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 | 1. If $2012^{2}+2010 \times 2011 \times 2013 \times 2014=k^{2}(k>0)$, then $k=$ | 4048142 | 50 | 7 |
math | Example 9 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. Then, what is the maximum number of elements that $S$ can have?
(43rd American High School Mathematics Examination) | 23 | 77 | 2 |
math | 2. ( $4 p$ ) a) Find the value of $x$ from the equality:
$$
(x+1)+(x+2)+(x+3)+\cdots+(x+50)=2525
$$
( 3 p ) b) How many natural numbers satisfy the relation:
$$
4 \cdot x+17<100 ?
$$ | 25 | 81 | 2 |
math | 1. Let $z$ be a complex number such that $|z|=1$ and $|z-1.45|=1.05$. Compute the real part of $z$. | \frac{20}{29} | 41 | 9 |
math | 1. (12 points) Solve the equation
$$
\sqrt[3]{(9-x)^{2}}-\sqrt[3]{(9-x)(7+x)}+\sqrt[3]{(7+x)^{2}}=4
$$ | 1 | 53 | 1 |
math | G10.4 If $S=\left(1-\frac{1}{2^{2}}\right)\left(1-\frac{1}{3^{2}}\right)\left(1-\frac{1}{4^{2}}\right) \cdots\left(1-\frac{1}{10^{2}}\right)$, find $S$. | \frac{11}{20} | 79 | 9 |
math | 4. The quadratic trinomial $p(x)=a x^{2}+b x+c, a>0$ when divided by ( $x-1$ ) gives a remainder of 4, and when divided by ( $x-2$ ) - a remainder of 15. Find the maximum possible value of the ordinate of the vertex of the parabola $y=p(x)$ under these conditions. For what value of $x$ is it achieved? | 4 | 98 | 1 |
math | Example 2 Find the positive integer solutions of the equation $x^{3}-y^{3}=z^{2}$. Here $y$ is a prime number, and neither 3 nor $y$ is a divisor of $z$. | (x,y,z)=(8,7,13) | 49 | 11 |
math | Find the maximum number of natural numbers $x_1,x_2, ... , x_m$ satisfying the conditions:
a) No $x_i - x_j , 1 \le i < j \le m$ is divisible by $11$, and
b) The sum $x_2x_3 ...x_m + x_1x_3 ... x_m + \cdot \cdot \cdot + x_1x_2... x_{m-1}$ is divisible by $11$. | 10 | 109 | 2 |
math | 17. The difference of the squares of two different real numbers is param 1 times greater than the difference of these numbers, and the difference of the cubes of these numbers is param 2 times greater than the difference of these numbers. By how many times is the difference of the fourth powers of these numbers greater... | 769 | 317 | 3 |
math | ## Zadatak B-4.1.
U razvoju binoma $\left(\sqrt[4]{x}+\frac{1}{2 \sqrt[8]{x}}\right)^{2019}$ odredite član koji ne sadrži $x$.
| \binom{2019}{1346}\cdot2^{-1346} | 62 | 22 |
math | # 2. Option 1
On a sheet, two rectangles are drawn. It is known that both the length and the width of the second rectangle are 3 cm larger than the length and width of the first rectangle, and the area of the second rectangle is 48 cm ${ }^{2}$ larger than the area of the first rectangle. Find the perimeter of the sec... | 38 | 80 | 2 |
math | 1.029. $\left(1 \frac{2}{5}+3.5: 1 \frac{1}{4}\right): 2 \frac{2}{5}+3.4: 2 \frac{1}{8}-0.35$. | 3 | 61 | 1 |
math | 3B. We have a mixture $AB$ composed of substances $A$ and $B$ in a ratio of $2:3$, a mixture $BC$, composed of substances $B$ and $C$ in a ratio of $1:2$, and a mixture $CA$, composed in a ratio of $1:1$. How many grams of each mixture should be taken to obtain $24 \mathrm{~g}$ of a mixture in which the ratio of substa... | 10\mathrm{~} | 115 | 7 |
math | 12. (6 points) A ship, on its first trip, travels 420 kilometers downstream and 80 kilometers upstream, taking 11 hours; on its second trip, it uses the same amount of time to travel 240 kilometers downstream and 140 kilometers upstream. How many hours does it take for this ship to travel 198 kilometers downstream? $\q... | 3.3 | 85 | 3 |
