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 | An equilateral triangle $ABC$ shares a side with a square $BCDE$. If the resulting pentagon has a perimeter of $20$, what is the area of the pentagon? (The triangle and square do not overlap). | 16 + 4\sqrt{3} | 48 | 10 |
math | $\left[\begin{array}{l}{[\text { Integer and fractional parts. Archimedes' principle ] }} \\ {[\quad \underline{\text { equations in integers }} \underline{\text { ] }}}\end{array}\right]$
How many solutions in natural numbers does the equation $\left[{ }^{x} / 10\right]=\left[{ }^{x} / 11\right]+1$ have? | 110 | 93 | 3 |
math | 1. Given the sequence $a_{0}, a_{1}, a_{2}, \cdots, a_{n}, \cdots$ satisfies the relation $\left(2-a_{n+1}\right)\left(4+a_{n}\right)=8$, and $a_{0}=2$, then $\sum_{i=0}^{n} \frac{1}{a_{i}}=$
$\qquad$ . $\qquad$ | \frac{1}{2}(2^{n+2}-n-3) | 95 | 17 |
math | 2. 2.9 * The range of negative values of $a$ that make the inequality $\sin ^{2} x+a \cos x+a^{2} \geqslant 1+\cos x$ hold for all $x \in \mathbf{R}$ is $\qquad$ . | \leqslant-2 | 65 | 7 |
math | Task 2. (10 points) Find the greatest value of the parameter $b$ for which the inequality $b \sqrt{b}\left(x^{2}-10 x+25\right)+\frac{\sqrt{b}}{\left(x^{2}-10 x+25\right)} \leq \frac{1}{5} \cdot \sqrt[4]{b^{3}} \cdot\left|\sin \frac{\pi x}{10}\right|$ has at least one solution. | 0.0001 | 110 | 6 |
math | 8-1. A beginner gardener planted daisies, buttercups, and marguerites on their plot. When they sprouted, it turned out that there were 5 times more daisies than non-daisies, and 5 times fewer buttercups than non-buttercups. What fraction of the sprouted plants are marguerites? | 0 | 76 | 1 |
math | 10.1. Different positive numbers $x, y, z$ satisfy the equations
$$
x y z=1 \quad \text{and} \quad x+y+z=\frac{1}{x}+\frac{1}{y}+\frac{1}{z}
$$
Find the median of them. Justify your answer. | 1 | 72 | 1 |
math | The 38th question, the sequence $\left\{x_{n}\right\}$ satisfies $x_{1}=1$, and for any $n \in Z^{+}$, it holds that $x_{n+1}=x_{n}+3 \sqrt{x_{n}}+\frac{n}{\sqrt{x_{n}}}$, try to find the value of $\lim _{n \rightarrow+\infty} \frac{n^{2}}{x_{n}}$. | \frac{4}{9} | 103 | 7 |
math | 1. Given $a>b>2$, and $a+b$, $a-b$, $a b$, $\frac{b}{a}$ form a geometric sequence in a certain order. Then $a=$ $\qquad$ | 7+5\sqrt{2} | 46 | 8 |
math | 663. Find all natural $n$ for which the number $n^{2}+3 n$ is a perfect square. | 1 | 28 | 1 |
math | Which pairs of numbers $(c ; d)$ are such that the equation $x^{3}-8 x^{2}+c x+d=0$ has three, not necessarily distinct, positive integer roots? | (13,-6),(17,-10),(19,-12),(20,-16),(21,-18) | 42 | 30 |
math | We want to color the three-element parts of $\{1,2,3,4,5,6,7\}$, such that if two of these parts have no element in common, then they must be of different colors. What is the minimum number of colors needed to achieve this goal? | 3 | 61 | 1 |
math | 4. A fair die is rolled twice in a row. What is the probability that the sum of the two numbers obtained is a prime number? | \frac{5}{12} | 29 | 8 |
math | 16. Given the sequence $\left\{a_{n}\right\}$ satisfies the condition $(n-1) a_{n+1}=(n+1)\left(a_{n}-1\right), a_{2}=6$, let $b_{n}=$ $a_{n}+n, n \in \mathbf{N}_{+}$.
