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31,873 | Find all $f: \mathbb{N}^{*} \longrightarrow \mathbb{N}^{*}$ such that $\forall m, n$ :
$$
f\left(f(m)^{2}+2 f(n)^{2}\right)=m^{2}+2 n^{2}
$$ | f(n)=n |
5,569 | Another trapezoid \(ABCD\) has \(AD\) parallel to \(BC\). \(AC\) and \(BD\) intersect at \(P\). If \(\frac{[ADP]}{[BCP]} = \frac{1}{2}\), find \(\frac{[ADP]}{[ABCD]}\). (Here, the notation \(\left[P_1 \cdots P_n\right]\) denotes the area of the polygon \(P_1 \cdots P_n\)). | 3 - 2\sqrt{2} |
61,874 | 4. Given a square $M N P Q$ with a side length of $1 m$. On the sides of this square, points $A, B, C$ are marked:
- point $A$ is at $\frac{1}{3}$ of the side $N P$ from point $N$,
- point $B$ is at $\frac{2}{3}$ of the side $M Q$ from point $M$,
- point $C$ is the midpoint of the side $M N$.
Triangle $A B C$ is not a right triangle. On which side and by how much should point $C$ be moved so that triangle $A B C$ becomes a right triangle? | \\frac{1}{6} |
61,630 | 51. What is the greatest number of squares with side 1 that can be placed ${ }^{1}$ ) next to a given unit square $K$ so that no two of them intersect?
$^{1}$) The superscript "1" is kept as is, since it might refer to a footnote or additional information in the original text. | 8 |
12,410 | Suppose the sequence $\left\{a_{n}\right\}$ satisfies the conditions: $a_{1}=\frac{1}{2}$ and $2 k a_{k}=(2 k-3) a_{k-1}$ for $k \geqslant 2$. Prove that for all $n \in \mathbf{N}$, we have $\sum_{k=1}^{n} a_{k}<1$. | \sum_{k=1}^{n} a_{k} < 1 |
21,845 | Let \( k(a) \) denote the number of points \((x, y)\) in the coordinate system such that \(1 \leq x \leq a\) and \(1 \leq y \leq a\) are relatively prime integers. Determine the following sum:
$$
\sum_{i=1}^{100} k\left(\frac{100}{i}\right)
$$ | 10000 |
22,921 | 10. In the Cartesian coordinate system, given the point set $I=\{(x, y) \mid x, y$ are integers, and $0 \leqslant x, y \leqslant 5\}$. Then the number of different squares with vertices in the set $I$ is . $\qquad$ | 105 |
53,681 | 6. Let the ellipse $C: \frac{x^{2}}{4}+\frac{y^{2}}{3}=1$, and the line $l: y=4x+m$. Then, the range of values for $m$ such that there are two points on the ellipse $C$ that are symmetric with respect to the line $l$ is $\qquad$. | (-\frac{2\sqrt{13}}{13},\frac{2\sqrt{13}}{13}) |
52,568 | 3. (3 points) Let $x_{1}, x_{2}, \ldots, x_{100}$ be natural numbers greater than 1 (not necessarily distinct). In an $80 \times 80$ table, the numbers are arranged as follows: at the intersection of the $i$-th row and the $k$-th column, the number $\log _{x_{k}} \frac{x_{i}}{16}$ is written. Find the smallest possible value of the sum of all numbers in the table. | -19200 |
59,293 | Exercise 9. Let $A B C$ be a triangle such that $\widehat{C A B}=20^{\circ}$. Let $D$ be the midpoint of the segment $[A B]$. Suppose that $\widehat{C D B}=40^{\circ}$. What is the value of the angle $\widehat{A B C}$? | 70 |
68,269 | 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} |
55,240 | 4. 155 To find the minimum value of \( n \) for which the following system of equations
\[
\left\{\begin{array}{l}
\sin x_{1}+\sin x_{2}+\cdots+\sin x_{n}=0, \\
\sin x_{1}+2 \sin x_{2}+\cdots+n \sin x_{n}=100 .
\end{array}\right.