math | Example 2 The line $l_{1}$ passes through the point $P(3,2)$, and its inclination angle is $\arctan \frac{3}{4}$. If $l_{1}$ intersects the line $l_{2}: x-2 y+11=0$ at point $Q$, find $|P Q|$. | 25 | 75 | 2 |
math | 4th ASU 1964 Problem 1 In the triangle ABC, the length of the altitude from A is not less than BC, and the length of the altitude from B is not less than AC. Find the angles. | 90,45,45 | 48 | 8 |
math | Let $P(x)=x^3+ax^2+bx+c$ be a polynomial where $a,b,c$ are integers and $c$ is odd. Let $p_{i}$ be the value of $P(x)$ at $x=i$. Given that $p_{1}^3+p_{2}^{3}+p_{3}^{3}=3p_{1}p_{2}p_{3}$, find the value of $p_{2}+2p_{1}-3p_{0}.$ | 18 | 113 | 2 |
math | Find the set of all real numbers $k$ with the following property: For any positive, differentiable function $f$ that satisfies $f^{\prime}(x) > f(x)$ for all $x,$ there is some number $N$ such that $f(x) > e^{kx}$ for all $x > N.$ | k < 1 | 72 | 5 |
math | 4. [5 points] Solve the system of equations
$$
\left\{\begin{array}{l}
\frac{1}{x^{2}+y^{2}}+x^{2} y^{2}=\frac{5}{4} \\
2 x^{4}+2 y^{4}+5 x^{2} y^{2}=\frac{9}{4}
\end{array}\right.
$$ | (\frac{1}{\sqrt{2}};\\frac{1}{\sqrt{2}}),(-\frac{1}{\sqrt{2}};\\frac{1}{\sqrt{2}}) | 91 | 45 |
math | 20. Given a regular pentagon; $M$ is an arbitrary point inside it (or on its boundary). Number the distances from point $M$ to the sides of the pentagon (or their extensions) in ascending order: $r_{1} \leqslant r_{2} \leqslant r_{3} \leqslant r_{4} \leqslant r_{5}$. Find all positions of point $M$ for which the value ... | r_{3} | 134 | 4 |
math | ## problem statement
Write the equation of the plane passing through point $A$ and perpendicular to the vector $\overrightarrow{B C}$.
$A(0; -8; 10)$
$B(-5; 5; 7)$
$C(-8; 0; 4)$ | 3x+5y+3z+10=0 | 65 | 13 |
math | Let $m, n, a, k$ be positive integers and $k>1$ such that the equality $$5^m+63n+49=a^k$$
holds. Find the minimum value of $k$. | 5 | 49 | 1 |
math | 11. Given the sequence $\left\{a_{n}\right\}$, the sum of the first $n$ terms $S_{n}$ satisfies $2 S_{n}-n a_{n}=n, n \in \mathbf{N}^{*}$, and $a_{2}=3$.
(1) Find the general term formula of the sequence $\left\{a_{n}\right\}$;
(2) Let $b_{n}=\frac{1}{a_{n} \sqrt{a_{n+1}}+a_{n+1} \sqrt{a_{n}}}, T_{n}$ be the sum of the... | 50 | 186 | 2 |
math | Find the biggest real number $ k$ such that for each right-angled triangle with sides $ a$, $ b$, $ c$, we have
\[ a^{3}\plus{}b^{3}\plus{}c^{3}\geq k\left(a\plus{}b\plus{}c\right)^{3}.\] | \frac{3\sqrt{2} - 4}{2} | 69 | 15 |
math | 7. Find all positive real numbers $a$ such that the equation
$$
a^{2} x^{2}+a x+1-13 a^{2}=0
$$
has two integer roots. $\qquad$ | 1,\frac{1}{3},\frac{1}{4} | 51 | 15 |
math | 5. Red Martians always tell the truth, while blue Martians lie and then turn red. In a company of 2018 Martians, each in turn answered the question of how many red ones there were among them. The answers were the numbers $1,2,3, \ldots, 2018$ (in that exact order). How many red ones could there have been initially? | 0or1 | 86 | 3 |
math | 12.010. An isosceles trapezoid with an acute angle \(\alpha\) at the base is circumscribed around a circle of radius \(R\). Find the perimeter of this trapezoid. | \frac{8R}{\sin\alpha} | 50 | 11 |
math | Alice and Bob are stuck in quarantine, so they decide to play a game. Bob will write down a polynomial $f(x)$ with the following properties:
(a) for any integer $n$, $f(n)$ is an integer;
(b) the degree of $f(x)$ is less than $187$.
