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 | 6. (5 points) There is a class of four-digit numbers, where the sum of any two adjacent digits is no greater than 2. If these numbers are arranged in ascending order, what is the second to last number? $\qquad$ . | 2011 | 52 | 4 |
math | 5. (4 points) An electrical circuit consists of a resistor with resistance $R$ and a capacitor with capacitance $C$ connected in series. A galvanic cell with electromotive force (emf) $\mathcal{E}$ and internal resistance $r$ is connected to the ends of the circuit. Determine the amount of heat released in the resistor... | Q_{R}=\frac{C\varepsilon^{2}R}{2(R+r)} | 269 | 20 |
math | 3. Let $[x]$ denote the greatest integer not exceeding the real number $x$. If $\left[x-\frac{1}{2}\right]\left[x+\frac{1}{2}\right]$ is a prime number, then the range of values for the real number $x$ is | -\frac{3}{2} \leqslant x<-\frac{1}{2} \text{ or } \frac{3}{2} \leqslant x<\frac{5}{2} | 61 | 47 |
math | 1. Using the digits 0 to 9, we form two-digit numbers $A B, C D, E F, G H, I J$, using each digit exactly once. Determine how many different values the sum $A B+C D+E F+G H+I J$ can take and what these values are. (Expressions like 07 are not considered two-digit numbers.)
(Jaroslav Zhouf) | 180to360 | 87 | 7 |
math | 6. Let $A B C$ be an equilateral triangle of side length 1 . For a real number $0<x<0.5$, let $A_{1}$ and $A_{2}$ be the points on side $B C$ such that $A_{1} B=A_{2} C=x$, and let $T_{A}=\triangle A A_{1} A_{2}$. Construct triangles $T_{B}=\triangle B B_{1} B_{2}$ and $T_{C}=\triangle C C_{1} C_{2}$ similarly.
There e... | (8,2) | 210 | 5 |
math | 2. If you take three different digits, form all six possible two-digit numbers using two different digits, and add these numbers, the result is 462. Find these digits. Provide all variants and prove that there are no others. | (6,7,8),(4,8,9),(5,7,9) | 49 | 19 |
math | 5. Xiao Ming and Xiao Hong independently and repeatedly roll a fair die until the first 6 appears. The probability that the number of rolls by Xiao Ming and Xiao Hong differs by no more than 1 is $\qquad$ | \frac{8}{33} | 46 | 8 |
math | Example 1 Given real numbers $x, y, z$ satisfy $x+y=5$ and $z^{2}=x y+y-9$.
Then $x+2 y+3 z=$ $\qquad$ . | 8 | 48 | 1 |
math | 15. Given the parabola $y=a x^{2}$ passes through the point $P(-1,1)$, a line $l$ with a positive slope is drawn through the point $Q\left(-\frac{1}{2}, 0\right)$ intersecting the parabola at points $M, N$ (point $M$ is between $Q$ and $N$). A line parallel to the $x$-axis is drawn through point $M$, intersecting $O P$... | S_{1}>3S_{2} | 175 | 9 |
math | Yakob and Baptiste are playing on a $20 \times 20$ grid where the cells are square and have a side length of 1. The distance between two cells is the distance between their centers. They take turns playing as follows: Yakob places a red stone on a cell, ensuring that the distance between any two cells with red stones i... | 100 | 165 | 3 |
math | 4. [5 points] Find the number of triples of natural numbers $(a ; b ; c)$ that satisfy the system of equations
$$
\left\{\begin{array}{l}
\operatorname{GCD}(a ; b ; c)=22 \\
\operatorname{LCM}(a ; b ; c)=2^{16} \cdot 11^{19}
\end{array}\right.
$$ | 9720 | 91 | 4 |
math | 1. Find all pairs of natural numbers $a, b$ greater than 1 such that both their sum and product are powers of prime numbers (with positive integer exponents). | (,b)=(2^{r},2^{r}), | 36 | 12 |
math | 9.114. For what values of $x$ is the following expression defined
$$
\log _{3}\left(1-\log _{0,5}\left(x^{2}-2 x-2.5\right)\right) ?
