_2=3$ since $(1)(23)$, $(2)(31)$, $(3)(12)$ are the only such permutations.) Evaluate
\[ a_n n^n + a_{n-1} n^{n-1} + \dots + a_1 n. \][i]Proposed by... | \binom{2n-1}{n-1} n! | 129 | 15 |
math | The struggle of the land, by 1996, the forest coverage rate of the whole county had reached 30% (becoming an oasis). From 1997 onwards, each year will see such changes: 16% of the original desert area will be converted, and 4% of it will be eroded, turning back into desert.
(1) Assuming the total area of the county is ... | 5 | 201 | 1 |
math | Example 6. Expand the function in powers of the difference $z-3$
$$
f(z)=\frac{1}{3-2 z}
$$ | -\frac{1}{3}+\frac{2}{3^{2}}(z-3)-\frac{2^{2}}{3^{3}}(z-3)^{2}+\frac{2^{3}}{3^{4}}(z-3)^{3}-\cdots | 33 | 64 |
math | Determine the number of pairs $(x, y)$ of positive integers for which $0<x<y$ and $2 x+3 y=80$. | 5 | 32 | 1 |
math | 211. $\int x^{2} \sin x d x$ | \intx^{2}\sinx=-x^{2}\cosx+2x\sinx+2\cosx+C | 16 | 27 |
math | 6. Given positive integers $m, n$ satisfying $m+n=19$. Then the maximum number of solutions to the equation $\cos m x=\cos n x$ in the interval $[0, \pi]$ is | 18 | 47 | 2 |
math | Find all sequences $\left(a_0,a_1,a_2,a_3\right)$, where for each $k$, $0\leq k\leq3$, $a_k$ is the number of times that the number $k$ appears in the sequence $\left(a_0,a_1,a_2,a_3\right)$. | (1, 2, 1, 0) | 75 | 12 |
math | 9. (16 points) Let real numbers $x, y, z > 0$. Find the minimum value of the following two expressions:
(1) $\max \left\{x, \frac{1}{y}\right\} + \max \left\{y, \frac{2}{x}\right\}$;
(2) $\max \left\{x, \frac{1}{y}\right\} + \max \left\{y, \frac{2}{z}\right\} + \max \left\{z, \frac{3}{x}\right\}$. | 2\sqrt{5} | 131 | 6 |
math | ## SUBIECTUL I
a) Demonstrați că $\frac{n}{k(k+n)}=\frac{1}{k}-\frac{1}{k+n}$, pentru orice $\mathrm{k}$ și $\mathrm{n}$ numere naturale.
b) Determinați numărul natural nenul $n$ pentru care
$$
\frac{1}{56 \cdot 57}+\frac{2}{57 \cdot 59}+\frac{3}{59 \cdot 62}+\cdots+\frac{62}{n(n+62)}=\frac{2015}{115976}
$$
Suplime... | 2009 | 158 | 4 |
math | Find the derivative of the solution of the equation $\ddot{x} =x + A\dot{x}^2$, with initial conditions $x(0) = 1$, $\dot{x}(0) = 0$, with respect to the parameter $A$ for $A = 0$. | g(t) = -\frac{1}{3} e^t - \frac{1}{3} e^{-t} + \frac{1}{6} \cosh(2t) + \frac{1}{2} | 62 | 51 |
math | 2. Given: $b_{1}, b_{2}, b_{3}, b_{4}$ are positive integers, the polynomial $g(z)=(1-z)^{b_{1}}\left(1-z^{2}\right)^{b_{2}}\left(1-z^{3}\right)^{b_{3}}\left(1-z^{4}\right)^{b_{4}}$ when expanded and terms higher than 4th degree are omitted, becomes $1-2 z$. Also, $\alpha$ is the largest root of the polynomial $f(x)=x^... | 5 | 180 | 1 |
math | 141 Write an $n$-digit number using only the digits 1 and 2, where no two adjacent digits are both 1, and let the number of such $n$-digit numbers be $f(n)$. Then $f(10)=$ $\qquad$ . | 144 | 62 | 3 |
math | 5. In $1,2,3, \cdots, 10$ a number $a$ is randomly selected, and in $-1,-2,-3, \cdots,-10$ a number $b$ is randomly selected, then the probability that $a^{2}+b$ is divisible by 3 is $\qquad$ | \frac{37}{100} | 75 | 10 |
math | 1. A clown is riding a bicycle around the circus arena, where the radius of the circumference of the front wheel is half the radius of the circumference of the rear wheel. If the circumference of the front wheel were increased by a meter, and the circumference of the rear wheel were decreased by a meter, then over 40 m... | 1;2 | 94 | 3 |