（1）Write out the first 4 terms of the sequence $\left\{a_{n}\right\}$;（2）Find $b_{n}$;（3）Does there exist non-zero constants $p, q$, such that the seque... | p+2q=0 | 181 | 6 |
math | On the diagonals $A C$ and $B D$ of trapezoid $A B C D$ with bases $B C=a$ and $A D=b$, points $K$ and $L$ are located respectively, such that
$C K: K A=B L: L D=7: 4$. Find $K L$.
# | \frac{1}{11}|7b-4a| | 76 | 14 |
math | . How many zeros are there in the number:
$12345678910111213141516171819202122 \ldots 20062007$ | 506 | 59 | 3 |
math | ## Task 3 - 040623
A rectangular school garden is to be fenced. On each of the shorter sides, which are each 40 m long, there are 21 concrete posts, and on the longer sides, there are 15 more each. The distance between any two adjacent posts is the same. A gate will be installed between two of these posts.
What are t... | 3345 | 138 | 4 |
math | 16. Let the complex number $z=x+i y(x, y \in \mathrm{R})$ satisfy $|z-i| \leqslant 1$, find the maximum and minimum values of $A=x\left(|z-i|^{2}-1\right)$ and the corresponding $z$ values. | -\frac{2\sqrt{3}}{9}\leqslantA\leqslant\frac{2\sqrt{3}}{9} | 68 | 34 |
math | 808. Solve the equation in integers
$$
(x+y)^{2}=x-y
$$ | [\frac{(1+)}{2};\frac{(1-)}{2}),\in\mathrm{Z} | 22 | 25 |
math | The numbers $a$ and $b$ are positive integers that satisfy $96 a^{2}=b^{3}$. What is the smallest possible value of $a$? | 12 | 37 | 2 |
math | 5. Let $a$ and $d$ be two positive integers. Construct a "triangle" similar to Pascal's triangle as follows:
$$
\begin{array}{l}
a \\
a+d \quad 2a \quad a+d \\
a+2d \quad 3a+d \quad 3a+d \quad a+2d \\
a+3d \quad 4a+3d \quad 6a+2d \quad 4a+3d \quad a+3d \\
\text{...... } \\
a+(n-1)d \\
\text{...... } \\
a+(n-1)d \\
\end... | n=6,=2,=30 | 334 | 10 |
math | 40. Compute
$$
\sum_{k=1}^{\infty} \frac{3 k+1}{2 k^{3}+k^{2}} \cdot(-1)^{k+1}
$$ | \frac{\pi^{2}}{12}+\frac{\pi}{2}-2+\ln2 | 48 | 22 |
math | Let $\mathcal{P}$ be the set of vectors defined by
\[\mathcal{P} = \left\{\begin{pmatrix} a \\ b \end{pmatrix} \, \middle\vert \, 0 \le a \le 2, 0 \le b \le 100, \, \text{and} \, a, b \in \mathbb{Z}\right\}.\]
Find all $\mathbf{v} \in \mathcal{P}$ such that the set $\mathcal{P}\setminus\{\mathbf{v}\}$ obtained by omitt... | \left\{\begin{pmatrix} 1 \\ 2n \end{pmatrix} \, \middle\vert \, n \in \{0, 1, 2, \ldots, 50\}\right\} | 164 | 55 |
math | 2.24. An isosceles trapezoid with bases 2 and 3 cm and an acute angle of $60^{\circ}$ rotates around the smaller base. Calculate the surface area and volume of the resulting solid of revolution. | 4\pi\sqrt{3} | 53 | 8 |
math | In an obtuse triangle, the largest side is 4, and the smallest side is 2. Can the area of the triangle be greater than $2 \sqrt{3}$? | No | 38 | 1 |
math | 6th Mexico 1992 Problem A2 Given a prime number p, how many 4-tuples (a, b, c, d) of positive integers with 0 < a, b, c, d < p-1 satisfy ad = bc mod p? | (p-2)(p^2-5p+7) | 57 | 13 |
math | 1. Let $f(x)$ be a function defined on $\mathbf{R}$, for any real number $x$ we have $f(x+3) f(x-4)=-1$. When $0 \leqslant x<7$, $f(x)=\log _{2}(9-x)$. Then the value of $f(-100)$ is $\qquad$ | -\frac{1}{2} | 84 | 7 |
math | Example 10. Given $a^{2}+2 a-5=0, b^{2}+2 b-5=0$, and $a \neq b$. Then the value of $a b^{2}+a^{2} b$ is what?
