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 | A triangle of numbers is constructed as follows. The first row consists of the numbers from $1$ to $2000$ in increasing order, and under any two consecutive numbers their sum is written. (See the example corresponding to $5$ instead of $2000$ below.) What is the number in the lowermost row?
1 2 3 4 5
3 5 7 9
8 12... | 2001 \cdot 2^{1998} | 112 | 14 |
math | 6. Let the complex number $z$ satisfy $\left|z+\frac{1}{z}\right| \leqslant 2$. Then the range of $|z|$ is $\qquad$ . | [\sqrt{2}-1, \sqrt{2}+1] | 45 | 15 |
math | 3. In $\triangle A B C$, the sides opposite to $\angle A, \angle B, \angle C$ are $a, b, c$ respectively. If
$$
a^{2}+2\left(b^{2}+c^{2}\right)=2 \sqrt{2} \text {, }
$$
then the maximum value of the area of $\triangle A B C$ is $\qquad$ . | \frac{1}{4} | 91 | 7 |
math | 18. The traffic inspector noticed that out of 20 cars that passed on the road to the airport, 14 were "Ladas," 15 were dark-colored, 17 were driven by men, and in 18 cars, there were passengers besides the driver. For what minimum number of cars could all 4 of these characteristics be true? | 4 | 76 | 1 |
math | 1. Let $\alpha, \beta \in$ $\left(0, \frac{\pi}{2}\right)$, and $\sin ^{2} \alpha+\sin ^{2} \beta-\frac{\sqrt{6}}{2} \sin \alpha-\frac{\sqrt{10}}{2} \sin \beta+1=0$, then $\alpha+\beta=$ $\qquad$ $ـ$. | \alpha+\beta=\frac{\pi}{2} | 90 | 11 |
math | 1. We have 3 kg of a copper-tin alloy, in which $40\%$ is copper, and 7 kg of another copper-tin alloy, in which $30\%$ is copper. What masses of these alloys should be taken to obtain 8 kg of an alloy containing $p\%$ copper after melting? Find all $p$ for which the problem has a solution. | 31.25\leqslantp\leqslant33.75 | 86 | 21 |
math | ## Task Condition
Find the derivative.
$$
y=\ln \left(e^{x}+\sqrt{1+e^{2 x}}\right)
$$ | \frac{e^{x}}{\sqrt{1+e^{2x}}} | 33 | 17 |
math | A4. In the sequence of positive integers, starting with $2018,121,16, \ldots$ each term is the square of the sum of digits of the previous term. What is the $2018^{\text {th }}$ term of the sequence? | 256 | 64 | 3 |
math | 7.4. Determine all natural numbers $n$, which have exactly four positive divisors, knowing that the sum of all positive divisors of the number $n$ is twice as large as $n$.
| 6 | 43 | 1 |
math | 8. (10 points) The difference between the minimum value of the sum of the squares of ten different odd numbers and the remainder when this minimum value is divided by 4 is $\qquad$
(Note: The product of the same two natural numbers is called the square of this natural number, such as $1 \times 1=1^{2}, 2 \times 2=2^{2}... | 1328 | 102 | 4 |
math | 9. The constant term in the expansion of $\left(|x|+\frac{1}{|x|}-2\right)^{3}$ is $\qquad$ . | -20 | 36 | 3 |
math | Let $\mathbb N$ be the set of positive integers. Determine all functions $f:\mathbb N\times\mathbb N\to\mathbb N$ that satisfy both of the following conditions:
[list]
[*]$f(\gcd (a,b),c) = \gcd (a,f(c,b))$ for all $a,b,c \in \mathbb{N}$.
[*]$f(a,a) \geq a$ for all $a \in \mathbb{N}$.
