task_type stringclasses 1
value | problem stringlengths 23 3.94k | answer stringlengths 1 231 | problem_tokens int64 8 1.39k | answer_tokens int64 1 200 |
|---|---|---|---|---|
math | A4. The equations $|x|^{2}-3|x|+2=0$ and $x^{4}-a x^{2}+4=0$ have the same roots. Determine the value of $a$. | 5 | 48 | 1 |
math | 5. The solution to the equation $\frac{x^{3}}{\sqrt{4-x^{2}}}+x^{2}-4=0$ is $\qquad$ | \sqrt{2} | 35 | 5 |
math | 8. Given the hyperbola $H: x^{2}-y^{2}=1$ and a point $M$ in the first quadrant on $H$, the line $l$ is tangent to $H$ at point $M$ and intersects the asymptotes of $H$ at points $P, Q$ (with $P$ in the first quadrant). If point $R$ is on the same asymptote as $Q$, then the minimum value of $\overrightarrow{R P} \cdot ... | -\frac{1}{2} | 117 | 7 |
math | Five, (17 points) Find the positive integer solutions of the equation
$$
x^{2}+6 x y-7 y^{2}=2009
$$ | (252,251),(42,35),(42,1) | 38 | 20 |
math | [Exponential Inequalities]
Arrange the numbers in ascending order: $222^{2} ; 22^{22} ; 2^{222} ; 22^{2^{2}} ; 2^{22^{2}} ; 2^{2^{22}} ; 2^{2^{2^{2}}}$. Justify your answer. | 222^{2}<2^{2^{2^{2}}}<22^{2^{2}}<22^{22}<2^{222}<2^{22^{2}}<2^{2^{22}} | 80 | 50 |
math | 4. Suppose that $\overline{A2021B}$ is a six-digit integer divisible by $9$. Find the maximum possible value of $A \cdot B$.
5. In an arbitrary triangle, two distinct segments are drawn from each vertex to the opposite side. What is the minimum possible number of intersection points between these segments?
6. Suppose... | 143 | 124 | 3 |
math | 11. (10 points) On Qingming Festival, students take a bus to the martyrs' cemetery to pay respects. If the bus increases its speed by one-fifth after driving for 1 hour, it can arrive 10 minutes earlier than the scheduled time; if the bus first drives 60 kilometers at the original speed and then increases its speed by ... | 180 | 107 | 3 |
math | 5. (3 points) From the ten digits $0 \sim 9$, the pair of different numbers with the largest product is $\qquad$
multiplied, the pair with the smallest sum is $\qquad$ and $\qquad$ added, pairs that add up to 10 are $\qquad$ pairs. | 9,8,0,1,4 | 68 | 9 |
math | 4B. If $\cos x+\cos ^{2} x+\cos ^{3} x=1$, calculate the value of the expression
$$
\sin ^{6} x-4 \sin ^{4} x+8 \sin ^{2} x
$$ | 4 | 60 | 1 |
math | Let $a_1=2021$ and for $n \ge 1$ let $a_{n+1}=\sqrt{4+a_n}$. Then $a_5$ can be written as $$\sqrt{\frac{m+\sqrt{n}}{2}}+\sqrt{\frac{m-\sqrt{n}}{2}},$$ where $m$ and $n$ are positive integers. Find $10m+n$. | 45 | 93 | 2 |
math | Example 1. Find the direction vector of the normal to the ellipsoid $x^{2}+2 y^{2}+3 z^{2}=6$ at the point $M_{0}(1,-1,1)$. | {2,-4,6} | 49 | 7 |
math | Find all functions $ f: \mathbb{Q}^{\plus{}} \rightarrow \mathbb{Q}^{\plus{}}$ which satisfy the conditions:
$ (i)$ $ f(x\plus{}1)\equal{}f(x)\plus{}1$ for all $ x \in \mathbb{Q}^{\plus{}}$
$ (ii)$ $ f(x^2)\equal{}f(x)^2$ for all $ x \in \mathbb{Q}^{\plus{}}$. | f(x) = x | 111 | 6 |
math | 41. Find the smallest constant $c$, such that for all real numbers $x, y$, we have $1+(x+y)^{2} \leqslant c\left(1+x^{2}\right)$ ( $\left.1+y^{2}\right)$ )(2008 German Mathematical Olympiad problem) | \frac{4}{3} | 70 | 7 |
math | 41. The road from home to school takes Seryozha 30 minutes. Once on the way, he remembered that he had forgotten a pen at home. Seryozha knew that if he continued on to school at the same speed, he would arrive there 9 minutes before the bell, but if he returned home for the pen, he would, walking at the same speed, be... | \frac{1}{3} | 103 | 7 |
math | 15. The inequality $\sin ^{2} x+a \cos x+a^{2} \geqslant 1+\cos x$ holds for all $x \in \mathbf{R}$, find the range of real number $a$.
