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1,635 | How many different positive values of $x$ will make this statement true: there are exactly $2$ positive two-digit multiples of $x$. | 16 |
31,650 | The National Bank of Hungary withdrew the 1 and 2 forint coins from circulation on March 1, 2008. In cash purchases in stores, the prices of individual items are not rounded separately, but the final amount is rounded to the nearest five. The following table can be seen in the issued information about the rounding rules:
amounts ending in 1 or 2 are rounded down to 0;
amounts ending in 3 or 4 are rounded up to 5;
amounts ending in 6 or 7 are rounded down to 5;
amounts ending in 8 or 9 are rounded up to 0.
Marci buys two croissants every morning at the local bakery. After a few days, due to rounding, he saves exactly the price of one croissant. We know that the price of a croissant is more than 10 Ft. What could be the unit price of the croissant if this saving occurs as quickly as possible? | 16 |
27,515 | 2.21. The height of the cylinder is $H$, and the radius of its base is $R$. Inside the cylinder, a pyramid is placed, the height of which coincides with the generatrix $A A_{1}$ of the cylinder, and the base is an isosceles triangle $A B C(A B=A C)$ inscribed in the base of the cylinder. Find the lateral surface area of the pyramid if $\angle A=120^{\circ}$. | 0.25R(4H+\sqrt{3R^{2}+12H^{2}}) |
57,550 | 8. Arrange $1,2, \cdots, n^{2}$ in a clockwise spiral pattern into an $n$ by $n$ table $T_{n}$, with the first row being $1,2, \cdots, n$. For example, $T_{3}=\left[\begin{array}{lll}1 & 2 & 3 \\ 8 & 9 & 4 \\ 7 & 6 & 5\end{array}\right]$. Let 2018 be in the $i$-th row and $j$-th column of $T_{100}$. Then $(i, j)=$ $\qquad$ | (34,95) |
15,387 | Let $A$ and $B$ be points on circle $\Gamma$ such that $AB=\sqrt{10}.$ Point $C$ is outside $\Gamma$ such that $\triangle ABC$ is equilateral. Let $D$ be a point on $\Gamma$ and suppose the line through $C$ and $D$ intersects $AB$ and $\Gamma$ again at points $E$ and $F \neq D.$ It is given that points $C, D, E, F$ are collinear in that order and that $CD=DE=EF.$ What is the area of $\Gamma?$ *Proposed by Kyle Lee* | \dfrac{38\pi}{15} |
66,622 | $\underline{\text { Frankin B.R. }}$
Given a polynomial $P(x)$ with real coefficients. An infinite sequence of distinct natural numbers $a_{1}, a_{2}, a_{3}, \ldots$ is such that
$P\left(a_{1}\right)=0, P\left(a_{2}\right)=a_{1}, P\left(a_{3}\right)=a_{2}$, and so on. What degree can $P(x)$ have? | 1 |
23,939 | Natural numbers \( x_{1}, x_{2}, \ldots, x_{13} \) are such that \( \frac{1}{x_{1}} + \frac{1}{x_{2}} + \ldots + \frac{1}{x_{13}} = 2 \). What is the minimum value of the sum of these numbers? | 85 |
65,883 | ## Task B-3.4.
In a convex quadrilateral $A B C D$, $|A B|=15,|B C|=20,|C D|=24$, and the angles at vertices $B$ and $D$ are right angles. Calculate the distance between the midpoints of the diagonals of the quadrilateral $A B C D$. | \frac{15}{2} |
8,134 | As shown in the figure, let points $E$ and $F$ be on the sides $AB$ and $AC$ of $\triangle ABC$, respectively. The line segments $CE$ and $BF$ intersect at point $D$. If the areas of $\triangle CDF$, $\triangle BCD$, and $\triangle BDE$ are 3, 7, and 7, respectively, what is the area of quadrilateral $AEDF$? | 18 |
63,488 | 3. In $\triangle A B C$, $\angle B A C=90^{\circ}$, points $D$ and $E$ are on the hypotenuse $B C$, satisfying $C D=C A, B E=B A$. $F$ is a point inside $\triangle A B C$ such that $\triangle D E F$ is an isosceles right triangle with $D E$ as the hypotenuse. Find $\angle B F C$. | 135 |
64,962 | Bob sends a secret message to Alice using her RSA public key $n = 400000001.$ Eve wants to listen in on their conversation. But to do this, she needs Alice's private key, which is the factorization of $n.$ Eve knows that $n = pq,$ a product of two prime factors. Find $p$ and $q.$ | p = 20201 |
68,079 | 5. If point $P$ is the circumcenter of $\triangle A B C$, and $\overrightarrow{P A}+\overrightarrow{P B}+\lambda \overrightarrow{P C}=\mathbf{0}, C=120^{\circ}$, then the value of the real number $\lambda$ is $\qquad$. | -1 |
29,541 | $\mathrm{Az} A B C$ is a right-angled triangle, and we construct squares on its sides. The vertices of these squares - which are not the vertices of the triangle - form a hexagon. The area of this hexagon needs to be calculated! Known: the hypotenuse $(c)$ and the sum of the two legs $(d)$! | ^2+^2 |
55,764 | 6. During an early morning drive to work, Simon encountered $n$ sets of traffic lights, each set being red, amber or green as he approached it. He noticed that consecutive lights were never the same colour.
