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 | Marta and Carmem have won many chocolates each. They mixed all the chocolates and now they no longer know how many chocolates each of them won. Let's help them find out these numbers? It is known that:
(a) together, they won 200 chocolates;
(b) Marta remembers that she won fewer than 100 chocolates, but more than $4 / 5$ of what Carmem won; and
(c) the number of chocolates each of them won is a multiple of 8. | Martawon96chocolatesCarmem104 | 106 | 14 |
math | Example 2 Arrange all positive integers that are coprime with 105 in ascending order, and find the 1000th term of this sequence. | 2186 | 35 | 4 |
math | 7. Given that the parabola $C$ has the center of the ellipse $E$ as its focus, the parabola $C$ passes through the two foci of the ellipse $E$, and intersects the ellipse $E$ at exactly three points. Then the eccentricity of the ellipse $E$ is $\qquad$ | \frac{2\sqrt{5}}{5} | 70 | 12 |
math | In an urn, there are 101 balls, among which exactly 3 are red. The balls are drawn one by one without replacement. At which position is it most likely that the second red ball will be drawn? | 51 | 46 | 2 |
math | (10 Given $\lg x_{1}, \lg x_{2}, \lg x_{3}, \lg x_{4}, \lg x_{5}$ are consecutive positive integers (in ascending or descending order), and $\left(\lg x_{4}\right)^{2}<\lg x_{1} \cdot \lg x_{5}$, then the minimum value of $x_{1}$ is $\qquad$ . | 100000 | 90 | 6 |
math | 16. From city $A$ to city $B$, one can travel by one of three types of transport, and from city $B$ to city $C$ - by one of four types of transport. In how many ways can one travel from city $A$ to city $C$, visiting city $B$ along the way? | 12 | 70 | 2 |
math | 7. In a lottery with 100000000 tickets, each ticket number consists of eight digits. A ticket number is called "lucky" if and only if the sum of its first four digits equals the sum of its last four digits. Then, the sum of all lucky ticket numbers, when divided by 101, leaves a remainder of | 0 | 77 | 1 |
math | 12. Four spheres with radii $6,6,6,7$ are pairwise externally tangent, and they are all internally tangent to a larger sphere. What is the radius of the larger sphere? $\qquad$ | 14 | 46 | 2 |
math | Consider rectangle $ABCD$ with $AB = 6$ and $BC = 8$. Pick points $E, F, G, H$ such that the angles $\angle AEB, \angle BFC, \angle CGD, \angle AHD$ are all right. What is the largest possible area of quadrilateral $EFGH$?
[i]Proposed by Akshar Yeccherla (TopNotchMath)[/i] | 98 | 97 | 2 |
math | 6. Given the sets $A=\{(x, y) \mid a x+y=1, x, y \in Z\}, B=\{(x, y) \mid x$ $+a y=1, x, y \in Z\}, C=\left\{(x, y) \mid x^{2}+y^{2}=1\right\}$, find the value of $a$ when $(A \cup B) \cap C$ is a four-element set. | -1 | 105 | 2 |
math | 1) Two nonnegative real numbers $x, y$ have constant sum $a$. Find the minimum value of $x^m + y^m$, where m is a given positive integer.
2) Let $m, n$ be positive integers and $k$ a positive real number. Consider nonnegative real numbers $x_1, x_2, . . . , x_n$ having constant sum $k$. Prove that the minimum value of the quantity $x^m_1+ ... + x^m_n$ occurs when $x_1 = x_2 = ... = x_n$. | \frac{k^m}{n^{m-1}} | 126 | 12 |
math | 894. Find three natural numbers whose sum equals their product. List all solutions. | 1,2,3 | 18 | 5 |
math | Find the value of the infinite continued fraction
$$
1+\frac{1}{2+\frac{1}{2+\frac{1}{2+\frac{1}{\ldots}}}} .
$$ | \sqrt{2} | 42 | 5 |
math | 2. $50 N$ is an integer, its base $b$ representation is 777, find the smallest positive integer $b$, such that $N$ is an integer to the fourth power. | 18 | 44 | 2 |
math | Find the set of values taken by the function $f:] 0,+\infty\left[{ }^{3} \mapsto \mathbb{R}\right.$ such that $f(x, y, z)=\frac{x \sqrt{y}+y \sqrt{z}+z \sqrt{x}}{\sqrt{(x+y)(y+z)(z+x)}}$. | ]0,\frac{3}{2\sqrt{2}}] | 80 | 14 |
math | When Paulo turned 15, he invited 43 friends to a party. The cake was in the shape of a regular 15-sided polygon and had 15 candles on it. The candles were placed in such a way that no three candles were in a straight line. Paulo divided the cake into triangular pieces where each cut connected two candles or connected a candle to a vertex. Moreover, no cut crossed another already made. Explain why, by doing this, Paulo was able to give a piece of cake to each of his guests, but he himself was left without any.
