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 | 5. For any positive integer $k$, let $f_{1}(k)$ be the sum of the squares of the digits of $k$, and for $n \geqslant 2, f_{n}(k)=$ $f_{1}\left(f_{n-1}(k)\right)$. Then $f_{2006}(2006)=$ $\qquad$ | 145 | 85 | 3 |
math | 4. Find all functions $f(x)$ and $g(x)$ that satisfy the following condition:
$$
f(x)+f(y)+g(x)-g(y)=\sin x+\cos y, \quad x, y \in \mathrm{R}.
$$ | f(x)=\frac{\sinx+\cosx}{2},(x)=\frac{\sinx-\cosx}{2}+ | 54 | 29 |
math | Write the digits $1,2,3,4,5,6,7,8,9$ in some order to form a nine-digit number $\overline{\operatorname{abcdefghi}}$. If $A=\overline{a b c}+\overline{b c d}+\overline{c d e}+\overline{d e f}+\overline{e f g}+\overline{f g h}+\overline{g h i}$, find the maximum possible value of $A$. | 4648 | 111 | 4 |
math | Example 3 Let $D$ be a point inside an acute $\triangle A B C$, $\angle A D B = \angle A C B + 90^{\circ}$, and $A C \cdot B D = A D \cdot B C$. Find the value of $\frac{A B \cdot C D}{A C \cdot B D}$. | \sqrt{2} | 77 | 5 |
math | Exercise 6. The strictly positive integers $x, y$, and $z$ satisfy the following two equations: $x+2y=z$ and $x^2-4y^2+z^2=310$. Find all possible values of the product $xyz$. | 2015 | 58 | 4 |
math | 6. Let $f(x)$ represent a fourth-degree polynomial in $x$. If
$$
f(1)=f(2)=f(3)=0, f(4)=6, f(5)=72 \text {, }
$$
then the last digit of $f(2010)$ is $\qquad$ | 2 | 71 | 1 |
math | Since 3 and $1 / 3$ are roots of the equation $a x^{2}-6 x+c=0$, we have:
$$
9 a-18+c=0 \Rightarrow 9 a+c=18 \text { and } \frac{a}{9}-2+c=0 \Rightarrow \frac{a}{9}+c=2
$$
Solving the system
$$
\left\{\begin{array}{l}
9 a+c=18 \\
\frac{a}{9}+c=2
\end{array} \quad \text { we obtain } \quad a=c=\frac{9}{5} . \text { T... | \frac{18}{5} | 161 | 8 |
math | Let $X_1,X_2,..$ be independent random variables with the same distribution, and let $S_n=X_1+X_2+...+X_n, n=1,2,...$. For what real numbers $c$ is the following statement true:
$$P\left(\left| \frac{S_{2n}}{2n}- c \right| \leqslant \left| \frac{S_n}{n}-c\right| \right)\geqslant \frac{1}{2}$$ | c \in \mathbb{R} | 116 | 10 |
math | Example 11. Find a particular solution of the equation
$$
y^{\prime \prime}-y=4 e^{x},
$$
satisfying the initial conditions
$$
y(0)=0, \quad y^{\prime}(0)=1
$$ | 2xe^{x}-\operatorname{sh}x | 58 | 12 |
math | 2. (2002 Romania IMO and Balkan Mathematical Olympiad Selection Test (First Round)) Find all pairs of sets $A$ and $B$ such that $A$ and $B$ satisfy:
(1) $A \cup B=\mathbf{Z}$;
(2) If $x \in A$, then $x-1 \in B$;
(3) If $x \in B, y \in B$, then $x+y \in A$. | A=2\boldsymbol{Z},B=2\boldsymbol{Z}+1 | 102 | 20 |
math | 4. Sasha went to visit his grandmother. On Saturday, he got on the train, and after 50 hours on Monday, he arrived at his grandmother's city. Sasha noticed that on this Monday, the date matched the number of the carriage he was in, that the number of his seat in the carriage was less than the number of the carriage, an... | CarriageNo.2,seatNo.1 | 113 | 10 |
math | ## Task $6 / 71$
The cubic equation with real coefficients $p, q$, and $r$
$$
x^{3}+p x^{2}+q x+r=0
$$
has three real solutions. What condition must the coefficients $p, q$, and $r$ satisfy if the solutions are to be the lengths of the sides of a plane triangle? | p^{3}-4pq+8r>0 | 81 | 11 |
