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 | 3. Let the integer $N>1, 1=d_{1}<d_{2}<\cdots<d_{s}=N$ be all the positive divisors of $N$. It is known that
$$
\left(d_{1}, d_{2}\right)+\left(d_{2}, d_{3}\right)+\cdots+\left(d_{s-1}, d_{s}\right)=N-2 \text {. }
$$
Find all possible values of $N$. | 3 | 104 | 1 |
math | 6.222.
$$
\left\{\begin{array}{l}
\sqrt{\frac{x+1}{x+y}}+\sqrt{\frac{x+y}{x+1}}=2 \\
\sqrt{\frac{x+1}{y+2}}-\sqrt{\frac{y+2}{x+1}}=1.5
\end{array}\right.
$$ | (11;1) | 81 | 6 |
math | 6. Probabilistic Voting (6-9). In the final of a play competition for March 8, two plays made it to the finals. In the first play, $n$ students from 5th grade A participated, and in the second play, $n$ students from 5th grade B participated. At the play, $2n$ mothers of all $2n$ students were present. The best play is... | 1-(\frac{1}{2})^{n} | 182 | 12 |
math | 3. Find the function $f: \mathbf{R}_{+} \rightarrow \mathbf{R}_{+}$, such that for any $x, y \in \mathbf{R}_{+}$, we have
$$
f\left(\frac{f(y)}{f(x)}+1\right)=f\left(x+\frac{y}{x}+1\right)-f(x) .
$$ | f(x)= | 90 | 3 |
math | 5・146 Find integer coefficient polynomials $P(x)$ and $Q(x)$, such that
$$
\frac{P(\sqrt{2}+\sqrt{3}+\sqrt{5})}{Q(\sqrt{2}+\sqrt{3}+\sqrt{5})}=\sqrt{3}+\sqrt{2} .
$$ | P(x)=3x^{4}-20x^{2}+24,Q(x)=4x^{3} | 74 | 25 |
math | Given the positive integer $m \geq 2$, $n \geq 3$. Define the following set
$$S = \left\{(a, b) | a \in \{1, 2, \cdots, m\}, b \in \{1, 2, \cdots, n\} \right\}.$$Let $A$ be a subset of $S$. If there does not exist positive integers $x_1, x_2, y_1, y_2, y_3$ such that $x_1 < x_2, y_1 < y_2 < y_3$ and
$$(x_1, y_1), (x_1,... | 2m + n - 2 | 193 | 7 |
math | 5. Place a sphere inside a cone, which is tangent to the cone's side and base. Then the maximum ratio of the sphere's surface area to the cone's surface area is $\qquad$
Translate the text above into English, please keep the original text's line breaks and format, and output the translation result directly. | \frac{1}{2} | 66 | 7 |
math | I am thinking of a five-digit number composed of even digits. If I swap the digit in the third position $\mathrm{s}$ with any other, the number decreases. Furthermore, I will reveal that the first digit is double the last, and the second digit is double the second to last.
What number am I thinking of?
(M. Mach)
Hin... | 88644 | 82 | 5 |
math | Example 7. Discrete independent random variables are given by the distribution laws:
| $X$ | 5 | 6 |
| :---: | :---: | :---: |
| $P$ | 0.4 | 0.6 |
| $Y$ | 7 | 8 |
| :---: | :---: | :---: |
| $P$ | 0.8 | 0.2 |
Formulate the distribution law of the random variable $Z=X+Y$. | \begin{pmatrix}Z&12&13&14\\\hlineP&0.32&0.56&0.12\\\end{pmatrix} | 110 | 41 |
math | Task A-4.6. (10 points)
Let $a$ and $b$ be natural numbers. Which digits can be in the units place of the decimal representation of the number $(a+b)^{5}-\left(a^{5}+b^{5}\right)$? | 0 | 60 | 1 |
math | 3rd ASU 1969 Problem 8 Find four different three-digit numbers (in base 10) starting with the same digit, such that their sum is divisible by three of the numbers. Solution | 108,117,135,180 | 44 | 15 |
math | 1. Find the sum of the coefficients of the polynomial obtained after expanding the brackets and combining like terms in the expression
$\left(2 x^{2021}-x^{2020}+x^{2019}\right)^{11}-29$. | 2019 | 59 | 4 |
math | suppose that $p$ is a prime number. find that smallest $n$ such that there exists a non-abelian group $G$ with $|G|=p^n$.