\]
has a solution. | 20 |
1,550 | Divide an \( m \)-by-\( n \) rectangle into \( m n \) nonoverlapping 1-by-1 squares. A polyomino of this rectangle is a subset of these unit squares such that for any two unit squares \( S \) and \( T \) in the polyomino, either
(1) \( S \) and \( T \) share an edge, or
(2) there exists a positive integer \( n \) such that the polyomino contains unit squares \( S_1, S_2, S_3, \ldots, S_n \) such that \( S \) and \( S_1 \) share an edge, \( S_n \) and \( T \) share an edge, and for all positive integers \( k < n \), \( S_k \) and \( S_{k+1} \) share an edge.
We say a polyomino of a given rectangle spans the rectangle if for each of the four edges of the rectangle the polyomino contains a square whose edge lies on it.
What is the minimum number of unit squares a polyomino can have if it spans a 128-by-343 rectangle? | 470 |
62,964 | Show that the planes $ACG$ and $BEH$ defined by the vertices of the cube shown in Figure are parallel. What is their distance if the edge length of the cube is $1$ meter?
[img]https://cdn.artofproblemsolving.com/attachments/c/9/21585f6c462e4289161b4a29f8805c3f63ff3e.png[/img] | \frac{1}{\sqrt{3}} |
17,902 | 3.186. $\frac{\sin ^{4} \alpha+\cos ^{4} \alpha-1}{\sin ^{6} \alpha+\cos ^{6} \alpha-1}=\frac{2}{3}$. | \dfrac{2}{3} |
18,213 | Example 1 Solve the functional equation $f(x)+f\left(\frac{x-1}{x}\right)=1+x \quad (x \neq 0,1)$ | \dfrac{x}{2} + \dfrac{1}{2x} - \dfrac{1}{2(x - 1)} |
15,954 | Replace the letters A, B, C, D, E with digits (different letters correspond to different digits) so that the difference of the three-digit numbers ABC - DEF takes the smallest possible positive value. In the answer, state the value of this difference. | 3 |
68,660 | $14 \cdot 41$ Try for any positive integer $n$, to calculate the sum $\sum_{k=0}^{\infty}\left[\frac{n+2^{k}}{2^{k+1}}\right]$.
(10th International Mathematical Olympiad, 1968) | n |
56,151 | 7・15 Given a regular $(2n+1)$-sided polygon, three different vertices are randomly selected from its vertices. If all such selections are equally likely, find the probability that the center of the regular polygon lies inside the triangle formed by the randomly chosen three points. | \frac{n+1}{2(2n-1)} |
60,814 | 1. The expression $1000 \sin 10^{\circ} \cos 20^{\circ} \cos 30^{\circ} \cos 40^{\circ}$ can be simplified as $a \sin b^{\circ}$, where $a$ and $b$ are positive integers with $0<b<90$. Find the value of $100 a+b$. | 12560 |
62,782 | ## Problem Statement
Find the indefinite integral:
$$
\int \frac{\sqrt[3]{1+\sqrt[4]{x^{3}}}}{x^{2}} d x
$$ | -\frac{\sqrt[3]{(1+\sqrt[4]{x^{3}})^{4}}}{x}+C |
52,726 | In one container, there is one liter of wine, in another, one liter of water. From the first container, we pour one deciliter into the second, mix it, then pour one deciliter of the mixture back into the first container. Calculate the limit of the amount of wine in the first container if the above procedure is repeated infinitely. (We assume perfect mixing during each transfer, and there is no fluid loss during the process.) | \frac{1}{2} |
50,931 | ## Task Condition
Find the derivative.