[/list] | f(a, b) = \gcd(a, b) | 108 | 13 |
math | 62 (985). In a chess tournament, more than 9 but fewer than 25 grandmasters and masters participated. At the end of the tournament, it turned out that each participant scored half of their points against grandmasters. How many people participated in the tournament? How many of them were masters? | 16 | 65 | 2 |
math | 2. The sum of the cubes of all roots of the equation $\left|x^{2}-x+\frac{1}{2010}\right|=\frac{1}{2010}$ is equal to | \frac{669}{335} | 45 | 11 |
math | Let $x$, $y$, and $z$ be real numbers such that $x+y+z=20$ and $x+2y+3z=16$. What is the value of $x+3y+5z$?
[i]Proposed by James Lin[/i] | 12 | 62 | 2 |
math | 1) Alice wants to color the integers between 2 and 8 (inclusive) using $k$ colors. She wishes that if $m$ and $n$ are integers between 2 and 8 such that $m$ is a multiple of $n$ and $m \neq n$, then $m$ and $n$ are of different colors. Determine the smallest integer $k$ for which Alice can color the integers $2,3, \ldo... | 4 | 217 | 1 |
math | ## Task 2 - 240512
Roland solved a division problem. He obtained the quotient 36.
To check his result, Roland multiplied the divisor by this quotient. However, he mistakenly read a 1 instead of a 7 in the divisor and obtained the result of this multiplication as 756 instead of the given dividend.
What was the divisi... | 972:27 | 87 | 6 |
math | 3. Let $G$ be the centroid of $\triangle A B C$, and $P Q$ passes through the centroid $G$, and satisfies
$$
\begin{array}{l}
\overrightarrow{C P}=m \overrightarrow{C A}, \overrightarrow{C Q}=n \overrightarrow{C B} . \\
\text { Then } \frac{1}{m}+\frac{1}{n}=
\end{array}
$$ | 3 | 98 | 1 |
math | How many ways are there to rearrange the letters of the word RAVEN such that no two vowels are consecutive?
[i]2015 CCA Math Bonanza Team Round #2[/i] | 72 | 42 | 2 |
math | Let $ABC$ be an acute triangle. $PQRS$ is a rectangle with $P$ on $AB$, $Q$ and $R$ on $BC$, and $S$ on $AC$ such that $PQRS$ has the largest area among all rectangles $TUVW$ with $T$ on $AB$, $U$ and $V$ on $BC$, and $W$ on $AC$. If $D$ is the point on $BC$ such that $AD\perp BC$, then $PQ$ is the harmonic mean of $\f... | 4 | 196 | 1 |
math | 1. Given $I$ is the incenter of $\triangle A B C$, $A C=2, B C=3$, $A B=4$. If $\overrightarrow{A I}=x \overrightarrow{A B}+y \overrightarrow{A C}$, then $x+y=$ $\qquad$ | \frac{2}{3} | 69 | 7 |
math | Eddie has a study block that lasts $1$ hour. It takes Eddie $25$ minutes to do his homework and $5$ minutes to play a game of Clash Royale. He can’t do both at the same time. How many games can he play in this study block while still completing his homework?
[i]Proposed by Edwin Zhao[/i]
[hide=Solution]
[i]Solution.[... | 7 | 135 | 1 |
math | Task 4. Masha wrote a three-digit number on the board, and Vera wrote the same number next to it, but she swapped the last two digits. After that, Polina added the obtained numbers and got a four-digit sum, the first three digits of which are 195. What is the last digit of this sum? (The answer needs to be justified.)
... | 4 | 83 | 1 |
math | 3. Calculate the value of the expression $\sin \frac{b \pi}{36}$, where $b$ is the sum of all distinct numbers obtained from the number $a=987654321$ by cyclic permutations of its digits (in a cyclic permutation, all digits of the number, except the last one, are shifted one place to the right, and the last one is move... | \frac{\sqrt{2}}{2} | 91 | 10 |
math | Laura and her grandmother Ana have just discovered that last year their ages were divisible by 8, and next year they will be divisible by 7. Grandma Ana is not yet a centenarian. What is Laura's age? | 41 | 46 | 2 |
math | 6. (2004 Croatian Mathematical Competition) Find all positive integers with more than two digits such that each pair of adjacent digits forms a square integer.