| \geqslant1or\leqslant-2 | 55 | 14 |
math | Let $\triangle ABC$ be a triangle with a right angle $\angle ABC$. Let $D$ be the midpoint of $\overline{BC}$, let $E$ be the midpoint of $\overline{AC}$, and let $F$ be the midpoint of $\overline{AB}$. Let $G$ be the midpoint of $\overline{EC}$. One of the angles of $\triangle DFG$ is a right angle. What is the le... | \frac{2}{3} | 111 | 7 |
math | 5. Let the line $y=k x-2$ intersect the parabola $y^{2}=8 x$ at points $A$ and $B$. If the x-coordinate of the midpoint of segment $A B$ is 2, then the length of segment $A B$ is $|A B|=$ $\qquad$ . | 2 \sqrt{15} | 72 | 7 |
math | 1. Find all integers $a$ for which the modulus of the number $\left|a^{2}-3 a-6\right|$ is a prime number. Answer. $-1 ; 4$. | -1,4 | 43 | 4 |
math | In an isosceles right-angled triangle AOB, points P; Q and S are chosen on sides OB, OA, and AB respectively such that a square PQRS is formed as shown. If the lengths of OP and OQ are a and b respectively, and the area of PQRS is 2 5 that of triangle AOB, determine a : b.
[asy]
pair A = (0,3);
pair B = (0,0);
pair C ... | 2 : 1 | 305 | 4 |
math | ## SUBJECT I
Solve the equation $\frac{x-2}{x}+\frac{x-4}{x}+\frac{x-6}{x}+\ldots . . .+\frac{2}{x}=12$ | 50 | 48 | 2 |
math | 3. A coin-flipping game, starting from the number $n$, if the coin lands heads up, subtract 1; if it lands tails up, subtract 2. Let $E_{n}$ be the expected number of coin flips before the number becomes zero or negative. If $\lim _{n \rightarrow \infty}\left(E_{n}-a n-b\right)=0$, then the pair $(a, b)=$ $\qquad$ | (\frac{2}{3},\frac{2}{9}) | 95 | 14 |
math | 4.1. Mom baked four raisin buns for breakfast for her two sons. $V$ In the first three buns, she put 7, 7, 23 raisins, and some more in the fourth. It turned out that the boys ate an equal number of raisins and did not divide any bun into parts. How many raisins could Mom have put in the fourth bun? List all the option... | 9,23,37 | 89 | 7 |
math | 10.334. Inside a square with side $a$, a semicircle is constructed on each side as a diameter. Find the area of the rosette bounded by the arcs of the semicircles. | \frac{^{2}(\pi-2)}{2} | 46 | 14 |
math | 5-4. Andrei, Boris, Vladimir, and Dmitry each made two statements. For each boy, one of his statements turned out to be true, and the other false.
Andrei: "Boris is not the tallest among us four." "Vladimir is the shortest among us four."