Given that he saw at least two red lights, find a simplified expression, in terms of $n$, for the number of possible sequences of colours Simon could have seen. | 3\times2^{n-1}-4n+2 |
68,193 | [ Processes and Operations ]
Between neighboring camps, it takes 1 day to travel. The expedition needs to transfer 1 can of food to the camp located 5 days away from the base camp and return. At the same time:
- each member of the expedition can carry no more than 3 cans of food;
- in 1 day, he consumes 1 can of food;
- food can only be left in camps.
What is the minimum number of cans of food that will need to be taken from the base camp for this purpose? | 243 |
25,087 | Square $A B C D$ has centre $O$. Points $P$ and $Q$ are on $A B, R$ and $S$ are on $B C, T$ and $U$ are on $C D$, and $V$ and $W$ are on $A D$, as shown, so that $\triangle A P W, \triangle B R Q, \triangle C T S$, and $\triangle D V U$ are isosceles and $\triangle P O W, \triangle R O Q, \triangle T O S$, and $\triangle V O U$ are equilateral. What is the ratio of the area of $\triangle P Q O$ to that of $\triangle B R Q$ ?
 | 1:1 |
55,754 | 6. Let $[x]$ denote the greatest integer not exceeding $x$. If $p, q, r$ are positive, find the minimum value of:
$$
\left[\frac{p+q}{r}\right]+\left[\frac{q+r}{p}\right]+\left[\frac{r+p}{q}\right]
$$ | 4 |
68,774 | 7. As shown in the figure: A table can seat 6 people, two tables put together can seat 10 people, three tables put together can seat 14 people. In this way, 10 tables arranged in two rows, with 5 tables in each row, can seat $\qquad$ people. | 44 |
53,074 | III A cylindrical container with a base radius of $1 \mathrm{~cm}$ contains four solid iron balls with a radius of $\frac{1}{2} \mathrm{~cm}$. The four balls are pairwise tangent, with the two balls at the bottom touching the bottom of the container. Now water is poured into the container so that the water level just covers all the iron balls. Then the amount of water needed is $\qquad$ $\mathrm{cm}^{3}$. | (\frac{1}{3}+\frac{\sqrt{2}}{2})\pi |
5,191 | ## Problem Statement
$$
\lim _{n \rightarrow \infty} \frac{\sqrt{n+2}-\sqrt{n^{2}+2}}{\sqrt[4]{4 n^{4}+1}-\sqrt[3]{n^{4}-1}}
$$ | 0 |
5,045 | A passenger train departs from point $A$ towards point $B$ at a speed of $60 \mathrm{~km/h}$. A high-speed train, which departs later, must catch up with the passenger train at point $B$ and travels at a speed of $120 \mathrm{~km/h}$. After completing $\frac{2}{3}$ of its journey, the passenger train continues at half its original speed. The high-speed train catches up with the passenger train 80 kilometers before point $B$. How far is point $B$ from point $A$? | 360 |
65,027 | B4. The infinite sequence of numbers
$$
0,1,2,2,1,-1,-2,-1,1,3, \ldots
$$
satisfies the following rule. For each quadruple of consecutive numbers $\ldots, a, b, c, d, \ldots$ in the sequence, it always holds that $d$ is equal to $c$ minus the smallest of the two numbers $a$ and $b$. Thus, the ninth number in the sequence is equal to $-1-(-2)=1$ and the tenth number is equal to $1-(-2)=3$. Calculate the 100th number in the sequence. | 2187 |
11,503 | 6. In triangle $ABC$, lines parallel to the sides of the triangle are drawn through an arbitrary point $O$. As a result, triangle $ABC$ is divided into three parallelograms and three triangles. The areas of the resulting triangles are $6 \text{ cm}^2$, $24 \text{ cm}^2$, and $54 \text{ cm}^2$. Find the area of triangle $ABC$.