# | 43 | 119 | 2 |
math | Let $ABCD$ be a quadrilateral with an inscribed circle $\omega$ and let $P$ be the intersection of its diagonals $AC$ and $BD$. Let $R_1$, $R_2$, $R_3$, $R_4$ be the circumradii of triangles $APB$, $BPC$, $CPD$, $DPA$ respectively. If $R_1=31$ and $R_2=24$ and $R_3=12$, find $R_4$. | 19 | 115 | 2 |
math |
Zadatak B-3.3. Riješite jednadžbu:
$$
\cos 2012 x-\cos 2010 x=0
$$
| k\pi,\quadk\in\mathbb{Z}\quad\text{ili}\quad\frac{k\pi}{2011},\quadk\in\mathbb{Z} | 42 | 43 |
math | A bucket full of milk weighed $35 \mathrm{~kg}$. The same bucket with half the amount of milk weighed $18 \mathrm{~kg}$.
How much does the empty bucket weigh?
(L. Hozová) | 1\mathrm{~} | 51 | 6 |
math | 16. Let $P$ be a moving point on the circle $C_{1}: x^{2}+y^{2}=2$, and let $Q$ be the foot of the perpendicular from $P$ to the $x$-axis. Point $M$ satisfies $\sqrt{2} \overrightarrow{M Q}=\overrightarrow{P Q}$.
(1) Find the equation of the trajectory $C_{2}$ of point $M$;
(2) Draw two tangents from a point $T$ on the line $x=2$ to the circle $C_{1}$, and let the points of tangency be $A$ and $B$. If the line $AB$ intersects the curve $C_{2}$ from (1) at points $C$ and $D$, find the range of $\frac{|C D|}{|A B|}$. | \left[\frac{\sqrt{2}}{2}, 1\right) | 187 | 17 |
math | 11. Let $x, y \in \mathbf{R}$, denote the minimum value of $2^{-x}, 2^{x-y}, 2^{y-1}$ as $P$. When $0<x<1, 0<y<1$, the maximum value of $P$ is $\qquad$. | 2^{-\frac{1}{3}} | 69 | 9 |
math | 2. Find the smallest prime number $p$ for which the number $p^{3}+2 p^{2}+p$ has exactly 42 positive divisors. | 23 | 37 | 2 |
math | A triangle has vertices $P=(-8,5)$, $Q=(-15,-19)$, and $R=(1,-7)$. The equation of the bisector of $\angle P$ can be written in the form $ax+2y+c=0$. Find $a+c$. | 89 | 62 | 2 |
math | Example 4.2.5. Let $n \in \mathbb{N}$. Find the minimum value of the following expression
$$f(x)=|1+x|+|2+x|+\ldots+|n+x|, \quad(x \in \mathbb{R})$$ | \left\{\begin{array}{l}
m(m+1) \text { if } n=2 m(m \in \mathbb{N}) \\
(m+1)^{2} \text { if } n=2 m+1(m \in \mathbb{N})
\end{array}\right.} | 63 | 69 |
math | Let $ABCD$ be a convex quadrilateral with positive integer side lengths, $\angle{A} = \angle{B} = 120^{\circ}, |AD - BC| = 42,$ and $CD = 98$. Find the maximum possible value of $AB$. | 69 | 63 | 2 |
math | 1. On a very long road, a race was organized. 20 runners started at different times, each running at a constant speed. The race continued until each runner had overtaken all the slower ones. The speeds of the runners who started first and last were the same, and the speeds of the others were different from theirs and distinct from each other.
What could be the number of overtakes, if each involved exactly two people? In your answer, indicate the largest and smallest possible numbers in any order, separated by a semicolon.