Boris: "Andrei is the oldest in the room." "Andrei is the shortest in the room.... | Vladimir | 155 | 3 |
math | 1. Fishermen caught several carp and pike. Each one caught as many carp as all the others caught pike. How many fishermen were there if the total number of carp caught was 10 times the number of pike?
## Answer: 11.
# | 11 | 56 | 2 |
math | 8. A positive integer $x \in$ $\{1,2, \cdots, 2016\}$ is generated with equal probability. Then the probability that the sum of the digits of $x$ in binary is no more than 8 is $\qquad$ | \frac{655}{672} | 59 | 11 |
math | 2. A barrel 1.5 meters high is completely filled with water and covered with a lid. The mass of the water in the barrel is 1000 kg. A long thin tube with a cross-sectional area of $1 \mathrm{~cm}^{2}$ is inserted vertically into the lid of the barrel, which is completely filled with water. Find the length of the tube i... | 1.5 | 125 | 3 |
math | ## Problem Statement
Find the derivative $y_{x}^{\prime}$.
$$
\left\{\begin{array}{l}
x=\ln \sqrt{\frac{1-t}{1+t}} \\
y=\sqrt{1-t^{2}}
\end{array}\right.
$$ | \cdot\sqrt{1-^{2}} | 61 | 10 |
math | ## Problem Statement
Calculate the limit of the numerical sequence:
$\lim _{n \rightarrow \infty} \frac{(2 n+1)^{2}-(n+1)^{2}}{n^{2}+n+1}$ | 3 | 52 | 1 |
math | Problem 13. Determine the length and width of a rectangular plot of land that will maximize its area given a fixed perimeter. | \frac{p}{4} | 26 | 7 |
math | 4. In $\triangle A B C$, $\angle B A C=60^{\circ}$, the angle bisector $A D$ of $\angle B A C$ intersects $B C$ at $D$, and $\overrightarrow{A D}=\frac{1}{4} \overrightarrow{A C}+t \overrightarrow{A B}$. If $A B=8$, then $A D=$ . $\qquad$ | 6\sqrt{3} | 95 | 6 |
math | 7. Let $A=\{1,2, \cdots, n\}$, and let $S_{n}$ denote the sum of the elements in all non-empty proper subsets of $A$; $B_{n}$ denote the number of subsets of $A$, then $\lim _{n \rightarrow+\infty} \frac{S_{n}}{n^{2} \cdot B_{n}}$ is $\qquad$ . | \frac{1}{4} | 94 | 7 |
math | \section*{Task 2 - 131222}
Each of the 41 students in a class had to participate in exactly three track and field running events.
Each of these students had to start once on lanes 1, 2, and 3.
Student A claims that due to these rules alone, there must be at least seven students in the class whose sequence of startin... | 7 | 114 | 1 |
math | 4A. Determine all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ for which
$$
f(f(x y))=x f(y)+3 f(x y)
$$ | f(x)=0f(x)=4x | 44 | 9 |
math | 14. Let the set $S=\{1,2, \cdots, n\}, A$ be an arithmetic sequence with at least two terms, a positive common difference, and each term in $S$. Additionally, adding any other element of $S$ to $A$ does not form an arithmetic sequence with the same common difference as $A$. Find the number of such $A$ (here, sequences ... | [\frac{n^{2}}{4}] | 96 | 9 |
math | In triangle $A B C$, side $A B$ is equal to 21, the bisector $B D$ is $8 \sqrt{7}$, and $D C=8$. Find the perimeter of triangle $A B C$. | 60 | 52 | 2 |
math | 6. Let $f_{1}(x)=-\frac{2 x+7}{x+3}, f_{n+1}(x)=f_{1}\left(f_{n}(x)\right), x \neq-2, x \neq-3$, then $f_{2022}(2021)=$ | 2021 | 73 | 4 |
math | Example 4. Given that $a, b, c$ satisfy $a+b+c=0, abc=8$. Then the range of values for $c$ is $\qquad$
(1st Hope Cup Junior High School Mathematics Invitational Competition) | c<0 \text { or } c \geqslant 2 \sqrt[3]{4} | 52 | 23 |
math | (Inspired by Indian Regional Mathematical Olympiad 2000, P1, untreated)
Find all pairs of positive integers $(x, y)$ such that $x^{3}+4 x^{2}-3 x+6=y^{3}$. | (x,y)=(1,2)(x,y)=(5,6) | 53 | 14 |
math | 【Question 7】
On Tree Planting Day, the students of Class 4(1) went to the park to plant trees. Along a 120-meter long road on both sides, they dug a hole every 3 meters. Later, due to the spacing being too small, they changed it to digging a hole every 5 meters. In this way, at most $\qquad$ holes can be retained. | 18 | 88 | 2 |
math | Proizvolov V.V.