math | Antoine, Benoît, Claude, Didier, Étienne, and Françoise go to the cinéma together to see a movie. The six of them want to sit in a single row of six seats. But Antoine, Benoît, and Claude are mortal enemies and refuse to sit next to either of the other two. How many different arrangements are possible? | 144 | 76 | 3 |
math | 184. The sequence $\left(a_{n}\right)$ satisfies the conditions:
$$
a_{1}=3, \quad a_{2}=9, \quad a_{n+1}=4 a_{n}-3 a_{n-1}(n>1)
$$
Find the formula for the general term of the sequence. | a_{n}=3^{n} | 71 | 8 |
math | 6. The sum of three fractions is $\frac{83}{72}$, where their numerators are in the ratio $5: 7: 1$. The denominator of the third fraction is in the ratio $1: 4$ to the denominator of the first fraction, and the denominator of the second fraction to the denominator of the third fraction is in the ratio $3: 2$. Determin... | \frac{5}{24},\frac{7}{9},\frac{1}{6} | 92 | 22 |
math | 13.374 Three candles have the same length but different thicknesses. The first candle was lit 1 hour earlier than the other two, which were lit simultaneously. At some point, the first and third candles had the same length, and 2 hours after this, the first and second candles had the same length. How many hours does it... | 16 | 99 | 2 |
math | 11. Find all positive integers $x, y, z$, such that: $x y \pmod{z} \equiv y z \pmod{x} \equiv z x \pmod{y} = 2$. | (3,8,22),(3,10,14),(4,5,18),(4,6,11),(6,14,82),(6,22,26) | 49 | 46 |
math | 8. Given a tetrahedron $ABCD$ where $AB=CD=2a$, $AC=BD=BC=AD=\sqrt{10}$, then the range of values for $a$ is $\qquad$. | 0<<\sqrt{5} | 50 | 7 |
math | Let $N \ge 5$ be given. Consider all sequences $(e_1,e_2,...,e_N)$ with each $e_i$ equal to $1$ or $-1$. Per move one can choose any five consecutive terms and change their signs. Two sequences are said to be similar if one of them can be transformed into the other in finitely many moves. Find the maximum number of pai... | 16 | 96 | 2 |
math | 5. In a convex quadrilateral $ABCD$, diagonal $BD$ is drawn, and a circle is inscribed in each of the resulting triangles $ABD$ and $BCD$. A line passing through vertex $B$ and the center of one of the circles intersects side $DA$ at point $M$. Here, $AM=\frac{8}{5}$ and $MD=\frac{12}{5}$. Similarly, a line passing thr... | AB=4,CD=5 | 176 | 7 |
math | 3-5. The city's bus network is organized as follows: 1) from any stop to any other stop, you can get without transferring;
2) for any pair of routes, there is, and only one, stop where you can transfer from one of these routes to the other;
3) on each route, there are exactly three stops.
How many bus routes are ther... | 7 | 82 | 1 |
math | Four. (50 points) Find all positive integers $a$ such that there exists an integer-coefficient polynomial $P(x)$ and
$$
Q(x)=a_{1} x+a_{2} x^{2}+\cdots+a_{2016} x^{2016},
$$
where, $a_{i} \in\{-1,1\}, i=1,2, \cdots, 2016$, satisfying
$$
\left(x^{2}+a x+1\right) P(x)=Q(x) \text {. }
$$ | 1 \text{ or } 2 | 127 | 8 |
math | ## problem statement
Calculate the volume of the tetrahedron with vertices at points $A_{1}, A_{2}, A_{3}, A_{4}$ and its height dropped from vertex $A_{4}$ to the face $A_{1} A_{2} A_{3}$.