(1989, Sichuan Province Junior High School Mathematics Competition) | 10 | 75 | 2 |
math | 11.159. The base of the pyramid is a parallelogram, the adjacent sides of which are 9 and $10 \mathrm{~cm}$, and one of the diagonals is $11 \mathrm{~cm}$. The opposite lateral edges are equal, and the length of each of the larger edges is 10.5 cm. Calculate the volume of the pyramid. | 200~^3 | 85 | 6 |
math | Example 10 Find all functions $f: \mathbf{Q} \rightarrow \mathbf{Q}$, satisfying the condition $f[x+f(y)]=f(x) \cdot f(y)(x, y \in \mathbf{Q})$.
| f(x)=0orf(x)=1 | 56 | 8 |
math | 1. A1 (GBR 3) ${ }^{1 \mathrm{MO}(}$ The function $f(n)$ is defined for all positive integers $n$ and takes on nonnegative integer values. Also, for all $m, n$,
$$
\begin{array}{c}
f(m+n)-f(m)-f(n)=0 \text { or } 1 ; \\
f(2)=0, \quad f(3)>0, \text { and } \quad f(9999)=3333 .
\end{array}
$$
Determine $f(1982)$. | 660 | 133 | 3 |
math | Given is a $n \times n$ chessboard. With the same probability, we put six pawns on its six cells. Let $p_n$ denotes the probability that there exists a row or a column containing at least two pawns. Find $\lim_{n \to \infty} np_n$. | 30 | 64 | 2 |
math | 7. To color 8 small squares on a $4 \times 4$ chessboard black, such that each row and each column has exactly two black squares, there are $\qquad$ different ways (answer with a number). | 90 | 48 | 2 |
math | 4. Given the function $f(x)=\frac{2 x^{3}-6 x^{2}+13 x+10}{2 x^{2}-9 x}$
Determine all positive integers $x$ for which $f(x)$ is an integer. | 5,10 | 57 | 4 |
math | IMO 1965 Problem A3 The tetrahedron ABCD is divided into two parts by a plane parallel to AB and CD. The distance of the plane from AB is k times its distance from CD. Find the ratio of the volumes of the two parts. | \frac{k^2(k+3)}{3k+1} | 56 | 15 |
math | 3. Find all natural numbers $n$ for which the number $2^{10}+2^{13}+2^{14}+3 \cdot 2^{n}$ is a square of a natural number.
$(16$ points) | 13,15 | 54 | 5 |
math | 6. (5 points) Find all pairs of integers $(x, y)$ for which the following equality holds:
$$
x(x+1)(x+7)(x+8)=y^{2}
$$
# | (1;12),(1;-12),(-9;12),(-9;-12),(0;0),(-8;0),(-4;-12),(-4;12),(-1;0),(-7;0) | 45 | 54 |
math | Let $\triangle XOY$ be a right-angled triangle with $\angle XOY=90^\circ$. Let $M$ and $N$ be the midpoints of legs $OX$ and $OY$, respectively. Find the length $XY$ given that $XN=22$ and $YM=31$. | 34 | 71 | 2 |
math | Kanel-Belov A.Y. A cube with a side of 20 is divided into 8000 unit cubes, and a number is written in each cube. It is known that in each column of 20 cubes, parallel to the edge of the cube, the sum of the numbers is 1 (columns in all three directions are considered). In some cube, the number 10 is written. Through th... | 333 | 126 | 3 |
math | Solve the following equation:
$$
\frac{3}{(x+2)(x-1)}=\frac{1}{x(x-1)^{2}}+\frac{3}{x(x-3)}
$$ | \frac{19}{13} | 46 | 9 |
math | 8.040. $\operatorname{ctg}\left(\frac{3 \pi}{2}-x\right)-\operatorname{ctg}^{2} x+\frac{1+\cos 2 x}{\sin ^{2} x}=0$. | \frac{\pi}{4}(4k+3),\quadk\inZ | 58 | 18 |
math | ## Task 14/84
For which natural numbers $n$ does no polyhedron exist with exactly $n$ edges? | Forthenatural\(n<6\)\(n=7\),theredoesnotexistpolyhedronwithexactly\(n\)edges | 29 | 31 |
math | The polynomial $x^3-kx^2+20x-15$ has $3$ roots, one of which is known to be $3$. Compute the greatest possible sum of the other two roots.