---
The translation maintains the original format and line breaks as requested. However, it's worth noting that "square integer" might be better phrased as "p... | 164,1649,364,3649,649,816,8164,81649 | 92 | 36 |
math | 9.2. Point $B$ is the midpoint of segment $A C$. Square $A B D E$ and equilateral triangle $B C F$ are located in the same half-plane relative to line $A C$. Find (in degrees) the measure of the acute angle between lines $C D$ and $A F$. | 75 | 69 | 2 |
math | 17.1.17 $\underset{\star}{\star \star}$ Let $a_{i}=\min \left\{k+\frac{i}{k}, k \right.$ is a positive integer $\}$. Try to find $S_{n^{2}}=\left[a_{1}\right]+\left[a_{2}\right]+\cdots$ $+\left[a_{n^{2}}\right]$, where $n \geqslant 2$. | S_{n^2}=\frac{8n^3-3n^2+13n-6}{6} | 100 | 27 |
math | The integers $a, b,$ and $c$ form a strictly increasing geometric sequence. Suppose that $abc = 216$. What is the maximum possible value of $a + b + c$? | 43 | 43 | 2 |
math | Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a continuous function with the following property: for all $\alpha \in \mathbb{R}_{>0}$, the sequence $(a_n)_{n \in \mathbb{N}}$ defined as $a_n = f(n\alpha)$ satisfies $\lim_{n \to \infty} a_n = 0$. Is it necessarily true that $\lim_{x \to +\infty} f(x) = 0$? | \lim_{x \to +\infty} f(x) = 0 | 112 | 17 |
math | 5. (6 points) Two identical air capacitors are charged to a voltage $U$ each. One of them is submerged in a dielectric liquid with permittivity $\varepsilon$ while charged, after which the capacitors are connected in parallel. The amount of heat released upon connecting the capacitors is $Q$. Determine the capacitance ... | \frac{2\varepsilon(\varepsilon+1)Q}{U^{2}(\varepsilon-1)^{2}} | 434 | 30 |
math | Three. (20 points) Let the domain of the function $f(x)$ be $(0,+\infty)$, and for any positive real numbers $x, y$, the following always holds:
$$
f(x y)=f(x)+f(y)
$$
It is known that $f(2)=1$, and when $x>1$, $f(x)>0$.
(1) Find the value of $f\left(\frac{1}{2}\right)$;
(2) Determine the monotonicity of $y=f(x)$ on $... | S_{n}=\frac{n(n+1)}{2}, a_{n}=n | 238 | 19 |
math | Example 6 Given that $a, b, c$ are three non-negative real numbers, and satisfy $3a+2b+c=5, 2a+b-3c=1$. If $S=$ $3a+b-7c$, then the sum of the maximum and minimum values of $S$ is $\qquad$ | -\frac{62}{77} | 71 | 9 |
math | For some fixed positive integer $n>2$, suppose $x_1$, $x_2$, $x_3$, $\ldots$ is a nonconstant sequence of real numbers such that $x_i=x_j$ if $i \equiv j \pmod{n}$. Let $f(i)=x_i + x_i x_{i+1} + \dots + x_i x_{i+1} \dots x_{i+n-1}$. Given that $$f(1)=f(2)=f(3)=\cdots$$ find all possible values of the product $x_1 x_2 ... | 1 | 136 | 1 |
