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 | [Decimal numeral system]
A two-digit number, when added to the number written with the same digits but in reverse order, results in a perfect square. Find all such numbers.
# | 29,38,47,56,65,74,83,92 | 37 | 23 |
math | 3. (43rd American High School Mathematics Examination) Let $S$ be a subset of the set $\{1,2, \cdots, 50\}$ with the following property: the sum of any two distinct elements of $S$ cannot be divisible by 7. What is the maximum number of elements that $S$ can have? | 23 | 74 | 2 |
math | A [i]strip[/i] is the region between two parallel lines. Let $A$ and $B$ be two strips in a plane. The intersection of strips $A$ and $B$ is a parallelogram $P$. Let $A'$ be a rotation of $A$ in the plane by $60^\circ$. The intersection of strips $A'$ and $B$ is a parallelogram with the same area as $P$. Let $x^\c... | 150^\circ | 135 | 5 |
math | 3. Given an integer $n \geqslant 2$. Find the smallest positive real number $c$, such that for any complex numbers $z_{1}, z_{2}, \cdots, z_{n}$, we have
$$
\left|\sum_{i=1}^{n} z_{i}\right|+c \sum_{1 \leqslant i<j \leqslant n}\left|z_{i}-z_{j}\right| \geqslant \sum_{i=1}^{n}\left|z_{i}\right| .
$$ | \frac{2}{n} | 128 | 7 |
math | Consider a rectangle $ABCD$ with $AB = a$ and $AD = b.$ Let $l$ be a line through $O,$ the center of the rectangle, that cuts $AD$ in $E$ such that $AE/ED = 1/2$. Let $M$ be any point on $l,$ interior to the rectangle.
Find the necessary and sufficient condition on $a$ and $b$ that the four distances from M to lines $A... | a = b | 116 | 3 |
math | 9. (NET 2) ${ }^{\text {IMO4 }}$ Find all solutions in positive real numbers $x_{i}(i=$ $1,2,3,4,5)$ of the following system of inequalities:
$$
\begin{array}{l}
\left(x_{1}^{2}-x_{3} x_{5}\right)\left(x_{2}^{2}-x_{3} x_{5}\right) \leq 0, \\
\left(x_{2}^{2}-x_{4} x_{1}\right)\left(x_{3}^{2}-x_{4} x_{1}\right) \leq 0, \... | x_{1}=x_{2}=x_{3}=x_{4}=x_{5} | 281 | 20 |
math | 1. Find the maximum value of the expression $x+y$, where $x, y-$ are integer solutions of the equation $3 x^{2}+5 y^{2}=345$ | 13 | 41 | 2 |
math | 9.2. Electronic clocks display time from 00.00.00 to 23.59.59. How much time during the day does the number on the display that reads the same from left to right and from right to left light up? | 96 | 57 | 2 |
math | 51 Let the set $S=\{100,101,102, \cdots, 999,1000\}, A=\left\{a_{1}, a_{2}, a_{3}, \cdots, a_{n-1}, a_{n}\right.$ $\mid a_{1}, a_{2}, \cdots, a_{n}$ are positive numbers, and $\left.\frac{a_{2}}{a_{1}}=\frac{a_{3}}{a_{2}}=\cdots=\frac{a_{n}}{a_{n-1}}=q>1\right\}$. Try to find the maximum possible number of elements in ... | 6 | 162 | 1 |
math | 5. Given real numbers $x, y$ satisfy
$$
\frac{4}{x^{4}}-\frac{2}{x^{2}}=3, y^{4}+y^{2}=3 \text {. }
$$
Then the value of $\frac{4}{x^{4}}+y^{4}$ is $\qquad$
(2008, "Mathematics Weekly Cup" National Junior High School Mathematics Competition) | 7 | 93 | 1 |
math | Example 3: On a plane, there are $n$ lines, no two of which are parallel, and no three of which intersect at the same point. How many intersection points do these $n$ lines have in total?