Example of answer format:
$10 ; 20$ | 18;171 | 123 | 6 |
math | 3rd Irish 1990 Problem 9 Let a n = 2 cos(t/2 n ) - 1. Simplify a 1 a 2 ... a n and deduce that it tends to (2 cos t + 1)/3. | \frac{2\cos+1}{3} | 55 | 11 |
math | 9. Let $n$ be an integer greater than 2, and $a_{n}$ be the largest $n$-digit number that is neither the sum of two perfect squares nor the difference of two perfect squares.
(1) Find $a_{n}$ (expressed as a function of $n$);
(2) Find the smallest value of $n$ such that the sum of the squares of the digits of $n$ is a perfect square. | a_{n}=10^{n}-2,\;n_{\}=66 | 96 | 18 |
math | 11.3. On one main diagonal and all edges of a cube, directions are chosen. What is the smallest length that the sum of the resulting 13 vectors can have, if the edge length is 1, and the length of the main diagonal is $\sqrt{3}$. | \sqrt{3} | 60 | 5 |
math | 20 Anna randomly picked five integers from the following list
$$
53,62,66,68,71,82,89
$$
and discover that the average value of the five integers she picked is still an integer. If two of the integers she picked were 62 and 89 , find the sum of the remaining three integers. | 219 | 79 | 3 |
math | ## Task A-3.2.
Determine all pairs $(x, y)$ of integers that satisfy the equation
$$
x^{2}(y-1)+y^{2}(x-1)=1
$$ | (1,2),(2,1),(2,-5),(-5,2) | 45 | 18 |
math | 4B. For the real numbers $a_{1}, a_{2}, a_{3}, \ldots, a_{2021}$, it is given that the sum of any three numbers whose indices are consecutive numbers $i, i+1, i+2, i=1, \ldots, 2019$, is 2021. If $a_{1}=1000$ and $a_{2021}=900$, determine the remaining numbers. | a_{1}=1000,a_{2}=900,a_{3}=121,a_{4}=1000,\ldots,a_{2018}=900,a_{2019}=121,a_{2020}=1000,a_{2021}=900 | 108 | 74 |
math | 7.1. The steamship "Rarity" is rapidly sinking. If Captain Alexei gives exactly $2017^{2017}$ instructions to his 26 sailors, the steamship can be saved. Each subsequent sailor can receive 2 fewer or 2 more instructions than the previous one. Can Alexei save the steamship? | No | 73 | 1 |
math | 3B. A boy needs to pass through three doors to reach the desired goal. At each door, there stands a guard. The boy carries a certain number of apples with him. To the first guard, he must give half of that number and another half apple. To the second guard, he must give half of the remainder and another half apple, and to the third guard, he must give half of the remainder and another half apple. Find the smallest number of apples the boy should carry with him to reach the desired goal without cutting any apple. | 7 | 111 | 1 |
math | Find all positive integers that can be represented as
$$\frac{abc+ab+a}{abc+bc+c}$$
for some positive integers $a, b, c$.
[i]Proposed by Oleksii Masalitin[/i] | 1 \text{ and } 2 | 52 | 8 |
math | ## Subject IV. (20 points)
Determine the function $f: \mathbb{R} \rightarrow \mathbb{R}_{+}^{*}$ such that the function $g: \mathbb{R} \rightarrow \mathbb{R}, g(x)=f(x)(\sin 2 x+4 \cos x)$
admits the primitive $G: \mathbb{R} \rightarrow \mathbb{R}, G(x)=\frac{f(x)}{2+\sin x}$.
Prof. Gheorghe Lobont, National College "Mihai Viteazul" Turda
All subjects are mandatory. 10 points are awarded by default.
SUCCESS!
Effective working time - 3 hours.
## Grading Scale for Grade 12 (OLM 2014 - local stage)
## Of. $10 \text{p}$ | f(x)=|\mathfrak{C}|\cdote^{\frac{2}{3}(2+\sinx)^{3}}\cdot(2+\sinx) | 190 | 37 |
math | Two workers are working on identical workpieces and both are currently meeting the standard. With the application of an innovation, one completes a workpiece 5 minutes faster than the standard time, while the other completes it 3 minutes faster. As a result, their average performance increases to 137.5%. What is the standard time for completing one workpiece? | 15 | 74 | 2 |
math | Example 2 Find the greatest common divisor of $48$, $60$, and $72$. | 12 | 22 | 2 |
math | 4. If $2016+3^{n}$ is a perfect square, then the positive integer $n=$ . $\qquad$ | 2 | 30 | 1 |
math | The NIMO problem writers have invented a new chess piece called the [i]Oriented Knight[/i]. This new chess piece has a limited number of moves: it can either move two squares to the right and one square upward or two squares upward and one square to the right. How many ways can the knight move from the bottom-left square to the top-right square of a $16\times 16$ chess board?