math | 5. Given $x, y, z \in \mathbf{Z}$, such that $\left\{\begin{array}{l}x+y+z=3 \\ x^{3}+y^{3}+z^{3}=3\end{array}\right.$, then the set of all possible values of $x^{2}+y^{2}+z^{2}$ is $\qquad$ . | 3,57 | 88 | 4 |
math | 40. Let $n \geqslant 2$ be a positive integer. Find the maximum value of the constant $C(n)$ such that for all real numbers $x_{1}, x_{2}, \cdots, x_{n}$ satisfying $x_{i} \in(0, 1)$ $(i=1,2, \cdots, n)$, and $\left(1-x_{i}\right)\left(1-x_{j}\right) \geqslant \frac{1}{4}(1 \leqslant i<j \leqslant n)$, we have $\sum_{i... | \frac{1}{n-1} | 208 | 9 |
math | 6.243. $\left\{\begin{array}{l}5 \sqrt{x^{2}-3 y-88}+\sqrt{x+6 y}=19 \\ 3 \sqrt{x^{2}-3 y-88}=1+2 \sqrt{x+6 y} .\end{array}\right.$ | (10;1),(-\frac{21}{2};\frac{53}{12}) | 71 | 24 |
math | 8,9
A trapezoid with lateral sides $a$ and $b$ is circumscribed around a circle. Find the sum of the squares of the distances from the center of the circle to the vertices of the trapezoid. | ^2+b^2 | 52 | 5 |
math | 2.1. The segment $A B$ is divided into three equal parts by points $C(3,4)$ and $D(5,6)$. Find the coordinates of points $A$ and $B$. | A(1,2),B(7,8) | 46 | 12 |
math | Circles with centers $O_{1}$ and $O_{2}$ have a common chord $A B, \angle A O_{1} B=60^{\circ}$. The ratio of the length of the first circle to the length of the second is $\sqrt{2}$. Find the angle $A O_{2} B$. | 90 | 72 | 2 |
math | 41. For a real number $x$, the symbol $[\mathrm{x}]$ represents the greatest integer not greater than $x$, for example $[0.3]=0,[\pi]=3,[-2.5]=-3$. If $\left[\frac{x+4}{10}\right]=5$, then the maximum value of $\left[\frac{6 x}{5}\right]$ is . $\qquad$ | 67 | 89 | 2 |
math | 382. Solve the system of equations:
$$
\left\{\begin{array}{l}
\frac{x y z}{x+y}=\frac{6}{5} \\
\frac{x y z}{y+z}=2 \\
\frac{x y z}{z+x}=\frac{3}{2}
\end{array}\right.
$$ | (3;2;1),(-3;-2;-1) | 74 | 14 |
math | 2 $*$ Find all non-negative integer solutions $(x, y, z, w)$ of the equation
$$
2^{x} \cdot 3^{y}-5^{z} \cdot 7^{w}=1
$$ | (x,y,z,w)=(1,0,0,0),(3,0,0,1),(1,1,1,0),(2,2,1,1) | 49 | 37 |
math | Task 4. (20 points) To get from the first to the second building of the university, Sasha took a car-sharing vehicle, while Zhenya rented a scooter. Sasha and Zhenya left the first building for the second at the same time, and at the same time, Professor Vladimir Sergeyevich left the second building for the first in a ... | 15,40,60 | 179 | 8 |
math | 3. The acute angle $x=$ $\qquad$ (in radians) that satisfies the equation $(\sin 2 x+\cos x)(\sin x-\cos x)=\cos x$.
