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 | \section*{Problem 12}
Two players alternately choose the sign for one of the numbers \(1,2, \ldots, 20\). Once a sign has been chosen it cannot be changed. The first player tries to minimize the final absolute value of the total and the second player to maximize it. What is the outcome (assuming both players play perf... | 30 | 162 | 2 |
math | 1. Given the sequence $\left\{a_{n}\right\}$ satisfies $a_{1}=32, a_{n+1}-a_{n}=2 n\left(n \in \mathbf{Z}_{+}\right)$. Then the minimum value of $\frac{a_{n}}{n}$ is $\qquad$. | \frac{31}{3} | 73 | 8 |
math | 30.4. In an arithmetic sequence, the third, fifth and eleventh terms are distinct and form a geometric sequence. If the fourth term of the arithmetic sequence is 6 , what is its 2007 th term? | 6015 | 49 | 4 |
math | [ Extreme properties of a triangle (miscellaneous).]
Consider all acute-angled triangles with a given side $a$ and angle $\alpha$.
What is the maximum value of the sum of the squares of the lengths of sides $b$ and $c$? | \frac{^{2}}{2\sin^{2}\frac{\alpha}{2}} | 54 | 19 |
math | Determine the least real number $k$ such that the inequality
$$\left(\frac{2a}{a-b}\right)^2+\left(\frac{2b}{b-c}\right)^2+\left(\frac{2c}{c-a}\right)^2+k \geq 4\left(\frac{2a}{a-b}+\frac{2b}{b-c}+\frac{2c}{c-a}\right)$$
holds for all real numbers $a,b,c$.
[i]Proposed by Mohammad Jafari[/i] | k = 8 | 118 | 4 |
math | ## Task 3 - 100523
The members of a working group "Young Botanists" supported their patron LPG in fruit growing.
$\mathrm{To}$ this end, they kept a 2.6 ha orchard free from pests, on which an average of 150 apple trees stood per hectare. Afterwards, an average of $50 \mathrm{~kg}$ of apples were harvested from each ... | 19.5 | 112 | 4 |
math | The vertices of a regular $2012$-gon are labeled $A_1,A_2,\ldots, A_{2012}$ in some order. It is known that if $k+\ell$ and $m+n$ leave the same remainder when divided by $2012$, then the chords $A_kA_{\ell}$ and $A_mA_n$ have no common points. Vasya walks around the polygon and sees that the first two vertices are lab... | A_{28} | 134 | 5 |
math | 4.9. Each diagonal of the convex pentagon $A B C D E$ cuts off a triangle of unit area from it. Calculate the area of the pentagon $A B C D E$. | \frac{\sqrt{5}+5}{2} | 42 | 12 |
math | 10. (20 points) Find the number of all positive integer solutions $(x, y, z)$ to the equation $\arctan \frac{1}{x}+\arctan \frac{1}{y}+\arctan \frac{1}{z}=\frac{\pi}{4}$. | 15 | 67 | 2 |
math | 11. Real numbers $x, y, z, w$ satisfy $x+y+z+w=1$, then the maximum value of $M=x w+2 y w+3 x y+3 z w+4 x z+5 y z$ is $\qquad$ . | \frac{3}{2} | 59 | 7 |
math | 13. Given the function
$$
f(x)=\sin ^{2} \omega x+\sqrt{3} \sin \omega x \cdot \sin \left(\omega x+\frac{\pi}{2}\right)
$$
has the smallest positive period of $\frac{\pi}{2}$, where $\omega>0$. Find the maximum and minimum values of $f(x)$ on $\left[\frac{\pi}{8}, \frac{\pi}{4}\right]$. | 1 \leqslant f(x) \leqslant \frac{3}{2} | 102 | 21 |
math | Three, (25 points) A chemical plant, starting from January this year, if it does not improve its production environment and continues to produce as it is, will earn 700,000 yuan per month. At the same time, it will receive penalties from the environmental protection department, with the first month's penalty being 30,0... | 9 | 310 | 1 |
math | Lazim rolls two $24$-sided dice. From the two rolls, Lazim selects the die with the highest number. $N$ is an integer not greater than $24$. What is the largest possible value for $N$ such that there is a more than $50$% chance that the die Lazim selects is larger than or equal to $N$? | 17 | 81 | 2 |
math | 7. If a die is thrown three times in succession, the probability that the three numbers that appear can form the side lengths of a triangle whose perimeter is divisible by 3 is $\qquad$ . . | \frac{11}{72} | 42 | 9 |
math | }
For which $n>3$ can a set of weights with masses $1,2,3, \ldots, n$ grams be divided into three equal-mass piles?