SL is an acronym for Special Lesson. this year our special lesson was Groups and Symmetries.
the exam time was 5 hours. | n = 3 | 67 | 4 |
math | 83 Passes through the fixed point $M(1,2)$. With the $y$-axis as the directrix, the eccentricity of the ellipse is $\frac{1}{2}$. The equation of the locus of the left vertex of the ellipse is | 9x^{2}-12x+4y^{2}-16y+19=0 | 56 | 22 |
math | $1.7 \quad \frac{a^{-1}-b^{-1}}{a^{-3}+b^{-3}}: \frac{a^{2} b^{2}}{(a+b)^{2}-3 a b}\left(\frac{a^{2}-b^{2}}{a b}\right)^{1}$;
$$
a=1-\sqrt{2}, b=1+\sqrt{2}
$$ | \frac{1}{4} | 91 | 7 |
math | 8. Determine all three-digit numbers $\overline{abc}$ divisible by 4 and 13, for which the sum of the digits is equal to 18. | 468,936 | 36 | 7 |
math | There are the following two number sequences: (1) $1,3,5,7, \cdots, 2013,2015,2017$; (2) 1, 4, 7, 10, , , 2011, 2014, 2017. The numbers that appear in both sequences are $\qquad$ in total. | 337 | 92 | 3 |
math | 【Example 3】 How many positive integers can 2160 be divided by? How many positive even integers can it be divided by? | 40 | 30 | 2 |
math | 3. The maximum value of the function $f(x)=\lg 2 \cdot \lg 5-\lg 2 x \cdot \lg 5 x$ is | \frac{1}{4} | 36 | 7 |
math | Example 4.24. Investigate the convergence of the functional series
$$
\sum_{n=1}^{\infty} \frac{\sin n x}{e^{n x}}
$$ | x\geqslant0 | 43 | 7 |
math | 3. The quadratic equation in $x$
$$
6 x^{2}-(2 m-1) x-(m+1)=0
$$
has a root $\alpha$, given that $\alpha$ satisfies $|\alpha| \leqslant 2000$, and makes $\frac{3}{5} \alpha$ an integer. Then the number of possible values for $m$ is $\qquad$. | 2401 | 89 | 4 |
math | Let $l, m$ be two skew lines, on $l$ there are three points $A, B, C$, and $A B=B C$. Draw perpendiculars from $A, B, C$ to $m$, denoted as $A D, B E, C F$, with feet of the perpendiculars being $D, E, F$ respectively. Given $A D=\sqrt{15}, B E=\frac{7}{2}, C F=\sqrt{10}$. Find the distance between $l$ and $m$. | \sqrt{6} | 116 | 5 |
math | 10. (GDR 1) Let $N=\{1,2, \ldots, n\}, n \geq 2$. A collection $F=\left\{A_{1}, \ldots, A_{t}\right\}$ of subsets $A_{i} \subseteq N, i=1, \ldots, t$, is said to be separating if for every pair $\{x, y\} \subseteq N$, there is a set $A_{i} \in F$ such that $A_{i} \cap\{x, y\}$ contains just one element. A collection $F... | t=\left[\log _{2} n\right]+1 | 210 | 14 |
math | ## Problem 4
Determine the differentiable function $f:(0, \infty) \rightarrow \mathbf{R}$ knowing that $f^{\prime}(x)=f(x)+\frac{f(x)}{x}+e^{x}$, for any $x>0$ and $f(1)=e$.