Find all natural numbers $a$ and $b$ such that $\left(a+b^{2}\right)\left(b+a^{2}\right)$ is an integer power of two. | =b=1 | 44 | 3 |
math | Example 1. Using the Green's function, solve the boundary value problem
$$
\begin{gathered}
y^{\prime \prime}(x)-y(x)=x \\
y(0)=y(1)=0 .
\end{gathered}
$$ | y(x)=\frac{\operatorname{sh}x}{\operatorname{sh}1}-x | 55 | 22 |
math | Let $a,b,c,d$ be distinct digits such that the product of the $2$-digit numbers $\overline{ab}$ and $\overline{cb}$ is of the form $\overline{ddd}$. Find all possible values of $a+b+c+d$. | 21 | 57 | 2 |
math | A circle in the first quadrant with center on the curve $y=2x^2-27$ is tangent to the $y$-axis and the line $4x=3y$. The radius of the circle is $\frac{m}{n}$ where $M$ and $n$ are relatively prime positive integers. Find $m+n$. | 11 | 73 | 2 |
math | 8. In $\triangle A B C$, the medians $A E, B F$, and $C D$ intersect at $M$. Given that $E, C, F$, and $M$ are concyclic, and $C D=n$. Find the length of segment $A B$.
(18th All-Russian Competition Problem) | AB=\frac{2}{3}\sqrt{3}n | 71 | 13 |
math | ## Task Condition
Find the derivative.
$y=x^{\arcsin x}$ | x^{\arcsinx}\cdot(\frac{\lnx}{\sqrt{1-x^{2}}}+\frac{\arcsinx}{x}) | 18 | 31 |
math | 4.2. In a right triangle $A B C$ with a right angle at $A$, the angle bisectors $B B_{1}$ and $C C_{1}$ are drawn. From points $B_{1}$ and $C_{1}$, perpendiculars $B_{1} B_{2}$ and $C_{1} C_{2}$ are dropped to the hypotenuse $B C$. What is the measure of angle $B_{2} A C_{2}$? | 45 | 104 | 2 |
math | 5. The quadratic function $f(x)=a+b x-x^{2}$ satisfies $f(1+x)=f(1-x)$ for any real number $x$, and $f(x+m)$ is an increasing function on $(-\infty, 4]$. Then the range of the real number $m$ is $\qquad$ | \in(-\infty,-3] | 71 | 9 |
math | 6.58. $\lim _{x \rightarrow+0} x^{x}$. | 1 | 20 | 1 |
math | Let's determine how many shuffles it takes to return 52 cards to their original order if we shuffle them as described in the article about Faro Shuffle. Solve this problem in the case where we start the shuffle with the bottom card of the right-hand deck, meaning the card originally in the 26th position ends up at the ... | 52 | 71 | 2 |
math | Shnol D.e.