[i] Proposed by Tony Kim and David Altizio [/i] | 252 | 104 | 3 |
math | [ [Decimal numeral system ]
The numbers $2^{2000}$ and $5^{2000}$ are written in sequence. How many digits are written in total?
# | 2001 | 39 | 4 |
math | Problem 11.1. In a basket, there are 41 apples: 10 green, 13 yellow, and 18 red. Alyona sequentially takes one apple at a time from the basket. If at any point she has pulled out fewer green apples than yellow ones, and fewer yellow ones than red ones, she will stop taking apples from the basket.
(a) (1 point) What is the maximum number of yellow apples Alyona can take from the basket?
(b) (3 points) What is the maximum number of apples Alyona can take from the basket? | 39 | 122 | 2 |
math | 10.3. Given a right triangle $A B C$. A point $M$ is taken on the hypotenuse $A C$. Let $K, L$ be the centers of the circles inscribed in triangles $A B M$ and $C B M$ respectively. Find the distance from point $M$ to the midpoint of $K L$, if the radius $R$ of the circle circumscribed around triangle $B K L$ is known. | \frac{R\sqrt{2}}{2} | 97 | 12 |
math | 2. (17 points) A tourist travels from point $A$ to point $B$ in 2 hours and 14 minutes. The route from $A$ to $B$ goes uphill first, then on flat terrain, and finally downhill. What is the length of the uphill road if the tourist's speed downhill is 6 km/h, uphill is 4 km/h, and on flat terrain is 5 km/h, and the total distance between $A$ and $B$ is 10 km? Additionally, the distances uphill and on flat terrain are whole numbers of kilometers. | 6 | 123 | 1 |
math | [b]5.[/b] Find the continuous solutions of the functional equation $f(xyz)= f(x)+f(y)+f(z)$ in the following cases:
(a) $x,y,z$ are arbitrary non-zero real numbers;
(b) $a<x,y,z<b (1<a^{3}<b)$.
[b](R. 13)[/b] | f(x) = \lambda \ln x | 77 | 10 |
math | 8. (10 points) The difference between the minimum value of the sum of the squares of ten different odd numbers and the remainder when this minimum value is divided by 4 is $\qquad$
(Note: The product of the same two natural numbers is called the square of this natural number, such as $1 \times 1=1^{2}, 2 \times 2=2^{2}, 3 \times 3$ $=3^{3}$, and so on) | 1328 | 102 | 4 |
math | One base of a trapezoid is 100 units longer than the other base. The segment that joins the midpoints of the legs divides the trapezoid into two regions whose areas are in the ratio $2: 3.$ Let $x$ be the length of the segment joining the legs of the trapezoid that is parallel to the bases and that divides the trapezoid into two regions of equal area. Find the greatest integer that does not exceed $x^2/100.$ | 181 | 111 | 3 |
math | [ Quadrilaterals (extreme properties) ]
The diagonals of a convex quadrilateral $A B C D$ intersect at point $O$. What is the smallest area that this quadrilateral can have if the area of triangle $A O B$ is 4 and the area of triangle $C O D$ is 9? | 25 | 68 | 2 |
math | 21. Each of the integers $1,2,3, \ldots, 9$ is assigned to each vertex of a regular 9 -sided polygon (that is, every vertex receives exactly one integer from $\{1,2, \ldots, 9\}$, and two vertices receive different integers) so that the sum of the integers assigned to any three consecutive vertices does not exceed some positive integer $n$. What is the least possible value of $n$ for which this assignment can be done? | 16 | 108 | 2 |
math | 5. [4] Let the functions $f(\alpha, x)$ and $g(\alpha)$ be defined as
$$
f(\alpha, x)=\frac{\left(\frac{x}{2}\right)^{\alpha}}{x-1} \quad g(\alpha)=\left.\frac{d^{4} f}{d x^{4}}\right|_{x=2}
$$
Then $g(\alpha)$ is a polynomial in $\alpha$. Find the leading coefficient of $g(\alpha)$. | \frac{1}{16} | 109 | 8 |