Gazeta Matematica
Selected by: Astalus Niculina, Ilie Stefan, Matefi Istvan
## INSPECTORATUL SCOLAR JUDETEAN MURES
S.S.M.R - BRANCH MURES
Ma... | f(x)=(1+\lnx)xe^{x},x\in(0,\infty) | 131 | 21 |
math | 1. Given $1 \leqslant a_{1} \leqslant a_{2} \leqslant a_{3} \leqslant a_{4} \leqslant a_{5} \leqslant a_{6} \leqslant$ 64. Then the minimum value of $Q=\frac{a_{1}}{a_{2}}+\frac{a_{3}}{a_{4}}+\frac{a_{5}}{a_{6}}$ is $\qquad$ . | \frac{3}{4} | 119 | 7 |
math | Task 3. Find all positive integers $k$ for which the equation
$$
\operatorname{lcm}(m, n)-\operatorname{gcd}(m, n)=k(m-n)
$$
has no positive integer solutions $(m, n)$ with $m \neq n$. | 2 | 62 | 1 |
math | ## Problem 1.
For four non-coplanar points, an equalizing plane is a plane such that the respective distances from each of the points to that plane are all equal. Given a set of four non-coplanar points, how many equalizing planes are there? | 7 | 56 | 1 |
math | 7. For four different integers, all their pairwise sums and pairwise products were calculated and written on the board. What is the smallest number of different numbers that could have appeared on the board? (I. Rubanov) | 6 | 45 | 1 |
math | [ Distance between skew lines]
On the line $l$ in space, points $A, B$ and $C$ are sequentially located, with $A B=18$ and $B C=14$. Find the distance between the lines $l$ and $m$, if the distances from points $A, B$ and $C$ to the line $m$ are 12, 15 and 20, respectively.
# | 12 | 94 | 2 |
math | Condition of the problem
Calculate the limit of the function:
$\lim _{x \rightarrow 1} \frac{\sqrt{x^{2}-x+1}-1}{\ln x}$ | \frac{1}{2} | 40 | 7 |
math | 21. The sequence $\left\{a_{n}\right\}$ is defined as follows: $a_{1}=0, a_{2}=1$ and for $n \geqslant 3$, $a_{n}=\frac{1}{2} n a_{n-1}+\frac{1}{2} n(n-1) a_{n-2}+(-1)^{n}\left(1-\frac{n}{2}\right)$. Try to find: $f_{n}=\sum_{k=0}^{n-1}(k+1) C_{n}^{k} a_{n-k}$ in its simplest form. | 2n!-(n+1) | 144 | 8 |
math | Let $I$ be the center of the incircle of triangle $ABC$ and let $A', B'$ and $C'$ be the symmetrics of $I$ with respect to the lines $(BC), (CA)$ and $(AB)$ respectively. The circumcircle of $A'B'C'$ passes through $B$. Find $\widehat{A B C}$. | 60 | 76 | 2 |
math | The base of a quadrilateral pyramid is a square, and all lateral faces are right triangles, with the vertices of the right angles lying on the base of the pyramid. Find the volume of the pyramid if its height is 1, and one of the dihedral angles at the vertex is $120^{\circ}$.