# | 8 | 39 | 1 |
math | Problem 6. Calculate $2 \operatorname{arctg} 3+\arcsin \frac{3}{5}$. | \pi | 29 | 2 |
math | $4 \cdot 140$ Solve the system of equations
$$
\left\{\begin{array}{l}
x y z=x+y+z, \\
y z t=y+z+t, \\
z t x=z+t+x, \\
t x y=t+x+y .
\end{array}\right.
$$ | (0,0,0,0),(\sqrt{3},\sqrt{3},\sqrt{3},\sqrt{3}),(-\sqrt{3},-\sqrt{3},-\sqrt{3},-\sqrt{3}) | 66 | 50 |
math | 921. Find the mass of a hemisphere if the surface density at each of its points is numerically equal to the distance of this point from the radius perpendicular to the base of the hemisphere. | \frac{\pi^{2}R^{3}}{2} | 40 | 14 |
math | Three, (15 points) Let the line $l: y=x+c$ intersect the ellipse $\frac{x^{2}}{2}+y^{2}=1$ at two points $A$ and $B$ (which can coincide), and intersect the circle $(x-2)^{2}+(y+2)^{2}=4$ at two points $C$ and $D$ (which can coincide). Find the maximum value of $9|A B|^{2}+|C D|^{2}$. | 256 \sqrt{2}-336 | 111 | 11 |
math | 11. (16 points) For an integer $k$, define the set
$$
S_{k}=\{n \mid 50 k \leqslant n<50(k+1), n \in \mathbf{Z}\} \text {. }
$$
How many of the 600 sets $S_{0}, S_{1}, \cdots, S_{599}$ do not contain any perfect squares? | 439 | 97 | 3 |
math | 11. Prime numbers $p$, $q$, $r$ satisfy $p+q=r$, and $(r-p) \cdot$ $(q-p)-27 p$ is a perfect square. Then all the triples $(p, q, r)=$ $\qquad$ | (2,29,31) | 58 | 9 |
math | 10. Find all prime numbers $p$ that satisfy the following condition: for any prime number $q<p$, if $p=k q+r, 0 \leqslant r<q$, then there does not exist an integer $a$ greater than 1 such that $a^{2} \mid r$. | 2,3,5,7,13 | 67 | 10 |
math | 10.076. Find the ratio of the radius of the circle inscribed in an isosceles right triangle to the height drawn to the hypotenuse. | \sqrt{2}-1 | 36 | 6 |
math | 2. Let $a$ and $b$ be real numbers such that $a>b>0$ and $a^{2}+b^{2}=6ab$. Determine the value of the expression $\frac{a+b}{a-b}$. | \sqrt{2} | 51 | 5 |
math | In a warehouse, the inventory is stored in packages weighing no more than 1 ton each. We have a 1-ton and a 4-ton truck. What is the maximum load that we can definitely deliver in one trip? | 4 | 46 | 1 |
math | Example 1.24. Perpendiculars are dropped from the point $P(2 ; 3 ;-5)$ to the coordinate planes. Find the equation of the plane passing through their feet. | 15x+10y-6z-60=0 | 41 | 15 |
math | Example 1. Solve the inequality
$$
\log _{7} \frac{x-2}{x-3}<0
$$ | -\infty<x<2 | 29 | 6 |
math | Mekkora az egyenló oldalú kúp, egyenló oldalú henger és gömb köbtartalmának aránya, ha felszíneik egyenlők?