math | Find all perfect squares $n$ such that if the positive integer $a \geqslant 15$ is some divisor of $n$ then $a+15$ is a prime power. | 1,4,9,16,49,64,196 | 43 | 18 |
math | 27.11. (Belgium, 77). Three shooters $A, B, C$ decided to duel simultaneously. They positioned themselves at the vertices of an equilateral triangle and agreed on the following: the first shot is taken by $A$, the second by $B$, the third by $C$, and so on in a circle; if one of the shooters is eliminated, the duel con... | 3 | 193 | 1 |
math | Problem 4. Point $O$ is the center of the circumcircle of triangle $ABC$ with sides $BC=5$ and $AB=4$. Find the length of side $AC$ if the length of the vector $3 \overrightarrow{OA}-4 \overrightarrow{OB}+\overrightarrow{OC}$ is 10. | 8 | 73 | 1 |
math | Example 1 Let the sequence $\left\{a_{n}\right\}$ satisfy
$$
a_{1}=a_{2}=1, a_{n}=\sqrt{3} a_{n-1}-a_{n-2}(n \geqslant 3) \text {. }
$$
Find $a_{2013}$. | 1-\sqrt{3} | 77 | 6 |
math | 5. (10 points) There is a basket of apples. If divided among Class A, each person gets 3 and there are 11 left; if divided among Class B, each person gets 4 and there are 10 left; if divided among Class C, each person gets 5 and there are 12 left. Then this basket of apples has at least $\qquad$.
将上面的文本翻译成英文，请保留源文本的换行... | 62 | 217 | 2 |
math | 1. Let $x>0$, planar vectors $\overrightarrow{A B}=(0,1), \overrightarrow{B C}=$ $(1,2), \overrightarrow{C D}=\left(\log _{3} x, \log _{9} x\right)$. If $\overrightarrow{A C} \cdot \overrightarrow{B D}=2$, then the value of $x$ is $\qquad$ . | \frac{1}{9} | 96 | 7 |
math | 6. (5 points) Using the digits $2$, $0$, $1$, and $5$, we can form $\qquad$ unique three-digit numbers where the digits do not repeat, and these three-digit numbers are multiples of 5. | 10 | 51 | 2 |
math | 2. Find all natural numbers $a, b, c, d, e$ such that
(1) $a \neq 2$,
(2) $a<b<c<d<e$
(3) $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}+\frac{1}{d}+\frac{1}{e}=1$. | 3,4,5,6,20 | 83 | 10 |
math | Let an integer $n > 1$ be given. In the space with orthogonal coordinate system $Oxyz$ we denote by $T$ the set of all points $(x, y, z)$ with $x, y, z$ are integers, satisfying the condition: $1 \leq x, y, z \leq n$. We paint all the points of $T$ in such a way that: if the point $A(x_0, y_0, z_0)$ is painted then p... | \left\lfloor \frac{3n^2 + 1}{4} \right\rfloor | 184 | 23 |
math | A4 Square $A B C D$ has side length $10 \mathrm{~cm}$. Points $W, X, Y$, and $Z$ are the midpoints of segments $A B, B C, C D$ and $W X$, respectively. Determine the area of quadrilateral $X C Y Z$ (in $\mathrm{cm}^{2}$ ). | 25 | 80 | 2 |
math | 8,9
What is the maximum number of numbers that can be chosen from the set $1,2, \ldots, 1963$, so that the sum of no two numbers is divisible by their difference? | 655 | 47 | 3 |
math | 13.139. Find a two-digit number, the quotient of which when divided by the product of its digits is $8 / 3$, and the difference between the number and the number written with the same digits but in reverse order is 18. | 64 | 55 | 2 |
math | $\left[\begin{array}{l}[\text { Modular arithmetic (other) })] \\ {[\text { Word problems (other). }]}\end{array}\right]$