Dima lives in a nine-story building. He descends from his floor to the first floor by elevator in 1 minute. Due to his small height, Dima cannot reach the button for his floor. Therefore, when going up, he presses the button he can reach, and then walks the rest of the way. The entire journey upwards takes ... | 7 | 115 | 1 |
math | One, (40 points) Find all positive real solutions of the equation
$$
17 x^{19}-4 x^{17}-17 x^{15}+4=0
$$ | x=1 | 44 | 3 |
math | Find all functions $f:\mathbb{R}\rightarrow\mathbb{R}$ such that for all real numbers $x$ and $y$ satisfies,
$$2+f(x)f(y)\leq xy+2f(x+y+1).$$ | f(x) = x + 2 | 52 | 9 |
math | 12. Let the foci of an ellipse be $F_{1}(-1,0)$ and $F_{2}(1,0)$ with eccentricity $e$. A parabola with vertex at $F_{1}$ and focus at $F_{2}$ intersects the ellipse at a point $P$. If $\frac{\left|P F_{1}\right|}{\left|P F_{2}\right|}=e$, then the value of $e$ is $\qquad$. | \frac{\sqrt{3}}{3} | 105 | 10 |
math | 5. If $m=1996^{3}-1995^{3}+1994^{3}-1993^{3}$ $+\cdots+4^{3}-3^{3}+2^{3}-1^{3}$, then the last digit of $m$ is | 0 | 67 | 1 |
math | ## Task 4 - 210814
A brigade of excellent quality had been tasked with completing a certain number of measuring instruments in the shortest possible time. The brigade consisted of an experienced worker as the brigadier and nine young workers who had just completed their training.
Over the course of a day, each of the... | 160 | 110 | 3 |
math | Question 8 Let $a, b, c$ be given positive real numbers. Find all positive real numbers $x, y, z$ that satisfy the system of equations
$$
\left\{\begin{array}{l}
x+y+z=a+b+c, \\
4 x y z-\left(a^{2} x+b^{2} y+c^{2} z\right)=a b c .
\end{array}\right.
$$
(36th IMO Shortlist) | x = \frac{b + c}{2}, y = \frac{c + a}{2}, z = \frac{a + b}{2} | 101 | 33 |
math | 5. (40th IMO Problem) $n$ is a given integer, $n \geqslant 2$, 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_... | \frac{1}{8} | 134 | 7 |
math | 10. Given that the four vertices of the tetrahedron $P-ABC$ lie on the surface of sphere $O$, $PA=PB=PC$. $\triangle ABC$ is an equilateral triangle with side length 2, $E$ and $F$ are the midpoints of $AC$ and $BC$ respectively, $\angle EPF=60^{\circ}$. Then the surface area of sphere $O$ is | 6\pi | 93 | 3 |
math | A triangle has sides of length 3, 4, and 5. Determine the radius of the inscribed circle (circle inside the triangle and tangent to all three sides). | 1 | 36 | 1 |
math | Example 4 Given that $a$ and $b$ are the two real roots of the quadratic equation $t^{2}-t-1=0$. Solve the system of equations
$$
\left\{\begin{array}{l}
\frac{x}{a}+\frac{y}{b}=1+x, \\
\frac{x}{b}+\frac{y}{a}=1+\hat{y} .
\end{array}\right.
$$
(1000th Zu Chongzhi Cup Junior High School Mathematics Invitational Competit... | x=-\frac{1}{2}, y=-\frac{1}{2} | 116 | 18 |
math | 15. Given the sequence $\left\{a_{n}\right\}$ satisfies: $a_{1}=1, a_{n+1}=\frac{1}{8} a_{n}^{2}+m\left(n \in \mathbf{N}^{*}\right)$, if for any positive integer $n$, we have $a_{n}<4$, find the maximum value of the real number $m$. | 2 | 93 | 1 |
math | Task B-2.2. Determine all complex numbers $z$ such that $z^{3}=\bar{z}$. | z_{1}=0,\quadz_{2}=i,\quadz_{3}=-i,\quadz_{4}=1,\quadz_{5}=-1 | 27 | 35 |
math | Problem 5. Solve the system of equations in real numbers:
$$
\left\{\begin{array}{l}
a+c=4 \\
a c+b+d=6 \\
a d+b c=5 \\
b d=2
\end{array}\right.