$$
\text { (20 points) }
$$ | 216 |
53,513 | 1. What is the greatest length of an arithmetic progression of natural numbers
$a_{1}, a_{2}, \ldots, a_{n}$ with a difference of 2, in which for all $k=1,2 \ldots, n$ all numbers $a_{k}^{2}+1$ are prime | 3 |
23,529 | In the quadrilateral pyramid \( S A B C D \):
- The areas of the lateral faces \( S A B, S B C, S C D, S D A \) are 9, 9, 27, 27, respectively;
- The dihedral angles at the edges \( A B, B C, C D, D A \) are equal;
- The quadrilateral \( A B C D \) is inscribed in a circle, and its area is 36.
Find the volume of the pyramid \( S A B C D \). | 54 |
26,478 | 31. [17] Given positive integers $a_{1}, a_{2}, \ldots, a_{2023}$ such that
$$
a_{k}=\sum_{i=1}^{2023}\left|a_{k}-a_{i}\right|
$$
for all $1 \leq k \leq 2023$, find the minimum possible value of $a_{1}+a_{2}+\cdots+a_{2023}$. | 2046264 |
34,275 | 8. Given $a \geqslant b \geqslant c \geqslant d \geqslant 0$,
$$
\frac{a^{2}+b^{2}+c^{2}+d^{2}}{(a+b+c+d)^{2}}=\frac{3}{8} \text {. }
$$
Then the maximum value of $\frac{a+c}{b+d}$ is | 3 |
19,353 | Let $\lfloor x \rfloor$ denote the greatest integer less than or equal to $x$ . How many real numbers $x$ satisfy the equation $x^2 + 10,000\lfloor x \rfloor = 10,000x$ | 199 |
7,924 | Let \( 0 < a < 1 \), the function \( f(x) = \log_a \frac{x-3}{x+3} \) and \( g(x) = 1 + \log_a (x-1) \) have a common domain \( D \). If \( [m, n] \subset D \) and the range of \( f(x) \) on \( [m, n] (m < n) \) is \( [g(n), g(m)] \), determine the range of values for \( a \). | \left(0, \dfrac{2 - \sqrt{3}}{4}\right) |
63,283 | 4. In a week of 7 days, it rained for 5 days, the number of ways it could have rained for exactly 3 consecutive days is $\qquad$.
将上面的文本翻译成英文，请保留源文本的换行和格式，直接输出翻译结果。
Note: The last sentence is a repetition of the instruction and should not be included in the translation output. Here is the correct translation:
4. In a week of 7 days, it rained for 5 days, the number of ways it could have rained for exactly 3 consecutive days is $\qquad$. | 9 |
60,468 | 10. How many ways can change be made for one dollar using
a) dimes and quarters
b) nickels. dimes, and quarters
c) pennies, nickels, dimes, and quarters? | 242 |
9,500 | On a rotating round table, there are 8 white teacups and 7 black teacups. Fifteen dwarves wearing hats (8 white hats and 7 black hats) are sitting around the table. Each dwarf picks a teacup of the same color as their hat and places it in front of them. After this, the table is rotated randomly. What is the maximum number of teacups that can be guaranteed to match the color of the dwarf's hat after the table is rotated? (The dwarves are allowed to choose their seating, but they do not know how the table will be rotated.) | 7 |
20,666 | The average age of grandpa, grandma, and their five grandchildren is 26 years. The average age of the grandchildren themselves is 7 years. Grandma is one year younger than grandpa.
How old is grandma?
(L. Hozová)
Hint. How old are all the grandchildren together? | 73 |
64,666 | 6. Find the sum of the integers that belong to the set of values of the function $f(x)=\log _{2}(5 \cos 2 x+11)$ for $x \in[1,25(\operatorname{arctg}(1 / 3)) \cos (\pi+\arcsin (-0.6)) ; \operatorname{arctg} 2] \quad$ (10 points) | 7 |
19,986 | 3. $P$ is a point inside $\triangle A B C$, and line segments $A P D, B P E$, and $C P F$ are drawn such that $D$ is on $B C$, $E$ is on $A C$, and $F$ is on $A B$. Given that $A P=6, B P=9, P D=6, P E=3, C F=20$, find the area of $\triangle A B C$.