# | \frac{1}{3} | 69 | 7 |
math | 6. (20 points) We will call a word any finite sequence of letters of the Russian alphabet. How many different five-letter words can be formed from the letters of the word САМСА? And from the letters of the word ПАСТА? In your answer, indicate the sum of the found numbers. | 90 | 67 | 2 |
math | 3. In how many ways can the number $6 k(k \in \mathbb{N})$ be written as the sum of three natural numbers? Records that differ only in the order of the addends are considered the same. | 3k^2 | 48 | 4 |
math | Example 2 The sports meet lasted for $n$ days $(n>1)$, and a total of $m$ medals were awarded. On the first day, 1 medal plus $\frac{1}{7}$ of the remaining medals were awarded, on the second day, 2 medals plus $\frac{1}{7}$ of the remaining medals were awarded, and so on. On the last day, the $n$-th day, exactly $n$ m... | 6 | 128 | 1 |
math | 10. $[8]$ Find the largest positive integer $n$ such that $n^{3}+4 n^{2}-15 n-18$ is the cube of an integer. | 19 | 42 | 2 |
math | Let $p$ be a monic cubic polynomial such that $p(0)=1$ and such that all the zeroes of $p^\prime (x)$ are also zeroes of $p(x)$. Find $p$. Note: monic means that the leading coefficient is $1$. | p(x) = (x + 1)^3 | 61 | 12 |
math | 16. If the integer $m$ makes the equation
$$
x^{2}-m x+m+2006=0
$$
have non-zero integer roots, then the number of such integers $m$ is
$\qquad$. | 5 | 53 | 1 |
math | Find all three real numbers $(x, y, z)$ satisfying the system of equations $$\frac{x}{y}+\frac{y}{z}+\frac{z}{x}=\frac{x}{z}+\frac{z}{y}+\frac{y}{x}$$ $$x^2 + y^2 + z^2 = xy + yz + zx + 4$$ | (x, x-2, x-2) | 82 | 12 |
math | Let $a, b, c, d$ be real numbers such that $b-d \geq 5$ and all zeros $x_{1}, x_{2}, x_{3}$, and $x_{4}$ of the polynomial $P(x)=x^{4}+a x^{3}+b x^{2}+c x+d$ are real. Find the smallest value the product $\left(x_{1}^{2}+1\right)\left(x_{2}^{2}+1\right)\left(x_{3}^{2}+1\right)\left(x_{4}^{2}+1\right)$ can take. | 16 | 140 | 2 |
math | 5. a square board consists of $2 n \times 2 n$ squares. You want to mark $n$ of these squares so that no two marked squares are in the same or adjacent rows, and so that no two marked squares are in the same or adjacent columns. In how many ways is this possible?
## Solution: | (n+1)^{2}n! | 68 | 9 |
math | # Task 8.2 (7 points)
The shares of the company "Nu-i-Nu" increase in price by 10 percent every day. Businessman Borya bought shares of the company for 1000 rubles every day for three days in a row, and on the fourth day, he sold them all. How much money did he make from this operation? | 641 | 80 | 3 |
math | In a fruit shop, Jaime noticed that an orange costs the same as half an apple plus half a real, and he also noticed that a third of an apple costs the same as a quarter of an orange plus half a real. With the value of 5 oranges plus 5 reals, how many apples can Jaime buy?
# | 5 | 67 | 1 |
math | Let $f : Z_{\ge 0} \to Z_{\ge 0}$ satisfy the functional equation $$f(m^2 + n^2) =(f(m) - f(n))^2 + f(2mn)$$ for all nonnegative integers $m, n$. If $8f(0) + 9f(1) = 2006$, compute $f(0)$. | 118 | 88 | 3 |
math | A coin is tossed 10 times. Find the probability that two heads never appear consecutively.
# | \frac{9}{64} | 22 | 8 |
math | Example 3 Let $a$ be a prime number, $b$ be a positive integer, and $9(2 a+b)^{2}=509(4 a+511 b)$. Find the values of $a$ and $b$.
Translate the above text into English, please retain the original text's line breaks and format, and output the translation result directly. | a=251, b=7 | 81 | 9 |
math | Alex and Katy play a game on an $8 \times 8$ square grid made of 64 unit cells. They take it in turns to play, with Alex going first. On Alex's turn, he writes 'A' in an empty cell. On Katy's turn, she writes ' $\mathrm{K}$ ' in two empty cells that share an edge. The game ends when one player cannot move. Katy's score... | 32 | 126 | 2 |
math | 10、One day, Qiqi went to the zoo, he saw three types of animals: monkeys, pandas, and lions, the total number of these three types of animals is between $26-32$. The total number of monkeys and lions is more than the number of pandas. The total number of pandas and lions is more than twice the number of monkeys. The to... | 13 | 116 | 2 |
math | 5. [5 points] Given the numbers $\log _{\sqrt{5 x-1}}(4 x+1), \log _{4 x+1}\left(\frac{x}{2}+2\right)^{2}, \log _{\frac{x}{2}+2}(5 x-1)$. For which $x$ are two of these numbers equal, and the third one less than them by 1? | 2 | 91 | 1 |
math | 12 (14 points) Given the function $f(x)=-2 x+4$, let
$$
S_{n}=f\left(\frac{1}{n}\right)+f\left(\frac{2}{n}\right)+\cdots+f\left(\frac{n-1}{n}\right)+f(1)\left(n \in \mathbf{N}^{*}\right) \text {, }
$$
If the inequality $\frac{a^{n}}{S_{n}}<\frac{a^{n+1}}{S_{n+1}}$ always holds, find the range of real number $a$. | (\frac{5}{2},+\infty) | 139 | 11 |
math | 4. (4 points) Solve the system of equations:
$$
\left\{\begin{aligned}
& \log _{2}(y-x)=\log _{8}(19 y-13 x) \\
& x^{2}+y^{2}=13
\end{aligned}\right.
