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. Let points $A(-2,0), B(2,0)$, and point $P$ be on the unit circle. Then the maximum value of $|P A||P B|$ is $\qquad$ | 5 | 47 | 1 |
math | 1. Determine the minimum value of the expression $\frac{x^{2}+y^{2}+z^{2}}{x y+y z}$, where $x>0, y>0, \quad z>0$. | \sqrt{2} | 48 | 5 |
math | Example 3 Find the smallest natural number $n$ such that $\frac{n-13}{5 n+6}$ is a non-zero reducible fraction.
(6th IMO) | 84 | 39 | 2 |
math | 9.166. $3^{\lg x+2}<3^{\lg x^{2}+5}-2$.
Translate the above text into English, please retain the original text's line breaks and format, and output the translation result directly.
9.166. $3^{\lg x+2}<3^{\lg x^{2}+5}-2$. | x\in(0.01;\infty) | 82 | 12 |
math | Example 3. Solve the equation $y^{\text {IV }}-16 y=0$. | C_{1}e^{2x}+C_{2}e^{-2x}+C_{3}\cos2x+C_{4}\sin2x | 22 | 34 |
math | 5. Let $A, B$ be two moving points on the ellipse $x^{2}+3 y^{2}=1$, and $O A \perp O B(O$ is the origin), then the product of the maximum and minimum values of $|A B|$ is $\qquad$ . | \frac{2\sqrt{3}}{3} | 64 | 12 |
math | 6-138 $S$ is the set of all non-negative integers. Find all functions $f: S \rightarrow S$, $g: S \rightarrow S, h: S \rightarrow S$ that satisfy the following two conditions:
(1) For any $m, n \in S, f(m+n)=g(m)+h(n)+2 m n$.
(2) $g(1)=h(1)=1$. | f(n)=n^{2}-a n+2 a, \\ g(n)=h(n)=n^{2}-a n+a, \\ a \in\{0,1,2,3,4\} | 92 | 45 |
math | 10. Given a positive integer $n$, let $p(n)$ be the product of the non-zero digits of $n$. For example, $p(7)=7, p(204)=2 \times 4=8$, etc. Let $S=p(1)+p(2)+\cdots+p(999)$. What is the largest prime factor of $S$ ?
(1 mark)
對於正整數 $n$, 設 $p(n)$ 為 $n$ 的所有非零數字之積。
例如: $p(7)=7$ 、 $p(204)=2 \times 4=8$ 等。
設 $S=p(1)+p(2)+\cdo... | 103 | 183 | 3 |
math | Question 216, Given $2^{2013}<5^{867}<2^{2014}$, how many integer pairs $(\mathrm{m}, \mathrm{n})$ satisfy: $5^{\mathrm{n}}<2^{\mathrm{m}}<$ $2^{\mathrm{m}+2}<5^{\mathrm{n}+1}$ ~ where $1 \leq \mathrm{m} \leq 2012$. | 279 | 103 | 3 |
math | 13. Determine the number of pairs $(a, b)$ of integers with $1 \leq b<a \leq 200$ such that the sum $(a+b)+(a-b)+a b+a / b$ is a square of a number. | 112 | 55 | 3 |
math | 10th Irish 1997 Problem B2 The quadrilateral ABCD has an inscribed circle. ∠A = ∠B = 120 o , ∠C = 30 o and BC = 1. Find AD. | \frac{1}{2}(\sqrt{3}-1) | 54 | 14 |
math | 10. (25 points) For any positive integers $m, n$, define the function $f(m, n)$ as follows:
(i) $f(1,1)=1$;
(ii) $f(m+1, n)=f(m, n)+2(m+n)$;
(iii) $f(m, n+1)=f(m, n)+2(m+n-1)$.