Alice knows that $f(x)$ satisfies (a) and (b), but she does not know $f(x)$. In every... | 187 | 173 | 3 |
math | 51st Putnam 1990 Problem B1 R is the real line. Find all possible functions f: R → R with continuous derivative such that f(α) 2 = 1990 + ∫ 0 α ( f(x) 2 + f '(x) 2 ) dx for all α. | f(x)=\\sqrt{1990}e^x | 70 | 14 |
math | 10-3-1. Non-negative integers $a, b, c, d$ are such that
$$
a b+b c+c d+d a=707
$$
What is the smallest value that the sum $a+b+c+d$ can take? | 108 | 57 | 3 |
math | ## Task B-1.5.
A sequence of $n$ white and $n$ black tiles is arranged such that black and white tiles alternate, starting with a white tile. The goal is to rearrange the tiles so that all black tiles are together at the beginning of the sequence, and all white tiles are at the end. The only allowed move is to swap ad... | \frac{n(n+1)}{2} | 92 | 10 |
math | Find all positive integers $x$ such that the product of all digits of $x$ is given by $x^2 - 10 \cdot x - 22.$ | 12 | 37 | 2 |
math | ## Task 4 - 190624
An automatic number stamp for a series production prints exactly one natural number every second. It starts with the number 0 and then continues to print the subsequent numbers $1,2,3, \ldots$ in sequence.
Determine the total number of the digit 1 that the stamp will print in the first quarter of a... | 280 | 81 | 3 |
math | Let $a_1, a_2, a_3, a_4$ be integers with distinct absolute values. In the coordinate plane, let $A_1=(a_1,a_1^2)$, $A_2=(a_2,a_2^2)$, $A_3=(a_3,a_3^2)$ and $A_4=(a_4,a_4^2)$. Assume that lines $A_1A_2$ and $A_3A_4$ intersect on the $y$-axis at an acute angle of $\theta$. The maximum possible value for $\tan \theta$ ca... | 503 | 179 | 3 |
math | On a piece of paper, we can read the following 100 statements:
1. On this piece of paper, exactly one statement is false. 2. On this piece of paper, exactly two statements are false.
2. On this piece of paper, exactly three statements are false. .
3. On this piece of paper, exactly 100 statements are false.
Determine... | 99 | 88 | 2 |
math | Solve the following equation:
$$
6 x^{5}-x^{4}-43 x^{3}+43 x^{2}+x-6=0 .
$$ | x_1=1,x_2=-3,x_3=-\frac{1}{3},x_4=2,x_5=\frac{1}{2} | 38 | 36 |
math | Example 7 The set of all real solutions to the equation $[2 x]+[3 x]=9 x-\frac{7}{4}$ is $\qquad$ . | x=-\frac{1}{36}, \frac{7}{36} | 36 | 18 |
math | Let $P(x)$ be a polynomial such that for all integers $x \geq 1$,
$$
P(x)=\sum_{n=1}^{x} n^{2012}
$$
(a) Find $P(-2)$.
(b) Find $P(1 / 2)$. | P(-2)=-1,\P(1/2)=\frac{1}{2^{2012}} | 67 | 25 |
math | 2. Dima and Seryozha decided to have a competition around a circular lake. They start simultaneously from the same point. Seryozha drives a motorboat at a constant speed of 20 km/h and somehow crosses the lake (not necessarily along the diameter) in 30 minutes. During this time, Dima runs along the shore of the lake fo... | 37.5 | 155 | 4 |
math | 5. Masha talked a lot on the phone with her friends, and the fully charged battery ran out exactly after 24 hours. It is known that the charge lasts for 5 hours of talking or 150 hours of standby. How long did Masha talk to her friends | \frac{126}{29} | 59 | 10 |
math | 7. The function $f$ is defined on the set of integers, satisfying
$$
f(n)=\left\{\begin{array}{ll}
n-3, & \text { when } n \geqslant 1000, \\
f(f(n+5)), & \text { when } n<1000 .
\end{array}\right.
$$
Find $f(84)$. | 997 | 91 | 3 |
math | 15. (6 points) The poetry lecture lasted for 2 hours $m$ minutes, and at the end, the positions of the hour and minute hands on the clock were exactly swapped compared to when it started. If $[x]$ represents the integer part of the decimal number $x$, then $[m]=$ $\qquad$ . | 46 | 71 | 2 |
math | 175. A divisor of its palindrome. In what base does 792 divide 297?