$$ | {(-\sqrt{13};0);(-3;-2);(-\frac{1}{\sqrt{2}};\frac{5}{\sqrt{2}})} | 68 | 37 |
math | 8. There are 10 cards, each card has two different numbers from $1,2,3,4,5$, and no two cards have the same pair of numbers. These 10 cards are to be placed into five boxes labeled $1,2,3,4,5$, with the rule that a card with numbers $i, j$ can only be placed in box $i$ or box $j$. A placement is called "good" if the nu... | 120 | 129 | 3 |
math | 16. On a plane, there are 100 lines. Can they contain 1985 different intersection points. | 1985 | 27 | 4 |
math | ## Task 4 - 060724
In a cylindrical container (a straight circular cylinder with a horizontal base), there is water. The water level is at $\frac{3}{4}$ of the height of the container. After exactly $2 \frac{1}{2}$ liters of water are poured out of this container, the water level is at $\frac{2}{5}$ of the container's... | \frac{50}{7} | 97 | 8 |
math | Example 1 Find the range of the function $y=x^{2}+x \sqrt{x^{2}-1}$. ${ }^{[1]}$ (2013, Hubei Provincial Preliminary of the National High School Mathematics League) | \left(\frac{1}{2},+\infty\right) | 53 | 15 |
math | ## Problem Statement
Calculate the limit of the function:
$$
\lim _{x \rightarrow 0}\left(\frac{\sin 4 x}{x}\right)^{\frac{2}{x+2}}
$$ | 4 | 46 | 1 |
math | 5. Given $O$ is the circumcenter of $\triangle A B C$, and $D$ is the midpoint of $B C$. If $\overrightarrow{A O} \cdot \overrightarrow{A D}=4, B C=2 \sqrt{6}$, then $A D=$ $\qquad$ | \sqrt{2} | 67 | 5 |
math | Let's assume that by removing a finite number of positive elements from the set of natural numbers, we obtained a set $S$ that is closed under addition. Let $k$ be an element of $S$. How many elements of $S$ are there such that subtracting $k$ from them results in a number that does not belong to $S$? | k | 74 | 1 |
math | 6. [45] Let $A B C D$ be an isosceles trapezoid with $A B=1, B C=D A=5, C D=7$. Let $P$ be the intersection of diagonals $A C$ and $B D$, and let $Q$ be the foot of the altitude from $D$ to $B C$. Let $P Q$ intersect $A B$ at
R. Compute $\sin \angle R P D$. | \frac{4}{5} | 104 | 7 |
math | 6. Solve the equation $\sqrt{\frac{x+3}{3 x-5}}+1=2 \sqrt{\frac{3 x-5}{x+3}}$. | 4 | 37 | 1 |
math | 6. Express $M=\frac{4 x^{2}+2 x+6}{x^{4}+x^{2}+1}$ as partial fractions. | M=\frac{x+2}{x^{2}+x+1}+\frac{-x+4}{x^{2}-x+1} | 35 | 31 |
math | 6. Find the ordered quadruple of digits $(A, B, C, D)$, with $A>B>C>D$, such that
$$
\begin{aligned}
& A B C D \\
-\quad & D C B A \\
= & B D A C .