(1) Find the analytical expression for $f(m, n)$;
(2) Let $a_{n}=\frac{\sqrt{f(n, n)}}{2^{n-1}}\left(n \in \mathbf{Z}_{+}\right), S_{n}$ be the... | S_{n}<6 | 178 | 5 |
math | For each integer $a_{0}>1$, define the sequence $a_{0}, a_{1}, \cdots$ as follows: for any $n \geqslant 0$,
$a_{n+1}=\left\{\begin{array}{ll}\sqrt{a_{n}}, & \sqrt{a_{n}} \text { is an integer; } \\ a_{n}+3, & \text { otherwise. }\end{array}\right.$
Find all $a_{0}$ such that there exists a number $A$ for which $a_{n}=A... | a_{0} \text{ is all multiples of 3} | 133 | 14 |
math | 7. In the tetrahedron $S-ABC$,
$$
SA=SB=SC=\sqrt{21}, BC=6 \text{.}
$$
If the projection of point $A$ onto the plane containing the side $SBC$ is exactly the orthocenter of $\triangle SBC$, then the volume of the inscribed sphere of the tetrahedron $S-ABC$ is | \frac{4 \pi}{3} | 87 | 9 |
math | ## Task B-1.1.
The sum of two numbers is 6. If the sum of their cubes is 90, what is the sum of their squares? | 22 | 36 | 2 |
math | One, (20 points) Given $t=\sqrt{2}-1$. If positive integers $a$, $b$, and $m$ satisfy
$$
(a t+m)(b t+m)=17 m
$$
find the value of $a b$. | 72 | 56 | 2 |
math | 6. Variant 1. In the kindergarten, 5 children eat porridge every day, 7 children eat porridge every other day, and the rest never eat porridge. Yesterday, 9 children ate porridge. How many children will eat porridge today? | 8 | 55 | 1 |
math | 1. $[\mathbf{3}]$ What is the sum of all of the distinct prime factors of $25^{3}-27^{2}$ ? | 28 | 34 | 2 |
math | On a clock, there are two instants between $12$ noon and $1 \,\mathrm{PM}$, when the hour hand and the minute hannd are at right angles. The difference [i]in minutes[/i] between these two instants is written as $a + \dfrac{b}{c}$, where $a, b, c$ are positive integers, with $b < c$ and $b/c$ in the reduced form. What i... | 51 | 108 | 2 |
math | 1. Given quadratic trinomials $f_{1}(x)=x^{2}-a x-3, f_{2}(x)=x^{2}+2 x-b, f_{3}(x)=3 x^{2}+(2-2 a) x-6-b$ and $f_{4}(x)=3 x^{2}+(4-a) x-3-2 b$. Let the differences of their roots be $A, B, C$ and $D$ respectively. It is known that $|C| \neq|D|$. Find the ratio $\frac{A^{2}-B^{2}}{C^{2}-D^{2}}$. The values of $A, B, C,... | 3 | 164 | 1 |
math | 14. If $a, b, c$ form an arithmetic sequence, then the midpoint of the line segment cut by the line $a x + b y + c = 0$ on the ellipse $\frac{x^{2}}{2} + \frac{y^{2}}{8} = 1$ has the trajectory equation $\qquad$. | 2\left(x-\frac{1}{2}\right)^{2}+\frac{(y+1)^{2}}{2}=1 | 74 | 30 |
math | 10. Let $a, b, c (1 \leqslant a < b < c \leqslant 9)$ be integers, and $\overline{a b c} \cdot \overline{b c a} \cdot \overline{c a b} + 1$ is divisible by 9. Then the minimum value of $a + b + c$ is $\qquad$, and the maximum value is $\qquad$. | 8, 23 | 98 | 5 |
math | For every $A \subset S$, let
$$
S_{\mathrm{A}}=\left\{\begin{array}{ll}
(-)^{\mid \mathrm{A}} \mid \sum_{\mathbf{a} \in \mathrm{A}} a, & A \neq \varnothing, \\
0, & A=\varnothing .