$$ | x\in(-\infty;-1)\cup(3;+\infty) | 56 | 18 |
math | 11. (15 points) The right focus of the hyperbola $3 x^{2}-y^{2}+24 x+36=0$ is $F$, and the right directrix is $l$. The ellipse $C$ has $F$ and $l$ as its corresponding focus and directrix. A line parallel to $y=x$ is drawn through $F$, intersecting the ellipse $C$ at points $A$ and $B$. It is known that the center of $... | 0<e<\sqrt{2-\sqrt{2}} | 136 | 13 |
math | Three, (25 points) Divide $1,2, \cdots, 9$ into three groups, each containing three numbers, such that the sum of the numbers in each group is a prime number.
(1) Prove that there must be two groups with equal sums;
(2) Find the number of all different ways to divide them.
| 12 | 73 | 2 |
math | 32nd CanMO 2000 Problem 2 How many permutations of 1901, 1902, 1903, ... , 2000 are such that none of the sums of the first n permuted numbers is divisible by 3 (for n = 1, 2, 3, ... , 2000)? | \frac{99!\cdot34!\cdot33!}{66!} | 82 | 19 |
math | 3. Determine the smallest natural number $n$ for which none of the fractions
$$
\frac{7}{n+9}, \frac{8}{n+10}, \frac{9}{n+11}, \ldots, \frac{2015}{n+2017}
$$
can be simplified. | 2015 | 73 | 4 |
math | 2. (8 points) Xiao Zhang has 200 pencils, and Xiao Li has 20 fountain pens. Each time Xiao Zhang gives Xiao Li 6 pencils, Xiao Li gives Xiao Zhang 1 fountain pen in return. After $\qquad$ such exchanges, the number of pencils Xiao Zhang has is 11 times the number of fountain pens Xiao Li has. | 4 | 77 | 1 |
math | I4.1 Let $a$ be a real number.
If $a$ satisfies the equation $\log _{2}\left(4^{x}+4\right)=x+\log _{2}\left(2^{x+1}-3\right)$, find the value of $a$ | 2 | 64 | 1 |
math | 5. Maja writes natural numbers on the board. If the number $n$ is written on the board, she writes $3n + 13$ on the board as well. If a perfect square is written on the board, she also writes its square root.
(a) Can Maja obtain the number 55 using the mentioned operations if the number $256$ is already written on the... | 55 | 151 | 2 |
math | Let $x_1,x_2,y_1,y_2$ be real numbers satisfying the equations $x^2_1+5x^2_2=10$, $x_2y_1-x_1y_2=5$, and $x_1y_1+5x_2y_2=\sqrt{105}$. Find the value of $y_1^2+5y_2^2$ | 23 | 96 | 2 |
math | Example 1. The density function of a random variable $X$ is given by
106
$$
p(x)=\frac{c}{1+x^{2}}
$$
Find the value of the parameter $c$. | \frac{1}{\pi} | 48 | 8 |
math | 2. Houses on the left side of the street have odd, and houses on the right side of the street have even house numbers. The sum of all house numbers on one side of the street is 1369, and on the other side 2162. How many houses are there on that street? | 83 | 66 | 2 |
math | Circles $C_1$ and $C_2$ intersect at points $X$ and $Y$ . Point $A$ is a point on $C_1$ such that the tangent line with respect to $C_1$ passing through $A$ intersects $C_2$ at $B$ and $C$, with $A$ closer to $B$ than $C$, such that $2016 \cdot AB = BC$. Line $XY$ intersects line $AC$ at $D$. If circles $C_1$ and $C_2$... | 2017 | 147 | 4 |
math | 9. (16 points) Given the ellipse $C: \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1(a>b>0)$, points $F_{1}, F_{2}$ are its left and right foci, respectively, and $A$ is the right vertex. A line $l$ passing through $F_{1}$ intersects the ellipse at points $P, Q$, and $\overrightarrow{A P} \cdot \overrightarrow{A Q}=\frac{1}{... | 1-\frac{\sqrt{2}}{2} | 142 | 11 |
math | ## Problem Statement