(Provided by An Zhenping) | \frac{\pi}{3} | 51 | 7 |
math | 3. In $\triangle A B C$, it is known that $A B=\sqrt{39}, B C=6$, $C A=\sqrt{3}$, $M$ is the midpoint of side $B C$, and a perpendicular line is drawn from point $B$ to the extension of $A M$, with the foot of the perpendicular being $D$. Then the length of segment $B D$ is $\qquad$ | \frac{3}{2} | 91 | 7 |
math | 28. For which $n \geqslant 2$ is the inequality
$$
x_{1}^{2}+x_{2}^{2}+\ldots+x_{n}^{2} \geqslant x_{n}\left(x_{1}+x_{2}+\ldots+x_{n-1}\right)
$$
106
valid for all real values of the variables $x_{i}$? | 2,3,4,5 | 96 | 7 |
math | 7.062. $8^{\frac{x-3}{3 x-7}} \sqrt[3]{\sqrt{0.25^{\frac{3 x-1}{x-1}}}}=1$. | \frac{5}{3} | 49 | 7 |
math | Example 1 Given $f(x)=\sqrt{25-x^{2}}(0 \leqslant x \leqslant 4)$, find the inverse function of $f(x)$. | f^{-1}(x)=\sqrt{25-x^{2}}(3\leqslantx\leqslant5) | 44 | 30 |
math | 13.298. Two athletes are running on the same closed track of a stadium. The speed of each is constant, but the first one takes 10 seconds less to run the entire track than the second one. If they start running from a common starting point in the same direction, they will meet again after 720 seconds. What fraction of t... | \frac{1}{80} | 87 | 8 |
math | 26th Putnam 1965 Problem A5 How many possible bijections f on {1, 2, ... , n} are there such that for each i = 2, 3, ... , n we can find j < n with f(i) - f(j) = ± 1? Solution | 2^{n-1} | 67 | 6 |
math | In $10$ years the product of Melanie's age and Phil's age will be $400$ more than it is now. Find what the sum of Melanie's age and Phil's age will be $6$ years from now. | 42 | 50 | 2 |
math | 6. $P$ is a point on the hyperbola $\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1(a, b>0)$ in the first quadrant, $Q$ is the point symmetric to $P$ with respect to the origin $O$, $P H \perp x$-axis at point $H$, and the line $H Q$ intersects the hyperbola at point $M$ (different from $Q$). If the slope of the angle bisect... | \frac{\sqrt{6}}{2} | 140 | 10 |
math | A man disposes of sufficiently many metal bars of length $2$ and wants to construct a grill of the shape of an $n \times n$ unit net. He is allowed to fold up two bars at an endpoint or to cut a bar into two equal pieces, but two bars may not overlap or intersect. What is the minimum number of pieces he must use? | n(n+1) | 75 | 6 |
math | 1. Which whole numbers from 1 to $8 \cdot 10^{20}$ (inclusive) are there more of, and by how many: those containing only even digits or those containing only odd digits? | \frac{5^{21}-5}{4} | 45 | 12 |
math | 57. Given an integer $a$, representing a leg of a right triangle, all sides of which are measured in integers; determine the length of the other leg $b$ and the hypotenuse $c$. How many solutions are there? Solve the problem, for example, for $a=15$ and for $a=12$. | =113,b=112;=25,b=20;=39,b=36;=17,b=8 | 72 | 32 |
math | 8. Given $a, b \in [1,3], a+b=4$. Then
$$
\left|\sqrt{a+\frac{1}{a}}-\sqrt{b+\frac{1}{b}}\right|
$$
the maximum value is $\qquad$. | \sqrt{\frac{10}{3}}-\sqrt{2} | 60 | 15 |
math | 15 Find the smallest positive integer $n(n \geqslant 3)$, such that in any set of $n$ points in the plane with no three points collinear, there must be three points that are the vertices of a non-isosceles triangle. | 7 | 57 | 1 |
math | Example 7 Find all non-zero real-coefficient polynomials $f(x)$ that satisfy $f\left(x^{2}\right)=f^{2}(x)$. | f(x)=x^{n}(n\in{N}) | 35 | 13 |
math | 5. Find the set of values of the expression $\frac{a \cos x + b \sin x + c}{\sqrt{a^{2} + b^{2} + c^{2}}}$, where $x, a, b, c$ are arbitrary numbers such that $a^{2} + b^{2} + c^{2} \neq 0$. (20 points) | [-\sqrt{2};\sqrt{2}] | 85 | 11 |
math | Given a regular triangular pyramid $S A B C$. Point $S$ is the vertex of the pyramid, $A B=1, A S=2, B M$ is the median of triangle $A B C$, $A D$ is the bisector of triangle $S A B$. Find the length of the segment $D M$.
