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 | Example 1 If the equations of the two asymptotes of a hyperbola are $y= \pm \frac{2}{3} x$, and it passes through the point $M\left(\frac{9}{2},-1\right)$, try to find its equation. | \frac{x^{2}}{18}-\frac{y^{2}}{8}=1 | 60 | 21 |
math | 7. The "Hua Luogeng" Golden Cup Junior Mathematics Invitational Competition, the first session was held in 1986, the second session in 1988, the third session in 1991, and thereafter every 2 years. The sum of the digits of the year in which the first "Hua Cup Competition" was held is: $A 1=1+9+8+6=24$.
The sum of the d... | 629 | 175 | 3 |
math | (2) If $f(g(x))=\sin 2 x, g(x)=\tan \frac{x}{2}(0<x<\pi)$, then $f\left(\frac{\sqrt{2}}{2}\right)=$ $\qquad$ . | \frac{4\sqrt{2}}{9} | 56 | 12 |
math | 1.5.2 * Let real numbers $a, x, y$ satisfy the following conditions
$$
\left\{\begin{array}{l}
x+y=2 a-1, \\
x^{2}+y^{2}=a^{2}+2 a-3 .
\end{array}\right.
$$
Find the minimum value that the real number $xy$ can take. | \frac{11-6\sqrt{2}}{4} | 84 | 15 |
math | We start with 5000 forints in our pocket to buy gifts. We enter three stores. In each store, we like a gift item, which we will buy if we can afford it. The prices of the items in each store, independently of each other, are 1000, 1500, or 2000 Ft with a probability of $\frac{1}{3}$. What is the probability that we wil... | \frac{17}{27} | 107 | 9 |
math | 13. (20 points) Two circles on the same side of the $x$-axis: a moving circle $C_{1}$ and the circle $4 a^{2} x^{2}+4 a^{2} y^{2}-4 a b x-2 a y+b^{2}=0$ are externally tangent $(a, b \in \mathbf{N}, a \neq 0)$, and the moving circle $C_{1}$ is tangent to the $x$-axis. Find
(1) the equation of the locus $\Gamma$ of the ... | a=686, b=784 | 196 | 11 |
math | 2. (HUN) For which real numbers $x$ does the following inequality hold:
$$
\frac{4 x^{2}}{(1-\sqrt{1+2 x})^{2}}<2 x+9 \text { ? }
$$ | -\frac{1}{2}\leqx<\frac{45}{8}x\neq0 | 53 | 23 |
math | 2. (5 points) Two different natural numbers end with 7 zeros and have exactly 72 divisors. Find their sum. | 70000000 | 28 | 8 |
math | 31st Putnam 1970 Problem A6 x is chosen at random from the interval [0, a] (with the uniform distribution). y is chosen similarly from [0, b], and z from [0, c]. The three numbers are chosen independently, and a ≥ b ≥ c. Find the expected value of min(x, y, z). Solution | /2-^2(1/(6a)+1/(6b))+^3/(12ab) | 77 | 23 |
math | On the whiteboard, the numbers are written sequentially: $1 \ 2 \ 3 \ 4 \ 5 \ 6 \ 7 \ 8 \ 9$. Andi has to paste a $+$ (plus) sign or $-$ (minus) sign in between every two successive numbers, and compute the value. Determine the least odd positive integer that Andi can't get from this process. | 43 | 85 | 2 |
math | For any function $f:\mathbb{N}\to\mathbb{N}$ we define $P(n)=f(1)f(2)...f(n)$ . Find all functions $f:\mathbb{N}\to\mathbb{N}$ st for each $a,b$ :
$$P(a)+P(b) | a! + b!$$ | f(n) = n | 75 | 6 |
math | Example 5 Let $a=\frac{20052005}{20062006}, b=\frac{20062006}{20072007}$, $c=\frac{20072007}{20082008}$.