Find the angle between the planes:
\[
\begin{aligned}
& x+y+3z-7=0 \\
& y+z-1=0
\end{aligned}
\] | \arccos2\sqrt{\frac{2}{11}}\approx3128'56'' | 44 | 25 |
math | ## 46. Math Puzzle $3 / 69$
Write down a three-digit number and subtract the number with the reversed digit order. Divide the result by the difference between the 1st and 3rd digit of the original number, then divide by 11 again. If you now take the square root, the result is always 3. This is a neat trick to amaze yo... | 3 | 95 | 1 |
math | Example 7 Given non-negative real numbers $x_{1}, x_{2}, \cdots, x_{n}(n \geqslant 3)$ satisfy the inequality: $x_{1}+x_{2}+\cdots+$ $x_{n} \leqslant \frac{1}{2}$, find the minimum value of $\left(1-x_{1}\right)\left(1-x_{2}\right) \cdots\left(1-x_{n}\right)$. | \frac{1}{2} | 108 | 7 |
math | Example 9 (15th All-Soviet Union Mathematical Olympiad) Find all positive integer solutions to the equation $x^{3}-y^{3}=x y+61$.
Translate the above text into English, please keep the original text's line breaks and format, and output the translation result directly. | (x,y)=(6,5) | 65 | 7 |
math | ## Task B-1.5.
During the French Revolution, there was an idea to divide the day (a period of 24 hours) into 10 hours, and each hour into 100 minutes. Although this idea did not catch on, in Marko's school, they sometimes use such a "decimal clock." When Marko started solving the problem on the "decimal clock," it was... | 1.8 | 166 | 3 |
math | In an arithmetic sequence with 20 terms, the sum of the first three terms is 15 and the sum of the last three terms is 12 . What is the sum of the 20 terms in the sequence?
(An arithmetic sequence is a sequence in which each term after the first is obtained from the previous term by adding a constant. For example, $3,... | 90 | 95 | 2 |
math | XXIX OM - II - Problem 4
From the vertices of a regular $2n$-gon, 3 different points are chosen randomly. Let $p_n$ be the probability that the triangle with vertices at the chosen points is acute. Calculate $\lim_{n\to \infty} p_n$.
Note. We assume that all choices of three different points are equally probable. | \frac{1}{4} | 81 | 7 |
math | 3. (3 points) $\star+\square=24, \boldsymbol{\square}+\bullet=30, \bullet+\star=36, \boldsymbol{\square}=\underline{ }, \bullet=\underline{ }, \star=$ | 9,21,15 | 53 | 7 |
math | 13.312. When unloading a barge, four lifting cranes of the same power initially worked for 2 hours. Then, two additional cranes of lesser but equal power were put into operation. After this, it took another 3 hours to complete the unloading. If all these cranes had started working simultaneously, the unloading would ha... | 14.4 | 111 | 4 |
math | In the case of Sándor Mátyás, the leader of the patriots sent an equation to his friends:
$$
2 x^{2}+4 x y+7 y^{2}-12 x-2 y+N=0
$$
The secret message was that the uprising should take place on the $N$-th day of the month. The patriots often said that the uprising was the only solution. This also characterized $N$: th... | 23 | 120 | 2 |
math | Four, (50 points) Find all integers $a$, such that there exist distinct positive integers $x, y$, satisfying $(a x y+1) \mid\left(a x^{2}+1\right)^{2}$.
| \geqslant-1 | 51 | 7 |
math | 9. (16 points) Given that the three interior angles of $\triangle A B C$ satisfy
$$
\begin{array}{l}
\angle A+\angle C=2 \angle B, \\
\cos A+\cos C=-\frac{\sqrt{2} \cos A \cdot \cos C}{\cos B} .