Example 4: Following the previous example, now we want to know how many non-overlapping regions will be formed by such $n$ lines div... | a_{n}=\frac{n(n+1)}{2}+1 | 80 | 16 |
math | $1.$A bottle in the shape of a cone lies on its base. Water is poured into the bottle until its level reaches a distance of 8 centimeters from the vertex of the cone (measured vertically). We now turn the bottle upside down without changing the amount of water it contains; This leaves an empty space in the upper part o... | 10 | 87 | 2 |
math | The points $P(2,0), Q(11,-3)$ and $R(x, 3)$ are the vertices of a triangle with $\angle P Q R=90^{\circ}$. What is the value of $x$ ? | 13 | 53 | 2 |
math | 2. (7 points) The kids were given the task to convert the turtle's speed from centimeters per second to meters per minute. Masha got an answer of 25 m/min, but she thought there were 60 cm in a meter and 100 seconds in a minute. Help Masha find the correct answer. | 9 | 70 | 1 |
math | $10 \cdot 81$ Find the last two digits of $\left[(\sqrt{29}+\sqrt{21})^{1984}\right]$.
(25th International Mathematical Olympiad Candidate Problem, 1984) | 71 | 56 | 2 |
math | 15. (15 points) 4 consecutive natural numbers, from smallest to largest, are multiples of 11, 7, 5, and 3, respectively. Find the minimum sum of these 4 natural numbers. | 1458 | 49 | 4 |
math | 7.4. In triangle $A B C$, the angles $A$ and $C$ at the base are $20^{\circ}$ and $40^{\circ}$, respectively. It is known that $A C - A B = 5$ (cm). Find the length of the angle bisector of angle $B$. | 5 | 72 | 1 |
math | 13. Fox Alice and Cat Basilio (8th grade. 2 points). Every day, Cat Basilio and Fox Alice visit all 20 courtyards of the capital of the Country of Fools, and in each courtyard, they either receive or do not receive one gold coin with a probability of $\frac{1}{2}$. If the number of gold coins collected by the end of th... | 5.25 | 137 | 4 |
math | 6. Let's call the distance between numbers the absolute value of their difference. It is known that the sum of the distances from thirty-three consecutive natural numbers to some number $a$ is 3168, and the sum of the distances from these same thirty-three numbers to some number $b$ is 924. Find all possible values of ... | =26,=-2,=122 | 86 | 11 |
math | \section*{Problem 3 - 021133}
In how many different ways can the number 99 be expressed as the sum of three distinct prime numbers?
(Two cases are considered the same if the same addends appear, merely in a different order.) | 21 | 58 | 2 |
math | How many ways are there to list the numbers 1 to 10 in some order such that every number is either greater or smaller than all the numbers before it? | 512 | 34 | 3 |
math | $4.1 \times 1+2 \times 2+3 \times 3+\ldots .2011 \times 2011+2012 \times 2012$ The last digit of the sum is $\qquad$ | 0 | 59 | 1 |
math | 13th Putnam 1953 Problem B3 k is real. Solve the differential equations y' = z(y + z) k , z' = y(y + z) k subject to y(0) = 1, z(0) = 0. | y=\frac{1}{2}((1-kx)^{1/k}+\frac{1}{(1-kx)^{1/k}}),\quadz=\frac{1}{2}((1-kx)^{1/k}-\frac{1}{(1-kx)^{1/k}})fork\neq0,\ | 58 | 72 |
math | 6. The sequence $a_{1}, a_{2}, a_{3}, \cdots$ satisfies (1) $a_{1}=\frac{1}{2}$, (2) $a_{1}+a_{2}+\cdots+a_{n}=n^{2} a_{n}(n \geqslant 2)$, then $a_{n}=$ $\qquad$ | \frac{1}{n(n+1)} | 87 | 10 |
math | Example 9 (1994 National High School League Question) Given $x, y \in\left[-\frac{\pi}{4}, \frac{\pi}{4}\right]$, and $\left\{\begin{array}{l}x^{3}+\sin x-2 a=0, \\ 4 y^{3}+\sin y \cdot \cos y+a=0,\end{array}\right.$ then $\cos (x+2 y)=$ $\qquad$ | 1 | 104 | 1 |