Note: The original problem statement is in Russian, but the provided translation is in English as requested. | 19 | 44 | 2 |
math | 5 Obtain the two integer values of $x$ closest to $2013^{\circ}$, both by default and by excess, that satisfy this trigonometric equation
$$
2^{\sin ^{2} x}+2^{\cos ^{2} x}=2 \sqrt{2}
$$ | x_{1}=1935 | 67 | 8 |
math | $(BEL 1)$ A parabola $P_1$ with equation $x^2 - 2py = 0$ and parabola $P_2$ with equation $x^2 + 2py = 0, p > 0$, are given. A line $t$ is tangent to $P_2.$ Find the locus of pole $M$ of the line $t$ with respect to $P_1.$ | x^2 + 2py = 0 | 94 | 11 |
math | 9. Let $A C$ be a diameter of a circle $\omega$ of radius 1 , and let $D$ be the point on $A C$ such that $C D=1 / 5$. Let $B$ be the point on $\omega$ such that $D B$ is perpendicular to $A C$, and let $E$ be the midpoint of $D B$. The line tangent to $\omega$ at $B$ intersects line $C E$ at the point $X$. Compute $A ... | 3 | 110 | 1 |
math | 10. (20 points) Find all possible positive integers $n \geqslant 3$, such that there exist distinct positive real numbers $a_{1}, a_{2}, \cdots, a_{n}$, for which all the products $a_{i} a_{j}(1 \leqslant i<j \leqslant n)$ can be rearranged in increasing order to form a geometric progression. | n=3orn=4 | 91 | 6 |
math | 3. Among $m$ students, it is known that in any group of three, two of them know each other, and in any group of four, two of them do not know each other. Then the maximum value of $m$ is $\qquad$ | 8 | 54 | 1 |
math | 43. Find all three-digit numbers that are equal to the arithmetic mean of all numbers obtained from the given number by all possible permutations of its digits (including, of course, the "identity permutation" that leaves all the digits of the number in place). | 111,222,333,444,555,666,777,888,999,407,518,629,370,481,592 | 52 | 59 |
math | ## Task 2 - 080612
Calculate the size of the smaller of the two angles formed by the hour and minute hands of a clock at 16:40! | 100 | 41 | 3 |
math | ## Task Condition
Compose the equation of the normal to the given curve at the point with abscissa $x_{0}$.
$y=\frac{x^{2}-3 x+6}{x^{2}}, x_{0}=3$ | 9x-26\frac{1}{3} | 50 | 12 |
math | 5. In the Cartesian coordinate system $x O y$, the moving point $A$ is on the circle $x^{2}+y^{2}=1$, and the coordinates of point $B$ are $(3,0)$. If point $C$ makes $\triangle A B C$ an equilateral triangle, then the maximum value of $|O C|$ is $\qquad$. | 4 | 81 | 1 |
math | 3. Let $\left\{a_{n}\right\}$ be an arithmetic sequence with the sum of the first $n$ terms denoted as $S_{n}$. If $S_{6}=26, a_{7}=2$, then the maximum value of $n S_{n}$ is $\qquad$ . | 338 | 69 | 3 |
math | Problem 2. Solve in the set of positive rational numbers the equation:
$$
\begin{array}{r}
\frac{x+1}{2}+\frac{x+5}{3}+\frac{x+11}{4}+\frac{x+19}{5}+\frac{x+29}{6}+\frac{x+41}{7}+\frac{x+55}{8}+\frac{x+71}{9}+\frac{x+89}{10}=45 . \\
\text { Mathematical Gazette nr. 10/2013 }
\end{array}
$$ | 1 | 130 | 1 |
math | 6. Let $f(x)$ be an odd function defined on $\mathbf{R}$, and when $x \geqslant 0$, $f(x)=x^{2}$. If for any $x \in[a, a+2]$, the inequality $f(x+a) \geqslant 2 f(x)$ always holds, then the range of the real number $a$ is | [\sqrt{2},+\infty) | 86 | 9 |
math | # Problem 1. (2 points)
Two given quadratic trinomials differ by the permutation of the free term and the second coefficient. The sum of these trinomials has a single root. What value does this sum take at $x=2$? | 8or32 | 55 | 4 |
math | Task B-1.4. A cattleman has secured food for his 24 cows for 18 weeks. How many cows would he have to sell after 8 weeks so that he would have enough food for another 12 weeks? | 4 | 51 | 1 |
math | 3. Let $A B C D$ be a rectangle with area 1 , and let $E$ lie on side $C D$. What is the area of the triangle formed by the centroids of triangles $A B E, B C E$, and $A D E$ ? | \frac{1}{9} | 58 | 7 |
math | 3. Postman Pechkin is riding a bicycle along a highway. He noticed that every 4.5 kilometers, a suburban bus overtakes him, and every 9 minutes, a suburban bus passes him in the opposite direction. The interval of bus movement in both directions is 12 minutes. At what speed is Pechkin riding? | 15 | 71 | 2 |