\end{array}
$$
Find the value of $\cos \frac{A-C}{2}$. | \frac{\sqrt{2}}{2} | 94 | 10 |
math | Let $n$ be a fixed integer, $n \geqslant 2$.
a) Determine the smallest constant $c$ such that the inequality
$$
\sum_{1 \leqslant i<j \leqslant n} x_{i} x_{j}\left(x_{i}^{2}+x_{j}^{2}\right) \leqslant c\left(\sum_{i=1}^{n} x_{i}\right)^{4}
$$
holds for all non-negative real numbers $x_{1}, x_{2}, \cdots, x_{n} \geqsla... | \frac{1}{8} | 270 | 7 |
math | ## Problem 1
Let $a_{n}$ be the number written with $2^{n}$ nines. For example, $a_{0}=9, a_{1}=99, a_{2}=9999$. Let $b_{n}=\Pi_{0}{ }^{n} a_{i}$. Find the sum of the digits of $b_{n}$.
| 9\cdot2^{n} | 83 | 7 |
math | 15.5. The area of a trapezoid is three times that of an equilateral triangle. If the heights of the trapezoid and the triangle are both equal to $8 \sqrt{3}$, what is the length of the median of the trapezoid? | 24 | 61 | 2 |
math | Find the largest positive integer $n$ such that $n\varphi(n)$ is a perfect square. ($\varphi(n)$ is the number of integers $k$, $1 \leq k \leq n$ that are relatively prime to $n$) | 1 | 57 | 1 |
math | 9.5. In a round-robin chess tournament, two boys and several girls participated. The boys scored a total of 8 points, while all the girls scored an equal number of points. How many girls could have participated in the tournament? (Win - 1 point, draw - 0.5 points, loss - 0 points.) | 7or14 | 71 | 4 |
math | 10.351. Let $B D$ be the height of triangle $A B C$, and point $E$ be the midpoint of $B C$. Calculate the radius of the circle circumscribed around triangle $B D E$, if $A B=30 \text{ cm}, B C=26 \text{ cm}$ and $A C=28 \text{ cm}$. | 16.9 | 86 | 4 |
math | The fifth question: If an infinite sequence of positive real numbers $\left\{x_{n}\right\}$ satisfies: $x_{0}=1, x_{i} \geq x_{i+1}(i \in N)$, then the sequence is called a “good sequence”. Find the smallest constant $c$ such that there exists a “good sequence” $\left\{x_{n}\right\}$ satisfying: $\sum_{i=0}^{n} \frac{x... | 4 | 128 | 1 |
math | An arithmetic sequence has a common difference, $d$, that is a positive integer and is greater than 1. The sequence includes the three terms 3, 468 and 2018. What is the sum of all of the possible values of $d$ ?
(An arithmetic sequence is a sequence in which each term after the first is obtained from the previous ter... | 191 | 115 | 3 |
math | 3. (10 points) Some integers, when divided by $\frac{3}{5}, \frac{5}{7}, \frac{7}{9}, \frac{9}{11}$ respectively, yield quotients that, when expressed as mixed numbers, have fractional parts of $\frac{2}{3}, \frac{2}{5}, \frac{2}{7}, \frac{2}{9}$ respectively. The smallest integer greater than 1 that satisfies these co... | 316 | 106 | 3 |
math | Juliana wants to assign each of the 26 letters $A, B, C, D, \ldots, W, X, Y, Z$ of the alphabet a different non-zero numerical value, such that $A \times C=B, B \times D=C, C \times E=D$, and so on, up to $X \times Z=Y$.