math | 5. (6 points) Two identical conducting spheres are located at a large distance from each other and have positive charges $Q_{1}$ and $Q_{2}$. A neutral metal ball on a non-conductive suspension is brought close to the first sphere and touches it. Then the ball is brought close to the second sphere and touches it. After... | Q_{2}^{\}=\frac{Q_{2}}{2}-q_{2}\\sqrt{\frac{Q_{2}^{2}}{4}+Q_1q_{2}} | 357 | 43 |
math | Find all functions $f: \mathbb N \to \mathbb N$ Such that:
1.for all $x,y\in N$:$x+y|f(x)+f(y)$
2.for all $x\geq 1395$:$x^3\geq 2f(x)$ | f(n) = kn | 67 | 6 |
math | 5. Seven fishermen stand in a circle. The fishermen have a professional habit of exaggerating numbers. Each fisherman has a measure of lying (each has their own, an integer) - how many times the number mentioned by the fisherman is greater than the true value. For example, if a fisherman with a lying measure of 3 catch... | 16 | 183 | 2 |
math | Problem 5.5. On some trees in the magical forest, coins grow. The number of trees that do not grow any coins at all is twice as many as the trees that grow three coins. On three trees, two coins grow, on four trees - four coins, and no tree grows more than four coins. By how much is the total number of coins in the mag... | 15 | 85 | 2 |
math | Six, (Full marks 12 points) On a circle, there are
12 points, one of which is painted red, and another is painted blue, with the remaining 10 points unpainted. Among the convex polygons formed by these points, those whose vertices include both the red and blue points are called bicolored polygons; those that include on... | 55 | 161 | 2 |
math | 4.3.12 ** Positive real numbers $a, b, c$ and non-negative real numbers $x, y$ satisfy the condition
$$
a^{6}+b^{6}+c^{6}=3 .(x+1)^{2}+y^{2} \leqslant 2 .
$$
Find the minimum value of $I=\frac{1}{2 a^{3} x+b^{3} y^{2}}+\frac{1}{2 b^{3} x+c^{3} y^{2}}+\frac{1}{2 c^{3} x+a^{3} y^{2}}$. | 3 | 137 | 1 |
math | 3. In a $999 \times 999$ grid, some cells are white, and the others are red. Let $T$ be the number of cell groups $\left(C_{1}, C_{2}, C_{3}\right)$ such that $C_{1}$ and $C_{2}$ are in the same row, $C_{2}$ and $C_{3}$ are in the same column, and $C_{1}$ and $C_{3}$ are white, while $C_{2}$ is red. Find the maximum va... | 148 \times 999^{3} | 122 | 12 |
math | 16. A polygon is said to be friendly if it is regular and it also has angles that when measured in degrees are either integers or half-integers (i.e. have a decimal part of exactly 0.5 ). How many different friendly polygons are there? | 28 | 54 | 2 |
math | Given a positive integer $n$. Find the smallest real number $\lambda$, such that there exist real numbers $a_{1}, a_{2}, \cdots, a_{n}$ in the interval $[0,1]$, for any real numbers $x_{i}(i=1,2, \cdots, n)$ satisfying $0 \leqslant x_{1} \leqslant x_{2} \leqslant \cdots \leqslant x_{n} \leqslant 1$, we have
$$
\min _{1... | \frac{1}{2n} | 154 | 8 |
math | 5. The function $f(x)=a^{2 x}+3 a^{x}-2(a>0, a \neq 1)$ has a maximum value of 8 on the interval $x \in[-1,1]$, then its minimum value on this interval is $\qquad$ | -\frac{1}{4} | 63 | 7 |
math | Find all functions $f: \mathbf{R} \rightarrow \mathbf{R}$, such that for all real numbers $x, y$, we have
$$
f\left(f(y)+x^{2}+1\right)+2 x=y+f^{2}(x+1) \text {. }
$$
(2014, Turkey National Team Selection Exam) | f(x)=x | 81 | 4 |
math | Ximena wishes to number the pages of a notebook. For this, she has a large quantity of stickers with the digits $0,1,3,4,5,6,7,8$ and 9, but she has only 100 stickers with the digit 2. Determine up to which page Ximena can number this notebook.