What is the ratio of the volumes of an equilateral cone, an equilateral cylinder, and a sphere if their surface areas are equal? | 2:\sqrt{2\cdot3}:3 | 79 | 10 |
math | 12. Let the continuous monotonic function $f: \mathbf{R} \rightarrow \mathbf{R}, f(0)=1$, and satisfy the inequality
$$
f(x+y) \geqslant f(x) f(y)-f(x y)+1 .
$$
Find $f(x)$. | f(x)=x+1 | 68 | 6 |
math | After a typist has written ten letters and had addressed the ten corresponding envelopes, a careless mailing clerk inserted the letters in the envelopes at random, one letter per envelope. What is the probability that [b]exactly[/b] nine letters were inserted in the proper envelopes? | 0 | 56 | 1 |
math | 1. Find the number of triples $(a, b, c)$ of positive integers such that $a+a b+a b c=11$. | 3 | 30 | 1 |
math | 2. For the visit to the Ivan Meštrović exhibition, $\frac{2}{7}$ more students registered than planned. Due to illness, one-sixth of the registered students canceled, and as a result, six more students went to the exhibition than planned. How many students went to the exhibition? | 90 | 64 | 2 |
math | Task B-4.2. If $z+z^{-1}=2 \cos \frac{\alpha}{2012}$, determine $\alpha$ for which $z^{2012}+z^{-2012}=1$ | \alpha=\\frac{\pi}{3}+2k\pi,\quadk\in\mathbb{Z} | 52 | 26 |
math | 3. Given that $A M$ is the median of $\triangle A B C$ on side $B C$, $P$ is the centroid of $\triangle A B C$, and a line $E F$ through point $P$ intersects sides $A B$ and $A C$ at points $E$ and $F$ respectively. Then $\frac{B E}{A E}+\frac{C F}{A F}=$ $\qquad$ | 1 | 95 | 1 |
math | 12. List all positive integers that are coprime with 105 in ascending order, and find the 100th term of this sequence. | 218 | 34 | 3 |
math | 8. A thin beam of light falls normally on a plane-parallel glass plate. Behind the plate, at some distance from it, stands an ideal mirror (its reflection coefficient is equal to one). The plane of the mirror is parallel to the plate. It is known that the intensity of the beam that has passed through this system is 16 ... | 0.5 | 129 | 3 |
math | 4. Masha chose a natural number $n$ and wrote down all natural numbers from 1 to 6 n on the board. Then, Masha halved half of these numbers, reduced a third of the numbers by a factor of three, and increased all the remaining numbers by a factor of six. Could the sum of all the resulting numbers match the sum of the or... | 78 | 84 | 2 |
math | 1. The rays forming a right angle with the vertex at the coordinate origin intersect the parabola $y^{2}=2 x$ at points $X$ and $Y$. Find the geometric locus of the midpoints of the segments $X Y$. | y^{2}=x-2 | 52 | 7 |
math | The incircle $\Gamma$ of a scalene triangle $ABC$ touches $BC$ at $D, CA$ at $E$ and $AB$ at $F$. Let $r_A$ be the radius of the circle inside $ABC$ which is tangent to $\Gamma$ and the sides $AB$ and $AC$. Define $r_B$ and $r_C$ similarly. If $r_A = 16, r_B = 25$ and $r_C = 36$, determine the radius of $\Gamma$. | 74 | 112 | 2 |
math | Problem 4. Solve in the set of natural numbers the equation $21^{x}+4^{y}=z^{2}$. | (1,1,5) | 29 | 7 |
math | Example 1. The vertex of the parabola is at the origin, and the focus F is the center of the circle given by $x^{2}+y^{2}-4 x=0$. A line passing through point $F$ with a slope of 2 intersects the parabola at points $A$ and $D$, and intersects the circle at points $B$ and $C$.
Find $|A B|+|C D|$. | 6 | 97 | 1 |
math | (55) Given the complex number $z_{1}$ satisfies $\left(z_{1}-2\right)(1+\mathrm{i})=1-\mathrm{i}$ (where $\mathrm{i}$ is the imaginary unit), and the imaginary part of the complex number $z_{2}$ is 2, then the condition for $z_{1} \cdot z_{2}$ to be a real number is $z_{2}=$ $\qquad$ . | 4+2\mathrm{i} | 95 | 7 |
math | 12.94 The equation $x^{n}+(2+x)^{n}+(2-x)^{n}=0$ has a rational solution, the necessary and sufficient condition regarding the positive integer $n$ is what?