In a store, there were 6 boxes weighing 15, 16, 18, 19, 20, and 31 kilograms, respectively. Two companies bought five boxes, with one company taking twice as much by weight as the other. Which box ... | 20 | 103 | 2 |
math | Two tangents $AT$ and $BT$ touch a circle at $A$ and $B$, respectively, and meet perpendicularly at $T$. $Q$ is on $AT$, $S$ is on $BT$, and $R$ is on the circle, so that $QRST$ is a rectangle with $QT = 8$ and $ST = 9$. Determine the radius of the circle. | 29 | 86 | 2 |
math | Given is a number with $1998$ digits which is divisible by $9$. Let $x$ be the sum of its digits, let $y$ be the sum of the digits of $x$, and $z$ the sum of the digits of $y$. Find $z$. | 9 | 61 | 1 |
math | Express in closed form the sum
$$
1 \cdot 2^{2} \cdot 3+2 \cdot 3^{2} \cdot 4+\ldots+(n-2)(n-1)^{2} n
$$
where $n$ is a natural number greater than three. | \frac{(n-2)(n-1)n(n+1)(2n-1)}{10} | 65 | 24 |
math | A triangle's sides are: $a=15 \mathrm{~cm}, b=20 \mathrm{~cm}, c=25 \mathrm{~cm}$. What is the length of the angle bisector $A A_{1}$? | 21.08 | 54 | 5 |
math | Find all pairs of positive integers $(a, b)$ such that $4b - 1$ is divisible by $3a + 1$ and $3a - 1$ is divisible by $2b + 1$. | (2, 2) | 48 | 6 |
math | $2 \cdot 41$ Given the set $\{1,2,3,4,5,6,7,8,9,10\}$. Find the number of subsets of this set that have the following property: each subset contains at least 2 elements, and the absolute difference between any two elements in each subset is greater than 1. | 133 | 76 | 3 |
math | 3.9 A crystal, during the formation stage, uniformly increases its mass. Observing the formation of two crystals, it was noticed that the first one gained the same mass increment over 3 months as the second one over 7 months. However, after a year, it turned out that the first crystal increased its initial mass by $4 \... | 35:12 | 95 | 5 |
math | ## 10. Distribution of Euros
Ana, Janko, and Tara have certain amounts of euros and want to redistribute them among themselves. First, Ana gives Janko and Tara a portion of her money so that after this, both Janko and Tara have twice as much money as they had before. Then, Janko gives Ana and Tara a portion of his mon... | 511 | 164 | 3 |
math | Example. Solve the equation
$$
\varphi(x)-\lambda \int_{0}^{\pi}\left(\cos ^{2} x \cos 2 t+\cos ^{3} t \cos 3 x\right) \varphi(t) d t=0
$$ | \begin{pmatrix}\varphi(x)=C\cos^{2}x,&\text{if}\lambda=\frac{4}{\pi}\\\varphi(x)=C\cos3x,&\text{if}\lambda=\frac{8}{\pi}\\\varphi(x)=0,&\text{if}\lambda\neq\frac{4}{\pi},\lambda\ne | 63 | 85 |
math | Three. (25 points) Given the quadratic trinomial $a x^{2}+b x+c$ $(a>0)$.
(1) When $c<0$, find the maximum value of the function
$$
y=-2\left|a x^{2}+b x+c\right|-1
$$
(2) For any real number $k$, the line $y=k(x-$
1) $-\frac{k^{2}}{4}$ intersects the parabola $y=a x^{2}+b x+c$ at exactly one point, find the value of ... | 0 | 132 | 1 |
math | 5. Let the function $f(x)=x-\ln (a x+2 a+1)+2$. If for any $x \geqslant-2$, $f(x) \geqslant 0$ always holds, then the range of the real number $a$ is $\qquad$ . | [0,1] | 67 | 5 |
math | What is the largest integer $n$ such that $n$ is divisible by every integer less than $\sqrt[3]{n}$? | 420 | 28 | 3 |
math | ## Task Condition
Find the derivative.