# | 244 | 75 | 3 |
math | Find an irreducible fraction such that the product of its numerator by its denominator is $2 \times 3 \times 4 \times 5 \times \cdots \times 10$. How many of these irreducible fractions exist? | 16 | 50 | 2 |
math | 15. Given the function $f(x)=a_{1} x+a_{2} x^{2}+\cdots+a_{n} x^{n}$
$\left(n \in \mathbf{N}_{+}\right), a_{1}, a_{2}, \cdots, a_{n}$ are terms of a sequence. If $f(1)=n^{2}+1$, (1) find the general term formula of the sequence $\left\{a_{n}\right\}$; (2) find $\lim _{n \rightarrow \infty}\left(1-\frac{1}{a_{n}}\right)... | e^{-\frac{1}{2}} | 142 | 9 |
math | 2. Given a positive integer $n$ less than 2006, and $\left[\frac{n}{3}\right]+\left[\frac{n}{6}\right]=\frac{n}{2}$.
Then the number of such $n$ is $\qquad$. | 334 | 57 | 3 |
math | 2. Given $a^{2}+b^{2}+c^{2}=1$,
$$
a\left(\frac{1}{b}+\frac{1}{c}\right)+b\left(\frac{1}{c}+\frac{1}{a}\right)+c\left(\frac{1}{a}+\frac{1}{b}\right)=-3 \text {. }
$$
Find the value of $a+b+c$. | +b+=0,-1,1 | 97 | 7 |
math | In a given circle, we draw two diameters that are perpendicular to each other, and from an arbitrary point $P$ on the circumference of the circle, we drop perpendiculars to these diameters. Determine the position of $P$ such that the sum of the lengths of these perpendiculars is maximized. | \alpha=45 | 63 | 5 |
math | A natural number $n$ is said to be $good$ if $n$ is the sum or $r$ consecutive positive integers, for some $r \geq 2 $. Find the number of good numbers in the set $\{1,2 \dots , 100\}$. | 93 | 62 | 2 |
math | Determine all positive integers $n \ge 2$ which have a positive divisor $m | n$ satisfying $$n = d^3 + m^3.$$
where $d$ is the smallest divisor of $n$ which is greater than $1$. | 16, 72, 520 | 54 | 11 |
math | 6. Given $p(x)=a x^{3}+b x^{2}+c x+d$ is a cubic polynomial, satisfying
$$
p\left(\frac{1}{2}\right)+p\left(-\frac{1}{2}\right)=1000 p(0) \text {. }
$$
Let $x_{1} 、 x_{2} 、 x_{3}$ be the three roots of $p(x)=0$. Then the value of $\frac{1}{x_{1} x_{2}}+\frac{1}{x_{2} x_{3}}+\frac{1}{x_{1} x_{3}}$ is $\qquad$ | 1996 | 149 | 4 |
math | 9. (16 points) Given that $F$ is the focus of the parabola $y^{2}=4 x$, $Q$ is the intersection point of its directrix with the $x$-axis, and line $l$ passes through point $Q$. Let line $l$ intersect the parabola at points $A$ and $B$.
(1) Let the slopes of lines $A F$ and $B F$ be $k_{1}$ and $k_{2}$, respectively. Fi... | x=1(-2<y<2, y \neq 0) | 186 | 16 |
math | 2. Let real numbers $x, y, z, w$ satisfy $x+y+z+w=x^{7}+y^{7}+z^{7}+w^{7}=0$, find the value of $w(w+x)(w+y)(w+z)$.