$$ | (3,2,1,1)(1,1,3,2) | 57 | 17 |
math | Let $S$ be the set of integers which are both a multiple of $70$ and a factor of $630{,}000$. A random element $c$ of $S$ is selected. If the probability that there exists an integer $d$ with $\gcd (c,d) = 70$ and $\operatorname{lcm} (c,d) = 630{,}000$ is $\frac mn$ for some relatively prime integers $m$ and $n$, comp... | 106 | 130 | 3 |
math | Example 5 Let the integer $a$ divided by 7 leave a remainder of 3, and the integer $b$ divided by 7 leave a remainder of 5. If $a^{2}>4 b$, find the remainder when $a^{2}-4 b$ is divided by 7.
(1994, Tianjin City Junior High School Mathematics Competition) | 3 | 79 | 1 |
math | 9.071. $\log _{0.3}\left(x^{2}-5 x+7\right)>0$. | x\in(2;3) | 28 | 8 |
math | Among the poor, 120 K was distributed. If the number of the poor was 10 less, then each would receive exactly as much more as they would receive less if the number of the poor was 20 more. How many poor were there? | 40 | 55 | 2 |
math | Three distinct real numbers form (in some order) a 3-term arithmetic sequence, and also form (in possibly a different order) a 3-term geometric sequence. Compute the greatest possible value of the common ratio of this geometric sequence. | -2 | 49 | 2 |
math | Let $x_{1}, x_{2}, \ldots, x_{n}$ be a sequence where each term is 0, 1, or -2. If
$$
\left\{\begin{array}{l}
x_{1}+x_{2}+\cdots+x_{n}=-5 \\
x_{1}^{2}+x_{2}^{2}+\cdots+x_{n}^{2}=19
\end{array}\right.
$$
determine $x_{1}^{5}+x_{2}^{5}+\cdots+x_{n}^{5}$. | -125 | 134 | 4 |
math | ## Task 4.
Find all natural numbers $a$ and $b$ such that
$$
\left(a^{2}+b\right) \mid\left(a^{2} b+a\right) \quad \text { and } \quad\left(b^{2}-a\right) \mid\left(a b^{2}+b\right)
$$ | (n,n+1) | 79 | 5 |
math | 10.309. The area of an isosceles trapezoid circumscribed about a circle is $32 \mathrm{~cm}^{2}$; the acute angle of the trapezoid is $30^{\circ}$. Determine the sides of the trapezoid. | 8 | 67 | 1 |
math | Task B-4.1. The lengths of the sides of 5 equilateral triangles form an arithmetic sequence. The sum of the perimeters of these triangles is $120 \mathrm{~cm}$. The sum of the areas of the smallest and the largest triangle is $10 \sqrt{3} \mathrm{~cm}^{2}$ less than the sum of the areas of the remaining three triangles... | 90\sqrt{3}\mathrm{~}^{2} | 109 | 14 |
math | 3.273. $\cos ^{6}\left(\alpha-\frac{\pi}{2}\right)+\sin ^{6}\left(\alpha-\frac{3 \pi}{2}\right)-\frac{3}{4}\left(\sin ^{2}\left(\alpha+\frac{\pi}{2}\right)-\cos ^{2}\left(\alpha+\frac{3 \pi}{2}\right)\right)^{2}$. | \frac{1}{4} | 96 | 7 |
math | Solve the equation
$mn =$ (gcd($m,n$))$^2$ + lcm($m, n$)
in positive integers, where gcd($m, n$) – greatest common divisor of $m,n$, and lcm($m, n$) – least common multiple of $m,n$. | (m, n) = (2, 4) \text{ or } (4, 2) | 66 | 22 |
math | Real numbers $x,y,z$ satisfy the inequalities
$$x^2\le y+z,\qquad y^2\le z+x\qquad z^2\le x+y.$$Find the minimum and maximum possible values of $z$. | [-\frac{1}{4}, 2] | 51 | 11 |
math | (with literal expressions)
Let $a$ and $b$ be positive real numbers. Factorize the following expressions:
$$
a^{4}-b^{4} \quad a+b-2 \sqrt{a b}
$$ | ^{4}-b^{4}=(-b)(+b)(^{2}+b^{2})\quad+b-2\sqrt{}=(\sqrt{}-\sqrt{b})^{2} | 46 | 41 |
math | 31. [15] Find the sum of all primes $p$ for which there exists a prime $q$ such that $p^{2}+p q+q^{2}$ is a square. | 8 | 44 | 1 |
math | 1. Let $a<b<c<d<e$ be real numbers. Among the 10 sums of the pairs of these numbers, the least three are 32,36 and 37 while the largest two are 48 and 51 . Find all possible values of $e$. | 27.5 | 63 | 4 |
math | Find all functions $f:\mathbb{R}^+\rightarrow\mathbb{R}^+$ such that for all $x, y\in\mathbb{R}^+$,
\[ \frac{f(x)}{y^2} - \frac{f(y)}{x^2} \le \left(\frac{1}{x}-\frac{1}{y}\right)^2\]
($\mathbb{R}^+$ denotes the set of positive real numbers.)