$A_{1}(0 ;-1 ;-1)$
$A_{2}(-2 ; 3 ; 5)$
$A_{3}(1 ;-5 ;-9)$
$A_{4}(-1 ;-6 ; 3)$ | \frac{37}{3\sqrt{5}} | 114 | 12 |
math | ## Task B-3.2.
If $\log _{2}\left[\log _{3}\left(\log _{4} x\right)\right]=\log _{3}\left[\log _{2}\left(\log _{4} y\right)\right]=\log _{4}\left[\log _{3}\left(\log _{2} z\right)\right]=0$, calculate $\log _{y}\left(\log _{z} x\right)$. | \frac{1}{4} | 108 | 7 |
math | 11. The product of the digits of a three-digit number is 18. The sum of all such three-digit numbers is | 5772 | 27 | 4 |
math | 2. Let $a$ be a real number, and the sequence $a_{1}, a_{2}, \cdots$ satisfies:
$$
\begin{array}{l}
a_{1}=a, \\
a_{n+1}=\left\{\begin{array}{ll}
a_{n}-\frac{1}{a_{n}}, & a_{n} \neq 0 ; \\
0, & a_{n}=0
\end{array}(n=1,2, \cdots) .\right.
\end{array}
$$
Find all real numbers $a$ such that for any positive integer $n$, w... | 0,\\frac{\sqrt{2}}{2} | 159 | 12 |
math | Let $n \geqslant 3$ be integer. Given convex $n-$polygon $\mathcal{P}$. A $3-$coloring of the vertices of $\mathcal{P}$ is called [i]nice[/i] such that every interior point of $\mathcal{P}$ is inside or on the bound of a triangle formed by polygon vertices with pairwise distinct colors. Determine the number of differen... | 2^n - 3 - (-1)^n | 112 | 10 |
math | 13.378 In a piece of alloy weighing 6 kg, copper is contained. In another piece of alloy weighing 8 kg, copper is contained in a different percentage than in the first piece. From the first piece, a certain part was separated, and from the second - a part twice as heavy as from the first. Each of the separated parts wa... | 2.4 | 116 | 3 |
math | 602. Find the greatest and the least values of the function $y=$ $=x^{5}-5 x^{4}+5 x^{3}+3$ on the interval $[-1,2]$. | y(1)=4,y(-1)=-8 | 47 | 11 |
math | 12th VMO 1974 Problem A2 (1) How many positive integers n are such that n is divisible by 8 and n+1 is divisible by 25? (2) How many positive integers n are such that n is divisible by 21 and n+1 is divisible by 165? (3) Find all integers n such that n is divisible by 9, n+1 is divisible by 25 and n+2 is divisible by 4... | 24\mod200,none,774\mod900 | 106 | 18 |
math | Let
\[I_n =\int_{0}^{n\pi} \frac{\sin x}{1+x} \, dx , \ \ \ \ n=1,2,3,4\]
Arrange $I_1, I_2, I_3, I_4$ in increasing order of magnitude. Justify your answer. | I_2 < I_4 < I_3 < I_1 | 73 | 15 |
math | 3. The irrational root of the fractional equation $\frac{1}{x-5}+\frac{2}{x-4}+\frac{3}{x-3}+\frac{4}{x-2}$ $=4$ is | \frac{7 \pm \sqrt{5}}{2} | 50 | 14 |
math | A right-angled triangle has side lengths that are integers, and the measurement of its perimeter is equal to the measurement of its area. Which triangle is this? | 6,8,10 | 32 | 6 |
math | Example 9 For the set $\{1,2, \cdots, n\}$ and each of its non-empty subsets, define a unique “alternating sum” as follows: arrange the numbers in the subset in decreasing order, then alternately subtract and add the successive numbers starting from the largest to get the alternating sum (for example, the alternating s... | 1024 | 116 | 4 |