$y=2 \sqrt{\frac{1-\sqrt{x}}{1+\sqrt{x}}}$ | -\frac{1}{\sqrt{x(1-x)}\cdot(1+\sqrt{x})} |
5,272 | Let \(ABC\) be a triangle inscribed in the circle below. Let \(I\) be the incenter of triangle \(ABC\) and \(D\) the point where the line \(AI\) intersects the circle. Show that \(|\overline{DB}| = |\overline{DC}| = |\overline{DI}|\). | |\overline{DB}| = |\overline{DC}| = |\overline{DI}| |
65,644 | $3+$ [ Evenness and Oddness $]$
In a square table $N \times N$, all integers are written according to the following rule: 1 is placed in any position, 2 is placed in the row with the number equal to the column number containing 1, 3 is placed in the row with the number equal to the column number containing 2, and so on. By how much does the sum of the numbers in the column containing $N^{2}$ differ from the sum of the numbers in the row containing 1. | N^2-N |
54,833 | 4. Find all three-digit numbers that are equal to the sum of the factorials of their digits! | 145 |
32,161 | 6. On the table, there are three cones standing on their bases, touching each other. The radii of their bases are 6, 24, and 24. A truncated cone is placed on the table with its smaller base down, and it shares a generatrix with each of the other cones. Find the radius of the smaller base of the truncated cone. | 2 |
62,707 | It is known that the point symmetric to the center of the inscribed circle of triangle $ABC$ with respect to side $BC$ lies on the circumcircle of this triangle. Find angle $A$.
# | 60 |
65,089 | Problem 1. Let's call a number small if it is a 10-digit number and there does not exist a smaller 10-digit number with the same sum of digits. How many small numbers exist | 90 |
11,145 | 3. In a watch repair shop, there is a certain number of electronic watches (more than one), displaying time in a 12-hour format (the number of hours on the watch face changes from 1 to 12). All of them run at the same speed, but show completely different times: the number of hours on the face of any two different watches is different, and the number of minutes is also different.
One day, the master added up the number of hours on the faces of all the watches, then added up the number of minutes on the faces of all the watches, and remembered the two resulting numbers. After some time, he did the same again and found that both the total number of hours and the total number of minutes had decreased by 1. What is the maximum number of electronic watches that could have been in the shop? | 11 |
56,244 | Example 2 Consider the following sequence:
$$101,10101,1010101, \cdots$$
Question: How many prime numbers are there in this sequence? | 1 |
17,370 | Find the smallest positive integer \( n \) such that the divisors of \( n \) can be partitioned into three sets with equal sums. | 120 |
56,478 | 6. For the numbers $1000^{2}, 1001^{2}, 1002^{2}, \ldots$, the last three digits are discarded. How many of the first terms of the resulting sequence form an arithmetic progression? | 32 |
53,748 | 11. A sequence $U_{1}, U_{2}, U_{3}, \ldots$ is defined as follows:
- $U_{1}=2$;
- if $U_{n}$ is prime then $U_{n+1}$ is the smallest positive integer not yet in the sequence;
- if $U_{n}$ is not prime then $U_{n+1}$ is the smallest prime not yet in the sequence.
The integer $k$ is the smallest such that $U_{k+1}-U_{k}>10$.
What is the value of $k \times U_{k}$ ? | 270 |
20,697 | Let \( G \) be an infinite complete graph. Show that if the edges of \( G \) are colored with a finite number of colors, then \( G \) contains an infinite monochromatic subgraph.
Hint: The idea is to construct a sequence of distinct vertices \(\left(u_{n}\right)_{n \in \mathbb{N}}\) and a sequence of colors (where a color may appear multiple times) \(\left(c_{n}\right)_{n \in \mathbb{N}}\) with the following property: if \( p > q \), then the edge between \( u_{p} \) and \( u_{q} \) is colored \( c_{q} \). | G \text{ contains an infinite monochromatic subgraph.} |
20,411 | 19. $\overline{P Q Q P Q Q}, \overline{P Q P Q P Q}, \overline{Q P Q P Q P}, \overline{P P P P P P}, \overline{P P P Q Q Q}$ are five six-digit numbers, where the same letters represent the same digits, and different letters represent different digits. Regardless of what $P$ and $Q$ are, among these five six-digit numbers, $\qquad$ of them are definitely multiples of 7. | 4 |
67,156 | 141. Calculate $x=\frac{2.48 \cdot 0.3665}{5.643}$. | 0.161 |
57,028 | 1. Determine the value of the parameter $n$ such that for every point $(x, y)$ on the graph of the function $y=-x+n$, the following equations hold: $y=\frac{5}{x}$ and $y^{2}=6-x^{2}$. Then, calculate the area of the region in the plane formed by the coordinate axes and the graph of the function. | 8 |
28,265 | Problem 9. In the decimal representation of an even number $M$, only the digits $0, 2, 4, 5, 7$, and 9 are used, and digits can repeat. It is known that the sum of the digits of the number $2M$ is 35, and the sum of the digits of the number $M / 2$ is 29. What values can the sum of the digits of the number $M$ take? List all possible answers. | 31 |
54,557 | 9. Find the result of the following binary subtraction operations:
(i) $(1010111)_{2}-(11001)_{2}-(11110)_{2}=$ ?