math | 5. (20 points) Alexei came up with the following game. First, he chooses a number $x$ such that $2017 \leqslant x \leqslant 2117$. Then he checks if $x$ is divisible by 3, 5, 7, 9, and 11 without a remainder. If $x$ is divisible by 3, Alexei awards the number 3 points, if by 5 - then 5 points, ..., if by 11 - then 11 p... | 2079 | 146 | 4 |
math | 6. Let the points be
$$
A\left(a, a+\frac{1}{2}\right), B\left(a+1, a+\frac{3}{2}\right) \text {, }
$$
A moving point $P$ is such that its distance to point $M(1,0)$ is 1 unit greater than its distance to the $y$-axis, and its trajectory forms a curve $C$. The line segment $AB$ intersects curve $C$. Then the range of ... | \left[\frac{1}{2}-\sqrt{2}, \frac{3}{2}-\sqrt{2}\right] \cup\left[\frac{1}{2}+\sqrt{2}, \frac{3}{2}+\sqrt{2}\right] | 118 | 58 |
math | ALB
3) If $x^{3}-3 \sqrt{3} x^{2}+9 x-3 \sqrt{3}-64=0$, find the value of $x^{6}-8 x^{5}+13 x^{4}-5 x^{3}+49 x^{2}-137 x+2015$. | 1898 | 79 | 4 |
math | 4. (8 points) In the following horizontal equation, the same Chinese characters represent the same digits, different Chinese characters represent different digits, and no Chinese character represents 7. "迎", "春", and "杯" are not equal to 1. Therefore, the sum of the three digits represented by "迎", "春", and "杯" is $\qq... | 15 | 107 | 2 |
math | 2. The general solution of the equation $\cos \frac{x}{4}-\cos x$ is ( ), within $(0,24 \pi)$, there are ( ) distinct solutions. | 20 | 40 | 2 |
math | Example 1. Given the sequence $\left\{a_{\mathrm{n}}\right\}$, where $a_{1}=1$. It satisfies the relation $a_{\mathrm{n}}=a_{\mathrm{n}-1}+2 n(n \geqslant 2, n \in N)$, find $a_{n}$. | a_{\mathrm{n}}=n^{2}+n-1 | 75 | 15 |
math | 14. A census taker stands in front of Aunt Wang's house and asks Aunt Wang: “Your age is 40, what are the ages of the three orphans you adopted?” Aunt Wang says: “The product of their ages equals my age, and the sum of their ages equals our house number.” The census taker looks at the house number and says: “I still ca... | 14 | 98 | 2 |
math | 1. Let $a>0, b>0, c>0$, and $a+b+c=1$, then the maximum value of $a^{3} b^{2} c$ is | \frac{1}{2^{4}\cdot3^{3}} | 41 | 14 |
math | a) Find all positive integers $g$ with the following property: for each odd prime number $p$ there exists a positive integer $n$ such that $p$ divides the two integers
\[g^n - n\quad\text{ and }\quad g^{n+1} - (n + 1).\]
b) Find all positive integers $g$ with the following property: for each odd prime number $p$ there ... | g = 2 | 137 | 5 |
math | 3. (17th Japan Mathematical Olympiad) Find the tens digit of $11^{12^{13}}$ (where $11^{12^{3}}$ represents 11 to the power of $12^{13}$) | 2 | 55 | 1 |
math | 7. In the Cartesian coordinate system, if the circle with center $(r+1,0)$ and radius $r$ has a point $(a, b)$ satisfying $b^{2} \geqslant 4 a$, then the minimum value of $r$ is $\qquad$
In the Cartesian coordinate system, if the circle with center $(r+1,0)$ and radius $r$ has a point $(a, b)$ satisfying $b^{2} \geqsl... | 4 | 120 | 1 |
math | Example 5 Let $x$ be a real number. Then
$$
|x-1|+|x+1|+|x+5|
$$
the minimum value is $\qquad$ (s) | 6 | 45 | 1 |
math | 4. Determine the largest natural number $n$ such that $n^{2}+2002 n$ is a perfect square of some natural number. | 500000 | 33 | 6 |
math | 4. The smallest positive integer $a$ that makes the inequality $\frac{1}{n+1}+\frac{1}{n+2}+\cdots+\frac{1}{2 n+1}<a-2007 \frac{1}{3}$ hold for all positive integers $n$ is $\qquad$. | 2009 | 70 | 4 |
math | 7. Given a triangle with sides as three consecutive natural numbers, the largest angle is twice the smallest angle. Then the perimeter of the triangle is $\qquad$ | 15 | 33 | 2 |
math | 3. Triangle $A B C$ is equilateral. On side $A C$, point $M$ is marked, and on side $B C$, point $N$, such that $M C=B N=2 A M$. Segments $M B$ and $A N$ intersect at point $Q$. Find the angle $C Q B$.