# | \frac{\sqrt{31}}{6} | 73 | 11 |
math | [Systems of linear equations]
Seven coins (weighing 1, 2, ... 7 grams) are laid out in a row in some order. For each coin (except the outermost ones), the sum of the weights of its neighbors is known.
What is the maximum number of coins whose weight can be guaranteed to be known?
# | 3 | 70 | 1 |
math | 6. (15 points) For a natural number $N$, if at least six numbers in the nine natural numbers from $1 \sim 9$ can be factors of $N$, then $N$ is called a “Six-Union Number”. Among the natural numbers greater than 2000, the smallest “Six-Union Number” is $\qquad$ | 2016 | 77 | 4 |
math | Example 5 Let non-negative real numbers $a_{1}, a_{2}, \cdots, a_{n}$ satisfy $a_{1}+a_{2}+\cdots+a_{n}=1$. Find
$$
\begin{array}{l}
\frac{a_{1}}{1+a_{2}+\cdots+a_{n}}+\frac{a_{2}}{1+a_{1}+a_{3}+\cdots+a_{n}}+ \\
\cdots+\frac{a_{n}}{1+a_{1}+\cdots+a_{n-1}}
\end{array}
$$
the minimum value. | \frac{n}{2 n-1} | 139 | 9 |
math | The point $M$ lies inside the rhombus $ABCD$. It is known that $\angle DAB=110^o$, $\angle AMD=80^o$, $\angle BMC= 100^o$. What can the angle $\angle AMB$ be equal? | 100^\circ \text{ or } 80^\circ | 62 | 15 |
math | 7. Given the elliptical region $D_{1}: \frac{x^{2}}{3}+\frac{y^{2}}{2} \leqslant 1$ and the circular region $D_{2}$ : $x^{2}+y^{2} \leqslant 2.5$. Then the area of the common region between $D_{1}$ and $D_{2}$ is $\qquad$ (accurate to 0.01).
Note: The following result can be used: In the ellipse $\frac{x^{2}}{a^{2}}+\f... | 7.27 | 216 | 4 |
math | 3.4. $\left\{\begin{array}{l}x+y+z=a, \\ x^{2}+y^{2}+z^{2}=a^{2}, \\ x^{3}+y^{3}+z^{3}=a^{3} .\end{array}\right.$ | (0,0,),(0,,0),(,0,0) | 65 | 15 |
math | Malmer Pebane's apartment uses a six-digit access code, with leading zeros allowed. He noticed that his fingers leave that reveal which digits were pressed. He decided to change his access code to provide the largest number of possible combinations for a burglar to try when the digits are known. For each number of dis... | 5 | 146 | 1 |
math | Problem 10.4. Roma thought of a natural number, the sum of the digits of which is divisible by 8. Then he added 2 to the thought number and again got a number, the sum of the digits of which is divisible by 8. Find the smallest number that Roma could have thought of. | 699 | 66 | 3 |
math | 59. Find the largest positive real number $a$, such that $\frac{x}{\sqrt{y^{2}+z^{2}}}+\frac{y}{\sqrt{z^{2}+x^{2}}}+\frac{z}{\sqrt{x^{2}+y^{2}}}>a$ holds for all positive real numbers $x, y, z$. (1994 Romanian National Training Team Problem) | 2 | 90 | 1 |
math | 1. The range of the function $f(x)=x^{2}-x^{2} \sqrt{1-x^{2}}$ is | [0,1] | 29 | 5 |
math | Let $ T$ be the set of all positive integer divisors of $ 2004^{100}$. What is the largest possible number of elements of a subset $ S$ of $ T$ such that no element in $ S$ divides any other element in $ S$? | 101^2 | 61 | 5 |
math | [ Rhombuses. Properties and Characteristics ] [ Two tangents drawn from one point ]
A rhombus and a triangle are circumscribed around a circle with a radius of 1, two sides of which are parallel to the diagonals of the rhombus, and the third side is parallel to one of the sides of the rhombus and is equal to 5. Find t... | \frac{25}{12} | 87 | 9 |