math | Problem 5. Determine all the functions $f: \mathbb{R} \rightarrow \mathbb{R}$ such that
$$
f\left(x^{2}+f(y)\right)=f(f(x))+f\left(y^{2}\right)+2 f(x y)
$$
for all real number $x$ and $y$. | f(x)=0 | 74 | 4 |
math | Find all ordered pairs $(a,b)$ of positive integers that satisfy $a>b$ and the equation $(a-b)^{ab}=a^bb^a$. | (4, 2) | 34 | 6 |
math | 8. (10 points) On the way from Xiaoming's house to the swimming pool, there are 200 trees. On the round trip, Xiaoming used red ribbons to mark some trees. On the way to the swimming pool, he marked the 1st tree, the 6th tree, the 11th tree, ... each time marking a tree after skipping 4 trees; on the way back, he marke... | 140 | 143 | 3 |
math | 13. Given $\sin ^{3} \theta+\cos ^{3} \theta=\frac{11}{16}$, find the value of $\sin \theta+\cos \theta$. | \frac{1}{2} | 43 | 7 |
math | 5. [5 points] Around a hook with a worm, in the same plane as it, a carp and a minnow are swimming along two circles. In the specified plane, a rectangular coordinate system is introduced, in which the hook (the common center of the circles) is located at the point $(0 ; 0)$. At the initial moment of time, the carp and... | (\sqrt{2}-4;-4-\sqrt{2}),(-4-\sqrt{2};4-\sqrt{2}),(4-\sqrt{2};4+\sqrt{2}),(4+\sqrt{2};\sqrt{2}-4) | 163 | 53 |
math | Example 5. Find the analytic function $w=f(z)$ given its imaginary part $v(x, y)=3 x+2 x y$ under the condition that $f(-i)=2$. | 3iz+z^2 | 41 | 5 |
math | 7. [6] Find all ordered pairs $(x, y)$ such that
$$
(x-2 y)^{2}+(y-1)^{2}=0 .
$$ | (2,1) | 38 | 5 |
math | Task 1. (5 points) Solve the equation $x^{9}-2022 x^{3}+\sqrt{2021}=0$.
# | {\sqrt[6]{2021};\sqrt[3]{\frac{-\sqrt{2021}\45}{2}}} | 36 | 31 |
math | 6.098. $\left\{\begin{array}{l}x^{3}+y^{3}=7 \\ x y(x+y)=-2\end{array}\right.$ | (2,-1),(-1,2) | 41 | 10 |
math | 1. Given that the domain of the function $f(x)$ is $[-1,1]$, find the domain of $f(a x)+f\left(\frac{x}{a}\right)$ (where $a>0$). | [-,]for\in(0,1);[-\frac{1}{},\frac{1}{}]for\in[1,+\infty) | 49 | 34 |
math | \section*{Problem 5 - 141035}
Determine all natural numbers \(n\) with \(n<40\) for which the number \(n^{2}+6 n-187\) is divisible by 19 without a remainder! | n\in{2,11,21,30} | 58 | 15 |
math | Authors: Kazzczzna T.V., Frankinn B.R., Shapovalov A.V.
Out of 100 members of the Council of Two Tribes, some are elves, and the rest are dwarves. Each member wrote down two numbers: the number of elves in the Council and the number of dwarves in the Council. Each member correctly counted their own tribe but made an e... | 66 | 138 | 2 |
math | 10、A, B, and C are guessing a two-digit number.
A says: It has an even number of factors, and it is greater than 50.
B says: It is an odd number, and it is greater than 60.
C says: It is an even number, and it is greater than 70.