\end{array}\right.
$$
Find $\sum_{\mathrm{A} \subset \mathrm{S}} S_{\mathrm{A}}$. | 0 | 109 | 1 |
math | Let $ABCD$ be an isosceles trapezoid with $AD=BC$ and $AB<CD.$ Suppose that the distances from $A$ to the lines $BC,CD,$ and $BD$ are $15,18,$ and $10,$ respectively. Let $K$ be the area of $ABCD.$ Find $\sqrt2 \cdot K.$ | 567 | 82 | 3 |
math | 3. There are three square pools, large, medium, and small, with inner side lengths of 6 meters, 3 meters, and 2 meters, respectively. Two piles of gravel are submerged in the medium and small pools, causing the water levels to rise by 6 cm and 4 cm, respectively. If these two piles of gravel are submerged in the large ... | 1\frac{17}{18} | 93 | 10 |
math | ## 172. Math Puzzle $9 / 79$
The Becker family from Halle has three children. They want to travel by train to visit relatives in Bitterfeld.
For Anja, this trip is still free of charge, as she is only two years old. Silke and Frank each have to pay half the adult fare. However, the family receives a one-third discoun... | 9.60\mathrm{M} | 135 | 9 |
math | 6. If the three sides $a, b, c$ of $\triangle A B C$ satisfy $a^{2}+b^{2}+3 c^{2}=7$, then the maximum value of the area of $\triangle A B C$ is | \frac{\sqrt{7}}{4} | 54 | 10 |
math | You use a lock with four dials, each of which is set to a number between 0 and 9 (inclusive). You can never remember your code, so normally you just leave the lock with each dial one higher than the correct value. Unfortunately, last night someone changed all the values to 5. All you remember about your code is that no... | 10 | 107 | 2 |
math | 7.3. On side $B C$ of triangle $A B C$, a point $M$ is marked such that $A B=B M$ and $A M=M C$. It is known that angle $B$ is five times angle $C$. Find the angles of the triangle. | \angleA=60,\angleB=100,\angleC=20 | 61 | 19 |
math | 4. Determine the remainder when
$$
\sum_{i=0}^{2015}\left\lfloor\frac{2^{i}}{25}\right\rfloor
$$
is divided by 100 , where $\lfloor x\rfloor$ denotes the largest integer not greater than $x$. | 14 | 71 | 2 |
math | 3. Factorize: $a+(a+b) x+(a+2 b) x^{2}+(a+3 b) x^{3}+3 b x^{4}+2 b x^{5}+b x^{6}$. | (1+x)(1+x^{2})(++^{2}+^{3}) | 53 | 17 |
math | Points $E$ and $C$ are chosen on a semicircle with diameter $AB$ and center $O$ such that $OE \perp AB$ and the intersection point $D$ of $AC$ and $OE$ is inside the semicircle. Find all values of $\angle{CAB}$ for which the quadrilateral $OBCD$ is tangent. | 30^\circ | 80 | 4 |
math | II. (This question is worth 25 points) Find all real numbers $p$ such that the cubic equation $5 x^{3}-5(p+1) x^{2}+(71 p-1) x+1=66 p$ has three roots that are all natural numbers. | 76 | 63 | 2 |
math | Consider a $2 \times n$ grid where each cell is either black or white, which we attempt to tile with $2 \times 1$ black or white tiles such that tiles have to match the colors of the cells they cover. We first randomly select a random positive integer $N$ where $N$ takes the value $n$ with probability $\frac{1}{2^n}$. ... | \frac{9}{23} | 117 | 8 |
math | 2 Find the smallest positive real number $k$ such that for any 4 distinct real numbers $a, b, c, d$ not less than $k$, there exists a permutation $p, q, r, s$ of $a, b, c, d$ such that the equation
$$
\left(x^{2}+p x+q\right)\left(x^{2}+r x+s\right)=0
$$
has 4 distinct real roots. (Supplied by Feng Zhigang) | 4 | 110 | 1 |
math | 1. For a finite set $A$, there exists a function $f: \mathbf{N}_{+} \rightarrow A$, with the following property: if $i, j \in \mathbf{N}_{+}$ and $|i-j|$ is a prime number, then $f(i) \neq f(j)$. How many elements does the set $A$ have at minimum? | 4 | 84 | 1 |
math | $4 \cdot 216$ Try to find the number of all such positive integers: in their representation in base $n$, all digits are different, and each digit, except the leftmost one, differs from some digit to its left by $\pm 1$ (express the answer as a simple explicit function of $n$), and prove your conclusion. | 2^{n+1}-2n-2 | 75 | 10 |
math | Let $m$ be a positive integer less than $2015$. Suppose that the remainder when $2015$ is divided by $m$ is $n$. Compute the largest possible value of $n$.