Try to compare the sizes of $a$, $b$, and $c$. | a<b<c | 90 | 3 |
math | 110. Find the general solution of the equation $y^{\prime \prime}=4 x$. | \frac{2}{3}x^{3}+C_{1}x+C_{2} | 22 | 21 |
math | Example 22 (Mathematical Problem 1433 from "Mathematical Bulletin") Solve the system of equations in the real numbers:
$$
\left\{\begin{array}{l}
y=x^{3}(3-2 x), \\
z=y^{3}(3-2 y), \\
x=z^{3}(3-2 z) .
\end{array}\right.
$$ | (0,0,0),(1,1,1),(-\frac{1}{2},-\frac{1}{2},-\frac{1}{2}) | 84 | 35 |
math | ## SUBJECT III
On the sides $A_{1} A_{2}, A_{2} A_{3}, \ldots, A_{n-1} A_{n}, A_{n} A_{1}$ of a regular polygon with side length $a$, consider the points $B_{1}, B_{2}, \ldots, B_{n}$, respectively, in the same direction and such that $\left[A_{1} B_{1}\right] \equiv\left[A_{2} B_{2}\right] \equiv \ldots \equiv\left[A... | \frac{}{2} | 174 | 6 |
math | $\begin{array}{l}\text { 1. In } \triangle A B C, A B=4, B C=7, C A=5, \\ \text { let } \angle B A C=\alpha. \text { Find } \sin ^{6} \frac{\alpha}{2}+\cos ^{6} \frac{\alpha}{2} \text {. }\end{array}$ | \frac{7}{25} | 88 | 8 |
math | Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a function such that
(i) For all $x, y \in \mathbb{R}$,
$$
f(x)+f(y)+1 \geq f(x+y) \geq f(x)+f(y),
$$
(ii) For all $x \in[0,1), f(0) \geq f(x)$,
(iii) $-f(-1)=f(1)=1$.
Find all such functions $f$.
Answer: $f(x)=\lfloor x\rfloor$, the largest integer that does not exce... | f(x)=\lfloorx\rfloor | 146 | 10 |
math | 4. 228 If the values $x=\sin \alpha, y=\sin \beta$ are given, then the expression
$$z=\sin (\alpha+\beta)$$
in general has four different values. Try to write the equation connecting $x, y$ and $z$ without using radicals or trigonometric functions. And find the values of $x$ and $y$ for which $z=\sin (\alpha+\beta)$ h... | z^{4}-2\left(x^{2}-2 x^{2} y^{2}+y^{2}\right) z^{2}+\left(x^{2}-y^{2}\right)^{2}=0 | 100 | 47 |
math | Beginning at 100, David counts down by 11 at a time. What is the first number less than 0 that David will count? | -10 | 32 | 3 |
math | 1. Given a finite set of cards. On each of them, either the number 1 or the number -1 is written (exactly one number on each card), and there are 100 more cards with -1 than cards with 1. If for each pair of different cards, the product of the numbers on them is found, and all these products are summed, the result is 1... | 3950 | 96 | 4 |
math | $N\geq9$ distinct real numbers are written on a blackboard. All these numbers are nonnegative, and all are less than $1$. It happens that for very $8$ distinct numbers on the board, the board contains the ninth number distinct from eight such that the sum of all these nine numbers is integer. Find all values $N$ for wh... | N = 9 | 90 | 5 |
math | Example 3 A person walks from place A to place B, and there are regular buses running between A and B, with equal intervals for departures from both places. He notices that a bus going to A passes by every 6 minutes, and a bus going to B passes by every 12 minutes. How often do the buses depart from their respective st... | 8 | 87 | 1 |
math | ## 300. Math Puzzle $5 / 90$
Today, you'll have to measure your cleverness against a sky-high "technikus" tower. The magazine has existed since 1963, and if the parents have already browsed through it to delve into the secrets of modern natural sciences and technology or to get smart tips, then a considerable collecti... | 72.38 | 191 | 5 |
math | 31. From the 100 numbers $1,2,3, \cdots, 100$, if 3 numbers are randomly selected, find the probability that their sum is divisible by 3. | \frac{817}{2450} | 46 | 12 |
math | 6.99 $k$, $l$ are natural numbers, find the number with the smallest absolute value in the form of $36^{k}-5^{l}$, and prove that the number found is indeed the smallest.