$$
y=\frac{5^{x}(\sin 3 x \cdot \ln 5-3 \cos 3 x)}{9+\ln ^{2} 5}
$$ | 5^{x}\cdot\sin3x | 49 | 9 |
math | 11. Sides $A B$ and $C D$ of quadrilateral $A B C D$ are perpendicular and are diameters of two equal tangent circles with radius $r$. Find the area of quadrilateral $A B C D$, if $\frac{|B C|}{|A D|}=k$. | 3r^{2}|\frac{1-k^{2}}{1+k^{2}}| | 66 | 20 |
math | (9) (16 points) Let the sequence $\left\{a_{n}\right\}$ satisfy $a_{1}=3, a_{2}=8, a_{n+2}=2 a_{n+1}+2 a_{n}$, $n \in \mathbf{N}^{*}$, find the general term formula of the sequence $\left\{a_{n}\right\}$. | a_{n}=\frac{2+\sqrt{3}}{2\sqrt{3}}(1+\sqrt{3})^{n}+\frac{\sqrt{3}-2}{2\sqrt{3}}(1-\sqrt{3})^{n} | 90 | 55 |
math | Points $C\neq D$ lie on the same side of line $AB$ so that $\triangle ABC$ and $\triangle BAD$ are congruent with $AB = 9$, $BC=AD=10$, and $CA=DB=17$. The intersection of these two triangular regions has area $\tfrac mn$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$. | 59 | 90 | 2 |
math | 23. From an urn containing $N$ balls, of which $n$ are white and $N-n$ are black, a selection without replacement is made ( $n \geqslant 2$ ). Consider ordered samples of size $M$. What is the probability of events $A_{i, j}$ and $A_{i, j, k}$, where $A_{i, j}$ is the event that white balls are in the $i$-th and $j$-th... | \frac{n(n-1)(n-2)}{N(N-1)(N-2)} | 157 | 21 |
math | A natural number $n$ is called [i]perfect [/i] if it is equal to the sum of all its natural divisors other than $n$. For example, the number $6$ is perfect because $6 = 1 + 2 + 3$. Find all even perfect numbers that can be given as the sum of two cubes positive integers.
| 28 | 74 | 4 |
math | Example 3. The line $\left\{\begin{array}{c}x=2+\frac{1}{2} t \\ y=1+\frac{\sqrt{3}}{2} t\end{array}\right.$ intersects the parabola $y^{2}=4 x$ to form a chord. Find the length of the chord. | \frac{8}{3} \sqrt{7-\sqrt{3}} | 74 | 16 |
math | \[
\left.\begin{array}{c}
\text { Division with remainder } \\
{[\text { Arithmetic of remainders (other).] }}
\end{array}\right]
\]
a) For which integers $n$ is the number $5 n^{2}+10 n+8$ divisible by 3?
b) And for which integers is it divisible by 4? | 2+3k | 83 | 4 |
math | To determine the volume of a rectangular parallelepiped, whose edges are the distances from the orthocenter of a triangle to its vertices, given that the circumradius of the triangle is $r=2.35$ and the angles are: $\alpha=63^{\circ} 18^{\prime} 13^{\prime \prime}, \beta=$ $51^{\circ} 42^{\prime} 19^{\prime \prime}$. | 12.2\,^3 | 102 | 8 |
math | 11. (14 points) Given the function
$$
f(x)=2 \sin ^{2}\left(\frac{\pi}{4}-x\right)-\sqrt{3} \cos 2 x \text {. }
$$
(1) Find the smallest positive period and the intervals of monotonic decrease of $f(x)$;
(2) If $f(x)<m+2$ holds for all $x \in\left[0, \frac{\pi}{6}\right]$, find the range of real numbers $m$. | (-1-\sqrt{3},+\infty) | 116 | 11 |
math | Subject (1).
a) Solve the equation $x^{4}-y^{4}=16$ in the set of integers.
b) Find the largest natural number $n$ for which $\frac{n^{2}-2012}{n+1} \in \mathbb{Z}$.
c) Find the real numbers $a$ and $b$ if $\max \left\{a^{2}-b+1, b^{2}+a+1\right\} \leq \frac{3}{4}$.