(ii) $(10110001)_{2}-(1101100)_{2}-(11110)_{2}=$ ? | (100111)_{2} |
15,625 | Find \( x_{1000} \) if \( x_{1} = 4 \), \( x_{2} = 6 \), and for any natural number \( n \geq 3 \), \( x_{n} \) is the smallest composite number greater than \( 2 x_{n-1} - x_{n-2} \). | 501500 |
61,477 | 3. Find all natural values of $n$ for which
$$
\cos \frac{2 \pi}{9}+\cos \frac{4 \pi}{9}+\cdots+\cos \frac{2 \pi n}{9}=\cos \frac{\pi}{9}, \text { and } \log _{2}^{2} n+45<\log _{2} 8 n^{13}
$$
In your answer, write the sum of the obtained values of $n$.
(6 points) | 644 |
54,136 | 8. (15 points) Fill in the 9 cells of the 3x3 grid with 9 different natural numbers, such that: in each row, the sum of the two left numbers equals the rightmost number; in each column, the sum of the two top numbers equals the bottom number. The smallest number in the bottom-right corner is . $\qquad$ | 12 |
66,831 | ## Problem Statement
A cylinder is filled with gas at atmospheric pressure (103.3 kPa). Assuming the gas is ideal, determine the work (in joules) during the isothermal compression of the gas by a piston moving inside the cylinder by $h$ meters (see figure).
Hint: The equation of state for the gas
$\rho V=$ const, where $\rho$ - pressure, $V$ - volume.
$$
H=2.0 \mathrm{~m}, h=1.0 \mathrm{~m}, R=0.4 \mathrm{~m}
$$
 | 72000 |
61,607 | 1. A 100-digit number has the form $a=1777 \ldots 76$ (with 98 sevens in the middle). The number $\frac{1}{a}$ is represented as an infinite periodic decimal. Find its period. Justify your answer. | 99 |
62,098 | 5. On a circle, 25 points are marked, painted either red or blue. Some of the points are connected by segments, with one end of each segment being blue and the other end red. It is known that there do not exist two red points that belong to the same number of segments. What is the maximum possible number of red points? | 13 |
23,693 | Let \( S = \{1, 2, 3, \ldots, 100\} \). Find the smallest positive integer \( n \) such that every \( n \)-element subset of \( S \) contains 4 pairwise coprime numbers. | 75 |
18,784 | Fomin S.B.
Two people toss a coin: one tossed it 10 times, the other - 11 times.
What is the probability that the second one got heads more times than the first one? | \dfrac{1}{2} |
62,176 | 8. (10 points) In the inscribed quadrilateral $A B C D$, the degree measures of the angles are in the ratio $\angle A: \angle B: \angle C=2: 3: 4$. Find the length of $A C$, if $C D=20, B C=24 \sqrt{3}-10$. | 52 |
53,165 | Shapovalov A.V.
A monkey becomes happy when it eats three different fruits. What is the maximum number of monkeys that can be made happy with 20 pears, 30 bananas, 40 peaches, and 50 tangerines? | 45 |
26,245 | 12・111 Given integers $m, n$ satisfying $m, n \in\{1,2, \cdots, 1981\}$ and
$$
\left(n^{2}-m n-m^{2}\right)^{2}=1 \text {. }
$$
Find the maximum value of $m^{2}+n^{2}$.