In the form of $36^{k}-5^{l}$, find the number with the smallest absolute value, and prove that the number found is indeed the smallest. | 11 | 83 | 2 |
math | ## Task Condition
Write the decomposition of vector $x$ in terms of vectors $p, q, r$:
$x=\{5 ; 15 ; 0\}$
$p=\{1 ; 0 ; 5\}$
$q=\{-1 ; 3 ; 2\}$
$r=\{0 ;-1 ; 1\}$ | 4p-q-18r | 75 | 7 |
math | 4. Find the number of pairs of integers $(x ; y)$ that satisfy the equation $x^{2}+x y=30000000$. | 256 | 36 | 3 |
math | In the acute-angled triangle $ABC$ the angle$ \angle B = 30^o$, point $H$ is the intersection point of its altitudes. Denote by $O_1, O_2$ the centers of circles inscribed in triangles $ABH ,CBH$ respectively. Find the degree of the angle between the lines $AO_2$ and $CO_1$. | 45^\circ | 86 | 4 |
math | 23. What is the maximum area that a triangle with sides \(a, b, c\) can have, given the following constraints:
\[
0 \leqslant a \leqslant 1 \leqslant b \leqslant 2 \leqslant c \leqslant 3 \text { ? }
\] | 1 | 77 | 1 |
math | The median to a 10 cm side of a triangle has length 9 cm and is perpendicular to a second median of the triangle. Find the exact value in centimeters of the length of the third median. | 3\sqrt{13} | 43 | 7 |
math | 6. Problem: Edward, the author of this test, had to escape from prison to work in the grading room today. He stopped to rest at a place 1,875 feet from the prison and was spotted by a guard with a crossbow. The guard fired an arrow with an initial velocity of $100 \mathrm{ft} / \mathrm{s}$. At the same time, Edward sta... | 75 | 166 | 2 |
math | ## Task B-2.5.
Determine the minimum value of the function $f: \mathbb{R} \backslash\left\{-\frac{1}{2}\right\} \rightarrow \mathbb{R}, f(x)=\frac{12 x^{2}+8 x+4}{(2 x+1)^{2}}$. For which $x$ does the function $f$ achieve its minimum? | \frac{8}{3} | 93 | 7 |
math | 7.268. $\left\{\begin{array}{l}9^{\sqrt[4]{x y^{2}}}-27 \cdot 3^{\sqrt{y}}=0, \\ \frac{1}{4} \lg x+\frac{1}{2} \lg y=\lg (4-\sqrt[4]{x}) .\end{array}\right.$
The system of equations is:
\[
\left\{\begin{array}{l}
9^{\sqrt[4]{x y^{2}}}-27 \cdot 3^{\sqrt{y}}=0, \\
\frac{1}{4} \lg x+\frac{1}{2} \lg y=\lg (4-\sqrt[4]{x}) ... | (1;9),(16;1) | 173 | 10 |
math | Example 6. Let $M=\frac{1}{1+\sqrt{2}}+\frac{1}{\sqrt{2}+\sqrt{3}}+$ $\cdots+\frac{1}{\sqrt{1993}+\sqrt{1994}}, N=1-2+3-4+\cdots$ $+1993-1994$. Then the value of $\frac{N}{(M+1)^{2}}$ is ( ). | -\frac{1}{2} | 104 | 7 |
math | 1. [3] Travis is hopping around on the vertices of a cube. Each minute he hops from the vertex he's currently on to the other vertex of an edge that he is next to. After four minutes, what is the probability that he is back where he started? | \frac{7}{27} | 56 | 8 |
math | For a two-digit number, the first digit is twice the second. If you add the square of its first digit to this number, you get the square of some integer. Find the original two-digit number.