Aurelia Caţaros \& Gabriela Ruse, Călăraşi | -\frac{1}{2},\frac{1}{2} | 134 | 14 |
math | 3. (3 points) Solve the system of equations: $\left\{\begin{array}{l}x y=6(x+y) \\ x z=4(x+z) \\ y z=2(y+z)\end{array}\right.$ | 0 | 51 | 1 |
math | ## Task 4 - 300614
The pages of a book are numbered from 1 to 235.
a) How many times was the digit 4 used in total for the numbering?
b) How many times was the digit 0 used in total for the numbering?
c) How many digits in total need to be printed for this numbering? | 44,43,597 | 78 | 9 |
math | ## Problem Statement
Find the coordinates of point $A$, which is equidistant from points $B$ and $C$.
$A(0 ; y ; 0)$
$B(0 ;-2 ; 4)$
$C(-4 ; 0 ; 4)$ | A(0;3;0) | 60 | 8 |
math | 25th ASU 1991 Problem 1 Find all integers a, b, c, d such that ab - 2cd = 3, ac + bd = 1. | (,b,,)=(1,3,1,0),(-1,-3,-1,0),(3,1,0,1),(-3,-1,0,-1) | 41 | 39 |
math | Example 7 Find the range of real number $a$ for which the equation $\sqrt{a+\sqrt{a+\sin x}}=\sin x$ has real solutions.
Translate the above text into English, please keep the original text's line breaks and format, and output the translation result directly. | [-\frac{1}{4},0] | 61 | 10 |
math | Let $S$ be the set of all ordered triples $\left(a,b,c\right)$ of positive integers such that $\left(b-c\right)^2+\left(c-a\right)^2+\left(a-b\right)^2=2018$ and $a+b+c\leq M$ for some positive integer $M$. Given that $\displaystyle\sum_{\left(a,b,c\right)\in S}a=k$, what is \[\displaystyle\sum_{\left(a,b,c\right)\in S... | 1009k | 152 | 5 |
math | 1. If $x+y+x y=1$, where $x, y$ are nonzero real numbers, find the value of
$$
x y+\frac{1}{x y}-\frac{y}{x}-\frac{x}{y} .
$$ | 4 | 54 | 1 |
math | $$
\begin{gathered}
\sqrt{2+\sqrt{3}}, \sqrt{2+\sqrt{2+\sqrt{3}}}, \sqrt{2+\sqrt{2+\sqrt{2+\sqrt{3}}}} \\
\sqrt{2-\sqrt{2+\sqrt{2+\sqrt{3}}}}
\end{gathered}
$$
Multiply the following nested radicals together:
$$
\begin{gathered}
\sqrt{2+\sqrt{3}}, \sqrt{2+\sqrt{2+\sqrt{3}}}, \sqrt{2+\sqrt{2+\sqrt{2+\sqrt{3}}}} \\
\sq... | 1 | 158 | 1 |
math | In a semicircle of radius $R$ with center at point $O$, a square $A B C D$ is inscribed such that points $A$ and $D$ lie on the diameter, and points $B$ and $C$ lie on the circumference. Find the radius of the circle inscribed in triangle $O B C$.
# | \frac{R(5-\sqrt{5})}{10} | 74 | 15 |
math | 2.92 The largest even number that cannot be expressed as the sum of two odd composite numbers is?
Will the text be translated into English, keeping the original text's line breaks and format, and output the translation result directly.