(22nd International Mathematical Olympiad, 1981) | 3524578 |
25,873 | 8. Given the hyperbola $x^{2}-y^{2}=t(t>0)$ with its right focus at $F$, any line passing through $F$ intersects the right branch of the hyperbola at points $M$ and $N$. The perpendicular bisector of $M N$ intersects the $x$-axis at point $P$. When $t$ is a positive real number not equal to zero, $\frac{|F P|}{|M N|}=$ $\qquad$ . | \frac{\sqrt{2}}{2} |
325 | From a six-digit phone number, how many seven-digit numbers can be obtained by removing one digit? | 70 |
28,419 | Example 6 Let $S=\{1,2,3, \cdots, 280\}$. Find the smallest natural number $n$ such that every subset of $S$ with $n$ elements contains 5 pairwise coprime numbers. | 217 |
28,899 | 6. Given a regular tetrahedron $\mathrm{ABCD}$ and a point $\mathrm{P}$ inside it, such that $P A=P B=\sqrt{11}, P C=P D=\sqrt{17}$. Then the edge length of the tetrahedron $\mathrm{ABCD}$ is
保留源文本的换行和格式，直接输出翻译结果。 | 6 |
9,285 | 1. On the Island of Knights and Liars, knights always tell the truth, while liars always lie. In a school on this island, both knights and liars study in the same class. One day, the teacher asked four children: Anu, Banu, Vanu, and Danu, who among them had completed their homework. They answered:
- Anu: Banu, Vanu, and Danu completed the homework.
- Banu: Anu, Vanu, and Danu did not complete the homework.
- Vanu: Don't believe them, sir! Anu and Banu are liars!
- Danu: No, sir, Anu, Banu, and Vanu are knights!
How many knights are among these children? | 1 |
21,151 | There is a stack of 52 face-down playing cards on the table. Mim takes 7 cards from the top of this stack, flips them over, and puts them back at the bottom, calling it one operation. The question is: how many operations are needed at least to make all the playing cards face-down again? | 112 |
24,857 | A regular hexagon \( K L M N O P \) is inscribed in an equilateral triangle \( A B C \) such that the points \( K, M, O \) lie at the midpoints of the sides \( A B, B C, \) and \( A C \), respectively. Calculate the area of the hexagon \( K L M N O P \) given that the area of triangle \( A B C \) is \( 60 \text{ cm}^2 \). | 30 |
68,718 | A hotel has 5 distinct rooms, all with single beds for up to 2 people. The hotel has no other guests and 5 friends want to spend the night there. In how many ways can the 5 friends choose their rooms? | 2220 |
56,231 | 9. (16 points) Let real numbers $x, y, z > 0$. Find the minimum value of the following two expressions:
(1) $\max \left\{x, \frac{1}{y}\right\} + \max \left\{y, \frac{2}{x}\right\}$;
(2) $\max \left\{x, \frac{1}{y}\right\} + \max \left\{y, \frac{2}{z}\right\} + \max \left\{z, \frac{3}{x}\right\}$. | 2\sqrt{5} |
50,320 | ## 162. Math Puzzle $11 / 78$
Assume that a fly lays 120 eggs at the beginning of summer, on June 21st, and after 20 days, fully developed insects emerge from these eggs, each of which then lays 120 eggs. How many "descendants" would this fly have in total by the beginning of autumn? | 209102520 |
53,776 | Example 6 Solve the system of equations:
$$
\left\{\begin{array}{l}
x=1+\ln y, \\
y=1+\ln z, \\
z=1+\ln x .
\end{array}\right.
$$ | (1,1,1) |
65,573 | 4. Let $k$ be one of the quotients of the roots of the quadratic equation $\left.p x^{2}-q x\right\lrcorner q=0$, where $p, q>0$. Express the roots of the equation $\sqrt{p} x^{2}-\sqrt{q} x+\sqrt{p}=0$ in terms of $k$ (not in terms of $p$ and $q$). | a_{1}=\sqrt{k},b_{1}=\frac{1}{\sqrt{k}} |
59,763 | 6. In the diagram, six squares form a $2 \times 3$ grid. The middle square in the top row is marked with an R. Each of the five remaining squares is to be marked with an $\mathrm{R}, \mathrm{S}$ or $\mathrm{T}$. In how many ways can the grid be completed so that it includes at least one pair of squares side-by-side in the same row or same column that contain the same letter? | 225 |
10,836 |
Let \( a_{1}, \ldots, a_{25} \) be non-negative integers, and let \( k \) be the smallest among them. Prove that
\[
\left[\sqrt{a_{1}}\right]+\left[\sqrt{a_{2}}\right]+\ldots+\left[\sqrt{a_{25}}\right] \geq\left[\sqrt{a_{1}+\ldots+a_{25}+200 k}\right]
\]
(As usual, \([x]\) denotes the greatest integer that does not exceed \( x \).)