a) If Juliana assigns the values 5 and 7 to $A$ and $B$, respectively, what will be the values of ... | 2010 | 207 | 4 |
math | G4.1 Regular tessellation is formed by identical regular $m$-polygons for some fixed $m$. Find the sum of all possible values of $m$. | 13 | 35 | 2 |
math | In a $4 \times 4$ chessboard composed of 16 small squares, 8 of the small squares are to be colored black, such that each row and each column has exactly 2 black squares. There are $\qquad$ different ways to do this. | 90 | 57 | 2 |
math | G3.1 Given that the solution of the equation $\sqrt{3 x+1}+\sqrt{3 x+6}=\sqrt{4 x-2}+\sqrt{4 x+3}$ is $a$, find the value of $a$. | 3 | 54 | 1 |
math | 7. Let the incircle of equilateral triangle $ABC$ have a radius of 2, with the center at $I$. If point $P$ satisfies $PI=1$, then the maximum value of the ratio of the areas of $\triangle APB$ to $\triangle APC$ is $\qquad$ . | \frac{3+\sqrt{5}}{2} | 65 | 12 |
math | 2. If $(n-1)$ ! is not divisible by $n^{2}$, find all possible values of $n$.
If $(n-1)$ ! cannot be divided by $n^{2}$, find all possible $n$. | n=8,9, p, 2p \text{ (where } p \text{ is a prime number)} | 51 | 26 |
math | 【Question 1】Calculate: $2 \times(999999+5 \times 379 \times 4789)=$ | 20150308 | 35 | 8 |
math | For example. Given that all edges of a parallelepiped are equal, and its four diagonals are $a$, $b$, $c$, and $d$, find the edge length of this parallelepiped. | \frac{\sqrt{3\left(a^{2}+b^{2}+c^{2}+d^{2}\right)}}{6} | 44 | 32 |
math | ## Problema 2
Să se calculeze $\int_{0}^{\frac{\pi}{2}} \frac{\sin ^{2} x+\sin x}{\sin x+\cos x+1} d x$.
| \frac{\pi}{4} | 51 | 7 |
math | Berpow S.l.
At the alumni meeting, 45 people attended. It turned out that any two of them who had the same number of acquaintances among those present were not acquainted with each other. What is the maximum number of pairs of acquaintances that could have been among those who attended the meeting? | 870 | 63 | 3 |
math | 4. Given $n$ distinct real numbers, the set of all their permutations is $A_{n}$. For $\left(a_{1}, a_{2}, \cdots, a_{n}\right) \in A_{n}$, if there are exactly two distinct integers $i, j \in\{1,2, \cdots, n-1\}$ such that $a_{i}>a_{i+1}, a_{j}>a_{j+1}$ hold, then the permutation is called a "good permutation". Find t... | 3^{n}-(n+1)\cdot2^{n}+\frac{1}{2}n(n+1)(n\in{N}^{*}) | 129 | 35 |
math | ## Task 15/70
Find all prime numbers that can be represented both as the sum and as the difference of two prime numbers. | 5 | 30 | 1 |
math | F6 (22-6, Finland) The function $f(x, y)$ satisfies the following conditions for all non-negative integers $x, y$:
(1) $f(0, y)=y+1$;
(2) $f(x+1,0)=f(x, 1)$;
(3) $f(x+1, y+1)=f(x, f(x+1, y))$.
Determine $f(4,1981)$. | f(4,1981)=-3+2^{2^{1984}} | 104 | 21 |
math | Let $m$ be a positive integer and let $A$, respectively $B$, be two alphabets with $m$, respectively $2m$ letters. Let also $n$ be an even integer which is at least $2m$. Let $a_n$ be the number of words of length $n$, formed with letters from $A$, in which appear all the letters from $A$, each an even number of times... | 2^{n-m} | 140 | 5 |
math | \left.\begin{array}{l}{[\quad \text { Pythagorean Theorem (direct and inverse). }} \\ \text { [ The ratio in which the bisector divides the side. }]\end{array}\right]
The height of a triangle, equal to 2, divides the angle of the triangle in the ratio 2:1, and the base of the triangle - into parts, the smaller of whic... | \frac{11}{3} | 101 | 8 |
math | 39. For $n \rightarrow+\infty$ find the limits of the following functions:
1) $S_{1}(n)=\frac{1}{n}+\frac{2}{n}+\frac{3}{n}+\ldots+\frac{n-1}{n}$;
2) $S_{2}(n)=\frac{1}{n^{2}}+\frac{2}{n^{2}}+\frac{3}{n^{2}}+\ldots+\frac{n-1}{n^{2}}$;
3) $S_{3}(n)=\frac{1}{n^{3}}+\frac{2}{n^{3}}+\frac{3}{n^{3}}+\ldots+\frac{n-1}{n^{3}}... | S_1arrow+\infty,\quadS_2arrow\frac{1}{2},\quadS_3arrow0 | 163 | 27 |
math | Example 1 Given the complex number $z_{1}=\mathrm{i}(1-\mathrm{i})^{3}$, when the complex number $z$ satisfies $|z|=1$, find the maximum value of $\left|z-z_{1}\right|$.