# | 21 | 43 | 2 |
math | ## Task 33/78
Determine the limit $\lim _{n \rightarrow \infty} \sqrt[n]{a^{n}+b^{n}}$, where $a$ and $b$ are arbitrary positive real numbers with $a \geq b$. | a | 59 | 1 |
math | 10. If $f(x)=f\left(\frac{1}{x}\right) \lg x+10$, then the value of $f(10)$ is $\qquad$ . | 10 | 43 | 2 |
math | $\begin{array}{l}\text { Example 4. Find } S=1+\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{3}}+\text {. } \\ +\frac{1}{\sqrt{1002001}} \text { the integer part of } S. \\\end{array}$ | 2000 | 76 | 4 |
math | Arrange the following four numbers from smallest to largest $ a \equal{} (10^{100})^{10}$, $ b \equal{} 10^{(10^{10})}$, $ c \equal{} 1000000!$, $ d \equal{} (100!)^{10}$ | a, d, c, b | 74 | 8 |
math | 14.3 If $\frac{6}{b}<x<\frac{10}{b}$, find the value of $c=\sqrt{x^{2}-2 x+1}+\sqrt{x^{2}-6 x+9}$.
If $\frac{6}{b}<x<\frac{10}{b}$, determine the value of $c=\sqrt{x^{2}-2 x+1}+\sqrt{x^{2}-6 x+9}$. | 2 | 100 | 1 |
math | 937. Find the points at which the function $z=e^{x}\left(x-y^{3}+3 y\right)$ is stationary (i.e., points where the derivative in any direction is zero). | (-3,1)(1,-1) | 45 | 9 |
math | Determine all quadruples $(a, b, c, d)$ of positive real numbers that satisfy $a+b+c+d=1$ and
$$
\max \left(\frac{a^{2}}{b}, \frac{b^{2}}{a}\right) \cdot \max \left(\frac{c^{2}}{d}, \frac{d^{2}}{c}\right)=(\min (a+b, c+d))^{4}
$$ | \left(\frac{1}{4}, \frac{1}{4}, \frac{1}{4}, \frac{1}{4}\right) | 99 | 32 |
math | 5th Swedish 1965 Problem 2 Find all positive integers m, n such that m 3 - n 3 = 999. | 10^3-1^3,12^3-9^3 | 33 | 17 |
math | Consider the set $E = \{5, 6, 7, 8, 9\}$. For any partition ${A, B}$ of $E$, with both $A$ and $B$ non-empty, consider the number obtained by adding the product of elements of $A$ to the product of elements of $B$. Let $N$ be the largest prime number amonh these numbers. Find the sum of the digits of $N$. | 17 | 96 | 2 |
math | For which positive integers $n$ is it true that $2^{n}$ is a divisor of $\left(5^{n}-1\right)$? | 1,2,4 | 32 | 5 |
math | Given that the polynomial $P(x) = x^5 - x^2 + 1$ has $5$ roots $r_1, r_2, r_3, r_4, r_5$. Find the value of the product $Q(r_1)Q(r_2)Q(r_3)Q(r_4)Q(r_5)$, where $Q(x) = x^2 + 1$. | 5 | 92 | 1 |
math | ## Task 2 - 030812
When asked by his parents about the result of the last math test, Klaus remembers that 5 students received a grade of 1, 8 students received a grade of 2, 4 students received a grade of 4, and the remaining students received a grade of 3. He also recalls that the average grade was exactly 2.5.