(7th AIME Problem) | 108 |
54,541 | 6. Let the side length of a regular $n$-sided polygon be $a$, and the longest and shortest diagonals be $b$ and $c$ respectively. If $a=b-c$, then $n=$ $\qquad$ | 9 |
9,394 | The circles \( S_{1} \) and \( S_{2} \) intersect at points \( A \) and \( P \). A tangent \( AB \) to the circle \( S_{1} \) is drawn through point \( A \), and a line \( CD \) is drawn through point \( P \), parallel to the line \( AB \) (points \( B \) and \( C \) lie on \( S_{2} \), point \( D \) lies on \( S_{1} \)). Prove that \( ABCD \) is a parallelogram. | ABCD \text{ is a parallelogram} |
19,661 | In triangle $ \triangle ABC $, $ AO $ is the median to the side $ BC $. Prove that $ m_{a}^{2} = \frac{1}{2}b^{2} + \frac{1}{2}c^{2} - \frac{1}{4}a^{2} $.
(Here, $ m_{a} = |AO|, |AC| = b, |AB| = c $) | m_{a}^{2} = \frac{1}{2}b^{2} + \frac{1}{2}c^{2} - \frac{1}{4}a^{2} |
55,750 | 4. Is it enough to make a closed rectangular box from all sides, enclosing no less than 1995 unit cubes, a) 962; b) 960; c) 958 square units of material? | 958 |
10,175 | Let \( P \) be any interior point of an equilateral triangle \( ABC \). Let \( D, E, \) and \( F \) be the feet of the perpendiculars dropped from \( P \) to the sides \( BC, CA, \) and \( AB \) respectively. Determine the value of the ratio
$$
\frac{PD + PE + PF}{BD + CE + AF}.
$$ | \dfrac{\sqrt{3}}{3} |
10,346 | From five number cards $0, 2, 4, 6, 8$, three different cards are chosen to form a three-digit number. How many different three-digit numbers can be formed (6 is considered as 9 when reversed)? | 78 |
25,411 | 8. Given that the parabola $P$ has the center of the ellipse $E$ as its focus, $P$ passes through the two foci of $E$, and $P$ intersects $E$ at exactly three points, then the eccentricity of the ellipse is $\qquad$
Translate the above text into English, please retain the original text's line breaks and format, and output the translation result directly. | \frac{2\sqrt{5}}{5} |
11,114 | Call a positive integer $n$ extra-distinct if the remainders when $n$ is divided by $2, 3, 4, 5,$ and $6$ are distinct. Find the number of extra-distinct positive integers less than $1000$ .
Please give the answer directly without any intermediate steps. | 49 |
8,196 | 302. Differentiate the function $y=2^{\sin x}$. | 2^{\sin x} \ln 2 \cos x |
28,962 | 88. The Chase (I). Ship $P$ has spotted ship $Q$, which is sailing in a direction perpendicular to $P Q$, maintaining its course. Ship $P$ is chasing $Q$, always heading directly towards $Q$; the speed of both ships is the same at any moment (but can vary over time). Without calculations, it is clear that $P$ is sailing along a curved path; if the chase lasts long enough, the trajectory of the pursuing ship and the trajectory of the fleeing ship will eventually become almost identical. What will then be the distance $P Q$, if initially it was 10 nautical miles? | 5 |
4,168 | Determine the largest square number that is not divisible by 100 and, when its last two digits are removed, is also a square number. | 1681 |