\end{aligned}
$$ | (7,6,4,1) | 63 | 9 |
math | 9.11. The numbers $a_{1}, a_{2}, \ldots, a_{n}$ are such that the sum of any seven consecutive numbers is negative, and the sum of any eleven consecutive numbers is positive. For what largest $n$ is this possible?
118 Chapter 9. Computation of Sums and Products
$$
\text { 9.3. Sums } S_{k}(n)=1^{k}+2^{k}+\ldots+n^{k}... | 16 | 222 | 2 |
math | 5. Let $f(x)$ be a function defined on $\mathbf{R}$ with a period of 2, which is even, strictly decreasing on the interval $[0,1]$, and satisfies $f(\pi)=1, f(2 \pi)=2$. Then the solution set of the inequality system
$$
\left\{\begin{array}{l}
1 \leqslant x \leqslant 2, \\
1 \leqslant f(x) \leqslant 2
\end{array}\right... | [\pi-2,8-2\pi] | 126 | 11 |
math | 5. If $a>b>c, a+b+c=0$, and $x_{1}, x_{2}$ are the two real roots of $a x^{2}+b x+c=0$. Then the range of $\left|x_{1}^{2}-x_{2}^{2}\right|$ is $\qquad$ | [0,3) | 70 | 5 |
math | 15. Find all natural numbers $a$ and $b$ that satisfy the following equation: $\left[\frac{a^{2}}{b}\right]+\left[\frac{b^{2}}{a}\right]=\left[\frac{a^{2}+b^{2}}{a b}\right]+a b$. | ^{2}+1,\in{N} | 70 | 10 |
math | 4. Solve the system of equations $\left\{\begin{array}{l}2 \cos ^{2} x+2 \sqrt{2} \cos x \cos ^{2} 4 x+\cos ^{2} 4 x=0, \\ \sin x=\cos y .\end{array}\right.$.
# | (\frac{3\pi}{4}+2\pik,\\frac{\pi}{4}+2\pin),(-\frac{3\pi}{4}+2\pik,\\frac{3\pi}{4}+2\pin),\quadk,n\inZ | 73 | 64 |
math | 5. In the office, each computer was connected by wires to exactly 5 other computers. After some computers were infected by a virus, all wires from the infected computers were disconnected (a total of 26 wires were disconnected). Now, each of the uninfected computers is connected by wires to only 3 others. How many comp... | 8 | 75 | 1 |
math | (a) Let's list the first 20092009 natural numbers. Then, we successively replace each number with the sum of its digits, until we obtain a list of numbers with only one digit. Does the list have more digits 4 or 5? How many 9s are there in the list?
(b) Applying the same process to the number $3^{2009}$, that is, repl... | 4,2232445,9,8 | 124 | 13 |
math | Find all positive integers $x$ such that $2x+1$ is a perfect square but none of the integers $2x+2, 2x+3, \ldots, 3x+2$ are perfect squares. | 4 | 50 | 1 |
math | 6. Find all nonnegative integer solutions of the equation
$$
\left(2^{2015}+1\right)^{x}+2^{2015}=2^{y}+1
$$
(Bojan Bašić)
Time allowed: 270 minutes.