(S. Berlov, A. Khrabrov) | \left[\sqrt{a_{1}}\right]+\left[\sqrt{a_{2}}\right]+\ldots+\left[\sqrt{a_{25}}\right] \geq\left[\sqrt{a_{1}+\ldots+a_{25}+200 k}\right] |
52,801 | 5. As shown in Figure 3, on each side of the square $A B C D$ with side length 1, segments $A E=B F$ $=C G=D H=x$ are cut off, and lines $A F 、 B G 、 C H 、 D E$ are drawn to form quadrilateral $P Q R S$. Express the area $S$ of quadrilateral $P Q R S$ in terms of $x$, then $S=$ | \frac{(1-x)^{2}}{1+x^{2}} |
12,732 | Once, I decided to take a chairlift ride. At a certain moment, I noticed that the chairlift coming towards me had the number 95, and the next one had the number 0, followed by 1, 2, and so on. I looked at the number of my chair, which turned out to be 66. Have I traveled half the distance? When I encounter which chair will I have traveled half the distance? | 18 |
53,795 | ## Task 2 - 230932
In the figure, a body $K$ is sketched. It consists of three cubes with an edge length of $1 \mathrm{~cm}$, which are firmly joined in the specified arrangement.
From a sufficient number of bodies of this shape $K$, a (completely filled) cube $W$ (edge length $n$ centimeters) is to be assembled.
Determine all natural numbers $n>0$ for which this is possible!
 | 3k |
52,359 | 6. Given that $a, b, c$ are all positive integers, and the parabola $y=a x^{2}+b x+c$ intersects the $x$-axis at two distinct points $A, B$. If the distances from $A, B$ to the origin are both less than 1. Then the minimum value of $a+b+c$ is $\qquad$ . | 11 |
31,655 | At least how many acute-angled triangles must be fitted together without gaps or overlaps to form an isosceles triangle with a $120^{\circ}$ vertex angle? | 7 |
28,009 | 18. A five-digit number $A B C D E$ is a multiple of 2014, and $C D E$ has exactly 16 factors. What is the smallest value of $A B C D E$? | 24168 |
60,076 | 1. Solve the inequality:
$$
\log _{9 x} 3 x+\log _{3 x^{2}} 9 x^{2}<\frac{5}{2}
$$ | x\in(0,\frac{1}{27\sqrt{3}})\cup(\frac{1}{9},\frac{1}{\sqrt{3}})\cup(1,\infty) |
51,901 | # Problem №3
A New Year's garland hanging along the school corridor consists of red and blue bulbs. Next to each red bulb, there is definitely a blue one. What is the maximum number of red bulbs that can be in this garland if there are 50 bulbs in total? | 33 |
18,995 | Compute the sum of all positive real numbers \( x \leq 5 \) satisfying
\[ x = \frac{\left\lceil x^{2} \right\rceil + \lceil x \rceil \cdot \lfloor x \rfloor}{\lceil x \rceil + \lfloor x \rfloor}. \] | 85 |
61,326 | 1. Let $n$ be the sum of all ten-digit numbers that have each of the digits $0,1, \ldots, 9$ in their decimal representation. Determine the remainder when $n$ is divided by seventy-seven. | 28 |
52,573 | 5. A school has 3 teachers who can teach English, 2 teachers who can teach Japanese, and 4 teachers who can teach both English and Japanese. Now, 3 English teachers and 3 Japanese teachers are to be selected to participate in off-campus tutoring during the holiday. How many ways are there to select them? | 216 |
6,688 | In an acute triangle \(ABC\), altitudes \(AP\) and \(CQ\) are drawn from vertices \(A\) and \(C\), respectively. Find the side \(AC\) if it is given that the perimeter of triangle \(ABC\) is 15, the perimeter of triangle \(BPQ\) is 9, and the circumradius of triangle \(BPQ\) is \(9/5\). | \dfrac{24}{5} |
63,166 | 6. Divide 2004 into the sum of $n(n>1)$ consecutive natural numbers, the number of ways to do this is ( )