54,558 | 4. Consider the sequence $\left(x_{n}\right)_{n \geqslant 1}$ of positive real numbers with $\lim _{n \rightarrow \infty} x_{n}=0$. Calculate $\lim _{n \rightarrow \infty} \frac{a_{n}}{n}$, where
$$
a_{n}=\sqrt{2015^{2} x_{1}^{2}+2015 x_{1} x_{2}+x_{2}^{2}}+\sqrt{2015^{2} x_{2}^{2}+2015 x_{2} x_{3}+x_{3}^{2}}+\ldots+\sqrt{2015^{2} x_{n}^{2}+2015 x_{n} x_{1}+x_{1}^{2}}
$$ | 0 |
15,224 | Let \( x \neq y \), and suppose the two sequences \( x, a_{1}, a_{2}, a_{3}, y \) and \( b_{1}, x, b_{2}, b_{3}, y, b_{1} \) are both arithmetic sequences. Determine the value of \( \frac{b_{4}-b_{3}}{a_{2}-a_{1}} \). | \dfrac{8}{3} |
13,764 | Example 1.25. Form the equation of the plane passing through the point $M(2 ; 3 ; 5)$ and perpendicular to the vector $\bar{N}=4 \bar{i}+3 \bar{j}+2 \bar{k}$ | 4x + 3y + 2z - 27 = 0 |
62,545 | In the diagram, the ratio of the number of shaded triangles to the number of unshaded triangles is
(A) $5: 2$
(B) $5: 3$
(C) $8: 5$
(D) $5: 8$
(E) $2: 5$
 | 5:3 |
20,556 | Given the ellipse \(\frac{x^{2}}{6}+\frac{y^{2}}{3}=1\) and a point \(P\left(1, \frac{1}{2}\right)\) inside the ellipse, draw a line through \(P\) that does not pass through the origin and intersects the ellipse at points \(A\) and \(B\). Find the maximum area of triangle \(OAB\), where \(O\) is the origin. | \dfrac{3\sqrt{6}}{4} |
67,381 | In a 12-hour interval (from $0^{\mathrm{h}}$ to $12^{\mathrm{h}}$), how many minutes are there when the value of the hours is greater than the value of the minutes? | 66 |
23,669 | Chester is traveling from Hualien to Lugang, Changhua, to participate in the Hua Luogeng Golden Cup Mathematics Competition. Before setting off, his father checked the car's odometer, which read a palindromic number of 69,696 kilometers (a palindromic number remains the same when read forward or backward). After driving for 5 hours, they arrived at their destination, and the odometer displayed another palindromic number. During the journey, the father's driving speed never exceeded 85 kilometers per hour. What is the maximum average speed (in kilometers per hour) at which Chester's father could have driven? | 82.2 |
63,067 | 7. As shown in Figure 3, in $\triangle A B C$, it is known that $\angle B=$ $40^{\circ}, \angle B A D=30^{\circ}$. If $A B$ $=C D$, then the size of $\angle A C D$ is $\qquad$ (degrees). | 40^{\circ} |
63,323 | 33rd Swedish 1993 Problem 6 For reals a, b define the function f(x) = 1/(ax+b). For which a, b are there distinct reals x 1 , x 2 , x 3 such that f(x 1 ) = x 2 , f(x 2 ) = x 3 , f(x 3 ) = x 1 . | -b^2 |
59,914 | ## Task 13/63
A mathematician had his bicycle stolen. When asked for his bicycle number, he replied: "You can calculate the number from the following information:
a) If you add the square of the first digit to the square of the second digit, you get the square of the third digit.
b) If you subtract the second digit from the first digit, you get the fifth digit increased by 1.
c) The second digit is equal to the fourth, the third digit is equal to the sixth and to the seventh."