[i]2015 CCA Math Bonanza Lightning Round #2.4[/i] | 5 | 65 | 1 |
math | 4. 212 The river water is flowing, entering a still lake at point $Q$, a swimmer travels downstream from $P$ to $Q$, then across the lake to $R$, taking a total of 3 hours. If he travels from $R$ to $Q$ and then to $P$, it takes a total of 6 hours. If the lake water also flows, at the same speed as the river, then trav... | \frac{15}{2} | 144 | 8 |
math | 2. A fraction in its simplest form was written on the board. Petya decreased its numerator by 1 and its denominator by 2. Vasya, on the other hand, increased the numerator by 1 and left the denominator unchanged. It turned out that the boys ended up with the same value. What exactly could their result be? | 1 | 71 | 1 |
math | 19. Let $p=a^{b}+b^{a}$. If $a, b$ and $p$ are all prime, what is the value of $p$ ? | 17 | 39 | 2 |
math | If $3 k=10$, what is the value of $\frac{6}{5} k-2$ ? | 2 | 25 | 1 |
math | What integers $a, b, c$ satisfy $a^2 + b^2 + c^2 + 3 < ab + 3b + 2c$ ? | (1, 2, 1) | 37 | 10 |
math | Determine all positive integers $k$, $\ell$, $m$ and $n$, such that $$\frac{1}{k!}+\frac{1}{\ell!}+\frac{1}{m!} =\frac{1}{n!} $$ | (k, \ell, m, n) = (3, 3, 3, 2) | 56 | 22 |
math | 7. Let $A=\{2,4, \cdots, 2014\}, B$ be any non-empty subset of $A$, and $a_{i} 、 a_{j}$ be any two elements in set $B$. There is exactly one isosceles triangle with $a_{i} 、 a_{j}$ as side lengths. Then the maximum number of elements in set $B$ is $\qquad$ | 10 | 95 | 2 |
math | 7. Given a sphere with a radius of 3. Then the maximum volume of the inscribed regular tetrahedron is $\qquad$ .
| 8\sqrt{3} | 31 | 6 |
math | Find all strictly increasing functions $f : \mathbb{N} \to \mathbb{N} $ such that $\frac {f(x) + f(y)}{1 + f(x + y)}$ is a non-zero natural number, for all $x, y\in\mathbb{N}$. | f(x) = ax + 1 | 66 | 9 |
math | Example 8. For a regular $n$-sided polygon $(n \geqslant 5)$ centered at point $O$, let two adjacent points be $A, B$. $\triangle X Y Z$ is congruent to $\triangle O A B$. Initially, let $\triangle X Y Z$ coincide with $\triangle O A B$, then move $\triangle X Y Z$ on the plane such that points $Y$ and $Z$ both move al... | d=\frac{a\left(1-\cos \frac{\pi}{n}\right)}{\sin ^{2} \frac{\pi}{n}} | 131 | 33 |
math | 6. Variant 1. In the kindergarten, 5 children eat porridge every day, 7 children eat porridge every other day, and the rest never eat porridge. Yesterday, 9 children ate porridge. How many children will eat porridge today? | 8 | 55 | 1 |
math | 6. Given the quadratic function $y=x^{2}-\frac{2 n+1}{n(n+2)} x+\frac{n+1}{n(n+2)^{2}}$ which intercepts a segment of length $d_{n}$ on the $x$-axis, then $\sum_{n=1}^{100} d_{n}=$ $\qquad$ . | \frac{7625}{10302} | 84 | 14 |
math | Find all positive integers $n$, for which the numbers in the set $S=\{1,2, \ldots, n\}$ can be colored red and blue, with the following condition being satisfied: the set $S \times S \times S$ contains exactly 2007 ordered triples $(x, y, z)$ such that (i) $x, y, z$ are of the same color and (ii) $x+y+z$ is divisible b... | 69 \text{ and } 84 | 123 | 10 |
math | 4. In a math test, the average score of a class is 78 points, and the average scores of boys and girls are 75.5 points and 81 points, respectively. What is the ratio of the number of boys to girls in the class? | 6:5 | 57 | 3 |
math | We will flip a symmetric coin $n$ times. Suppose heads appear $m$ times. The number $\frac{m}{n}$ is called the frequency of heads. The number $m / n - 0.5$ is called the deviation of the frequency from the probability, and the number $|m / n - 0.5|$ is called the absolute deviation. Note that the deviation and the abs... | In\\series\of\10\tosses | 184 | 11 |
math | 9. In $\triangle A B C, \angle B=60^{\circ}, \angle C=90^{\circ}$ and $A B=1$. $B C P, C A Q$ and $A B R$ are equilateral triangles outside $\triangle A B C$. $Q R$ meets $A B$ at $T$. Find the area of $\triangle P R T$.