A. $\theta$
B. 1
C. 2
1). 3 | 3 |
65,424 | (3) Let $a, b$ be positive real numbers, $\frac{1}{a}+\frac{1}{b} \leqslant 2 \sqrt{2}, (a-b)^{2}=4(a b)^{3}$, then $\log _{a} b=$ | -1 |
54,907 | 3. Four different numbers $a, b, c, d$, greater than one and not divisible by 5, are such that $\operatorname{GCD}(a, b)=\operatorname{GCD}(c, d)$ and $\operatorname{LCM}(a, b)=\operatorname{LCM}(c, d)$. What is the smallest possible value of $a+b+c+d$? | 24 |
28,987 | 9. Given that the line $l$ with slope $k$ passing through the point $P(3,0)$ intersects the right branch of the hyperbola $C: x^{2}-\frac{y^{2}}{3}=1$ at points $A$ and $B$, and $F$ is the right focus of the hyperbola $C$, and $|A F|+|B F|=16$, find the value of $k$. | \3 |
23,816 | A square field is enclosed by a wooden fence, which is made of 10-meter-long boards placed horizontally. The height of the fence is four boards. It is known that the number of boards in the fence is equal to the area of the field, expressed in hectares. Determine the dimensions of the field. | 16000 |
34,659 | 10. (10 points) Two people, A and B, take turns selecting numbers from the integers 1 to 17, with the rule: they cannot select numbers that have already been chosen by either party, they cannot select a number that is twice an already chosen number, and they cannot select a number that is half of an already chosen number. The person who cannot select a number loses. Now, A has already chosen 8, and B wants to ensure a certain win. The number B should choose next is $\qquad$
---
Please note that the blank space at the end (indicated by $\qquad$) is part of the original format and has been retained in the translation. | 6 |
69,068 | 3. Four congruent right triangles can be assembled to form the square string figure shown in Figure 2(a), or the rhombus shown in Figure 2(b). If the area of the large square in Figure 2(a) is 100, and the area of the small square is 4, then the cosine value of one acute angle in the rhombus in Figure 2(b) is | \frac{7}{25} |
15,169 | Given that \(0 \leq a_{k} \leq 1 \) for \(k=1,2, \ldots, 2020\), and defining \(a_{2021} = a_{1}\), \(a_{2022} = a_{2}\), find the maximum value of \(\sum_{k=1}^{2020}\left(a_{k} - a_{k+1}a_{k+2}\right)\). | 1010 |
53,366 | Problem 10.5. Greedy Vovochka has 25 classmates. For his birthday, he brought 200 candies to the class. Vovochka's mother, to prevent him from eating them all himself, told him to distribute the candies so that any 16 of his classmates would collectively have at least 100 candies. What is the maximum number of candies Vovochka can keep for himself while fulfilling his mother's request? | 37 |
6,449 | The method of substitution ${ }^{[4]}$ refers to assigning one or several special values to the variables in a functional equation within the domain of the function, thereby achieving the purpose of solving the problem.
Example $\mathbf{3}$ Find all functions $f: \mathbf{R} \rightarrow \mathbf{R}$ such that for all $x, y \in \mathbf{R}$, we have
$$
f(x y+1)=f(x) f(y)-f(y)-x+2 \text {. }
$$
(2016, Croatian Mathematical Competition) | f(x) = x + 1 |
18,624 | In how many ways can the number 1500 be represented as a product of three natural numbers? Variations where the factors are the same but differ in order are considered identical. | 32 |
54,640 | 1.27 Find the largest positive integer $n$, such that the inequality $\frac{8}{15}<\frac{n}{n+k}<\frac{7}{13}$ holds for exactly one integer $k$.