math | $p(n) $ is a product of all digits of n.Calculate:
$ p(1001) + p(1002) + ... + p(2011) $ | 91125 | 42 | 5 |
math | 1. Answer: $2011^{2011}+2009^{2009}>2011^{2009}+2009^{2011}$. | 2011^{2011}+2009^{2009}>2011^{2009}+2009^{2011} | 48 | 42 |
math | Example 1 (2005 China Southeast Mathematical Olympiad Winter Camp Pre-competition Training $A$ Paper) Find the function $f: \mathbf{R} \rightarrow \mathbf{R}$, satisfying:
$$
x[f(x+1)-f(x)]=f(x), \forall x \in \mathbf{R} \text { and }|f(x)-f(y)| \leqslant|x-y|, \forall x, y \in \mathbf{R} \text {. }
$$ | f(x)=kx,x>0 | 112 | 8 |
math | $1 \cdot 40$ form "words" by taking $n$ numbers from the alphabet $\{0,1,2,3,4\}$, such that the difference between every two adjacent numbers is 1. How many words can be formed in total? | m_{n}={\begin{pmatrix}14\times3^{k-1},n=2k+1,\\8\times3^{k-1},n=2k0\end{pmatrix}.} | 57 | 50 |
math | Given a positive integer $n$, suppose that $P(x,y)$ is a real polynomial such that
\[P(x,y)=\frac{1}{1+x+y} \hspace{0.5cm} \text{for all $x,y\in\{0,1,2,\dots,n\}$} \] What is the minimum degree of $P$?
[i]Proposed by Loke Zhi Kin[/i] | 2n | 92 | 4 |
math | 19.5. Given 8 objects, one of which is marked. It is required to ask 3 questions, to which only "yes" and "no" answers are given, and find out which object is marked. | \varepsilon_{1}+2\varepsilon_{2}+4\varepsilon_{3} | 47 | 22 |
math | For any natural number $n$, expressed in base $10$, let $S(n)$ denote the sum of all digits of $n$. Find all positive integers $n$ such that $n^3 = 8S(n)^3+6S(n)n+1$. | 17 | 57 | 2 |
math | Find all polynomials $P \in \mathbb{R}[X]$ such that $16 P\left(X^{2}\right)=P(2 X)^{2}$
Hint: use the previous question | P(x)=16(\frac{x}{4})^{i} | 45 | 14 |
math | 6. Find the minimum value of the function $f(x)=\sqrt{x^{2}-8 x+25}+\sqrt{x^{2}-4 x+13}$. | 2\sqrt{10} | 38 | 7 |
math | 13. Find the smallest integer greater than $(1+\sqrt{2})^{3}$. | 15 | 20 | 2 |
math | Which is the smallest natural number ending in 4, such that when its last digit is written at the beginning of the number, we get four times the original number? | 102564 | 34 | 6 |
math | 3. The equation $a \cos (x+1)+b \cos (x+2)+c \cos (x+3)$ $=0$ has at least two roots in the open interval $(0, \pi)$. Then all the roots of this equation are $\qquad$ . | All real numbers | 62 | 3 |
math | 4. Given $a_{0}=-1, a_{1}=1, a_{n}=2 a_{n-1}+3 a_{n-2}+3^{n}(n \geqslant 3)$, find $a_{n}$. | a_{n}=\frac{1}{16}[(4n-3)\cdot3^{n+1}-7(-1)^{n}] | 58 | 32 |
math |
6. Find all pairs $(x, y)$ of real numbers such that
$$
16^{x^{2}+y}+16^{x+y^{2}}=1
$$
| (x,y)=(-\frac{1}{2},-\frac{1}{2}) | 43 | 18 |
math | Find the largest value of the expression
$$
x y+x \sqrt{1-y^{2}}+y \sqrt{1-x^{2}}-\sqrt{\left(1-x^{2}\right)\left(1-y^{2}\right)}
$$ | \sqrt{2} | 53 | 5 |
math | 15. For two functions $f(x)$ and $g(x)$ defined on the interval $[p, q]$, if for any $x \in [p, q]$, we have $|f(x) - g(x)| \leqslant 1$, then $f(x)$ and $g(x)$ are said to be close functions on the interval $[p, q]$. Otherwise, they are said to be non-close functions on $[p, q]$. Now, consider two functions $f_{1}(x) ... | \in(0,\frac{9-\sqrt{57}}{12}] | 264 | 18 |
math | ## Problem Statement
Find the distance from point $M_{0}$ to the plane passing through three points $M_{1}, M_{2}, M_{3}$.