(15th Putnam Mathematical Competition, 1955) | 1 | 63 | 1 |
math | Find all pairs $(a, b)$ of coprime positive integers, such that $a<b$ and $$b \mid (n+2)a^{n+1002}-(n+1)a^{n+1001}-na^{n+1000}$$ for all positive integers $n$. | (3, 5) | 69 | 7 |
math | Out of five numbers, the first four form an arithmetic progression, the sum of which is 40. The last three form a geometric progression, in which the product of the two outer terms is 32 times the second of the five numbers. The five numbers are to be determined. | \frac{420}{9},\frac{200}{9},-\frac{20}{9},-\frac{240}{9},-320 | 59 | 39 |
math | ## 229. Math Puzzle $6 / 84$
A company has set itself the goal of reducing its energy consumption by 5 percent of the previous year's value each year.
After how many years will the energy consumption be only about 77 percent of the initial value? | 5 | 60 | 1 |
math | Determine all functions $f:\mathbb{R}\rightarrow \mathbb{R}$ such that the set
\[\left \{ \frac{f(x)}{x}: x \neq 0 \textnormal{ and } x \in \mathbb{R}\right \}\]
is finite, and for all $x \in \mathbb{R}$
\[f(x-1-f(x)) = f(x) - x - 1\] | f(x) = x | 100 | 6 |
math | 4. How many decimal numbers $\overline{a_{1} a_{2} a_{3} a_{4} a_{5} a_{6} a_{7} a_{8} a_{9} a_{10}}$, for which $a_{1}=1$ and each of the digits $a_{2}, a_{3}, \ldots, a_{10}$ is equal to 0 or 1, satisfy the condition
$$
a_{1}+a_{3}+a_{5}+a_{7}+a_{9}=a_{2}+a_{4}+a_{6}+a_{8}+a_{10} ?
$$ | 126 | 150 | 3 |
math | Two (not necessarily different) numbers are chosen independently and at random from $\{1, 2, 3, \dots, 10\}$. On average, what is the product of the two integers? (Compute the expected product. That is, if you do this over and over again, what will the product of the integers be on average?) | 30.25 | 74 | 5 |
math | For the complex-valued function $f(x)$ which is continuous and absolutely integrable on $\mathbb{R}$, define the function $(Sf)(x)$ on $\mathbb{R}$: $(Sf)(x)=\int_{-\infty}^{+\infty}e^{2\pi iux}f(u)du$.
(a) Find the expression for $S(\frac{1}{1+x^2})$ and $S(\frac{1}{(1+x^2)^2})$.
(b) For any integer $k$, let $f_k(x)=(... | (c_1, c_2) = (-2k, -4\pi^2) | 193 | 22 |
math | 3. (10 points) The simplest fraction $\frac{\mathrm{a}}{\mathrm{b}}$ satisfies $\frac{1}{5}<\frac{\mathrm{a}}{\mathrm{b}}<\frac{1}{4}$, and $b$ does not exceed 19, then the product of the maximum possible value and the minimum possible value of $a+b$ is | 253 | 82 | 3 |
math | 89. The triangle is defined by points $A(5 ; 2), B(-1 ;-4), C(-5$; -3). Form the equation of the line passing through point $B$ and parallel to $A C$. | x-2y-7=0 | 50 | 8 |
math | H4. The points $A, B$ and $C$ are the centres of three faces of a cuboid that meet at a vertex. The lengths of the sides of the triangle $A B C$ are 4,5 and 6 .
What is the volume of the cuboid? | 90\sqrt{6} | 61 | 7 |
math | # Assignment 1. (10 points)
Each of the 2017 middle school students studies English or German. English is studied by $70 \%$ to $85 \%$ of the total number of students, and both languages are studied by $5 \%$ to $8 \%$. What is the maximum number of students who can study German. | 766 | 75 | 3 |
math | Let $z$ be a complex number. If the equation \[x^3 + (4-i)x^2 + (2+5i)x = z\] has two roots that form a conjugate pair, find the absolute value of the real part of $z$.
[i]Proposed by Michael Tang[/i] | 423 | 68 | 3 |
math | II. (40 points) Find all real-coefficient polynomials $P(x)$ such that $P(x)$ is non-constant and
$$
P\left(x^{2}-2 x+4\right) \mid P\left(x^{3}+8\right) \text {. }
$$ | P(x)=^{n}(\in{R},\neq0,n\in{Z}_{+}) | 65 | 23 |
math | 4. 100 balls of the same mass move along a trough towards a metal wall with the same speed. After colliding with the wall, a ball bounces off it with the same speed. Upon collision of two balls, they scatter with the same speed. (The balls move only along the trough). Find the total number of collisions between the bal... | 4950 | 74 | 4 |
math | Find all natural numbers $n> 1$ for which the following applies:
The sum of the number $n$ and its second largest divisor is $2013$.