How ... | 28 | 92 | 2 |
math | 2. The coordinates $(x ; y)$ of points in the square $\{(x ; y):-2 \pi \leq x \leq 3 \pi,-\pi \leq y \leq 4 \pi\}$ satisfy the system of equations $\left\{\begin{array}{l}\sin x+\sin y=\sin 4 \\ \cos x+\cos y=\cos 4\end{array}\right.$. How many such points are there? Find the coordinates $(x ; y)$ of the points with th... | {\begin{pmatrix}4+\frac{5\pi}{3}\\4+\frac{\pi}{3}\end{pmatrix};{\begin{pmatrix}4+\frac{5\pi}{3}\\4+\frac{7\pi}{3}\end{pmatrix};{\begin{pmatrix}4+\frac{5\pi}{} | 117 | 75 |
math | Find the least positive integer $n$ such that the decimal representation of the binomial coefficient $\dbinom{2n}{n}$ ends in four zero digits. | 313 | 34 | 3 |
math | 3. In a lake, a stream flows in, adding the same amount of water to the lake every day. 183 horses can drink all the water today, i.e., they would empty the lake in 24 hours, and 37 horses, starting from today, can drink all the water in 5 days. How many days, starting from today, would it take for one horse to drink a... | 365 | 89 | 3 |
math | Five. (20 points) Let $N=\{0,1,2, \cdots\}$, and given a $k \in \mathbf{N}$. Try to find all functions $f: \mathbf{N} \rightarrow \mathbf{N}$, such that for every $n \in \mathbf{N}$, we have
$$
f(n)<f(n+1) \text {, and } f(f(n))=n+2 k \text {. }
$$ | f(n)=n+k | 109 | 5 |
math | 5. 147 Find all polynomials in two variables that satisfy the following conditions:
(1) $P$ is homogeneous of degree $n$. That is, for all real numbers $t, x, y$, we have
$$P(t x, t y)=t^{n} P(x, y)$$
(2) For all real numbers $a, b, c$,
$$P(a+b, c)+P(b+c, a)+P(c+a, b)=0$$
(3) $P(1,0)=1$. | P(x, y) = (x+y)^{n-1}(x-2y) | 117 | 20 |
math | Let's determine all integers $a, b, c$ such that $1<a<b<c$, and $(a-1)(b-1)(c-1)$ is a divisor of $(abc-1)$. | =2,b=4,=8=3,b=5,=15 | 44 | 17 |
math | Example 4 Given that $a$ is an integer, the equation concerning $x$
$$
\frac{x^{2}}{x^{2}+1}-\frac{4|x|}{\sqrt{x^{2}+1}}+2-a=0
$$
has real roots. Then the possible values of $a$ are $\qquad$
(2008, I Love Mathematics Junior High School Summer Camp Mathematics Competition) | 0, 1, 2 | 92 | 7 |
math | 1. Given the function $f(f(f(f(f(x)))))=16 x+15$, find the analytical expression of the linear function $f(x)$. | f(x)=2x+1orf(x)=-2x+3 | 35 | 15 |
math | Find all functions $f(x)$ such that $f(2 x+1)=4 x^{2}+14 x+7$.
# | f(x)=x^{2}+5x+1 | 31 | 12 |
math | 【Question 6】Using 2 unit squares (unit squares) can form a 2-connected square, which is commonly known as a domino. Obviously, dominoes that can coincide after translation, rotation, or symmetry transformation should be considered as the same one, so there is only one domino.
Similarly, the different 3-connected square... | 12 | 145 | 2 |
math | 7. In a free-throw test, a person only needs to make 3 shots to pass and does not need to continue shooting, but each person can shoot at most 5 times. The probability that a player with a shooting accuracy of $\frac{2}{3}$ passes the test is . $\qquad$ | \frac{64}{81} | 65 | 9 |
math | 18. (3 points) Li Shuang rides a bike at a speed of 320 meters per minute from location $A$ to location $B$. On the way, due to a bicycle malfunction, he pushes the bike and walks for 5 minutes to a place 1800 meters from $B$ to repair the bike. After 15 minutes, he continues towards $B$ at 1.5 times his original ridin... | 72 | 124 | 2 |
math | 4- $124 m, n$ are two distinct positive integers, find the common complex roots of the equations
$$x^{m+1}-x^{n}+1=0$$
and
$$x^{n+1}-x^{m}+1=0$$ | x=\frac{1}{2} \pm \frac{\sqrt{3}}{2} i | 60 | 21 |
math | 2. Milan wrote down several of the first natural numbers, omitting only the numbers 4, $9,14,19,24,29, \ldots$ Then he inserted the minus and plus signs alternately between the written numbers, so he got the expression
$$
1-2+3-5+6-7+8-10+11-12+13-15+\ldots
$$
Finally, he inserted a left parenthesis after each minus ... | 69,412,418 | 184 | 10 |
math | On the sides of the right triangle, outside are constructed regular nonagons, which are constructed on one of the catheti and on the hypotenuse, with areas equal to $1602$ $cm^2$ and $2019$ $cm^2$, respectively. What is the area of the nonagon that is constructed on the other cathetus of this triangle?