$M_{1}(-2 ; 0 ;-4)$
$M_{2}(-1 ; 7 ; 1)$
$M_{3}(4 ;-8 ;-4)$
$M_{0}(-6 ; 5 ; 5)$ | \frac{23\sqrt{2}}{5} | 90 | 13 |
math | 8. The function $f$ defined on ordered pairs of positive integers satisfies the following three properties:
$f(x, x)=x, f(x, y)=f(y, x)$ and $(x+$ $y) f(x, y)=y f(x, x+y)$. Try to compute $f(14$, 52 ). | 364 | 70 | 3 |
math | Problem 11.2. Vera has a set of weights of different masses, each weighing an integer number of grams. It is known that the lightest weight in the set weighs 71 times less than all the other weights combined. It is also known that the two lightest weights in the set together weigh 34 times less than all the other weigh... | 35 | 90 | 2 |
math | Consider the function $$f(x)=\sum_{k=1}^{m}(x-k)^{4}~, \qquad~ x \in \mathbb{R}$$ where $m>1$ is an integer. Show that $f$ has a unique minimum and find the point where the minimum is attained.
| \frac{m+1}{2} | 67 | 9 |
math | 39th BMO 2003 Problem 1 Find all integers 0 < a < b < c such that b - a = c - b and none of a, b, c have a prime factor greater than 3. | (,b,)=(k,2k,3k),(2k,3k,4k)or(2k,9k,16k),wherek=2^3^n | 50 | 41 |
math | 4. What is the greatest power of 7 that divides the product $1 \cdot 2 \cdot 3 \cdot 4 \cdot \ldots \cdot 999 \cdot 1000$? | 164 | 48 | 3 |
math | A5. Determine all functions $f:(0, \infty) \rightarrow \mathbb{R}$ satisfying
$$
\left(x+\frac{1}{x}\right) f(y)=f(x y)+f\left(\frac{y}{x}\right)
$$
for all $x, y>0$. | f(x)=C_{1}x+\frac{C_{2}}{x} | 69 | 18 |
math | 2. The coefficient of $x^{150}$ in the expansion of $\left(1+x+x^{2}+\cdots+x^{100}\right)^{3}$, after combining like terms, is $\qquad$
$\qquad$ . (Answer with a number) | 7651 | 61 | 4 |
math | 6.368 Solve the equation $x^{4}-4 x^{3}+3 x^{2}+8 x-10=0$, given that two of its roots differ only in sign. | x_{12}=\\sqrt{2} | 44 | 10 |
math | 13.229. Find four numbers forming a proportion, given that the sum of the extreme terms is 14, the sum of the middle terms is 11, and the sum of the squares of these four numbers is 221. | 12,8,3,2 | 54 | 8 |
math | (The Towers of Hanoi) Ada is playing a game consisting of three vertical needles on which disks of different sizes can be placed. In the initial configuration, $n$ disks are placed on the left needle, the widest at the base and the others, increasingly narrower, stacked to the top. Ada can move the disks from one needl... | 2^{n}-1 | 120 | 5 |
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