# | 90 | 75 | 2 |
math | 3. Two equal rectangles $P Q R S$ and $P_{1} Q_{1} R_{1} S_{1}$ are inscribed in triangle $A B C$ (with points $P$ and $P_{1}$ lying on side $A B$, points $Q$ and $Q_{1}$ lying on side $B C$, and points $R, S, R_{1}$ and $S_{1}$ lying on side $A C$). It is known that $P S=12, P_{1} S_{1}=3$. Find the area of triangle $... | \frac{225}{2} | 131 | 9 |
math | 4.8 There is a four-digit number less than 2000, which has exactly 14 positive divisors (including 1 and itself), and one of its prime divisors ends with the digit 1. Find this four-digit number.
(Shanghai Junior High School Mathematics Competition, 1984) | 1984 | 68 | 4 |
math | 11.1. Find the largest term of the sequence a) $a_{n}=\frac{n}{n^{2}+2020}$, б) $a_{n}=\frac{2020^{n}}{n!}$ (where $n!=1 \cdot 2 \cdot 3 \cdots n)$ | )\frac{45}{4045},b)\frac{2020^{2019}}{2019!} | 74 | 32 |
math | Let $A = (a_1, a_2, \ldots, a_{2001})$ be a sequence of positive integers. Let $m$ be the number of 3-element subsequences $(a_i,a_j,a_k)$ with $1 \leq i < j < k \leq 2001$, such that $a_j = a_i + 1$ and $a_k = a_j + 1$. Considering all such sequences $A$, find the greatest value of $m$. | 667^3 | 112 | 5 |
math | 5.1. (12 points) The equation $x^{2}+5 x+1=0$ has roots $x_{1}$ and $x_{2}$. Find the value of the expression
$$
\left(\frac{x_{1} \sqrt{6}}{1+x_{2}}\right)^{2}+\left(\frac{x_{2} \sqrt{6}}{1+x_{1}}\right)^{2}
$$ | 220 | 98 | 3 |
math | 4. In the cells of a $3 \times 3$ square, the numbers $1,2,3, \ldots, 9$ are arranged. It is known that any two consecutive numbers are located in adjacent (by side) cells. Which number can be in the central cell if the sum of the numbers in the corner cells is $18?$
# | 7 | 78 | 1 |
math | 7. Given the function $f(x)=\sqrt{x^{2}+1}-\frac{2}{5} a x$, where $a>0$, is a monotonic function on the interval $[0,+\infty)$, then the range of values for $a$ is $\qquad$ . | [\frac{5}{2},+\infty) | 66 | 11 |
math | 19. Let the three sides of $\triangle ABC$ be $a, b, c$, and $a+b+c=3$. Find the minimum value of $f(a, b, c)=a^{2}+$ $b^{2}+c^{2}+\frac{4}{3} a b c$. (2007 Northern Mathematical Olympiad Problem) | \frac{13}{3} | 78 | 8 |
math | (15) Find the smallest positive real number $k$, such that the inequality
$$
a b+b c+c a+k\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right) \geqslant 9,
$$
holds for all positive real numbers $a, b, c$. | 2 | 75 | 1 |
math | 1. Consider the set $G=(1, \infty)$ and the function $f: G \rightarrow \mathbb{R}_{+}^{*}, f(x)=x-1$.