Note: The note above is not part of the translation but is provided to clarify th... | 38 | 71 | 2 |
math | João calculated the product of the non zero digits of each integer from $1$ to $10^{2009}$ and then he summed these $10^{2009}$ products. Which number did he obtain? | 46^{2009} | 49 | 8 |
math | G10.4 $A$ number $X$ consists of 2 digits whose product is 24 . By reversing the digits, the new number formed is 18 greater than the original one. What is $X$ ? (Reference: 1991 FG6.1-2) | 46 | 64 | 2 |
math | Ana & Bruno decide to play a game with the following rules.:
a) Ana has cards $1, 3, 5,7,..., 2n-1$
b) Bruno has cards $2, 4,6, 8,...,2n$
During the first turn and all odd turns afterwards, Bruno chooses one of his cards first and reveals it to Ana, and Ana chooses one of her cards second. Whoever's card is higher gain... | \left\lfloor \frac{n}{2} \right\rfloor | 193 | 17 |
math | Determine all values of $x$ for which $\left(2 \cdot 4^{x^{2}-3 x}\right)^{2}=2^{x-1}$. | \frac{1}{4},3 | 38 | 8 |
math | $$
\begin{array}{l}
\text { 6. Given the quadratic equation in } x \text { is } \\
x^{2}+a x+(m+1)(m+2)=0
\end{array}
$$
For any real number $a$, the equation has real roots. Then the range of real number $m$ is . $\qquad$ | -2 \leqslant m \leqslant -1 | 81 | 15 |
math | Find all non-negative integers $x, y$ such that:
$$
2^{x}=y^{2}+y+1
$$ | 0 | 29 | 1 |
math | Example 7. Let $a, b, m$ be integers, if $a$ and $m$ are coprime, find the value of the sum
$$
\begin{array}{l}
\left\{\frac{b}{m}\right\}+\left\{\frac{a+b}{m}\right\}+\left\{\frac{2 a+b}{m}\right\}+\cdots \\
+\left\{\frac{(m-1)}{m} \frac{a+b}{m}\right\} \text{. }
\end{array}
$$ | \frac{m-1}{2} | 125 | 9 |
math | 3. In the first quarter, Masha and Lena together received 23 fives, Svetlana and Masha together received 18 fives, and Svetlana and Lena together received 15 fives. How many fives did each of the girls receive? | Masha-13,Lena-10,Svetlana-5 | 60 | 16 |
math | 11. (20 points) In the Cartesian coordinate system $x O y$, the curve $C_{1}: y^{2}=4 x$,
curve $C_{2}:(x-4)^{2}+y^{2}=8$, a line $l$ with an inclination angle of $45^{\circ}$ is drawn through a point $P$ on curve $C_{1}$, intersecting curve $C_{2}$ at two different points $Q$ and $R$. Find the range of $|P Q||P R|$. | [4,8)\cup(8,200) | 119 | 13 |
math | 4.9. Form the equation of the plane passing through the line $(x-1) / 2=(y-3) / 4=z /(-1)$ and the point $P_{1}(1,5,2)$. | 5x-2y+2z+1=0 | 50 | 12 |
math | 3. Neighbor Ana raises chickens. Four years ago, she had 32 chickens. The number of chickens increased by $25 \%$ each of the first two years, and then decreased by $20 \%$ each of the next two years. How many chickens does neighbor Ana have this year? | 32 | 62 | 2 |
math | a) Determine the parameters $p, q$ such that the expressions
$$
(x+3)^{2}+(7 x+p)^{2} \text { and }(3 x+5)^{2}+(p x+q)^{2}
$$
are equal to the square of a first-degree polynomial.