If each of them is only half right, then the number is $\qquad$ | 64 | 89 | 2 |
math | 3、Given $2014=\left(a^{2}+b^{2}\right) \times\left(c^{3}-d^{3}\right)$, where $\mathbf{a} 、 \mathbf{b} 、 \mathbf{c} 、 \mathbf{d}$ are four positive integers, please write a multiplication equation that satisfies the condition: $\qquad$ | 2014=(5^{2}+9^{2})\times(3^{3}-2^{3}) | 86 | 25 |
math | [ Rectangular parallelepipeds ] [ Pythagorean theorem in space ] The diagonals of three different faces of a rectangular parallelepiped are $m$, $n$, and $p$. Find the diagonal of the parallelepiped. | \sqrt{\frac{^{2}+n^{2}+p^{2}}{2}} | 49 | 21 |
math | 6. (3 points) Starting with 100, perform the operations "add 15, subtract 12, add 3, add 15, subtract 12, add 3, ... " in a repeating sequence. After 26 steps (1 step refers to each "add" or "subtract" operation), the result is $\qquad$ . | 151 | 80 | 3 |
math | ## problem statement
Calculate the area of the parallelogram constructed on vectors $a$ and $b$.
$a=3 p-q$
$b=p+2 q$
$|p|=3$
$|q|=4$
$(\widehat{p, q})=\frac{\pi}{3}$ | 42\sqrt{3} | 63 | 7 |
math | ## Task $8 / 72$
Determine all real numbers $a$ for which the equation
$$
\sin x + \sin (x + a) + \sin (x + 2a) = 0
$$
holds for any real $x$. | a_{1}=2(\frac{1}{3}+k)\pi;a_{2}=2(\frac{2}{3}+k)\pi | 58 | 32 |
math | 10. (10 points) On September 21, 2015, the Ministry of Education announced the list of national youth campus football characteristic schools. Xiao Xiao's school was on the list. To better prepare for the elementary school football league to be held in the city next year, the school's team will recently play a football ... | 7 | 202 | 1 |
math | Determine for how many natural numbers greater than 900 and less than 1001 the digital sum of the digital sum of their digital sum is equal to 1.
(E. Semerádová)
Hint. What is the largest digital sum of numbers from 900 to 1001? | 12 | 69 | 2 |
math | Players A and B play a game. They are given a box with $n=>1$ candies. A starts first. On a move, if in the box there are $k$ candies, the player chooses positive integer $l$ so that $l<=k$ and $(l, k) =1$, and eats $l$ candies from the box. The player who eats the last candy wins. Who has winning strategy, in terms of... | n \equiv 1 \pmod{2} | 94 | 12 |
math | Let $a$ be a real number and $P(x)=x^{2}-2 a x-a^{2}-\frac{3}{4}$. Find all values of $a$ such that $|P(x)| \leqslant 1$ for all $x \in[0,1]$. | [-\frac{1}{2},\frac{\sqrt{2}}{4}] | 66 | 18 |
math | ## Problem Statement
Calculate the limit of the function:
$\lim _{x \rightarrow \pi} \frac{\left(x^{3}-\pi^{3}\right) \sin 5 x}{e^{\sin ^{2} x}-1}$ | -15\pi^{2} | 54 | 8 |
math | 3. Given that for any $m \in\left[\frac{1}{2}, 3\right]$, we have $x^{2}+m x+4>2 m+4 x$.
Then the range of values for $x$ is $\qquad$ . | x>2orx<-1 | 60 | 7 |
math | 7. At the ends of a vertically positioned homogeneous spring, two small loads are fixed. Above is a load with mass $m_{1}$, and below is $-m_{2}$. A person grabbed the middle of the spring and held it vertically in the air. In this case, the upper half of the spring was deformed by $x_{1}=8 \mathrm{~cm}$, and the lower... | 30\, | 161 | 4 |
math | Let $A=\{1,2,\ldots, 2006\}$. Find the maximal number of subsets of $A$ that can be chosen such that the intersection of any 2 such distinct subsets has 2004 elements. | 2006 | 53 | 4 |
math | Example 4.3.3 $n$ points, no three of which are collinear, are connected by line segments, and these segments are colored with two colors, red and blue. If for any coloring, there must exist 12 monochromatic triangles, find the minimum value of $n$.
| 9 | 64 | 1 |
math | ..... ]
[ Auxiliary area. The area helps to solve the problem ]
[ Thales' theorem and the theorem of proportional segments ]
In a right triangle with legs of 6 and 8, a square is inscribed, sharing a right angle with the triangle. Find the side of the square.