(IMO - 26 Shortlist) | 0 | 66 | 1 |
math | Suppose points $F_1, F_2$ are the left and right foci of the ellipse $\frac{x^2}{16}+\frac{y^2}{4}=1$ respectively, and point $P$ is on line $l:$, $x-\sqrt{3} y+8+2\sqrt{3}=0$. Find the value of ratio $\frac{|PF_1|}{|PF_2|}$ when $\angle F_1PF_2$ reaches its maximum value. | \sqrt{3} - 1 | 109 | 8 |
math | Triangle $ ABC$ is isosceles with $ AC \equal{} BC$ and $ \angle{C} \equal{} 120^o$. Points $ D$ and $ E$ are chosen on segment $ AB$ so that $ |AD| \equal{} |DE| \equal{} |EB|$. Find the sizes of the angles of triangle $ CDE$. | 60^\circ, 60^\circ, 60^\circ | 80 | 16 |
math | 11. Let $\triangle A B C$ be inscribed in a circle $\odot O$ with radius $R$, and $A B=A C, A D$ be the altitude from $A$ to the base $B C$. Then the maximum value of $A D + B C$ is $\qquad$ . | R+\sqrt{5} R | 68 | 7 |
math | The sequence $\{c_{n}\}$ is determined by the following equation. \[c_{n}=(n+1)\int_{0}^{1}x^{n}\cos \pi x\ dx\ (n=1,\ 2,\ \cdots).\] Let $\lambda$ be the limit value $\lim_{n\to\infty}c_{n}.$ Find $\lim_{n\to\infty}\frac{c_{n+1}-\lambda}{c_{n}-\lambda}.$ | 1 | 112 | 1 |
math | 9. If $a \in A$, and $a-1 \notin A, a+1 \notin A$, then $a$ is called an isolated element of set $A$. Therefore, the number of four-element subsets of set $M=\{1,2, \cdots, 9\}$ without isolated elements is $\qquad$ . | 21 | 74 | 2 |
math | 448. A cyclist set off from point A to point B, and 15 minutes later, a car set off after him. Halfway from A to B, the car caught up with the cyclist. When the car arrived at B, the cyclist still had to cover another third of the entire distance. How long will it take the cyclist to travel the distance from A to B? | 45 | 80 | 2 |
math | 3. (10 points) Car $A$ departs from station A heading to station B, while cars $B$ and $C$ depart from station B heading towards station A at the same time. On the way, $A$ meets $B$ 20 minutes after meeting $C$. It is known that the speeds of $A$, $B$, and $C$ are 90 km/h, 80 km/h, and 60 km/h, respectively. The dista... | 425 | 117 | 3 |
math | $16(x-1)-(x-1)^{2}+(x-1)^{3}-(x-1)^{4}+(x-1)^{5}$ The coefficient of the $x^{2}$ term in the expanded form is $\qquad$ | -20 | 57 | 3 |
math | ## Task B-1.5.
Determine all three-digit numbers that are three times greater than the square of the sum of their digits. | 243,972 | 29 | 7 |
math | 6. On the interval $\left[\frac{1}{2}, 2\right]$, the function $f(x)=x^{2}+p x+q$ and $g(x)=2 x+\frac{1}{x^{2}}$ achieve the same minimum value at the same point. The maximum value of $f(x)$ on $\left[\frac{1}{2}, 2\right]$ is $\qquad$ . | 4 | 92 | 1 |
math | 5.12. a) Represent as a sum of squares
\[
\left(a_{1}^{2}+a_{2}^{2}+a_{3}^{2}\right)\left(b_{1}^{2}+b_{2}^{2}+b_{3}^{2}\right)-\left(a_{1} b_{1}+a_{2} b_{2}+a_{3} b_{3}\right)^{2}
\]
b) Represent as a sum of squares
\[
\left(a_{1}^{2}+\ldots+a_{n}^{2}\right)\left(b_{1}^{2}+\ldots+b_{n}^{2}\right)-\left(a_{1} b_{1}+\... | \sum(a_{i}b_{j}-a_{j}b_{i})^{2} | 198 | 21 |
math | 1. Eva, Igor, Marko, and Maruša each wrote a natural number on a piece of paper. If they erased the last digit of Eva's number, they would get Igor's number. If they erased the last digit of Igor's number, they would get Marko's number. If they erased the last digit of Marko's number, they would get Maruša's number. Th... | 3456,345,34,3 | 115 | 13 |
math | 5.27. The function $f(x)$ is not defined at $x=0$. Determine the value of $f(0)$ so that $f(x)$ is continuous at $x=0$, if
$$
f(x)=\frac{\sqrt[3]{1+x}-1}{\sqrt{4+x}-2}
$$ | \frac{4}{3} | 71 | 7 |
math | Bob has $30$ identical unit cubes. He can join two cubes together by gluing a face on one cube to a face on the other cube. He must join all the cubes together into one connected solid. Over all possible solids that Bob can build, what is the largest possible surface area of the solid?