(5th American Invitational Mathematics Examination, 1987) | 112 |
13,909 |
A finite sequence of numbers \( x_{1}, x_{2}, \ldots, x_{N} \) possesses the following property:
\[ x_{n+2} = x_{n} - \frac{1}{x_{n+1}} \text{ for all } 1 \leq n \leq N-2. \]
Find the maximum possible number of terms in this sequence if \( x_{1} = 20 \) and \( x_{2} = 16 \). | 322 |
58,712 | 6.1. Find the largest six-digit number, all digits of which are different, and each of the digits, except for the extreme ones, is either the sum or the difference of the adjacent digits. | 972538 |
28,928 | 8 non-negative real numbers $a, b, c$ satisfy $a b + b c + c a = 1$. Find the minimum value of $\frac{1}{a+b} + \frac{1}{b+c} + \frac{1}{c+a}$. | \frac{5}{2} |
53,770 | 11. (12 points) If a three-digit number $\overline{\mathrm{abc}}$ (where $a, b, c$ are all non-zero digits) satisfies $\overline{\mathrm{ab}}>\overline{\mathrm{bc}}>\overline{\mathrm{ca}}$, then the three-digit number is called a “Longteng number”. How many “Longteng numbers” are there? | 120 |
32,804 | 7. For any point $P$ on the ellipse $C: \frac{x^{2}}{3}+\frac{y^{2}}{2}=1$, draw a perpendicular line $P H$ (where $H$ is the foot of the perpendicular) to the directrix of $C$. Extend $P H$ to $Q$ such that $H Q=\lambda P H(\lambda \geqslant 1)$. As point $P$ moves along $C$, the range of the eccentricity of the locus of point $Q$ is $\qquad$. | [\frac{\sqrt{3}}{3},1) |
2,264 | In the diagram, there is a karting track layout. Start and finish are at point $A$, but the go-kart driver can return to point $A$ any number of times and resume the lap.
The path from $A$ to $B$ and vice versa takes one minute. The loop also takes one minute to complete. The loop can only be traversed counterclockwise (the arrows indicate the possible direction of travel). The driver does not turn back halfway and does not stop. The racing duration is 10 minutes. Find the number of possible different routes (sequences of segments traversed). | 34 |
25,166 | ##
Side $A B$ of parallelogram $A B C D$ is equal to $2, \angle A=45^{\circ}$. Points $E$ and $F$ are located on diagonal $B D$, such that
$\angle A E B=\angle C F D=90^{\circ}, B F=\frac{3}{2} B E$.
Find the area of the parallelogram. | 3 |
31,341 | 11.1. Each of 10 people is either a knight, who always tells the truth, or a liar, who always lies. Each of them thought of some integer. Then the first said: “My number is greater than 1”, the second said: “My number is greater than $2 ”, \ldots$, the tenth said: “My number is greater than 10”. After that, all ten, speaking in some order, said: “My number is less than 1”, “My number is less than $2 ”, \ldots$, “My number is less than 10” (each said exactly one of these ten phrases). What is the maximum number of knights that could have been among these 10 people? | 8 |
22,863 | In an alphabet with $n>1$ letters, a word is defined as any finite sequence of letters where any two consecutive letters are different. A word is called "good" if it is not possible to delete all but four letters from it to obtain a sequence of the form $a a b b$, where $a$ and $b$ are different letters. Find the maximum possible number of letters in a "good" word. | 2n + 1 |
21,406 | 28. Given a package containing 200 red marbles, 300 blue marbles and 400 green marbles. At each occasion, you are allowed to withdraw at most one red marble, at most two blue marbles and a total of at most five marbles out of the package. Find the minimal number of withdrawals required to withdraw all the marbles from the package. | 200 |
53,461 | 8. If $a$, $b$, $c$ are distinct integers, then
$$
3 a^{2}+2 b^{2}+4 c^{2}-a b-3 b c-5 c a
$$
the minimum value is $\qquad$ | 6 |
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