b) Let $a, b, c, A, B, C$ be given, non-zero numbers such that the polynomials
$$
(a x+b)^{2}+(A x+B)^{2} \text { and }(b x+c)^{2}+(B x+... | p=21,q=35 | 173 | 8 |
math | ## Problem Statement
Calculate the limit of the function:
$$
\lim _{x \rightarrow u} \frac{2^{x}-1}{\ln (1+2 x)}
$$ | \frac{\ln2}{2} | 40 | 8 |
math | 5. Each of the three Greeks brought the same number of wreaths. Meeting nine Muses, they divided the wreaths in such a way that each Greek and each Muse had the same number of wreaths. How many wreaths did each Greek have at first? | 4m | 54 | 2 |
math | For what value of $\lambda$ does the equation
$$
\lambda x^{2}+4 x y+y^{2}-4 x-2 y-3=0
$$
represent a pair of lines? | 4 | 45 | 1 |
math | 7.5. 16 travelers, each of whom is either a liar or a knight (liars always lie, knights always tell the truth), settled into 3 rooms of a hotel. When everyone gathered in their rooms, Basil, who was staying in the first room, said: "There are more liars than knights in this room right now. Although no - there are more ... | 9 | 138 | 1 |
math | There are $2n$ different numbers in a row. By one move we can interchange any two numbers or interchange any $3$ numbers cyclically (choose $a,b,c$ and place $a$ instead of $b$, $b$ instead of $c$, $c$ instead of $a$). What is the minimal number of moves that is always sufficient to arrange the numbers in increasing or... | n | 83 | 1 |
math | Let $ \mathcal P$ be a parabola, and let $ V_1$ and $ F_1$ be its vertex and focus, respectively. Let $ A$ and $ B$ be points on $ \mathcal P$ so that $ \angle AV_1 B \equal{} 90^\circ$. Let $ \mathcal Q$ be the locus of the midpoint of $ AB$. It turns out that $ \mathcal Q$ is also a parabola, and let $ V_2$ and $ F_2... | \frac{7}{8} | 141 | 8 |
math |
Problem 1. Find all triples $(a, b, c)$ of real numbers such that the following system holds:
$$
\left\{\begin{array}{l}
a+b+c=\frac{1}{a}+\frac{1}{b}+\frac{1}{c} \\
a^{2}+b^{2}+c^{2}=\frac{1}{a^{2}}+\frac{1}{b^{2}}+\frac{1}{c^{2}}
\end{array}\right.
$$
| (,b,)=(,\frac{1}{},1),(,\frac{1}{},-1) | 112 | 22 |
math | $1 \cdot 2$ If an ordered pair of non-negative integers $(m, n)$ does not require a carry when adding $m+n$ (in decimal), then it is called "simple". Find the number of all simple ordered pairs of non-negative integers whose sum is 1492. | 300 | 63 | 3 |
math | $3 \cdot 1$ If the numbers $p, p+10, p+14$ are all prime, find $p$.
(Kiev Mathematical Olympiad, 1981) | 3 | 44 | 1 |
math | In a triangle $ABC$, the median $AD$ divides $\angle{BAC}$ in the ratio $1:2$. Extend $AD$ to $E$ such that $EB$ is perpendicular $AB$. Given that $BE=3,BA=4$, find the integer nearest to $BC^2$. | 29 | 65 | 2 |
math | Example 14 Find all $f: \mathbf{N} \rightarrow \mathbf{N}$, such that for any $n \in \mathbf{N}$, we have
$$
\frac{1}{f(1) f(2)}+\frac{1}{f(2) f(3)}+\cdots+\frac{1}{f(n) f(n+1)}=\frac{f(f(n))}{f(n+1)}
$$ | f(n)=n,n\in{N} | 100 | 10 |
math | Find the real numbers $x$ and $y$ such that
$$
\left\{\begin{array}{l}
x+y=1 \\
x^{3}+y^{3}=19
\end{array}\right.
$$ | (3,-2)(-2,3) | 51 | 10 |
math | 7. Let
$$
f(x)=\ln x-\frac{1}{2} a x^{2}-2 x(a \in[-1,0)) \text {, }
$$
and $f(x)<b$ holds for all $x$ in the interval $(0,1]$. Then the range of the real number $b$ is $\qquad$ . | (-\frac{3}{2},+\infty) | 79 | 12 |
math | If
$$
x=\left(1-\frac{1}{12}\right)\left(1-\frac{1}{11}\right)\left(1-\frac{1}{10}\right)\left(1-\frac{1}{9}\right)\left(1-\frac{1}{8}\right)\left(1-\frac{1}{7}\right)\left(1-\frac{1}{6}\right)\left(1-\frac{1}{5}\right)\left(1-\frac{1}{4}\right)\left(1-\frac{1}{3}\right)\left(1-\frac{1}{2}\right)
$$
what is the value... | \frac{1}{12} | 152 | 8 |
math | 14. A line passing through the left focus $F_{1}$ of a hyperbola and perpendicular to the real axis of the hyperbola intersects the hyperbola at points $A$ and $B$. If there exists a point $C$ on the line of the imaginary axis of the hyperbola such that $\angle A C B=90^{\circ}$, then the range of the eccentricity $e$ ... | e \geqslant \frac{\sqrt{5}+1}{2} | 101 | 18 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.