[i] Proposed by Michael Ren [/i] | 1007 | 57 | 4 |
math | Solve the system in positive numbers:
$$
\begin{cases}x^{y} & =z \\ y^{z} & =x \\ z^{x} & =y\end{cases}
$$ | 1 | 44 | 1 |
math | 11. Use the digits 1-9 each once to form a two-digit perfect square, a three-digit perfect square, and a four-digit perfect square. What is the smallest four-digit perfect square among them? $\qquad$ | 1369 | 48 | 4 |
math | 7. Let the pair of positive integers $(x, y)$ satisfy $\frac{1}{\sqrt{x}}+\frac{1}{\sqrt{y}}=\frac{1}{\sqrt{20}}$. Then $xy$ has $\qquad$ different possible values. | 2 | 58 | 1 |
math | Find the number of ordered triples of sets $(T_1, T_2, T_3)$ such that
1. each of $T_1, T_2$, and $T_3$ is a subset of $\{1, 2, 3, 4\}$,
2. $T_1 \subseteq T_2 \cup T_3$,
3. $T_2 \subseteq T_1 \cup T_3$, and
4. $T_3\subseteq T_1 \cup T_2$. | 625 | 116 | 3 |
math | Example 2. Consider the sequence $\left\{a_{n}\right\}$ defined by $a_{1}=2, a_{n+1}=\frac{a_{n}}{a_{n}+3}$ $(n \geqslant 1)$. Solve the following problems:
(1) If $b_{n}=\frac{1}{a_{n}}$, find the relationship between $b_{n+1}$ and $b_{n}$;
(2) Find the general term of the sequence $\left\{a_{n}\right\}$. | a_{n}=\frac{2}{2 \cdot 3^{n-1}-1} | 123 | 21 |
math | 3. Let points $A$ and $B$ lie on the parabola $y^{2}=6 x$ and the circle $\odot C:(x-4)^{2}+y^{2}=1$, respectively. Then the range of $|A B|$ is | [\sqrt{15}-1,+\infty) | 59 | 12 |
math | Find the largest constant $K$ such that for all positive real numbers $a, b$, and $c$, we have
$$
\sqrt{\frac{a b}{c}}+\sqrt{\frac{b c}{a}}+\sqrt{\frac{a c}{b}} \geqslant K \sqrt{a+b+c}
$$ | \sqrt{3} | 72 | 5 |
math | Let $n$ be a nonnegative integer less than $2023$ such that $2n^2 + 3n$ is a perfect square. What is the sum of all possible $n$?
[i]Proposed by Giacomo Rizzo[/i] | 444 | 58 | 3 |
math | Let $a$ and $b$ be nonzero real numbers such that $\tfrac{1}{3a}+\tfrac{1}{b}=2011$ and $\tfrac{1}{a}+\tfrac{1}{3b}=1$. What is the quotient when $a+b$ is divided by $ab$? | 1509 | 73 | 4 |
math | 2.3.18 ** On a horizontal plane, three points are 100 meters, 200 meters, and 300 meters away from the base of an antenna. The sum of the angles of elevation from these three points to the antenna is $90^{\circ}$. What is the height of the antenna? | 100 | 72 | 3 |
math | 5. Real numbers $a, b, c$ and a positive number $\lambda$ make $f(x)=x^{3}+a x^{2}+b x+c$ have three real roots $x_{1}, x_{2}, x_{3}$, and satisfy
(1) $x_{2}-x_{1}=\lambda$;
(2) $x_{3}>\frac{1}{2}\left(x_{1}+x_{2}\right)$.