(R. Henner, Vienna) | n = 1342 | 44 | 8 |
math | 13.376 The desired three-digit number ends with the digit 1. If it is erased and then written as the first digit of the number, the new three-digit number obtained will be less than the desired one by $10 a^{\log _{\sqrt{a}} 3}$. Find this number. | 211 | 68 | 3 |
math | Find all $p, q$ primes such that $p q$ divides $2^{p}+2^{q}$. | (2,2),(2,3),(3,2) | 26 | 13 |
math | ## Problem Statement
Find the derivative.
$$
y=\frac{x+2}{x^{2}+4 x+6}+\frac{1}{\sqrt{2}} \cdot \operatorname{arctg} \frac{x+2}{\sqrt{2}}
$$ | \frac{4}{(x^{2}+4x+6)^{2}} | 59 | 19 |
math | 5. A5 (KOR) Let $\mathbb{R}^{+}$ be the set of all positive real numbers. Find all functions $f: \mathbb{R}^{+} \rightarrow \mathbb{R}^{+}$ that satisfy the following conditions: (i) $f(x y z) + f(x) + f(y) + f(z) = f(\sqrt{x y}) f(\sqrt{y z}) f(\sqrt{z x})$ for all $x, y, z \in \mathbb{R}^{+}$. (ii) $f(x) < f(y)$ for ... | f(x)=x^{k}+x^{-k} | 144 | 12 |
math | Solve the following system of equations:
$$
\log _{2 x} z=3 ; \log _{5 y} z=6 ;(1) \log _{x y} z=\frac{2}{3}
$$ | \frac{1}{2\sqrt[3]{10}},\quad\frac{1}{5\sqrt[6]{10}},\quad\frac{1}{10} | 51 | 40 |
math | Let's determine the sum of the following sequence:
$$
S_{n}=2\binom{n}{2}+6\binom{n}{3}+\cdots+(n-2)(n-1)\binom{n}{n-1}+(n-1) n\binom{n}{n}
$$ | s_{n}=n(n-1)2^{n-2} | 67 | 15 |
math | ## Task $7 / 78$
The smallest natural number of the form $2^{n}-1$ (with $n \in N, n>0$) that is divisible by 1001 without a remainder is sought. | 2^{60}-1 | 51 | 6 |
math | Example 6 When the volume of a cylindrical metal beverage can is fixed, how should its height and base radius be selected to minimize the material used? | 2R | 30 | 2 |
math | G4.3 If $c=2 \sqrt{3} \times \sqrt[3]{1.5} \times \sqrt[6]{12}$, find the value of $c$. | 6 | 43 | 1 |
math | 17. How many solutions does equation ||$|x-1|-1|-1|=1$ have?
The modulus function $|x|$ evaluates the absolute value of a number; for example $|6|=|-6|=6$. | 4 | 49 | 1 |
math | 48. A person's average speed going up the mountain (from the foot to the top) is $V_{1}$, and the average speed going down the mountain (from the top to the foot, returning the same way) is $V_{2}$,
From the start of the climb to the top and immediately descending back to the foot, the average speed for the entire jou... | \frac{2k}{1+k};2 | 145 | 10 |
math | 4. Find
$$
\frac{\sqrt{31+\sqrt{31+\sqrt{31+\ldots}}}}{\sqrt{1+\sqrt{1+\sqrt{1+\ldots}}}} .
$$ | 6-\sqrt{5} | 47 | 6 |
math | 5. Select some numbers without repetition from $1,2, \ldots 15$, such that the sum of any two numbers is not a perfect square of a natural number, then the maximum number of numbers that can be selected is ( ). | 8 | 51 | 1 |
math | For $n \in \mathbf{N}$, let $S_{n}$ be
$$
\sum_{k=1}^{n} \sqrt{(2 k-1)^{2}+a_{k}^{2}}
$$
the minimum value, where $a_{1}, a_{2}, \cdots, a_{n}$ are positive real numbers, and $a_{1}+a_{2}+\cdots+a_{n}=17$. If there exists a unique $n$ such that $S_{n}$ is also an integer, find the value of $n$. | 12 | 128 | 2 |
math | Find the least three digit number that is equal to the sum of its digits plus twice the product of its digits. | 397 | 23 | 3 |
math | ## Task 2 - 010922
a) A hemp rope with a diameter of $15 \mathrm{~mm}$ can withstand a load of $175 \mathrm{kp}$ without breaking.