[i]Proposed by Nathan Xiong[/i] | 122 | 76 | 3 |
math | 34. A company gathered for a meeting. Let's call a person sociable if in this company they have at least 20 acquaintances, and at least two of them are acquainted with each other. Let's call a person shy if in this company they have at least 20 strangers, and at least two of them are strangers to each other. It turned ... | 40 | 117 | 2 |
math | 4. Choose any two numbers from $2,4,6,7,8,11,12,13$ to form a fraction. Then, there are $\qquad$ irreducible fractions among these fractions. | 36 | 47 | 2 |
math | We thought of 5 numbers. Adding them in pairs, we got the following numbers: 0, 2, 4, 4, 6, 8, 9, 11, $13,15$. Determine the numbers. | =-1,b=1,=3,=5,e=10 | 54 | 15 |
math | Let $ A \equal{} \{(a_1,\dots,a_8)|a_i\in\mathbb{N}$ , $ 1\leq a_i\leq i \plus{} 1$ for each $ i \equal{} 1,2\dots,8\}$.A subset $ X\subset A$ is called sparse if for each two distinct elements $ (a_1,\dots,a_8)$,$ (b_1,\dots,b_8)\in X$,there exist at least three indices $ i$,such that $ a_i\neq b_i$.
Find the maxi... | 7! | 144 | 4 |
math | 6. Problem: How many pairs of positive integers $(a, b)$ with $\leq b$ satisfy $\frac{1}{a}+\frac{1}{b}=\frac{1}{6}$ ? | 5 | 44 | 1 |
math | What's the largest number of elements that a set of positive integers between $1$ and $100$ inclusive can have if it has the property that none of them is divisible by another? | 50 | 40 | 4 |
math | 8. If $(x+1)(y+1)=2$, then $\arctan x+\arctan y=$ | \frac{\pi}{4} | 26 | 7 |
math | On Fridays, the price of a ticket to a museum is $\$ 9$. On one particular Saturday, there were 200 visitors to the museum, which was twice as many visitors as there were the day before. The total money collected from ticket sales on that particular Saturday was $\frac{4}{3}$ as much as the day before. The price of tic... | 6 | 90 | 1 |
math | ([b]3[/b]) Let $ \ell$ be the line through $ (0,0)$ and tangent to the curve $ y \equal{} x^3 \plus{} x \plus{} 16$. Find the slope of $ \ell$. | 13 | 54 | 2 |
math | Example 5 Given that $a, b, x, y$ satisfy the system of equations
$$
\left\{\begin{array}{l}
a x+b y=3, \\
a x^{2}+b y^{2}=7, \\
a x^{3}+b y^{3}=16, \\
a x^{4}+b y^{4}=42 .
\end{array}\right.
$$
Find the value of $a x^{5}+b y^{5}$. | 20 | 109 | 2 |
math | Find all functions $f$ from the real numbers to the real numbers such that $f(xy) \le \frac12 \left(f(x) + f(y) \right)$ for all real numbers $x$ and $y$. | f(x) = c \quad \forall x \neq 0 \quad \text{and} \quad f(0) = a \le c | 50 | 34 |
math | IMO 1981 Problem A3 Determine the maximum value of m 2 + n 2 , where m and n are integers in the range 1, 2, ... , 1981 satisfying (n 2 - mn - m 2 ) 2 = 1. | 1597^2+987^2 | 62 | 12 |
math | 6. Find all values of the parameter $a$, for each of which there exists a number $b$ such that the system
$$
\left\{\begin{array}{l}
x=|y+a|+\frac{4}{a} \\
x^{2}+y^{2}+24+b(2 y+b)=10 x
\end{array}\right.
$$
has at least one solution $(x ; y)$. | \in(-\infty,0)\cup[\frac{2}{3};+\infty) | 95 | 21 |
math | In a set of 20 elements there are $2 k+1$ different subsets of 7 elements such that each of these subsets intersects exactly $k$ other subsets. Find the maximum $k$ for which this is possible.
The answer is $k=2$. | 2 | 56 | 1 |
math | 4. If $x_{1}, x_{2}, x_{3}, x_{4}, x_{5}, x_{6}$ are six different positive integers, taking values from $1, 2, 3, 4, 5, 6$. Let
$$
\begin{aligned}
S= & \left|x_{1}-x_{2}\right|+\left|x_{2}-x_{3}\right|+\left|x_{3}-x_{4}\right|+ \\
& \left|x_{4}-x_{5}\right|+\left|x_{5}-x_{6}\right|+\left|x_{6}-x_{1}\right| .