(1 mark)
在 $\triangle A B C$ 中, $\angle B=60^{\circ} 、 \angle C=90^{\circ} 、 A B=1 \circ B C P 、 C ... | \frac{9\sqrt{3}}{32} | 188 | 13 |
math | A point $P$ is chosen in an arbitrary triangle. Three lines are drawn through $P$ which are parallel to the sides of the triangle. The lines divide the triangle into three smaller triangles and three parallelograms. Let $f$ be the ratio between the total area of the three smaller triangles and the area of the given tri... | f \geq \frac{1}{3} | 103 | 12 |
math | \section*{Problem 2 - 131212}
From a straight circular frustum, a cone is to be cut out whose apex is the center of the (larger) base of the frustum and whose base coincides with the top face of the frustum.
Determine the values of the ratio of the radius of the base to the radius of the top face of the frustum, for ... | \frac{r_{1}}{r_{2}}=2 | 106 | 14 |
math | 1. Find the smallest natural number $N$ such that $N+2$ is divisible (without remainder) by 2, $N+3$ by 3, ..., $N+10$ by 10.
ANSWER: 2520. | 2520 | 57 | 4 |
math | 4 Try to find a set of positive integer solutions for the equation $x^{2}-51 y^{2}=1$ | x=50, y=7 | 26 | 8 |
math | 2. Positive numbers $a, b, c$ are such that $a+b+c=3$. Find the minimum value of the expression
$$
A=\frac{a^{3}+b^{3}}{8 a b+9-c^{2}}+\frac{b^{3}+c^{3}}{8 b c+9-a^{2}}+\frac{c^{3}+a^{3}}{8 c a+9-b^{2}}
$$ | \frac{3}{8} | 99 | 7 |
math | 10,11
Three spheres of radius $R$ touch each other and a certain plane. Find the radius of the sphere that touches the given spheres and the same plane. | \frac{R}{3} | 37 | 7 |
math | 1. Find all ordered pairs $(x, y)$ of positive integers such that:
$$
x^{3}+y^{3}=x^{2}+42 x y+y^{2}
$$
(Moldova) | (1,7),(7,1),(22,22) | 47 | 15 |
math | 14. How many four-digit numbers can be formed using the digits $1$, $9$, $8$, $8$ that leave a remainder of 8 when divided by 11? | 4 | 40 | 1 |
math | 2.2. For any positive integer $n$, evaluate
$$
\sum_{i=0}^{\left\lfloor\frac{n+1}{2}\right\rfloor}\binom{n-i+1}{i},
$$
where $\binom{m}{k}=\frac{m!}{k!(m-k)!}$ and $\left\lfloor\frac{n+1}{2}\right\rfloor$ is the greatest integer less than or equal to $\frac{n+1}{2}$. | \frac{5+3\sqrt{5}}{10}(\frac{1+\sqrt{5}}{2})^{n}+\frac{5-3\sqrt{5}}{10}(\frac{1-\sqrt{5}}{2})^{n} | 110 | 60 |
math | 11. A wooden stick of length $L$, is divided into 8, 12, and 18 equal segments using red, blue, and black lines respectively. The stick is then cut at each division line. How many segments will be obtained in total? What is the length of the shortest segment? | \frac{L}{72} | 65 | 8 |
math | Example. Find the general solution of the differential equation
$$
\left(1+x^{2}\right) y^{\prime \prime}+2 x y^{\prime}=12 x^{3}
$$ | C_{1}\operatorname{arctg}x+x^{3}-3x+C_{2} | 45 | 22 |
math | 8. Given $0<x<\frac{\pi}{2}, \sin x-\cos x=\frac{\pi}{4}$. If $\tan x+\frac{1}{\tan x}$ can be expressed in the form $\frac{a}{b-\pi^{c}}$ ($a$, $b$, $c$ are positive integers), then $a+b+c=$ $\qquad$ . | 50 | 83 | 2 |
math | 6. Let $x_{k} 、 y_{k} \geqslant 0(k=1,2,3)$. Calculate:
$$
\begin{array}{l}
\sqrt{\left(2018-y_{1}-y_{2}-y_{3}\right)^{2}+x_{3}^{2}}+\sqrt{y_{3}^{2}+x_{2}^{2}}+ \\
\sqrt{y_{2}^{2}+x_{1}^{2}}+\sqrt{y_{1}^{2}+\left(x_{1}+x_{2}+x_{3}\right)^{2}}
\end{array}
$$
the minimum value is | 2018 | 155 | 4 |
math | 8. The number of positive integer pairs $(x, y)$ that satisfy $y=\sqrt{x+51}+\sqrt{x+2019}$ is. $\qquad$ | 6 | 39 | 1 |