Find the maximum value of $\frac{2 a^{3}+27 c-9 a b}{\lambda^{3}}$. | \frac{3\sqrt{3}}{2} | 131 | 12 |
math | Grandpa forgot the four-digit code of his mobile phone. He only remembered that the first digit was not zero, that in the middle were either two fours or two sevens or a four and a seven (in an unknown order), and that the number was divisible by 15. How many possibilities are there for the forgotten code? What digit c... | 24 | 102 | 2 |
math | [ Triangle inequality (miscellaneous). ] [ Minimum or maximum distance (length).]
Petya bought a "Constructor" set, which contained 100 sticks of different lengths. The instructions for the "Constructor" state that any three sticks from the set can form a triangle. Petya decided to test this statement by forming trian... | 1 | 118 | 1 |
math | 8. Given the sequences $\left\{a_{n}\right\}$ and $\left\{b_{n}\right\}$ with the general terms $a_{n}=2^{n}, b_{n}=5 n-2$. Then the sum of all elements in the set
$$
\left\{a_{1}, a_{2}, \cdots, a_{2019}\right\} \cap\left\{b_{1}, b_{2}, \cdots, b_{2019}\right\}
$$
is $\qquad$ | 2184 | 123 | 4 |
math | 2A. Determine all pairs of integers $x$ and $y$ such that
$$
1 + 2026x + 2028y = xy
$$ | (x,y)\in{(2029,2027^{2}+1),(2027,-2027^{2}+2026),(4055,4053),(1,-1),(2027^{2}+2028,2027),} | 39 | 71 |
math | 13. (10 points) There are two warehouses, A and B. Warehouse B originally had 1200 tons of inventory. When $\frac{7}{15}$ of the goods in Warehouse A and $\frac{1}{3}$ of the goods in Warehouse B are moved, and then 10% of the remaining goods in Warehouse A are moved to Warehouse B, the weights of the goods in Warehous... | 1875 | 107 | 4 |
math | 1. Given that $x$ and $y$ are real numbers, and satisfy
$$
\left(x+\sqrt{x^{2}+2008}\right)\left(y+\sqrt{y^{2}+2008}\right)=2008 \text {. }
$$
Then the value of $x^{2}-3 x y-4 y^{2}-6 x-6 y+2008$
is $\qquad$ | 2008 | 98 | 4 |
math | Let's determine the maximum or minimum value of the function
$$
y=\left(x+\frac{a}{2}\right)^{2}-(3 x+2 a)^{2}
$$
knowing that $y$ takes its extreme value when
$$
x=-\frac{11}{8}
$$ | \frac{1}{8} | 67 | 7 |
math | Example 11. What is the probability of event $A$ occurring in each trial, if the most probable number of occurrences of event $A$ in 120 trials is 32? | \frac{32}{121}\leqp\leq\frac{33}{121} | 42 | 25 |
math | ## Task 30/76
Determine all pairs $(n ; m)$ of natural numbers that satisfy the equation $4^{n}+65=9^{m}$. | n==2 | 39 | 3 |
math | 13.386 On the sides $AB, BC, AC$ of an equilateral triangle $ABC$, points $A_1, B_1, C_1$ are located such that $AA_1 = BB_1 = CC_1 = x$. The side of the triangle is $a$. Find such $x$ for which the ratio of the areas of triangles $A_1B_1C_1$ and $ABC$ is equal to $m$. Within what limits can the value of $m$ vary? | \frac{}{6}(3\\sqrt{12-3})for\frac{1}{4}\leq<1 | 113 | 27 |
math | (10) Given positive real numbers $a$ and $b$ satisfying $a^{2}+b^{2}=1$, and $a^{3}+b^{3}+1=m(a+b+1)^{3}$, find the range of values for $m$. | [\frac{3\sqrt{2}-4}{2},\frac{1}{4}) | 60 | 20 |
math | Find all functions $f,g: \mathbb{R} \to \mathbb{R}$ such that satisfies
$$f(x^2-g(y))=g(x)^2-y$$
for all $x,y \in \mathbb{R}$ | f(x) = g(x) = x \ \ \forall x \in \mathbb{R} | 54 | 22 |
math | 3. The volume of the circumsphere of the smaller tetrahedron formed by the centers of the four faces of a regular tetrahedron with edge length 1 is $\qquad$ | \frac{\sqrt{6}\pi}{216} | 41 | 13 |