(2001 National College Entrance Examination Question) | 2\sqrt{2}+1 | 66 | 8 |
math | Let's determine the digits $A, B, C, D$ if
$$
A^{\overline{A B}}=\overline{C C B B D D C A}
$$ | A=2,B=5,C=3,D=4 | 41 | 12 |
math | Cla
Two circles with centers $O$ and $Q$, intersecting each other at points $A$ and $B$, intersect the bisector of angle $O A Q$ at points $C$ and $D$ respectively. Segments $A D$ and $O Q$ intersect at point $E$, and the areas of triangles $O A E$ and $Q A E$ are 18 and 42 respectively. Find the area of quadrilateral... | 200;3:7 | 113 | 7 |
math | ## 14. Math Puzzle $7 / 66$
While the truck driver was checking his odometer during the drive, it showed $15951 \mathrm{~km}$. He noticed that this reading was a symmetrical number, a number that reads the same forwards and backwards.
After two hours of driving, the odometer again displayed a number that reads the sa... | 55\frac{\mathrm{}}{\mathrm{}} | 97 | 12 |
math | 8、We know that the number of divisors of $2013$, $2014$, and $2015$ are the same. Therefore, for three consecutive natural numbers $n$, $n+1$, and $n+2$ that have the same property (the same number of divisors), the smallest value of $\mathbf{n}$ is | 33 | 78 | 2 |
math | [ Decimal numeral system ]
Can you find two consecutive numbers, where the sum of the digits of the first one is 8, and the second one is divisible by 8?
# | 7172 | 37 | 4 |
math | Problem 18. (4 points)
By producing and selling 4000 items at a price of 6250 rubles each, a budding businessman earned 2 million rubles in profit. Variable costs for one item amounted to 3750 rubles. By what percentage should the businessman reduce the production volume to make his revenue equal to the cost? (Provide... | 20 | 103 | 2 |
math | ## Task Condition
Approximately calculate using the differential.
$y=x^{6}, x=2.01$ | 65.92 | 24 | 5 |
math | 6.168. $\sqrt{3 x^{2}-2 x+15}+\sqrt{3 x^{2}-2 x+8}=7$. | x_{1}=-\frac{1}{3},x_{2}=1 | 35 | 17 |
math | Let $a, b, c$ be positive integers such that $a + 2b +3c = 100$.