Each problem is worth 7 points. | (0,2015)(1,2016) | 73 | 15 |
math | 1. We are given a trapezoid with bases of lengths 1 and 4, respectively. We divide it into two trapezoids by a cut parallel to the bases, of length 3. We now want to further divide these two new trapezoids, always by cuts parallel to the bases, into $m$ and $n$ trapezoids, respectively, so that all $m+n$ trapezoids obt... | 15 | 123 | 2 |
math | 2.274. $\frac{8-m}{\sqrt[3]{m}+2}:\left(2+\frac{\sqrt[3]{m^{2}}}{\sqrt[3]{m}+2}\right)+\left(\sqrt[3]{m}+\frac{2 \sqrt[3]{m}}{\sqrt[3]{m}-2}\right) \cdot \frac{\sqrt[3]{m^{2}}-4}{\sqrt[3]{m^{2}}+2 \sqrt[3]{m}}$. | 2 | 116 | 1 |
math | 2. Solve the inequality $\log _{x}(5 x-4)>2$. | x\in(4/5;1)\cup(1;4) | 18 | 16 |
math | 1. Calculate the value of the expression
$$
A=\frac{1}{a(a-b)(a-c)}+\frac{1}{b(b-a)(b-c)}+\frac{1}{c(c-a)(c-b)}
$$
if $abc=1$ and $a \neq b \neq c \neq a$. | 1 | 72 | 1 |
math | 7.272. $\left\{\begin{array}{l}3^{1+2 \log _{3}(y-x)}=48, \\ 2 \log _{5}(2 y-x-12)-\log _{5}(y-x)=\log _{5}(y+x) .\end{array}\right.$ | (16;20) | 76 | 7 |
math | If $P_{1} P_{2} \ldots P_{100}$ is a regular 100 -gon, what is the measure of the angle $\angle P_{20} P_{2} P_{1}$ in degrees? | 145.8 | 54 | 5 |
math | 5. Given that $f(x)$ is a linear function, and $f(f[\underbrace{f \cdots f(x)]} \geqslant 1024 x+1023$, find $f(x)$. | f(x)=2x+,(b\geqslant1)orf(x)=-2x+,(b\leqslant-3) | 52 | 32 |
math | Example 4 Given that $x, y$ are integers, and
$$
15 x^{2} y^{2}=35 x^{2}-3 y^{2}+412 \text {. }
$$
then $15 x^{2} y^{2}=$ $\qquad$ . | 960 | 66 | 3 |
math | ## Task 1 - 310621
a) A six-digit number is to be written using the digits 1, 1, 2, 2, 3, 3. The order of these six digits should be chosen such that the following conditions are met:
(1) Between the two digits 1, exactly one other digit should stand.
(2) Between the two digits 2, exactly two other digits should sta... | 231213,312132,23421314,41312432 | 208 | 31 |
math | Let $P$ be a function defined by $P(t)=a^t+b^t$, where $a$ and $b$ are complex numbers. If $P(1)=7$ and $P(3)=28$, compute $P(2)$.
[i] Proposed by Justin Stevens [/i] | 19 | 66 | 2 |
math | Suppose $ P(x) \equal{} a_nx^n\plus{}\cdots\plus{}a_1x\plus{}a_0$ be a real polynomial of degree $ n > 2$ with $ a_n \equal{} 1$, $ a_{n\minus{}1} \equal{} \minus{}n$, $ a_{n\minus{}2} \equal{}\frac{n^2 \minus{} n}{2}$ such that all the roots of $ P$ are real. Determine the coefficients $ a_i$. | a_i = (-1)^{n-i} \binom{n}{i} | 113 | 18 |
math | Example 3 Let the temperature $T$ of an object be a function of time $t$: $T(t)=a t^{3}+b t^{2}+c t+d(a \neq 0)$, where the temperature is in ${ }^{\circ} \mathrm{C}$ and the time is in hours, with $t=0$ representing 12:00, and positive $t$ values representing times after 12:00. If the temperature of the object is meas... | 62\mathrm{C} | 258 | 7 |
math | . Let $x, y \in \mathbb{R}$ be such that $x=y(3-y)^{2}$ and $y=x(3-x)^{2}$. Find all possible values of $x+y$. | {0,3,4,5,8} | 48 | 11 |
math | 2. Function
$$
y=\sin ^{2} x+\sin x \cdot \cos x-2 \cos ^{2} x
$$
The range of the function is . $\qquad$ | \left[-\frac{\sqrt{10}+1}{2}, \frac{\sqrt{10}-1}{2}\right] | 45 | 30 |
math | ## Problem Statement
Are the vectors $c_{1 \text { and }} c_{2}$, constructed from vectors $a \text{ and } b$, collinear?