(Vladislav Ki... | 417 \, \text{cm}^2 | 91 | 12 |
math | In triangle $ABC$, $AB=14$, $BC=6$, $CA=9$. Point $D$ lies on line $BC$ such that $BD:DC=1:9$. The incircles of triangles $ADC$ and $ADB$ touch side $AD$ at points $E$ and $F$. Find the length of segment $EF$.
# | 4.9 | 78 | 3 |
math | Example 2. Find the equation of the circle that passes through the intersection points of the circles $\mathrm{x}^{2}+\mathrm{y}^{2}+6 \mathrm{y}-4=0$ and $x^{2}+y^{2}+6 y-28=0$, and whose center lies on the line $\mathrm{x}-\mathrm{y}-4=0$. | x^{2}+y^{2}-x+7 y-32=0 | 85 | 18 |
math | 11. Vera bought 6 fewer notebooks than Misha and Vasya together, and Vasya bought 10 fewer notebooks than Vera and Misha together. How many notebooks did Misha buy? | 8 | 43 | 1 |
math | Beto plays the following game with his computer: initially the computer randomly picks $30$ integers from $1$ to $2015$, and Beto writes them on a chalkboard (there may be repeated numbers). On each turn, Beto chooses a positive integer $k$ and some if the numbers written on the chalkboard, and subtracts $k$ from each ... | 11 | 152 | 2 |
math | 360. The sums of the terms of each of the arithmetic progressions, having $n$ terms, are equal to $n^{2}+p n$ and $3 n^{2}-2 n$. Find the condition under which the $n$-th terms of these progressions will be equal. | 4(n-1) | 65 | 5 |
math | For a real number $\alpha>0$, consider the infinite real sequence defined by $x_1=1$ and
\[ \alpha x_n = x_1+x_2+\cdots+x_{n+1} \mbox{\qquad for } n\ge1. \] Determine the smallest $\alpha$ for which all terms of this sequence are positive reals.
(Proposed by Gerhard Woeginger, Austria) | 4 | 92 | 1 |
math | 7. Given the sets $M=\{x \mid x=3 n, n=1,2,3,4\}, P=\left\{x \mid x=3^{k}, k=\right.$ $1,2,3\}$. If there is a set $S$ that satisfies the condition $(M \cap P) \subseteq S \subseteq(M \cup P)$, then there are $\qquad$ such $S$. | 8 | 95 | 1 |
math | 5. Determine the angle between the tangents to the parabola $y^{2}=4 x$ at the points of its intersection $\mathrm{s}$
(8) with the line $2 x+y-12=0$. | 45 | 49 | 2 |
math | 21. Let $f$ be a function so that
$$
f(x)-\frac{1}{2} f\left(\frac{1}{x}\right)=\log x
$$
for all $x>0$, where log denotes logarithm base 10 . Find $f(1000)$. | 2 | 70 | 1 |
math | 4. For a positive integer $n$, if there exist positive integers $a$ and $b$ such that $n=a+b+a b$, then $n$ is called a "good number". For example, $3=1+1+1 \times 1$, so 3 is a good number. Then, among the 20 positive integers from $1 \sim 20$, the number of good numbers is $\qquad$ . | 12 | 94 | 2 |
math | [ Maximum/Minimum Distance ]
The side of the base $ABCD$ of a regular prism $ABCD A1B1C1D1$ is $2a$, and the lateral edge is $a$. Segments are considered with endpoints on the diagonal $AD1$ of the face $AA1D1D$ and the diagonal $DB1$ of the prism, parallel to the plane $AA1B1B$. a) One of these segments passes throug... | \frac{\sqrt{5}}{3},\frac{}{\sqrt{2}} | 138 | 18 |
math | Sonkin $M$.