# | \frac{24}{7} | 61 | 8 |
math | Example 1. Expand the function $y=\arccos x+1$ into a Fourier series on the interval $(-1,1)$ using the system of Chebyshev polynomials. | \arccosx+1=\frac{\pi+2}{2}+\frac{2}{\pi}\sum_{n=1}^{\infty}\frac{(-1)^{n}-1}{n^{2}}T_{n}(x),x\in(-1,1) | 41 | 63 |
math | 6. Find the smallest natural number $n$, such that in any two-coloring of $K_{n}$ there are always two monochromatic triangles that share exactly one vertex.
---
The translation maintains the original text's format and line breaks as requested. | 9 | 52 | 1 |
math | In one American company, every employee is either a Democrat or a Republican. After one Republican decided to become a Democrat, the number of each became equal. Then, three more Republicans decided to become Democrats, and the number of Democrats became twice the number of Republicans. How many employees are there in ... | 18 | 62 | 2 |
math | 3. Let set $A=\{2,0,1,3\}$, set $B=\left\{x \mid -x \in A, 2-x^{2} \notin A\right\}$. Then the sum of all elements in set $B$ is $\qquad$. | -5 | 64 | 2 |
math | ## Task A-2.5.
Let $A$ be the number of six-digit numbers whose product of digits is 105, and $B$ be the number of six-digit numbers whose product of digits is 147. Determine the ratio $A: B$. | 2:1 | 58 | 3 |
math | 4. Given a regular quadrilateral pyramid $P-A B C D$ with all edges of equal length. Taking $A B C D$ as one face, construct a cube $A B C D-E F G H$ on the other side of the pyramid. Then, the cosine value of the angle formed by the skew lines $P A$ and $C F$ is $\qquad$ | \frac{2+\sqrt{2}}{4} | 81 | 12 |
math | $1 \cdot 34$ Let $a_{1}=1, a_{2}=3$, and for all positive integers $n$,
$$
a_{n+2}=(n+3) a_{n+1}-(n+2) a_{n} .
$$
Find all values of $n$ for which $a_{n}$ is divisible by 11.
(7th Balkan Mathematical Olympiad, 1990) | n=4,n=8,n\geqslant10 | 98 | 14 |
math | 6.112. $\left\{\begin{array}{l}\sqrt{x}+\sqrt{y}=10 \\ \sqrt[4]{x}+\sqrt[4]{y}=4\end{array}\right.$ | (1;81),(81;1) | 49 | 11 |
math | 7. Given a cube with edge length 6, there is a regular tetrahedron with edge length $x$ inside it, and the tetrahedron can rotate freely within the cube. Then the maximum value of $x$ is $\qquad$ . | 2\sqrt{6} | 55 | 6 |
math | 3. In the cells of a $3 \times 3$ square, the numbers $0,1,2, \ldots, 8$ are arranged. It is known that any two consecutive numbers are located in adjacent (by side) cells. Which number can be in the central cell if the sum of the numbers in the corner cells is $18?$ | 2 | 76 | 1 |
math | 8. Given the set $A=\{1,2, \cdots, 104\}, S$ is a subset of $A$. If $x \in S$, and at the same time $x-1 \notin S$ and $x+1 \notin S$, then $x$ is called an "isolated point" of $S$. The number of all 5-element subsets of $A$ with no "isolated points" is . $\qquad$ | 10000 | 102 | 5 |
math | 4.2. 12 * Given that $x, y, z$ are positive numbers, and satisfy $x y z(x+y+z)=1$. Find the minimum value of the expression $(x+y)(x+z)$.