math | [ Number of divisors and their sum of a number] $[\quad \underline{\text { equations in integers }}]$
A certain natural number $n$ has two prime divisors. Its square has a) 15; b) 81 divisors. How many divisors does the cube of this number have? | 160or169 | 68 | 7 |
math | 2. In the Cartesian coordinate system $x O y$, a circle passes through $(0,2)$ and $(3,1)$, and is tangent to the $x$-axis. Then the radius of this circle is $\qquad$ . | 15 \pm 6 \sqrt{5} | 51 | 11 |
math | 9. Find the smallest positive integer $n$, such that when the positive integer $k \geqslant n$, in the set $M=$ $\{1,2, \cdots, k\}$ of the first $k$ positive integers, for any $x \in M$, there always exists another number $y \in M(y \neq x)$, such that $x+y$ is a perfect square. | 7 | 89 | 1 |
math | 2.1. Trapezoid $A B C D$ with base $A D=6$ is inscribed in a circle. The tangent to the circle at point $A$ intersects lines $B D$ and $C D$ at points $M$ and $N$ respectively. Find $A N$, if $A B \perp M D$ and $A M=3$. | 12 | 82 | 2 |
math | ## Task Condition
Calculate the area of the parallelogram constructed on vectors $a_{\text {and }} b$.
\[
\begin{aligned}
& a=4 p-q \\
& b=p+2 q \\
& |p|=5 \\
& |q|=4 \\
& (\widehat{p, q})=\frac{\pi}{4}
\end{aligned}
\] | 90\sqrt{2} | 82 | 7 |
math | Task B-4.7. Vectors $\vec{a}=2 \vec{m}+\vec{n}$ and $\vec{b}=\vec{m}-\vec{n}$ determine a parallelogram. Here, $\vec{m}$ and $\vec{n}$ are unit vectors, and the measure of the angle between $\vec{m}$ and $\vec{n}$ is $\frac{\pi}{3}$. Determine the area of the parallelogram determined by vectors $\vec{a}$ and $\vec{b}$. | \frac{3\sqrt{3}}{2} | 110 | 12 |
math | Example 3 Given positive integers $k, l$. Find all functions $f: \mathbf{Z}_{+} \rightarrow \mathbf{Z}_{+}$, such that for any $m, n \in \mathbf{Z}_{+}$, we have $(f(m)+f(n)) \mid (m+n+l)^{k}$. | f(n)=n+\frac{}{2}(n\in{Z}_{+}) | 75 | 18 |
math | 13. (15 points) A motorcycle traveling 120 kilometers takes the same time as a car traveling 180 kilometers. In 7 hours, the distance traveled by the motorcycle is 80 kilometers less than the distance traveled by the car in 6 hours. If the motorcycle starts 2 hours earlier, and then the car starts from the same starting point to catch up, how many hours after the car starts can it catch up to the motorcycle?
| 4 | 97 | 1 |
math | 7. Find the minimum value of the function $f(x, y)=\frac{2015(x+y)}{\sqrt{2015 x^{2}+2015 y^{2}}}$ and specify all pairs $(x, y)$ at which it is achieved. | -\sqrt{4030} | 61 | 8 |
math | Example. Find the indefinite integral
$$
\int \frac{x^{2}}{\sqrt{9-x^{2}}} d x
$$ | \int\frac{x^{2}}{\sqrt{9-x^{2}}}=\frac{9}{2}\arcsin\frac{x}{3}-\frac{x}{2}\sqrt{9-x^{2}}+C | 29 | 47 |
math | he sequence of real number $ (x_n)$ is defined by $ x_1 \equal{} 0,$ $ x_2 \equal{} 2$ and $ x_{n\plus{}2} \equal{} 2^{\minus{}x_n} \plus{} \frac{1}{2}$ $ \forall n \equal{} 1,2,3 \ldots$ Prove that the sequence has a limit as $ n$ approaches $ \plus{}\infty.$ Determine the limit. | L = 1 | 106 | 5 |
math | 5. The terms of the sequence $\left\{a_{n}\right\}$ are all positive, and the sum of the first $n$ terms $S_{n}$ satisfies
$$
S_{n}=\frac{1}{2}\left(a_{n}+\frac{1}{a_{n}}\right) .
$$
Then $a_{n}=$ | a_{n}=\sqrt{n}-\sqrt{n-1} | 79 | 14 |
math | 15. Suppose there are 128 ones written on a blackboard. In each step, you can erase any two numbers \(a\) and \(b\) and write \(ab + 1\). After performing this operation 127 times, only one number remains. Let the maximum possible value of this remaining number be \(A\). Determine the last digit of \(A\).