Solve the equation $\left(x^{2}-y^{2}\right)^{2}=1+16 y$ in integers. | (\1,0),(\4,3),(\4,5) | 34 | 15 |
math | 9.185. $x^{2}\left(x^{4}+36\right)-6 \sqrt{3}\left(x^{4}+4\right)<0$. | x\in(-\sqrt[4]{12};\sqrt[4]{12}) | 40 | 20 |
math | 2. Given the lengths of line segments $A B$ and $C D$ are $a$ and $b$ $(a, b > 0)$. If line segments $A B$ and $C D$ slide on the $x$-axis and $y$-axis respectively, and such that points $A, B, C, D$ are concyclic, then the equation of the locus of the centers of these circles is $\qquad$ . | 4 x^{2}-4 y^{2}-a^{2}+b^{2}=0 | 97 | 20 |
math | Aileen plays badminton where she and her opponent stand on opposite sides of a net and attempt to bat a birdie back and forth over the net. A player wins a point if their opponent fails to bat the birdie over the net. When Aileen is the server (the first player to try to hit the birdie over the net), she wins a point ... | 73 | 174 | 2 |
math | 4. $P$ is inside rectangle $A B C D$. $P A=2, P B=3$, and $P C=10$. Find $P D$. | \sqrt{95} | 38 | 6 |
math | 19th APMC 1996 Problem 3 The polynomials p n (x) are defined by p 0 (x) = 0, p 1 (x) = x, p n+2 (x) = x p n+1 (x) + (1 - x) p n (x). Find the real roots of each p n (x). | 0 | 81 | 1 |
math | To what does the following sequence converge?
$$
a_{n}=\frac{\sum_{j=0}^{\infty}\binom{n}{2 j} \cdot 2^{j}}{\sum_{j=0}^{\infty}\binom{n}{2 j+1} \cdot 2^{j}}
$$ | \sqrt{2} | 71 | 5 |
math | ## Task Condition
Find the derivative.
$y=\left(2 x^{2}+6 x+5\right) \operatorname{arctg} \frac{x+1}{x+2}-x$ | (4x+6)\operatorname{arctg}\frac{x+1}{x+2} | 46 | 22 |
math | ## Task 4 - 180934
In a right-angled triangle $A B C$ with the right angle at $C$, the height dropped from $C$ to the hypotenuse $A B$ divides it in the ratio $1: 3$.
Calculate the size of the interior angles at $A$ and $B$ of the triangle $A B C$! | \alpha=30,\beta=60 | 84 | 10 |
math | Consider digits $\underline{A}, \underline{B}, \underline{C}, \underline{D}$, with $\underline{A} \neq 0,$ such that $\underline{A} \underline{B} \underline{C} \underline{D} = (\underline{C} \underline{D} ) ^2 - (\underline{A} \underline{B})^2.$ Compute the sum of all distinct possible values of $\underline{A} + \under... | 21 | 128 | 2 |
math | 5. One side of a certain triangle is twice as large as another, and the perimeter of this triangle is 60. The largest side of the triangle, when added to four times the smallest side, equals 71. Find the sides of this triangle.
(7 points) | 11,22,27 | 58 | 8 |
math | Find the largest $K$ satisfying the following:
Given any closed intervals $A_1,\ldots, A_N$ of length $1$ where $N$ is an arbitrary positive integer. If their union is $[0,2021]$, then we can always find $K$ intervals from $A_1,\ldots, A_N$ such that the intersection of any two of them is empty. | K = 1011 | 87 | 8 |
math | $$
\begin{array}{l}
\text { Example } 4 \text { Given } \frac{x-a-b}{c}+\frac{x-b-c}{a}+ \\
\frac{x-c-a}{b}=3 \text {, and } \frac{1}{a}+\frac{1}{b}+\frac{1}{c} \neq 0 \text {. Then } x-a- \\
b-c=
\end{array}
$$
Example 4 Given $\frac{x-a-b}{c}+\frac{x-b-c}{a}+$ $\frac{x-c-a}{b}=3$, and $\frac{1}{a}+\frac{1}{b}+\frac{... | x-a-b-c=0 | 183 | 6 |
math | 11.5. Find the set of values of the function $y=\sqrt{x}-\sqrt{2-x}+2 \sin x$. | [-\sqrt{2};\sqrt{2}+2\sin2] | 31 | 17 |
math | ## Task 3 - 211213
A set $M$ contains exactly 55 elements. For each natural number $k$ with $0 \leq k \leq 55$, let $A_{k}$ denote the number of all those subsets of $M$ that contain exactly $k$ elements.