math | Consider a cube with a fly standing at each of its vertices. When a whistle blows, each fly moves to a vertex in the same face as the previous one but diagonally opposite to it. After the whistle blows, in how many ways can the flies change position so that there is no vertex with 2 or more flies? | 81 | 66 | 2 |
math | Determine all primes $p$ and all non-negative integers $m$ and $n$, such that $$1 + p^n = m^3. $$ | (p, n, m) = (7, 1, 2) | 32 | 18 |
math | ## Problem 2
8 players compete in a tournament. Everyone plays everyone else just once. The winner of a game gets 1 , the loser 0 , or each gets $1 / 2$ if the game is drawn. The final result is that everyone gets a different score and the player placing second gets the same as the total of the four bottom players. Wh... | 3\wins\against\7 | 93 | 7 |
math | 24th Eötvös 1917 Problem 2 A square has 10s digit 7. What is its units digit? | 6 | 32 | 1 |
math | 2. Jovan and Goce were fishing with their sons. Jovan caught as many fish as his son, and Goce caught three times as many as his son. Together, they caught 25 fish. Jovan's son is named Risto. What is the name of Goce's son? | Jovan | 63 | 2 |
math | We call a set “sum free” if no two elements of the set add up to a third element of the set. What is the maximum size of a sum free subset of $\{ 1, 2, \ldots , 2n - 1 \}$. | n | 57 | 1 |
math | The value of $ 21!$ is $ 51{,}090{,}942{,}171{,}abc{,}440{,}000$, where $ a$, $ b$, and $ c$ are digits. What is the value of $ 100a \plus{} 10b \plus{} c$? | 709 | 88 | 3 |
math | 2. If $(2 x+4)^{2 n}=a_{0}+a_{1} x+a_{2} x^{2}+\cdots+a_{2 n} x^{2 n}\left(n \in \mathbf{N}^{+}\right)$, then the remainder when $a_{2}+a_{4}+\cdots+a_{2 n}$ is divided by 3 is . $\qquad$ | 1 | 93 | 1 |
math | 3.18. Calculate
$$
\int_{L}\left(z^{2}+2 z \bar{z}\right) d z
$$
where $L-$ is the arc of the circle $|z|=1, \arg z \in[0, \pi]$. | -\frac{14}{3} | 62 | 8 |
math | 6. Let complex numbers $a, b, c$ satisfy:
$$
|a|=|b|=|c|=2, a+b+c=0 \text {. }
$$
Define $f(z)=|z-a|+|z-b|+|z-c|(z$ being any complex number). Then the minimum value of $f(z)$ is $\qquad$ | 6 | 78 | 1 |
math | 1. The maximum value of the function $y=5 \sqrt{x-1}+\sqrt{10-2 x}$ is | 6\sqrt{3} | 28 | 6 |
math | Find the limit of the following sequence:
$$
u_{n}=\sum_{i=1}^{n} \frac{1}{F_{i} F_{i+2}}
$$ | 1 | 40 | 1 |
math | Given a grid strip (one cell wide), infinite in both directions. Two cells of the strip are traps, with $-N$ cells between them, one of which is occupied by a grasshopper. On each move, we call out a natural number, after which the grasshopper jumps that number of cells to the left or right (at its choice). For which $... | 2^{k}-1 | 132 | 5 |
math | Given $0<a<1,0<b<1$, and $a b=\frac{1}{36}$. Find the minimum value of $u=\frac{1}{1-a}+\frac{1}{1-b}$. | \frac{12}{5} | 49 | 8 |
math | 5.085 Six boxes of different materials are delivered to eight floors of a construction site. In how many ways can the materials be distributed across the floors? In how many of these ways will at least two materials be delivered to the eighth floor? | 8^{6};8^{6}-13\cdot7^{5} | 51 | 16 |
math | 5. The sequence $\left\{x_{n}\right\}$ satisfies
$$
\begin{array}{l}
x_{1}=1, \\
x_{i+1}-x_{i}=\sqrt{x_{i+1}+x_{i}}(i=1,2, \cdots) .