\end{alig... | 10 | 163 | 2 |
math | Solve the following equation!
$$
\frac{(39-x) \sqrt[5]{x-6}-(x-6) \sqrt[5]{39-x}}{\sqrt[5]{39-x}-\sqrt[5]{x-6}}=30
$$ | x_1=38,x_2=7 | 61 | 11 |
math | 102. Find the sum of the squares of the distances from the points of tangency of the inscribed circle with the sides of a given triangle to the center of the circumscribed circle, if the radius of the inscribed circle is $r$, and the radius of the circumscribed circle is $R$. | 3R^{2}-4Rr-r^{2} | 66 | 12 |
math | ## Problem Statement
Calculate the volume of the tetrahedron with vertices at points \( A_{1}, A_{2}, A_{3}, A_{4} \) and its height dropped from vertex \( A_{4} \) to the face \( A_{1} A_{2} A_{3} \).
\( A_{1}(7 ; 2 ; 4) \)
\( A_{2}(7 ;-1 ;-2) \)
\( A_{3}(3 ; 3 ; 1) \)
\( A_{4}(-4 ; 2 ; 1) \) | \frac{43}{\sqrt{105}} | 126 | 13 |
math | 4. Find all real numbers that satisfy the equation $\sqrt{x^{2}-p}+2 \sqrt{x^{2}-1}=x$, where $p$ is a parameter. | \frac{4-p}{\sqrt{8(2-p)}} | 38 | 14 |
math | 1. Let the set $M=\{1,2, \cdots, 12\}$, and the three-element set $A=$ $\{a, b, c\}$ satisfies $A \subset M$, and $a+b+c$ is a perfect square. Then the number of sets $A$ is $\qquad$. | 26 | 71 | 2 |
math | 5. $p, q$ are two distinct prime numbers, and the natural number $n \geqslant$ 3. Find all integers $a$, such that the polynomial $f(x)=x^{n}+$ $a x^{n-1}+p q$ can be factored into the product of two integer-coefficient polynomials, each of degree at least one. (Supplied by Huang Qiguo, Fudan University) | a=-(1+p q), a=1+(-1)^{n-2} p q | 95 | 21 |
math | Solve the following equation on the set of real numbers:
$$
\log _{19}(x-3)+\log _{93}(x-3)=3-\lg \left(x^{5}-24\right)
$$ | 4 | 51 | 1 |
math | 2. What is the greatest possible value of the expression $x y - x^{3} y - x y^{3}$, where $x$, $y$ are positive real numbers? For which $x$, $y$ is this value achieved?
(Mária Dományová, Patrik Bak) | \frac{1}{8} | 63 | 7 |
math | Five. (20 points) Let $a_{1}=\frac{1}{2}$,
$$
a_{n+1}=\frac{a_{n}}{(1-\sqrt{2})^{n+1} a_{n}+\sqrt{2}+1}(n=1,2, \cdots) \text {. }
$$
Find $\lim _{n \rightarrow \infty} \sqrt[n]{a_{n}}$. | \sqrt{2}-1 | 97 | 6 |
math | 3. Let non-zero real numbers $a, b$ satisfy $a^{2}+b^{2}=25$. If the function $y=\frac{a x+b}{x^{2}+1}$ has a maximum value $y_{1}$ and a minimum value $y_{2}$, then $y_{1}-y_{2}=$ $\qquad$ | 5 | 79 | 1 |
math | ## Task 22/88
Determine all pairs $(p ; n)$, where $p$ is a prime number and $n$ is a natural number, for which the solutions of the equation
$$
x^{2}+2\left(p^{n}+2\right) x+p^{2 n}=0
$$
are integers! | (3;1)(2;3) | 76 | 9 |
math | A prison has $2004$ cells, numbered $1$ through $2004$. A jailer, carrying out the terms of a partial amnesty, unlocked every cell. Next he locked every second cell. Then he turned the key in every third cell, locking the opened cells, and unlocking the locked ones. He continued this way, on $n^{\text{th}}$ trip, turni... | 44 | 120 | 2 |
math | Solve the equation $1 / a+1 / b+1 / c=1$ in integers.