Find the greatest value of $M = abc$ | 6171 | 39 | 4 |
math | Example 10. At an enterprise, products of a certain type are manufactured on three production lines. The first line produces $30 \%$ of the products from the total production volume, the second line - $25 \%$, and the third line produces the remaining part of the products. Each line is characterized by the following pe... | 0.032 | 105 | 5 |
math | Example 14 (Question from the 10th "Hope Cup" Invitational Competition) Given that real numbers $x, y$ satisfy the equation $(x+2)^{2}+y^{2}=1$, what is the minimum value of $\frac{y-1}{x-2}$? | 0 | 65 | 1 |
math | 2. The product of two two-digit numbers is 4032. The second number is written with the same digits as the first, but in reverse order. What are these numbers? | 8448 | 39 | 4 |
math | 4. A trapezoid $ABCD (AD \| BC)$ and a rectangle $A_{1}B_{1}C_{1}D_{1}$ are inscribed in a circle $\Omega$ with radius 13, such that $AC \perp B_{1}D_{1}, BD \perp A_{1}C_{1}$. Find the ratio of the area of $ABCD$ to the area of $A_{1}B_{1}C_{1}D_{1}$, given that $AD=10, BC=24$. | \frac{289}{338} | 124 | 11 |
math | 3. For the cyclic quadrilateral $ABCD$, the lengths of the four sides in sequence are $AB=2, BC=7, CD=6, DA=9$. Then the area of the quadrilateral is $\qquad$ . | 30 | 50 | 2 |
math | Let $n\ge 1$ be a fixed integer. Calculate the distance $\inf_{p,f}\, \max_{0\le x\le 1} |f(x)-p(x)|$ , where $p$ runs over polynomials of degree less than $n$ with real coefficients and $f$ runs over functions $f(x)= \sum_{k=n}^{\infty} c_k x^k$ defined on the closed interval $[0,1]$ , where $c_k \ge 0$ and $\... | 2^{-2n+1} | 135 | 9 |
math |
Opgave 3. Vind alle functies $f: \mathbb{Z} \rightarrow \mathbb{Z}$ die voldoen aan
$$
f(-f(x)-f(y))=1-x-y
$$
voor alle $x, y \in \mathbb{Z}$.
| f(x)=x-1 | 67 | 6 |
math | Exercise 4. Let $n \geqslant 3$ be an integer. For each pair of prime numbers $p$ and $q$ such that $p<q \leqslant n$, Morgane has written the sum $p+q$ on the board. She then notes $\mathcal{P}(n)$ as the product of all these sums. For example, $\mathcal{P}(5)=(2+3) \times(2+5) \times(3+5)=280$.
Find all values of $n... | 7 | 197 | 1 |
math | 67 If $x>0, y>0, z>0$, and $x^{2}+y^{2}+z^{2}=1$, then the minimum value of $\frac{y z}{x}+\frac{x z}{y}+\frac{x y}{z}$ is | \sqrt{3} | 62 | 5 |
math | 23.1. (SRP, 62). Under what restrictions on the integers $p$ and $q$:
a) the polynomial $P(x)=x^{2}+p x+q$ takes even (odd) values for all $x \in \mathbf{Z}$;
b) the polynomial $Q(x)=x^{3}+p x+q$ takes values divisible by 3 for all $x \in \mathbf{Z}$? | q\equiv0(\bmod3),\quadp\equiv2(\bmod3) | 101 | 20 |
math | 1.5.2 * Let real numbers $a, x, y$ satisfy the following conditions
$$
\left\{\begin{array}{l}
x+y=2 a-1, \\
x^{2}+y^{2}=a^{2}+2 a-3 .
\end{array}\right.
$$
Find the minimum value that the real number $xy$ can take. | \frac{11-6\sqrt{2}}{4} | 84 | 15 |
math | 4. Solve the inequality $4+x^{2}+2 x \sqrt{2-x^{2}}<8 \sqrt{2-x^{2}}+5 x$. (20 points) | (-1;\sqrt{2}] | 41 | 7 |
math | 49*. Factor the polynomial with integer coefficients:
$$
x^{5}+x+1
$$ | (x^{2}+x+1)(x^{3}-x^{2}+1) | 22 | 20 |
math | Let $z=z(x,y)$ be implicit function with two variables from $2sin(x+2y-3z)=x+2y-3z$. Find $\frac{\partial z}{\partial x}+\frac{\partial z}{\partial y}$. | 1 | 55 | 1 |
math | For real number $x$ let $\lfloor x\rfloor$ be the greatest integer less than or equal to $x$, and define $\{x\}=x-\lfloor x\rfloor$ to be the fractional part of $x$. For example, $\{3\}=0$ and $\{4.56\}=0.56$. Define $f(x)=x\{x\}$, and let $N$ be the number of real-valued solutions to the equation $f(f(f(x)))=17$ for $... | 10 | 146 | 2 |
math | 19. A bank issues ATM cards to its customers. Each card is associated with a password, which consists of 6 digits with no three consecutive digits being the same. It is known that no two cards have the same password. What is the maximum number of ATM cards the bank has issued?