What was the mathematician's bicycle number? | 4353055 |
57,567 | 275. Maximum Number. Let a set of distinct complex numbers $z_{i}, i=1,2, \ldots, n$, be given, satisfying the inequality
$$
\min _{i \neq j}\left|z_{i}-z_{j}\right| \geqslant \max _{i}\left|z_{i}\right|
$$[^16]
Find the maximum possible $n$ and for this $n$ all sets satisfying the condition of the problem. | 7 |
65,266 | Folklore
In Chicago, there are 36 gangsters, some of whom are at odds with each other. Each gangster is a member of several gangs, and there are no two gangs with the same membership. It turned out that gangsters who are in the same gang do not feud, but if a gangster is not a member of a certain gang, then he feuds with at least one of its members. What is the maximum number of gangs that could exist in Chicago? | 3^{12} |
26,019 | Task B-2.4. (20 points) Points $A, B$ and $C$ lie on the same line and $B$ is between $A$ and $C$. On the same side of the line $A C$, three semicircles with diameters $|A B|=2 R,|B C|=2 r$ and $|A C|$ are constructed. Determine the radius of the circle that is tangent to all three semicircles. | \frac{Rr(R+r)}{R^2+Rr+r^2} |
18,797 | Factorize $n^{5}-5 n^{3}+4 n$. What can we conclude in terms of divisibility? | 120 |
15,899 | In triangle \( \triangle ABC \), the angles \( \angle A, \angle B, \angle C \) opposite to the sides \( BC = a, CA = b, AB = c \) respectively. Prove that \( 2b \cos \frac{C}{2} + 2c \cos \frac{B}{2} > a + b + c \). | 2b \cos \frac{C}{2} + 2c \cos \frac{B}{2} > a + b + c |
63,931 | 95. Four people, A, B, C, and D, made predictions about the rankings before the competition. The table below shows their predictions:
\begin{tabular}{|c|c|c|c|c|}
\hline & A's prediction & B's prediction & C's prediction & D's prediction \\
\hline 1st place & B & A & C & B \\
\hline 2nd place & C & B & D & C \\
\hline 3rd place & A & D & A & D \\
\hline 4th place & D & C & B & A \\
\hline
\end{tabular}
After the competition, when comparing the predictions with the actual results, it was found that B and D correctly predicted the rankings of two people, while A and C only correctly predicted the ranking of one person. If A, B, C, and D placed in positions $A$, $B$, $C$, and $D$ respectively, then the four-digit number $\overline{A B C D}=$ . $\qquad$ | 1423 |
62,140 | 9. (5 points) Xiaohong brought 5 RMB notes of 50 yuan, 20 yuan, and 10 yuan denominations, 6 notes of 20 yuan, and 7 notes of 10 yuan. She bought a commodity worth 230 yuan. Therefore, there are $\qquad$ ways to pay. | 11 |
53,482 | 4. In the Cartesian coordinate plane, the number of integer points that satisfy the system of inequalities $\left\{\begin{array}{l}y \leqslant 3 x \\ y \geqslant \frac{x}{3} \\ x+y \leqslant 100\end{array}\right.$ is | 2551 |
23,134 | 1. What relation that does not depend on $m$ exists between the solutions of the equation
$$
\left(x^{2}-6 x+5\right)+m\left(x^{2}-5 x+6\right)=0 ?
$$ | x_1 + x_2 + x_1 x_2 = 11 |
20,981 | In triangle \(ABC\), it is known that \(AB = BC\) and \(AC = 4 \sqrt{3}\). The radius of the inscribed circle is 3. Line \(AE\) intersects the altitude \(BD\) at point \(E\), and the inscribed circle at points \(M\) and \(N\) (with \(M\) being between \(A\) and \(E\)). Given that \(ED = 2\), find \(EN\). | \dfrac{1 + \sqrt{33}}{2} |
10,723 | Problem 7.4. In triangle $ABC$, the median $CM$ and the bisector $BL$ were drawn. Then, all segments and points were erased from the drawing, except for points $A(2 ; 8)$, $M(4 ; 11)$, and $L(6 ; 6)$. What were the coordinates of point $C$?
 | (14; 2) |
52,332 | 8. Let $A(2,0)$ be a fixed point on the plane, and $P\left(\sin \left(2 t-60^{\circ}\right), \cos \left(2 t-60^{\circ}\right)\right)$ be a moving point. Then, as $t$ varies from $15^{\circ}$ to $45^{\circ}$, the area of the figure swept by the line segment $A P$ is $\qquad$ . | \frac{\pi}{6} |
61,174 | Determine the third-degree equation admitting the roots: $2 \sin \frac{2 \pi}{9}, 2 \sin \frac{8 \pi}{9}, 2 \sin \frac{14 \pi}{9}$. | X^{3}-3X+\sqrt{3}=0 |
54,104 | 7. [5] Compute
$$
\sum_{n=1}^{\infty} \frac{1}{n \cdot(n+1) \cdot(n+1)!}
$$ | 3-e |
42 | Let \( x \) and \( y \) satisfy the conditions:
\[
\left\{
\begin{array}{l}
2x + y \geq 4 \\
x - y \geq 1 \\
x - 2y \leq 2
\end{array}
\right.