# | (3,3,3),(2,3,6),(2,4,4),(1,,-) | 23 | 23 |
math | We define a sequence of natural numbers by the initial values $a_0 = a_1 = a_2 = 1$ and the recursion
$$ a_n = \bigg \lfloor \frac{n}{a_{n-1}a_{n-2}a_{n-3}} \bigg \rfloor $$
for all $n \ge 3$. Find the value of $a_{2022}$. | 674 | 93 | 3 |
math | ## Problem Statement
Calculate the volume of the tetrahedron with vertices at points \( A_{1}, A_{2}, A_{3}, A_{4} \) and its height dropped from vertex \( A_{4} \) to the face \( A_{1} A_{2} A_{3} \).
\( A_{1}(1 ; 5 ;-7) \)
\[
\begin{aligned}
& A_{2}(-3 ; 6 ; 3) \\
& A_{3}(-2 ; 7 ; 3) \\
& A_{4}(-4 ; 8 ;-12)
\end{a... | 7 | 139 | 1 |
math | ## Task 36/77
The smallest natural number $n>1$ is sought, such that
$$
z_{n}=1977+n^{462}
$$
is divisible by 1978. | 3 | 51 | 1 |
math | In a scalene triangle one angle is exactly two times as big as another one and some angle in this triangle is $36^o$. Find all possibilities, how big the angles of this triangle can be. | (36^\circ, 48^\circ, 96^\circ) | 43 | 19 |
math | 1. [4] A hundred friends, including Petya and Vasya, live in several cities. Petya learned the distance from his city to the city of each of the remaining 99 friends and added these 99 numbers. Vasya did the same. Petya got 1000 km. What is the largest number Vasya could have obtained? (Consider the cities as points on... | 99000 | 117 | 5 |
math | Example 1. The following numbers are all approximate numbers obtained by rounding. Find their absolute error bounds and relative error bounds:
$$
\begin{array}{l}
\text { (1) } \mathrm{a}_{1} \approx 12.5, \quad(2) \mathrm{a}_{2} \approx 1.25 \times 10^{4}, \\
\text { (3) } \mathrm{a}_{3} \approx 0.0125
\end{array}
$$ | 0.4\% | 114 | 5 |
math | A rectangular storage bin measures $10$ feet by $12$ feet, is $3$ feet tall, and sits on a flat plane. A pile of dirt is pushed up against the outside of the storage bin so that it slants down from the top of the storage bin to points on the ground $4$ feet away from the base of the storage bin as shown. The number o... | 280 | 124 | 3 |
math | Example 6 Given the sequence $\left\{a_{n}\right\}$, the sum of the first $n$ terms $S_{n}$ satisfies $S_{n}=2 a_{n}+(-1)^{n}, n \geqslant 1$.
(1) Write down the first 3 terms of the sequence $\left\{a_{n}\right\}$, $a_{1}, a_{2}, a_{3}$;
(2) Find the general term formula of the sequence $\left\{a_{n}\right\}$;
(3) Pro... | \frac{1}{a_{4}}+\frac{1}{a_{5}}+\cdots+\frac{1}{a_{}}<\frac{7}{8} | 175 | 37 |
math | 5. Solve the equation $\sqrt[3]{15 x+1-x^{2}}+\sqrt[3]{x^{2}-15 x+27}=4$. | 0,2,13,15 | 37 | 9 |
math | 3. In $\triangle A B C$, it is known that $O$ is the circumcenter, the three altitudes $A D, B E, C F$ intersect at point $H$, line $E D$ intersects $A B$ at point $M$, and $F D$ intersects $A C$ at point $N$. Then $\overrightarrow{O H} \cdot \overrightarrow{M N}=$ $\qquad$ | 0 | 93 | 1 |
math | 10,11
During an interview, ten people were offered a test consisting of several questions. It is known that any five people together answered all the questions (that is, at least one of the five gave the correct answer to each question), but any four did not. What is the minimum number of questions for which this coul... | 210 | 74 | 3 |
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