(1992 Saint Petersburg City Team Selection Test) | 2 | 93 | 1 |
math | 1. The range of the function $y=|\cos x|-\cos 2 x(x \in \mathbf{R})$ is | [0,\frac{9}{8}] | 30 | 9 |
math | Solve the following equation:
$$
1+a+a^{2}+\ldots+a^{2 x-1}+a^{2 x}=(1+a)\left(1+a^{2}\right)\left(1+a^{4}\right)\left(1+a^{8}\right)
$$ | 7.5 | 62 | 3 |
math | 1. In the set of natural numbers, solve the system of equations
$$
\left\{\begin{array}{l}
a b+c=13 \\
a+b c=23
\end{array}\right.
$$ | (,b,)=(1,2,11),(2,3,7),(1,11,2) | 49 | 25 |
math | 17. Let $p$ and $q$ be positive integers such that $\frac{72}{487}<\frac{p}{q}<\frac{18}{121}$. Find the smallest possible value of $q$.
(2 marks)
設 $p \vee q$ 為滿足 $\frac{72}{487}<\frac{p}{q}<\frac{18}{121}$ 的正整數。求 $q$ 的最小可能值。
(2 分) | 27 | 115 | 2 |
math | 225. Find the condition under which the difference of the roots in the equation $x^{2}+p x+$ $+q=0$ would be equal to $a$. | ^{2}-p^{2}=-4q | 39 | 10 |
math | 10. (This sub-question is worth 15 points) Given an ellipse with its center at the origin $O$, foci on the $x$-axis, eccentricity $\frac{\sqrt{3}}{2}$, and passing through the point $\left(\sqrt{2}, \frac{\sqrt{2}}{2}\right)$. Let a line $l$ that does not pass through the origin $O$ intersect the ellipse at points $P$ and $Q$, and the slopes of the lines $O P, P Q, O Q$ form a geometric sequence. Find the range of the area of $\triangle O P Q$. | (0,1) | 136 | 5 |
math | 40.3. From point $A$ along the circumference, two bodies start moving simultaneously in opposite directions: the first with a constant speed $v$, and the second with a constant linear acceleration $a$ and an initial speed of 0. After what time did the bodies meet for the first time, if they met again for the second time at point $A$? | t_{1}=\frac{v}{}(\sqrt{5}-1) | 77 | 17 |
math | (Moscow 2000, exercise A1) Let $x$ and $y$ be two different real numbers such that $x^{2}-2000 x=y^{2}-2000 y$. What is the value of $x+y$? | 2000 | 58 | 4 |
math | I2.3 Given that $T=\sin 50^{\circ} \times\left(S+\sqrt{3} \times \tan 10^{\circ}\right)$, find the value of $T$. | 1 | 48 | 1 |
math | $$
\begin{array}{l}
a^{2}+b^{2}+c^{2}+d^{2}+(1-a)^{2}+(a-b)^{2}+ \\
(b-c)^{2}+(c-d)^{2}+(d-1)^{2}
\end{array}
$$
When the value of the above algebraic expression is minimized, what is the value of $a+b+c+d$? Prove your conclusion. | \frac{6}{5} | 101 | 7 |
math | 3. Given real numbers $a, b, c, d$ satisfy $a d - b c = 1$, then the minimum value of $a^{2} + b^{2} + c^{2} + d^{2} + a c + b d$ is | \sqrt{3} | 58 | 5 |
math | 6.42 Given the parabola $y=a x^{2}+b x+c$ has a line of symmetry at $x=-2$, it is tangent to a certain line at one point, this line has a slope of 2, and a y-intercept of 1, and the parabola intersects the $y=0$ at two points, the distance between which is $2 \sqrt{2}$. Try to find the equation of this parabola. | y=x^{2}+4 x+2 \text{ or } y=\frac{1}{2} x^{2}+2 x+1 | 101 | 32 |
math | 6. $a$ and $b$ are both positive real numbers, the equations $x^{2}+a x+2 b=0$ and $x^{2}+2 b x+a=0$ both have real roots, the minimum value of $a+b$ is $\qquad$ . | 6 | 64 | 1 |