保留源文本的换行和格式，翻译结果如下：
4.2. 12 * Given that $x, y, z$ are positive numbers, and satisfy $x y z(x+y+z)=1$. Find the minimum value of the expression $(x+y)(x+z)$. | 2 | 109 | 1 |
math | In convex quadrilateral $ABCD$, $\angle BAD = \angle BCD = 90^o$, and $BC = CD$. Let $E$ be the intersection of diagonals $\overline{AC}$ and $\overline{BD}$. Given that $\angle AED = 123^o$, find the degree measure of $\angle ABD$. | 78^\circ | 77 | 4 |
math | ## Task $9 / 66$
Calculate
$$
\sum_{k=1}^{n}\left(k x^{k-1}\right)=1+2 x+3 x^{2}+\ldots+n x^{n-1}
$$ | \frac{1-(n+1)x^{n}+nx^{n+1}}{(1-x)^{2}} | 54 | 26 |
math | 2. Solve the equation
$$
\log _{3 x+4}(2 x+1)^{2}+\log _{2 x+1}\left(6 x^{2}+11 x+4\right)=4
$$ | \frac{3}{4} | 53 | 7 |
math | 1. Naomi has a broken calculator. All it can do is either add one to the previous answer, or square the previous answer. (It performs the operations correctly.) Naomi starts with 2 on the screen. In how many ways can she obtain an answer of 1000 ? | 128 | 59 | 3 |
math | Example 3 Let $n$ be a positive integer,
$$
\begin{aligned}
S= & \{(x, y, z) \mid x, y, z \in\{0,1, \cdots, n\}, \\
& x+y+z>0\}
\end{aligned}
$$
is a set of $(n+1)^{3}-1$ points in three-dimensional space. Try to find the minimum number of planes whose union contains $S$ but does not contain $(0,0,0)$. ${ }^{[4]}$ | 3n | 121 | 2 |
math | A boy tried to align 480 cans in the shape of a triangle with one can in the $1^{\text{st}}$ row, 2 cans in the $2^{\text{nd}}$ row, and so on. In the end, 15 cans were left over. How many rows does this triangle have? | 30 | 72 | 2 |
math | 3. Petya bought one cupcake, two muffins, and three bagels, Anya bought three cupcakes and a bagel, and Kolya bought six muffins. They all paid the same amount of money for their purchases. Lena bought two cupcakes and two bagels. How many muffins could she have bought for the same amount she spent? | 5 | 74 | 1 |
math | 8.5. In Nastya's room, 16 people gathered, each pair of whom either are friends or enemies. Upon entering the room, each of them wrote down the number of friends who had already arrived, and upon leaving - the number of enemies remaining in the room. What can the sum of all the numbers written down be, after everyone h... | 120 | 80 | 3 |
math | 7.095. $\sqrt{\log _{a} x}+\sqrt{\log _{x} a}=\frac{10}{3}$. | \sqrt[9]{};^{9} | 35 | 10 |
math | 8. A math competition problem: the probabilities of A, B, and C solving it alone are $\frac{1}{a}$, $\frac{1}{b}$, and $\frac{1}{c}$, respectively, where $a$, $b$, and $c$ are all single-digit numbers. Now A, B, and C are solving this problem independently at the same time. If the probability that exactly one of them s... | \frac{4}{15} | 118 | 8 |
math | Four. (50 points) The International Mathematical Olympiad Chief Committee has $n$ countries participating, with each country being represented by a team leader and a deputy leader. Before the meeting, the participants shake hands with each other, but the team leader does not shake hands with their own deputy leader. Af... | =0,n=50 | 137 | 6 |
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. If from the set $S=\{1,2, \cdots, 20\}$, we take a three-element subset $A=\left\{a_{1}, a_{2}, a_{3}\right\}$, such that it simultaneously satisfies: $a_{2}-a_{1} \geqslant 5,4 \leqslant a_{3}-a_{2} \leqslant 9$, then the number of all such subsets $A$ is $\qquad$ (answer with a specific number). | 251 | 119 | 3 |
math |
Problem N2. Find all positive integers $n$ such that $36^{n}-6$ is a product of two or more consecutive positive integers.