A bank issues ATM cards to its customers.... | 963090 | 117 | 6 |
math | Let $\Phi$ and $\Psi$ denote the Euler totient and Dedekind‘s totient respectively. Determine all $n$ such that $\Phi(n)$ divides $n +\Psi (n)$. | \{1, 2^{n_1}, 2^{n_1}3^{n_2}, 2^{n_1}5^{n_2} : n_1, n_2 \in \mathbb{N}\} | 43 | 55 |
math | II. (40 points) Let positive real numbers $a, b, c$ satisfy
$$
\begin{array}{l}
a+b=\sqrt{a b+9}, \\
b+c=\sqrt{b c+16}, \\
c+a=\sqrt{c a+25} .
\end{array}
$$
Find $a+b+c$. | \sqrt{25+12 \sqrt{3}} | 77 | 13 |
math | 10. Given that $f(x)$ is a function defined on
$\mathbf{R}$, $f(1)=1$ and for any $x \in \mathbf{R}$ we have
$$
f(x+5) \geqslant f(x)+5, f(x+1) \leqslant f(x)+1 \text {. }
$$
If $g(x)=f(x)+1-x$, then $g(2002)=$ | 1 | 103 | 1 |
math | 6.7 The denominator of the geometric progression is $\frac{1}{3}$, the fourth term of this progression is $\frac{1}{54}$, and the sum of all its terms is $\frac{121}{162}$. Find the number of terms in the progression. | 5 | 63 | 1 |
math | 15. Find the maximum value of the positive real number $A$ such that for any real numbers $x, y, z$, the inequality
$$
\begin{array}{l}
x^{4}+y^{4}+z^{4}+x^{2} y z+x y^{2} z+x y z^{2}- \\
A(x y+y z+z x)^{2} \geqslant 0
\end{array}
$$
holds. | \frac{2}{3} | 102 | 7 |
math | Let $d(n)$ denote the number of positive divisors of the positive integer $n$. Solve the equation
$$
d\left(n^{3}\right)=n
$$ | 1,28,40 | 37 | 7 |
math | 28. It is given that $a, b, c$ and $d$ are four positive prime numbers such that the product of these four prime numbers is equal to the sum of 55 consecutive positive integers. Find the smallest possible value of $a+b+c+d$. (Remark: The four numbers $a, b, c, d$ are not necessarily distinct.) | 28 | 77 | 2 |
math | At 7:00 a.m. yesterday, Sherry correctly determined what time it had been 100 hours before. What was her answer? (Be sure to include "a.m." or "p.m." in your answer.) | 3:000 | 50 | 5 |
math | 4. A reservoir contains 400 tons of water. At midnight every day, the inlet and outlet gates are opened simultaneously. The amount of water $w$ (tons) flowing out through the outlet gate is a function of time $t$ (hours): $w=120 \sqrt{6 t}(0 \leqslant t \leqslant 24)$.
(1) If the reservoir is to contain 400 tons of wat... | 40 | 172 | 2 |
math | 6. The number of triangles with different shapes that can be formed using the vertices of a regular tridecagon is $\qquad$ (Note: Congruent triangles are considered to have the same shape). | 14 | 43 | 2 |
math | 1. Determine all triples $(a, b, c)$ of natural numbers for which
$$
2^{a}+4^{b}=8^{c} .
$$ | (6n-4,3n-2,2n-1) | 35 | 16 |
math | 7. If the function $y=a^{2 x}+2 a^{x}-9(a>0, a \neq 1)$, has a maximum value of 6 on the interval $[-1,1]$, then $a=$ | 3 | 52 | 1 |
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