What length of the rope corresponds to this load, i.e., when does the rope break under its own weight, if a rope of $1 \mathrm{~m}$ length weighs $1 \mathrm{p}$ per $\mathrm{mm}^{2}$ cros... | 990.3 | 182 | 5 |
math | 3-4. Find the integer $a$, for which
$$
(x-a)(x-10)+1
$$
can be factored into the product $(x+b)(x+c)$ of two factors with integer $b$ and $c$. | =8or=12 | 53 | 6 |
math | A $b^{2} x^{2}+a^{2} y^{2}=a^{2} b^{2}$ ellipse, draw a rectangle whose area $t$ is given. Determine the coordinates of the vertices of the rectangle. Furthermore, determine the coordinates of the vertices of the largest area rectangle that can be inscribed in the ellipse. | \xi=\\frac{}{2}\sqrt{2}\quad | 72 | 14 |
math | Given a line segment $|AB|=2a$, a moving point $P$ has a right angle view of $AB$, a moving point $Q$ is equidistant from $A$ and $B$, and $\triangle QAB$ has the same area as $\triangle PAB$ (but not zero). Point $O$ is the midpoint of $AB$, and $OP$ intersects $BQ$ at $M$. Taking $O$ as the origin and the perpendicul... | x^{2}+y^{2}=^{2}(y\neq0),\quad0(|y|\leqslanty\neq0),\quady^{2}=-2(x-\frac{}{2}) | 349 | 49 |
math | Example. Calculate the area of the parallelogram constructed on the vectors $\vec{a}=3 \vec{p}+2 \vec{q}$ and $\vec{b}=2 \vec{p}-\vec{q}$, given that $|\vec{p}|=4,|\vec{q}|=3$ and the angle between the vectors $\vec{p}$ and $\vec{q}$ is $3 \pi / 4$. | 42\sqrt{2} | 95 | 7 |
math | 10. Simplify: (1) $\sin 5 A-5 \sin 3 A+10 \sin A$;
(2) $1+C_{n}^{1} \cos x+C_{n}^{2} \cos 2 x+C_{n}^{3} \cos 3 x+\cdots+C_{n}^{n} \cos n x \quad\left(n \in \mathbf{N}^{*}\right)$. | 16\sin^{5}A | 101 | 8 |
math | 4. Find all natural numbers $a$ for which the number
$$
\frac{a+1+\sqrt{a^{5}+2 a^{2}+1}}{a^{2}+1}
$$
is also a natural number. | 1 | 54 | 1 |
math | Let $\mathcal{S}$ be the set of real numbers that can be represented as repeating decimals of the form $0.\overline{abc}$ where $a, b, c$ are distinct digits. Find the sum of the elements of $\mathcal{S}$. | 360 | 58 | 3 |
math | What is the probability with two dice
a) when rolled once, to roll more than 9?
b) when rolled three times in a row, to roll less than 6 each time?
(For the example, see the article titled 》Elements of Probability Theory《 in this issue.) | \frac{1}{6} | 60 | 7 |
math | Complex numbers $a,$ $b,$ and $c$ are zeros of a polynomial $P(z) = z^3 + qz + r,$ and $|a|^2 + |b|^2 + |c|^2 = 250.$ The points corresponding to $a,$ $b,$ and $c$ in the complex plane are the vertices of a right triangle with hypotenuse $h.$ Find $h^2.$ | 375 | 91 | 3 |
math | $9.133 \frac{|x+2|-|x|}{\sqrt{4-x^{3}}}>0$.
Translate the above text into English, please retain the original text's line breaks and format, and output the translation result directly.
$9.133 \frac{|x+2|-|x|}{\sqrt{4-x^{3}}}>0$. | x\in(-1;\sqrt[3]{4}) | 82 | 12 |
math | Compute the largest possible number of distinct real solutions for $x$ to the equation \[x^6+ax^5+60x^4-159x^3+240x^2+bx+c=0,\] where $a$, $b$, and $c$ are real numbers.