$a=\{-1 ; 2 ;-1\}$
$b=\{2 ;-7 ; 1\}$
$c_{1}=6 a-2 b$
$c_{2}=b-3 a$ | c_{1}=-2\cdotc_{2} | 79 | 12 |
math | Solve the equation $[x / 2]+[x / 4]=x$. ([x], the integer part of $x$ is the greatest integer not exceeding $x$.) | 0,-3,-2,-5 | 39 | 7 |
math | ## Problem Statement
Calculate the limit of the function:
$\lim _{x \rightarrow 4} \frac{\sqrt[3]{16 x}-4}{\sqrt{4+x}-\sqrt{2 x}}$ | -\frac{4\sqrt{2}}{3} | 47 | 12 |
math | ## Zadatak B-1.2.
Izračunajte
$$
\frac{1-\frac{1}{x^{2}}}{1+\frac{6}{x}+\frac{5}{x^{2}}} \cdot \frac{x+5}{x-1}
$$
| 1 | 65 | 1 |
math | 15. Form an $n$-digit number using the digits $1,2,3$, such that each of $1,2,3$ appears at least once. Find the number of such $n$-digit numbers. | 3^{n}-3\times2^{n}+3 | 49 | 13 |
math | 8、Two candidates, A and B, participate in an election, with A receiving $n$ votes and B receiving $m$ votes $(n>m)$. The probability that A's cumulative vote count always exceeds B's cumulative vote count during the vote counting process is $\qquad$ . | \frac{-n}{+n} | 59 | 8 |
math | In the triangle $ABC$ it is known that$\angle A = 75^o, \angle C = 45^o$. On the ray $BC$ beyond the point $C$ the point $T$ is taken so that $BC = CT$. Let $M$ be the midpoint of the segment $AT$. Find the measure of the $\angle BMC$.
(Anton Trygub) | 45^\circ | 85 | 4 |
math | Let positive integers $p,q$ with $\gcd(p,q)=1$ such as $p+q^2=(n^2+1)p^2+q$. If the parameter $n$ is a positive integer, find all possible couples $(p,q)$. | (p, q) = (n+1, n^2 + n + 1) | 55 | 21 |
math | Find the largest natural number in which each non-edge digit is less than the arithmetic mean of the adjacent digits.
# | 96433469 | 23 | 8 |
math | 7. [6] For any positive real numbers $a$ and $b$, define $a \circ b=a+b+2 \sqrt{a b}$. Find all positive real numbers $x$ such that $x^{2} \circ 9 x=121$. | \frac{31-3\sqrt{53}}{2} | 59 | 16 |
math | ## Problem 3
For the non-zero natural number $n$, there exists a natural number $k, k \geq 2$, and positive rational numbers $a_{1}, a_{2}, \ldots, a_{k}$ such that $a_{1}+a_{2}+\ldots+a_{k}=a_{1} a_{2} \cdot \ldots \cdot a_{k}=n$. Determine all possible values of the number $n$. | n\in\mathbb{N}^{*}-{1,2,3,5} | 100 | 20 |
math | 11.2. A triplet of real numbers $A, B, C$ is such that $\operatorname{Sin} A+\operatorname{Sin} B+\operatorname{Sin} C=0$ and $\cos A+\operatorname{Cos} B+\operatorname{Cos} C=0$. Find the value of the expression
$\operatorname{Cos}(A-B)+\operatorname{Cos}(B-C)+\operatorname{Cos}(C-A)$. | -\frac{3}{2} | 100 | 7 |
math | 4. A sphere is inscribed in a regular triangular prism, touching all three sides and both bases of the prism. Find the ratio of the surface areas of the sphere and the prism. | \frac{2\pi}{9\sqrt{3}} | 38 | 13 |
math | Consider the following sequence
$$(a_n)_{n=1}^{\infty}=(1,1,2,1,2,3,1,2,3,4,1,2,3,4,5,1,\dots)$$
Find all pairs $(\alpha, \beta)$ of positive real numbers such that $\lim_{n\to \infty}\frac{\displaystyle\sum_{k=1}^n a_k}{n^{\alpha}}=\beta$.
(Proposed by Tomas Barta, Charles University, Prague) | \left(\frac{3}{2}, \frac{\sqrt{2}}{3}\right) | 121 | 22 |
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