[i](Proposed by Ivan Chan Guan Yu)[/i] | f(x) = \frac{C}{x^2} | 120 | 14 |
math | 249. Bonus Fund. In a certain institution, there was a bonus fund. It was planned to distribute it so that each employee of the institution would receive 50 dollars. But it turned out that the last person on the list would receive only 45 dollars. To maintain fairness, it was then decided to give each employee 45 dolla... | 995 | 110 | 3 |
math | Gwen, Eli, and Kat take turns flipping a coin in their respective order. The first one to flip heads wins. What is the probability that Kat will win? | \frac{1}{7} | 34 | 7 |
math | 189*. The sequence $\left(a_{n}\right)$ satisfies the conditions:
$$
a_{1}=0, \quad a_{n+1}=\frac{n}{n+1}\left(a_{n}+1\right)
$$
Find the formula for $a_{n}$. | a_{n}=\frac{n-1}{2} | 64 | 12 |
math | 14th Mexico 2000 Problem B1 Given positive integers a, b (neither a multiple of 5) we construct a sequence as follows: a 1 = 5, a n+1 = a a n + b. What is the largest number of primes that can be obtained before the first composite member of the sequence? | 5 | 72 | 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 | 5. Find all subsequences $\left\{a_{1}, a_{2}, \cdots, a_{n}\right\}$ of the sequence $\{1,2, \cdots, n\}$, such that for $1 \leqslant i \leqslant n$, the property $i+1 \mid 2\left(a_{1}+a_{2}+\cdots+a_{j}\right)$ holds. | (1,2,\cdots,n)or(2,1,3,4,\cdots,n) | 96 | 23 |
math | 6. Let $\left(x_{1}, x_{2}, \cdots, x_{20}\right)$ be a permutation of $(1,2, \cdots, 20)$, and satisfy
$$
\sum_{i=1}^{20}\left(\left|x_{i}-i\right|+\left|x_{i}+i\right|\right)=620 \text {. }
$$
Then the number of such permutations is. $\qquad$ | (10!)^{2} | 103 | 7 |
math | 5. Two natural numbers $x$ and $y$ sum to 111, such that the equation
$$
\sqrt{x} \cos \frac{\pi y}{2 x}+\sqrt{y} \sin \frac{\pi x}{2 y}=0
$$
holds. Then a pair of natural numbers $(x, y)$ that satisfies the condition is
$\qquad$ | (37,74) | 84 | 7 |
math | 3. Find the sum
$$
\frac{2^{1}}{4^{1}-1}+\frac{2^{2}}{4^{2}-1}+\frac{2^{4}}{4^{4}-1}+\frac{2^{8}}{4^{8}-1}+\cdots .