[i]Proposed by Tristan Shin | 4 | 74 | 1 |
math | 13. Two cleaning vehicles, A and B, are tasked with cleaning the road between East City and West City. Vehicle A alone would take 10 hours to clean the road, while Vehicle B alone would take 15 hours. The two vehicles start from East City and West City respectively, heading towards each other. When they meet, Vehicle A... | 60 | 93 | 2 |
math | Example 2.74. Calculate the work done by a body of mass $m$ when falling to the surface of the Earth from a height $h$. | \frac{}{R+} | 33 | 7 |
math | 4. In the infinite sequence $\left\{a_{n}\right\}$, $a_{1}=0, a_{n}=\frac{a_{n-1}+4}{a_{n-1}-2}(n \geq 2)$. If $\lim a_{n}=A$, then $A=$ | -1 | 70 | 2 |
math | 5. Three numbers are stored in a computer's memory. Every second, the following operation is performed: each number in this triplet is replaced by the sum of the other two. For example, the triplet $(1 ; 3 ; 7)$ turns into $(10 ; 8 ; 4)$. What will be the difference between the largest and smallest number in the triple... | 19 | 99 | 2 |
math | The sum of the three smallest distinct divisors of some number $A$ is 8. How many zeros can the number $A$ end with?
# | 1 | 32 | 1 |
math | ## Task A-4.5.
Given is a board of dimensions $2020 \times 2022$. Two fields of this board are said to be adjacent if they share a common side or if they are at the beginning and end of the same row or column. Thus, each field has exactly four adjacent fields.
Viktor, in each step, chooses one field of the board and ... | 5 | 132 | 1 |
math | Find all integers $n$ such that $2^{n}+3$ is a perfect square. Same question with $2^{n}+1$.
## - Solution - | n=0for2^n+3,\;n=3for2^n+1 | 37 | 18 |
math | 5. Xiao Hua plays a certain game, in each round he can play as many times as he wants, each time the score is one of the three numbers: $8, a$ (a natural number), 0. The total score of each round is called the total points of that round. Xiao Hua has once achieved the following total points: $103,104,105,106,107,108,10... | 13 | 129 | 2 |
math | Example 16 Polynomial $(1-z)^{b_{1}} \cdot\left(1-z^{2}\right)^{b_{2}} \cdot\left(1-z^{3}\right)^{b_{3}} \cdots \cdots \cdot\left(1-z^{32}\right)^{b_{32}}$, where $b_{i}$ are positive integers $(i=1,2, \cdots, 32)$, has the following remarkable property: when expanded, and terms with powers of $z$ higher than 32 are de... | 2^{27}-2^{11} | 166 | 10 |
math | ## A5.
Let $a, b, c$ and $d$ be real numbers such that $a+b+c+d=2$ and $a b+b c+c d+d a+a c+b d=0$.
Find the minimum value and the maximum value of the product $a b c d$.
| \()=-1 | 65 | 4 |
math | 3. In July, Volodya and Dima decided to start their own business producing non-carbonated bottled mineral water called "Dream," investing 1,500,000 rubles, and used these funds to purchase equipment for 500,000 rubles. The technical passport for this equipment indicates that the maximum production capacity is 100,000 b... | 372500 | 274 | 6 |
math | 2. (8 points) This year is 2014, and 2014 is not a perfect square, but the digits can be rearranged to form a new four-digit number that is a perfect square, for example, $1024=32^{2}$. It is known that using the digits $2, 0, 1, 4$ once each, another four-digit perfect square can be formed. What is this new four-digit... | 2401 | 107 | 4 |
math | 1. Calculate the value of the expression: $\frac{1}{\sqrt{15}+\sqrt{13}}+\frac{1}{\sqrt{13}-\sqrt{11}}-\frac{\sqrt{5.5}-\sqrt{7.5}}{\sqrt{2}}$. | \sqrt{15} | 65 | 6 |
math | How many positive common divisors do $10^{100}$ and $10^{121}+10^{813}+10$ have? | 4 | 38 | 1 |
math | Example 7 Given that $x_{1}, x_{2}, \cdots, x_{40}$ are all positive integers, and $x_{1}+x_{2}+\cdots+x_{40}=58$. If the maximum value of $x_{1}^{2}+x_{2}^{2}+\cdots+x_{40}^{2}$ is $A$, and the minimum value is $B$, then $A+B=$ $\qquad$ | 494 | 103 | 3 |
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