math | 10. Given that the largest angle of an isosceles triangle is 4 times the smallest angle, then the difference between the largest and smallest angles is $(\quad)$ degrees. | 90 | 39 | 2 |
math | 2B. In a quiz, the contestant answers 24 questions. If the contestant answers a question correctly, they earn 4 points, and if they answer incorrectly, they lose 1.4 points. How many questions did the contestant not know the answer to if they ended up with 69 points at the end of the quiz? | 5 | 70 | 1 |
math | 3. 2.9 * Let the sequence $\left\{a_{n}\right\}$ satisfy $a_{1}=a_{2}=1, a_{3}=2,3 a_{n+3}=4 a_{n+2}+a_{n+1}-2 a_{n}$, $n=1,2, \cdots$. Find the general term of the sequence $\left\{a_{n}\right\}$. | a_{n}=\frac{1}{25}[1+15n-\frac{27}{2}(-\frac{2}{3})^{n}] | 96 | 36 |
math | Example 3. Given points A, B and line l in the same plane, try to find a point P on l such that
$$
P A+P B \text { is minimal. }
$$ | P | 43 | 1 |
math | 9. Given the sequence $a_{n}=\sqrt{4+\frac{1}{n^{2}}}+\sqrt{4+\frac{2}{n^{2}}}+\cdots+\sqrt{4+\frac{n}{n^{2}}}-2 n, n$ is a positive integer, then the value of $\lim _{n \rightarrow \infty} a_{n}$ is $\qquad$ | \frac{1}{8} | 86 | 7 |
math | Example 4. Find the general solution of the equation
$$
y^{\prime \prime}+3 y^{\prime}+2 y=2 x^{2}-4 x-17
$$ | C_{1}e^{-x}+C_{2}e^{-2x}+x^{2}-5x-2 | 44 | 27 |
math | 2. 79 In a game, scoring is as follows: answering an easy question earns 3 points, and answering a difficult question earns 7 points. Among the integers that cannot be the total score of a player, find the maximum value.
| 11 | 51 | 2 |
math |
A2. Two ants start at the same point in the plane. Each minute they choose whether to walk due north, east, south or west. They each walk 1 meter in the first minute. In each subsequent minute the distance they walk is multiplied by a rational number $q>0$. They meet after a whole number of minutes, but have not taken exactly the same route within that time. Determine all possible values of $q$.
(United Kingdom)
| 1 | 95 | 1 |
math | 6.369 Solve the equation $2 x^{5}-x^{4}-2 x^{3}+x^{2}-4 x+2=0$, given that it has three roots, two of which differ only in sign. | x_{1,2}=\\sqrt{2};x_{3}=\frac{1}{2} | 50 | 22 |
math | 1. If $a+\log _{3} 2016, a+\log _{9} 2016, a+\log _{27} 2016(a \in \mathbf{R})$ form a geometric sequence, then its common ratio is
$\qquad$ (Yang Yunxin provided the problem) | \frac{1}{3} | 76 | 7 |
math | 11. Given that the domain of the function $f(x)$ is $\mathbf{R}$, for any real number $x$, it always holds that $f(1+x)=f(3-x)$, and $f(2+x)=-f(1-x)$. Then $f(1)+f(2)+\cdots+f(100)=$ $\qquad$ . | 0 | 84 | 1 |
math | 2. 47 If $a<b<c<d<e$ are consecutive positive integers, $b+c+d$ is a perfect square, and $a+b+c+d+e$ is a perfect cube, what is the minimum value of $c$? | 675 | 53 | 3 |
math | [ Equations in integers ]
Solve the equation $x^{2}-5 y^{2}=1$ in integers.
# | \(x_{n},\y_{n}),wherex_{n}+y_{n}\sqrt{5}=(9+4\sqrt{5})^{n},n=0,1,2,3,\ldots | 26 | 49 |
math | Determine if there exists a subset $E$ of $Z \times Z$ with the properties:
(i) $E$ is closed under addition,
(ii) $E$ contains $(0,0),$
(iii) For every $(a,b) \ne (0,0), E$ contains exactly one of $(a,b)$ and $-(a,b)$.