Determine all those natural numbers $k$ for which $A_{k}$ is the largest! | k=27k=28 | 91 | 8 |
math | 7. There is a class of numbers that are multiples of 7 and 5, and when 9 is added to them, they become prime numbers. The smallest number in this class is $\qquad$ . | 70 | 44 | 2 |
math | Given a circle with center $O$ and radius $r$. Write a variable triangle $ABC$ around this circle such that the product $OA \cdot OB \cdot OC = p^3$ is constant. Find the geometric locus of the center of the circumcircle of triangle $ABC$. | OP=\frac{p}{4r^{2}}\sqrt{p(p^{3}-8r^{3})} | 59 | 25 |
math | 3. For two two-digit, positive integers $a$ and $b$, it is known that 1) one of them is 12 greater than the other; 2) in their decimal representation, one digit is the same; 3) the sum of the digits of one number is 3 greater than the sum of the digits of the other. Find these numbers. | =11+10,b=11-2,=2,3,\ldots,8;\quad\tilde{}=11+1,\quad\tilde{b}=11+13,\quad=1 | 78 | 50 |
math | [ Sorting in ascending (descending) order. ]
[ Linear inequalities and systems of inequalities ] On each of four cards, a natural number is written. Two cards are randomly drawn and the numbers on them are added. With equal probability, this sum can be less than 9, equal to 9, and greater than 9. What numbers can be w... | (1,2,7,8),(1,3,6,8),(1,4,5,8),(2,3,6,7),(2,4,5,7),(3,4,5,6) | 76 | 49 |
math | N22 (22-3, Netherlands) Given integers $m, n$ satisfying $m, n \in\{1,2, \cdots, 1981\}$ and $\left(n^{2}-m n-m^{2}\right)^{2}=1$, find the maximum value of $m^{2}+n^{2}$. | 3524578 | 78 | 7 |
math | ## Task 12/69
All pairs $(x ; y)$ are to be found that satisfy the system of equations
$$
\begin{aligned}
& |y-x|=|x+1| \\
& \frac{y-3}{4}=\left[\frac{x-1}{5}\right]
\end{aligned}
$$
where $[a]$ is an integer with $a-1<[a] \leq a$. | (1;3)(x_{0};-1)with-4\leqx_{0}<1 | 95 | 22 |
math | 11.175. A right parallelepiped is described around a sphere, with the diagonals of the base being $a$ and $b$. Determine the total surface area of the parallelepiped. | 3ab | 44 | 2 |
math | ## Task 17/87
Determine all pairs $(p ; q)$ of prime numbers $p$ and $q$ that satisfy the following conditions:
1. Their sum $P$ is also a prime number.
2. The product of the three prime numbers $p ; q ; P$ is divisible by 10. | (2;3),(3;2),(2;5),(5;2) | 70 | 17 |
math | Example 4. The random variable $X$ has a variance $D(X)=0.001$. What is the probability that the random variable $X$ differs from $M(X)=a$ by more than 0.1? | 0.1 | 50 | 3 |
math | 5. The engine of a car traveling at a speed of $v_{0}=72 \mathrm{km} / \mathbf{h}$ operates with a power of $P=50$ kW. Determine the distance from the point of engine shutdown at which the car will stop, if the resistance force is proportional to the car's speed. The mass of the car is m=1500 kg. (15
## points) | 240 | 93 | 3 |
math | How many ordered triplets $(a, b, c)$ of positive integers such that $30a + 50b + 70c \leq 343$. | 30 | 39 | 2 |
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