math | Let's find four-digit numbers that, when divided by $2, 3, 5, 7$, and 11, respectively give remainders of $0, 1, 3, 5$, and 9. | 2308,4618,6928,9238 | 49 | 19 |
math | Task 4 - 120624 Manfred reported in the Circle of Young Mathematicians about a visit to the Rostock Overseas Harbor:
"I saw a total of 21 ships from five different countries there. The number of ships from the GDR was half as large as that of all the foreign ships lying in the harbor. These came from the Soviet Union,... | 7 | 166 | 1 |
math | 638*. Find all exact powers of the number 2 that occur among numbers of the form $6 n+8(n=0,1,2, \ldots)$. | powers\of\2\with\any\odd\natural\exponent,\greater\than\1 | 38 | 22 |
math | The two numbers $0$ and $1$ are initially written in a row on a chalkboard. Every minute thereafter, Denys writes the number $a+b$ between all pairs of consecutive numbers $a$, $b$ on the board. How many odd numbers will be on the board after $10$ such operations?
[i]Proposed by Michael Kural[/i] | 683 | 78 | 3 |
math | 1. Vasya can get the number 100 using ten threes, parentheses, and arithmetic operation signs: $100=(33: 3-3: 3) \cdot(33: 3-3: 3)$. Improve his result: use fewer threes and get the number 100. (It is sufficient to provide one example). | 100=33\cdot3+3:3 | 82 | 13 |
math | A $(2^n - 1) \times (2^n +1)$ board is to be divided into rectangles with sides parallel to the sides of the board and integer side lengths such that the area of each rectangle is a power of 2. Find the minimum number of rectangles that the board may be divided into. | 2n | 64 | 4 |
math | 12.126. Find the angle in the axial section of a cone if a sphere with its center at the vertex of the cone, touching its base, divides the volume of the cone in half. | 2\arccos\frac{1+\sqrt{17}}{8} | 43 | 18 |
math | Problem 6.4. Find any solution to the puzzle
$$
\begin{array}{r}
\mathrm{ABA} \\
+\mathrm{ABC} \\
\mathrm{ACC} \\
\hline 1416
\end{array}
$$
where $A, B, C$ are three different non-zero digits.
Enter the values of the digits $A, B, C$.
Instead of the letter $A$ should stand the digit:
Instead of the letter $B$ sho... | A=4,B=7,C=6 | 121 | 9 |
math | Example 2. Let $p \neq 0$, and the quadratic equation with real coefficients $z^{2}-2 p z+q=0$ has imaginary roots $z_{1}$ and $z_{2}$. The corresponding points of $z_{1}$ and $z_{2}$ in the complex plane are $Z_{1}$ and $Z_{2}$. Find the length of the major axis of the ellipse with foci at $Z_{1}$ and $Z_{2}$ and pass... | 2 \sqrt{q} | 119 | 6 |
math | The admission fee for an exhibition is $ \$25$ per adult and $ \$12$ per child. Last Tuesday, the exhibition collected $ \$1950$ in admission fees from at least one adult and at least one child. Of all the possible ratios of adults to children at the exhibition last Tuesday, which one is closest to $ 1$? | \frac{27}{25} | 77 | 9 |
math | Suppose $x$ is a positive real number such that $\{x\}, [x]$ and $x$ are in a geometric progression. Find the least positive integer $n$ such that $x^n > 100$. (Here $[x]$ denotes the integer part of $x$ and $\{x\} = x - [x]$.) | 10 | 76 | 2 |
math | Task 1. (5 points) Find $\frac{a^{8}+256}{16 a^{4}}$, if $\frac{a}{2}+\frac{2}{a}=3$. | 47 | 45 | 2 |
math | One. (20 points) Given the equation in terms of $x$
$$
\frac{x^{2}+(5-2 m) x+m-3}{x-1}=2 x+m
$$
has no positive real roots. Find the range or value of $m$.