$$ | 1 | 69 | 1 |
math | 12. (VIE 2) The polynomial $1976\left(x+x^{2}+\cdots+x^{n}\right)$ is decomposed into a sum of polynomials of the form $a_{1} x+a_{2} x^{2}+\ldots+a_{n} x^{n}$, where $a_{1}, a_{2}, \cdots, a_{n}$ are distinct positive integers not greater than $n$. Find all values of $n$ for which such a decomposition is possible. | 1 < n < 3951 \text{ and } n+1 \mid 2 \cdot 1976 | 113 | 28 |
math | 4. For which integer values of $a$ and $b$ does the system
$$
\left\{\begin{array}{l}
m^{2}+n^{2}=b \\
\frac{m^{n}-1}{m^{n}+1}=a
\end{array}\right.
$$
have a solution (for $m$ and $n$) in the set $\mathbb{Z}$ of integers. | =-1,b=n^{2},n\neq0;=0,b=1+n^{2};=2,b=10;=3,b=5 | 93 | 34 |
math | Three. (25 points) Given the equation
$$
\left(m^{2}-1\right) x^{2}-3(3 m-1) x+18=0
$$
has two positive integer roots, where $m$ is an integer.
(1) Find the value of $m$;
(2) The sides of $\triangle A B C$, $a$, $b$, and $c$, satisfy $c=2 \sqrt{3}$, $m^{2}+a^{2} m-8 a=0$, $m^{2}+b^{2} m-8 b=0$, find the area of $\tria... | 1 \text{ or } \sqrt{9+12 \sqrt{2}} | 145 | 18 |
math | 5.10. Given vectors $\bar{a}(6 ;-8 ; 5 \sqrt{2})$ and $\bar{b}(2 ;-4 ; \sqrt{2})$. Find the angle formed by the vector $\bar{a}-\bar{b}$ with the $O z$ axis. | 45 | 65 | 2 |
math | 17. Calculate $\frac{3}{1!+2!+3!}+\frac{4}{2!+3!+4!}+\cdots+\frac{2001}{1999!+2000!+2001!}$ The value is $\qquad$ . | \frac{1}{2}-\frac{1}{2001!} | 69 | 18 |
math | 4. At the vertices of a cube, the numbers $1^{2}, 2^{2}, \ldots, 8^{2}$ (one number per vertex) are placed. For each edge, the product of the numbers at its ends is calculated. Find the maximum possible sum of all these products. | 9420 | 64 | 4 |
math | Task B-3.6. Solve the system of equations on the interval $\left\langle 0, \frac{\pi}{2}\right\rangle$
$$
\left\{\begin{array}{l}
\frac{\cos x}{\cos y}=2 \cos ^{2} y \\
\frac{\sin x}{\sin y}=2 \sin ^{2} y
\end{array}\right.
$$ | \frac{\pi}{4},\frac{\pi}{4} | 91 | 14 |
math | A sphere tangent to the edges of a $4 \mathrm{~cm}$ cube is inscribed. One vertex of the cube is $A$. What is the volume of the region consisting of points that are inside the cube, outside the sphere, and closest to $A$ among the vertices of the cube? | 8-\frac{4}{3}\pi | 63 | 9 |
math | Example 6. Find $\lim _{x \rightarrow+\infty} x\left(\operatorname{arctg} x-\frac{\pi}{2}\right)$. | -1 | 38 | 2 |
math | 7.5. Find the largest natural number $n$ for which $3^{n}$ divides the number $a=1 \cdot 2 \cdot 3 \cdot \ldots \cdot 2020$. | 1005 | 47 | 4 |
math | Find all functions $f: (1,\infty)\text{to R}$ satisfying
$f(x)-f(y)=(y-x)f(xy)$ for all $x,y>1$.
[hide="hint"]you may try to find $f(x^5)$ by two ways and then continue the solution.
I have also solved by using this method.By finding $f(x^5)$ in two ways
I found that $f(x)=xf(x^2)$ for all $x>1$.[/hide] | f(x) = \frac{k}{x} | 107 | 11 |
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