\end{array}
$$
Then the general term formula $x_{n}=$ . $\qquad$ | x_{n}=\frac{n^{2}+n}{2} | 91 | 15 |
math | LI OM - I - Task 5
Determine all pairs $ (a,b) $ of natural numbers for which the numbers $ a^3 + 6ab + 1 $ and $ b^3 + 6ab + 1 $ are cubes of natural numbers. | (1,1) | 57 | 5 |
math | 3. Problem: Find all real numbers $x$ and $y$ such that
$$
\begin{aligned}
x^{2}+y^{2} & =2, \\
\frac{x^{2}}{2-y}+\frac{y^{2}}{2-x} & =2 .
\end{aligned}
$$ | 1 | 70 | 1 |
math | 5. Let $d_{1}, d_{2}, \ldots, d_{n}$ be all the natural divisors of the number $10!=1 \cdot 2 \cdot \ldots \cdot 10$. Find the sum
$$
\frac{1}{d_{1}+\sqrt{10!}}+\frac{1}{d_{2}+\sqrt{10!}}+\ldots+\frac{1}{d_{n}+\sqrt{10!}}
$$ | \frac{270}{2\sqrt{10!}} | 107 | 15 |
math | 21.3.7 ** From the sequence of positive integers $1,2,3,4, \cdots$, remove the multiples of 3 and 4, but retain all multiples of 5 (for example, 15 and 120 are not removed). The remaining numbers form a new sequence: $a_{1}=1, a_{2}=2, a_{3}=5, a_{4}=7, \cdots$. Find $a_{1999}$. | 3331 | 106 | 4 |
math | Let $\{a_n\}_{n\geq 1}$ be a sequence defined by $a_n=\int_0^1 x^2(1-x)^ndx$.
Find the real value of $c$ such that $\sum_{n=1}^{\infty} (n+c)(a_n-a_{n+1})=2.$ | 22 | 75 | 2 |
math | 6. How many solutions in natural numbers does the equation $(a+1)(b+1)(c+1)=2 a b c$ have? | 27 | 31 | 2 |
math | (4) Given an integer $n \geqslant 3$, real numbers $a_{1}, a_{2}, \cdots, a_{n}$ satisfy $\min _{1 \leqslant i<j \leqslant n} \mid a_{i}-$ $a_{j} \mid=1$. Find the minimum value of $\sum_{k=1}^{n}\left|a_{k}\right|^{3}$. | \frac{1}{32}(n^{2}-1)^{2} | 99 | 17 |
math | Let $AB$ be a segment of unit length and let $C, D$ be variable points of this segment. Find the maximum value of the product of the lengths of the six distinct segments with endpoints in the set $\{A,B,C,D\}.$ | \frac{\sqrt{5}}{125} | 53 | 12 |
math | 3. In a right-angled triangle $ABC$, with the right angle at vertex $C$, the ratio of the lengths of the altitude and the median drawn from the vertex of the right angle is equal to 12 : 13. Determine the ratio of the lengths of the legs of the triangle $a: b$ if $a > b$. | 3:2 | 73 | 3 |
math | 12.8 $f(x)=\sqrt{x^{2}+3}+\frac{2 x}{x+1} ; f^{\prime}(1)=?$ | 1 | 36 | 1 |
math | 2. For the quadratic equation in $x$
$$
m^{2} x^{2}+(2 m+3) x+1=0
$$
there are two real roots whose product is 1; for the quadratic equation in $x$
$$
x^{2}+(2 a+m) x+2 a+1-m^{2}=0
$$
there is a real root that is greater than 0 and less than 4. Then the integer value of $a$ is . $\qquad$ | -1 | 109 | 2 |
math | Example 8 Try to find
$$
\begin{aligned}
p= & (1-1993)\left(1-1993^{2}\right) \cdots\left(1-1993^{1993}\right)+1993\left(1-1993^{2}\right)\left(1-1993^{3}\right) \cdots\left(1-1993^{1993}\right)+ \\
& 1993^{2}\left(1-1993^{3}\right) \cdots\left(1-1993^{1993}\right)+1993^{3}\left(1-1993^{4}\right) \cdots\left(1-1993^{19... | 1 | 242 | 1 |