Determine a composition law ,,*' defined on $G$ such that $(G, *)$ is a group, and the function $f$ is an isomorphism from the group $(G, *)$ to the group $\left(\mathbb{R}_{+}^{*}, \cdot\right)$. | x*y=xy-x-y+2 | 107 | 8 |
math | Let $x, y, z, w \in [0,1]$. Find the maximum value of $S=x^{2} y+y^{2} z+z^{2} w+w^{2} x-x y^{2}-$ $y z^{2}-z w^{2}-w x^{2}$. | \frac{8}{27} | 67 | 8 |
math | Find the greatest real number $C$ such that, for all real numbers $x$ and $y \neq x$ with $xy = 2$ it holds that
\[\frac{((x + y)^2 - 6)((x - y)^2 + 8)}{(x-y)^2}\geq C.\]
When does equality occur? | 18 | 76 | 2 |
math | $10 \cdot 17$ A bus ticket number is a six-digit number. If the sum of the first three digits equals the sum of the last three digits, the ticket is called a "lucky ticket." How many consecutive ticket numbers must be bought from the ticket office to ensure that at least one of them is a "lucky ticket"?
(St. Petersburg... | 1001 | 88 | 4 |
math | 6. Let the hyperbola $\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1(a>0, b>0)$ have its left and right foci as $F_{1}$ and $F_{2}$, respectively, and let $A$ be a point on the asymptote of the hyperbola such that $A F_{2} \perp F_{1} F_{2}$. The distance from the origin $O$ to the line $A F_{1}$ is $\frac{1}{3}\left|O F_{1... | \frac{\sqrt{6}}{2} | 144 | 10 |
math | Exercise 2. Find all pairs of digits $(a, b)$ such that the integer whose four digits are $a b 32$ is divisible by 99. | 6,7 | 36 | 3 |
math | Given an odd $n\in\mathbb N$. In an $n\times n$ chessboard, you may place many $2\times2$ squares. How many grids, at most, are covered by exactly one square? | n(n-1) | 49 | 6 |
math | Find all integers $k$ such that all roots of the following polynomial are also integers: $$f(x)=x^3-(k-3)x^2-11x+(4k-8).$$ | k = 5 | 43 | 5 |
math | 15. If $a=1.69, b=1.73$ and $c=0.48$, find the value of
$$
\frac{1}{a^{2}-a c-a b+b c}+\frac{2}{b^{2}-a b-b c+a c}+\frac{1}{c^{2}-a c-b c+a b}
$$ | 20 | 84 | 2 |
math | 4. Find the minimum value of the sum
$$
\left|x-1^{2}\right|+\left|x-2^{2}\right|+\left|x-3^{2}\right|+\ldots+\left|x-10^{2}\right|
$$ | 275 | 57 | 3 |
math | 13.4. 19 ** In $\triangle A B C$ with a fixed perimeter, it is known that $|A B|=6$, and when vertex $C$ is at a fixed point $P$, $\cos C$ has a minimum value of $\frac{7}{25}$.
（1）Establish an appropriate coordinate system and find the equation of the locus of vertex $C$;
(2) Draw a line through point $A$ that interse... | 16 | 137 | 2 |
math |
Problem 6. Find the number of non-empty sets of $S_{n}=\{1,2, \ldots, n\}$ such that there are no two consecutive numbers in one and the same set.
| f_{n}=\frac{1}{\sqrt{5}}((\frac{1+\sqrt{5}}{2})^{n+2}-(\frac{1-\sqrt{5}}{2})^{n+2})-1 | 46 | 52 |
math | Each of the thirty sixth-graders has one pen, one pencil, and one ruler. After their participation in the Olympiad, it turned out that 26 students lost a pen, 23 - a ruler, and 21 - a pencil. Find the smallest possible number of sixth-graders who lost all three items. | 10 | 68 | 2 |
math | Find all positive integer $N$ which has not less than $4$ positive divisors, such that the sum of squares of the $4$ smallest positive divisors of $N$ is equal to $N$. | 130 | 44 | 3 |
math | 4. Let $A_{1} A_{2} \cdots A_{21}$ be a regular 21-sided polygon inscribed in a circle. Select $n$ different vertices from $A_{1}, A_{2}, \cdots, A_{21}$ and color them red, such that the distances between any two of these $n$ red points are all different. Then the maximum value of the positive integer $n$ is $\qquad$ ... | 5 | 99 | 1 |
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