| 3 | 61 | 1 |
math | 2.60. Pentagon $A B C D E$ is inscribed in a circle. The distances from point $E$ to the lines $A B, B C$ and $C D$ are $a, b$ and $c$ respectively. Find the distance from point $E$ to the line $A D$. | \frac{ac}{b} | 69 | 7 |
math | 1. 152 For which natural numbers $n$ is the number $3^{2 n+1}-2^{2 n+1}-6^{n}$ composite. | n>1 | 37 | 3 |
math | 4. Let $[x]$ denote the greatest integer not exceeding the real number $x$. Then the set
$$
\{[x]+[2 x]+[3 x] \mid x \in \mathbf{R}\} \cap\{1,2, \cdots, 100\}
$$
has elements. | 67 | 73 | 2 |
math | ## N3.
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,,-),(,0,-),(,-,0):\in\mathbb{Z}} | 54 | 25 |
math | 13.4.33 **The ellipse $C: \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1(a>b>0)$ has its left vertex at $A(-a, 0)$. Points $M$ and $N$ are on the ellipse $C$, and $M A \perp N A$. Does the line $M N$ pass through a fixed point? If so, find the coordinates of this point; if not, explain the reason. | (\frac{(b^{2}-^{2})}{^{2}+b^{2}},0) | 112 | 21 |
math | 9. The constant term in the expansion of $\left(|x|+\frac{1}{|x|}-2\right)^{3}$ is $\qquad$ . | -20 | 36 | 3 |
math | In a right-angled triangle with hypotenuse $c$ given, six times the tangent of one acute angle is 1 greater than the tangent of the other acute angle. What are the lengths of the legs? | =\frac{2\sqrt{5}}{5},\quadb=\frac{\sqrt{5}}{5} | 44 | 25 |
math | 4. Through a point $A$ on the line $x=-4$, draw a tangent line $l$ to the parabola $C$: $y^{2}=2 p x(p>0)$, with the point of tangency being $B(1,2)$. Line $l$ intersects the $x$-axis at point $D$. $P$ is a moving point on the parabola $C$ different from point $B$. $\overrightarrow{P E}=\lambda_{1} \overrightarrow{P A}... | y^{2}=\frac{8}{3}(x+\frac{1}{3})(y\neq\frac{4}{3}) | 210 | 30 |
math | ### 9.305 Find $a$, for which the inequality
$x^{2}-2^{a+2} \cdot x-2^{a+3}+12>0$ is true for any $x$. | \in(-\infty;0) | 49 | 9 |
math | Bogosnov I.I.
On the plane, the curves $y=\cos x$ and $x=100 \cos (100 y)$ were drawn, and all points of their intersection with positive coordinates were marked. Let $a$ be the sum of the abscissas, and $b$ be the sum of the ordinates of these points. Find $a / b$. | 100 | 84 | 3 |
math | \section*{Problem 7}
What is the largest possible value of \(|\ldots|\left|a_{1}-a_{2}\right|-a_{3}\left|-\ldots-a_{1990}\right|\), where \(\mathrm{a}_{1}, a_{2}, \ldots, a_{1990}\) is a permutation of \(1,2,3, \ldots, 1990\) ?
Answer \(\quad 1989\)
| 1989 | 109 | 4 |
math | Example 1.1. Calculate
$$
\left|\begin{array}{rrr}
2 & -1 & 1 \\
3 & 2 & 2 \\
1 & -2 & 1
\end{array}\right|
$$ | 5 | 52 | 1 |
math | Example + Given that $f(x)$ is an even function defined on $\mathbf{R}$, if $g(x)$ is an odd function, and $g(x)=f(x-1)$, $g(1)=2003$, find the value of $f(2004)$. | 2003 | 65 | 4 |
math | 2. If real numbers $x, y$ satisfy $y^{2}=4 x$, then the range of $\frac{y}{x+1}$ is $\qquad$ . | [-1,1] | 38 | 5 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.