math | 1. The number of five-digit numbers where the sum of any two adjacent digits is divisible by 3 is $\qquad$ (answer with a number) . | 1254 | 33 | 4 |
math | [ Regular pyramid ] [ Sections, unfoldings, and other frameworks. ]
On the extension of edge $S T$ beyond point $T$ of the regular quadrilateral pyramid $S P Q R T$ with vertex $S$, a point $B$ is taken such that the distance from it to the plane $S P Q$ is $\frac{9 \sqrt{7}}{2}$. Find the segment $B T$, if $Q R=12$, ... | 5 | 105 | 1 |
math | I live on a very short street, consisting of 14 small family houses. The odd-numbered houses are on one side of the street, and the even-numbered ones are on the other (e.g., 1 and 2 are opposite each other). On one side, families with surnames representing colors live, and on the other side, families with surnames rep... | 13 | 231 | 2 |
math | 210. Find the variance and standard deviation of the discrete random variable $X$, given by the distribution law:
$$
\begin{array}{ccccc}
X & -5 & 2 & 3 & 4 \\
p & 0.4 & 0.3 & 0.1 & 0.2
\end{array}
$$ | 3.9 | 76 | 3 |
math | 3rd ASU 1963 Problem 6 Find the smallest value x such that, given any point inside an equilateral triangle of side 1, we can always choose two points on the sides of the triangle, collinear with the given point and a distance x apart. Solution | \frac{2}{3} | 59 | 7 |
math | We know about an 80-term sequence that any intermediate term is equal to the product of its neighbors, and the product of the first 40 terms is 8, as well as the product of all terms is 8. Determine the first and second elements of the sequence! | a_{1}=2,a_{2}=4 | 58 | 10 |
math | The rhombus $ABCD$ is given. Let $E$ be one of the points of intersection of the circles $\Gamma_B$ and $\Gamma_C$, where $\Gamma_B$ is the circle centered at $B$ and passing through $C$, and $\Gamma_C$ is the circle centered at $C$ and passing through $B$. The line $ED$ intersects $\Gamma_B$ at point $F$.
Find the val... | 60^\circ | 110 | 4 |
math | 2. Two bottles of equal volume are filled with a mixture of water and juice. In the first bottle, the ratio of the quantities of water and juice is $2: 1$, and in the second bottle, it is $4: 1$. If we pour the contents of both bottles into a third bottle, what will be the ratio of the quantities of water and juice in ... | 11:4 | 80 | 4 |
math | If $n$ is a positive integer, what is the smallest value that the sum of the digits in the decimal representation of $3 n^{2}+n+1$ can take?
# | 3 | 40 | 1 |
math | 1. (6 points) During New Year's Day, the class committee used 730 yuan to buy a souvenir worth 17 yuan for each student, with 16 yuan left over. Therefore, the class has $\qquad$ students. | 42 | 53 | 2 |
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