| 1 | 33 | 1 |
math | For the following equation:
$$
m x^{2}-(1-m) x+(m-1)=0
$$
For which values of $m$ are the roots of the equation real? Examine the signs of the roots as $m$ varies from $-\infty$ to $+\infty$. | \begin{pmatrix}\text{if}-\infty\leqq<-\frac{1}{3}\quad\text{then}&\text{therootscomplex,}\\=-\frac{1}{3}&x_{1}=x_{2}=-2,\\-\frac{1}{3}<<0,&\text{bothrootsnegative,}\\=0&x_{1} | 65 | 84 |
math | ## Problem Statement
Calculate the limit of the function:
$\lim _{x \rightarrow 1} \frac{x^{2}-1}{\ln x}$ | 2 | 33 | 1 |
math | 3. Determine the largest positive integer $n$ for which there exist pairwise different sets $S_{1}, S_{2}, \ldots, S_{n}$ with the following properties:
$1^{\circ}\left|S_{i} \cup S_{j}\right| \leq 2004$ for any two indices $1 \leq i, j \leq n$, and
$2^{\circ} S_{i} \cup S_{j} \cup S_{k}=\{1,2, \ldots, 2008\}$ for an... | 32 | 186 | 2 |
math | Putnam 1995 Problem B4 Express (2207-1/(2207-1/(2207-1/(2207- ... )))) 1/8 in the form (a + b√c)/d, where a, b, c, d are integers. Solution | \frac{3+\sqrt{5}}{2} | 69 | 12 |
math | 9. An electronic watch at 9:15:12 AM shows the time as $09: 15: 12$, and at 1:11:29 PM, it shows the time as 13:11:29. Then, within 24 hours a day, the number of times the six digits on the electronic watch form a symmetrical time (i.e., reading from left to right is the same as reading from right to left, such as 01:3... | 96 | 121 | 2 |
math | Example 1 Calculate
$$
S\left(9 \times 99 \times 9999 \times \cdots \times \underset{2^{\circ} \uparrow}{99 \cdots 9}\right) \text {. }
$$
(1992, USA Mathematical Olympiad)
[Analysis] If you are familiar with Lemma 1 and have a bit of a sense of magnitude, you can get the answer right away. | 9 \times 2^{n} | 100 | 8 |
math | Find all integers $n$ such that $2^{n}+3$ is a perfect square. Same question with $2^{n}+1$.
## - Solution - | n=0for2^n+3,\;n=3for2^n+1 | 37 | 18 |
math | 29th Putnam 1968 Problem B5 Let F be the field with p elements. Let S be the set of 2 x 2 matrices over F with trace 1 and determinant 0. Find |S|. Solution | p^2+p | 50 | 4 |
math | Three boxes each contain an equal number of hockey pucks. Each puck is either black or gold. All 40 of the black pucks and exactly $\frac{1}{7}$ of the gold pucks are contained in one of the three boxes. Determine the total number of gold hockey pucks. | 140 | 62 | 3 |
math | What digits should be placed instead of the asterisks so that the number 454** is divisible by 2, 7, and 9?
# | 45486 | 33 | 5 |
math | 13.7.3 * $M$ is a moving point on the circle $x^{2}+y^{2}-6 x-8 y=0$, $O$ is the origin, $N$ is a point on the ray $O M$, $|O M| \cdot|O N|=150$. Find the equation of the locus of point $N$.
Translate the above text into English, please retain the original text's line breaks and format, and output the translation resu... | 3x+4y-75=0 | 107 | 10 |
math | 7. Easter Problem ${ }^{3}$. (From 6th grade, 2 points) Two players are playing an "Easter egg fight." In front of them is a large basket of eggs. They randomly take an egg each and hit them against each other. One of the eggs breaks, the loser takes a new egg, while the winner keeps their egg for the next round (the o... | \frac{11}{12} | 129 | 9 |
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