Remark: We define $(a,b)+(a',b') = (a+a',b+b')$ and $-(a,b) = (-a,-b)$. | E = (\mathbb{Z}_{>0} \times \mathbb{Z}) \cup (\{0\} \times \mathbb{Z}_{\ge 0}) | 110 | 40 |
math | 12: Use 6 equal-length thin iron rods to weld into a regular tetrahedral frame, ignoring the thickness of the iron rods and welding errors. Let the radius of the largest sphere that this frame can contain be $R_{1}$, and the radius of the smallest sphere that can enclose this frame be $R_{2}$. Then $\frac{R_{1}}{R_{2}}$ equals $\qquad$ | \frac{\sqrt{3}}{3} | 90 | 10 |
math | 2. It is known that $\frac{\sin 3 x}{(2 \cos 2 x+1) \sin 2 y}=\frac{1}{5}+\cos ^{2}(x-2 y)$ and $\frac{\cos 3 x}{(1-2 \cos 2 x) \cos 2 y}=\frac{4}{5}+\sin ^{2}(x-2 y)$. Find all possible values of the expression $\cos (x-6 y)$, given that there are at least two. Answer: 1 or $-\frac{3}{5}$. | -\frac{3}{5}or1 | 130 | 9 |
math | G10.1 If $b+c=3$
(1), $c+a=6$
(2), $a+b=7$
(3) and $P=a b c$, find $P$. | 10 | 44 | 2 |
math | There is a positive real number $x$ not equal to either $\tfrac{1}{20}$ or $\tfrac{1}{2}$ such that\[\log_{20x} (22x)=\log_{2x} (202x).\]The value $\log_{20x} (22x)$ can be written as $\log_{10} (\tfrac{m}{n})$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$. | 112 | 111 | 3 |
math | 7. Given moving points $P$, $M$, and $N$ are on the $x$-axis, circle $\Gamma_{1}$: $(x-1)^{2}+(y-2)^{2}=1$, and circle $\Gamma_{2}$: $(x-3)^{2}+$ $(y-4)^{2}=3$, respectively. Then the minimum value of $|P M|+|P N|$ is $\qquad$. | 2\sqrt{10}-\sqrt{3}-1 | 98 | 13 |
math | ## Zadatak A-1.1.
Odredi $x_{1006}$ ako je
$$
\begin{aligned}
& \frac{x_{1}}{x_{1}+1}= \frac{x_{2}}{x_{2}+3}=\frac{x_{3}}{x_{3}+5}=\ldots=\frac{x_{1006}}{x_{1006}+2011} \\
& x_{1}+x_{2}+\ldots+x_{1006}=503^{2}
\end{aligned}
$$
| \frac{2011}{4} | 133 | 10 |
math | ## 6. Math Puzzle 11/65
In a pursuit race, the drivers start one minute apart.
How fast is a driver going if he catches up to the driver in front of him, who is traveling at 36 $\frac{\mathrm{km}}{\mathrm{h}}$, after $6 \mathrm{~km}$? | 40\frac{\mathrm{}}{\mathrm{}} | 73 | 12 |
math | 5. Find all values of the parameter $b$, for each of which there exists a number $a$ such that the system
$$
\left\{\begin{array}{l}
x=\frac{7}{b}-|y+b| \\
x^{2}+y^{2}+96=-a(2 y+a)-20 x
\end{array}\right.
$$
has at least one solution $(x ; y)$. | b\in(-\infty;-\frac{7}{12}]\cup(0;+\infty) | 95 | 25 |
math | Sally rolls an $8$-sided die with faces numbered $1$ through $8$. Compute the probability that she gets a power of $2$. | \frac{1}{2} | 33 | 7 |
math | 7.1. Solve the equation
$$
3 \cos \frac{4 \pi x}{5}+\cos \frac{12 \pi x}{5}=2 \cos \frac{4 \pi x}{5}\left(3+\operatorname{tg}^{2} \frac{\pi x}{5}-2 \operatorname{tg} \frac{\pi x}{5}\right)
$$
In the answer, write the sum of its roots on the interval $[-11 ; 19]$. | 112.5 | 111 | 5 |
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