\]
If \( z = ax + y \) achieves its minimum value only at point \( A(2,0) \), determine the range of the real number \( a \). | (-\frac{1}{2}, 2) |
20,165 | The domain of the function \( f(x) \) is \(\mathbf{R}\) and it satisfies the following conditions: for \( x \in [0, 1) \), \( f(x) = 2^x - x \), and for any real number \( x \), \( f(x) + f(x+1) = 1 \). Let \( a = \log_2 3 \). Find the value of the expression \( f(a) + f(2a) + f(3a) \). | \dfrac{17}{16} |
20,859 | Four identical pieces, in the shape of a right triangle, are arranged in two different ways as shown in the given figures. The squares $ABCD$ and $EFGH$ have sides measuring $3 \text{ cm}$ and $9 \text{ cm}$ respectively. Determine the side length of the square $IJKL$. | 3\sqrt{5} |
69,037 | 5. Let $A B C D E F$ be a regular hexagon. A frog starts at vertex $A$, and each time it can randomly jump to one of the two adjacent vertices. If it reaches point $D$ within 5 jumps, it stops jumping; if it does not reach point $D$ within 5 jumps, it stops after 5 jumps. How many different jumping sequences can the frog have from the start to the stop? $\qquad$ | 26 |
67,329 | 2. In the Cartesian coordinate system $x O y$, a circle passes through $(0,2)$ and $(3,1)$, and is tangent to the $x$-axis. Then the radius of this circle is $\qquad$ . | 15 \pm 6 \sqrt{5} |
68,083 | 5. For a positive integer $n$, denote by $\varphi(n)$ the number of positive integers $k \leq n$ relatively prime to $n$. How many positive integers $n$ less than or equal to 100 are divisible by $\varphi(n)$ ? | 16 |
29,913 | A $20 \times 20 \times 20$ cube is divided into 8000 unit cubes. A number is written in each unit cube. In each row and in each column of 20 small cubes, parallel to one of the cube's edges, the sum of the numbers is 1. In one of the small cubes, the number written is 10. Through this small cube pass three layers $1 \times 20 \times 20$ parallel to the faces of the cube. Find the sum of all the numbers outside these three layers.
## - Solution of the exercises -
untranslated portion:
将上面的文本翻译成英文，请保留源文本的换行和格式，直接输出翻译结果。
---
A $20 \times 20 \times 20$ cube is divided into 8000 unit cubes. A number is written in each unit cube. In each row and in each column of 20 small cubes, parallel to one of the cube's edges, the sum of the numbers is 1. In one of the small cubes, the number written is 10. Through this small cube pass three layers $1 \times 20 \times 20$ parallel to the faces of the cube. Find the sum of all the numbers outside these three layers.
## - Solution of the exercises - | 333 |
4,612 | Let \( a \) and \( b \) be positive integers. The quotient of \( a^{2} + b^{2} \) divided by \( a + b \) is \( q \), and the remainder is \( r \), such that \( q^{2} + r = 2010 \). Find the value of \( ab \). | 1643 |
62,901 | 6.4. A natural number is called a palindrome if it remains unchanged when its digits are written in reverse order (for example, the numbers 4, 55, 626 are palindromes, while 20, 201, 2016 are not). Represent the number 2016 as a product of two palindromes (find all options and explain why there are no others). | 8\cdot252 |
62,476 | 9. Four black $1 \times 1 \times 1$ cubes and four white $1 \times 1 \times 1$ cubes can form $\qquad$ different $2 \times 2 \times 2$ cubes (cubes that are the same after rotation are considered the same type). | 7 |
53,313 | 4B. Given a square $A B C D$. The points $X$ and $Y$ lie on the sides $B C$ and $C D$, respectively. The lengths of the segments $X Y, A X$ and $A Y$ are 3, 4, and 5, respectively. Calculate the length of the side of the square. | \frac{16\sqrt{17}}{17} |
57,646 | 3. (17 points) The cross-section of a regular triangular pyramid passes through the midline of the base and is perpendicular to the base. Find the area of the cross-section if the side of the base is 8 and the height of the pyramid is 12. | 18 |
32,255 | 12.11. (CSSR, 80). The set $M$ is obtained from the plane by removing three distinct points $A, B$, and $C$. Find the smallest number of convex sets whose union is the set $M$. | 4 |
12,130 | Let \( x = \frac{a}{a+2b} + \frac{b}{b+2c} + \frac{c}{c+2a} \), where \( a, b, c > 0 \). Prove that \( x \geq 1 \). | x \geq 1 |
32,897 | 13. (2005 National High School Mathematics League Additional Question) Let positive numbers $a$, $b$, $c$, $x$, $y$, $z$ satisfy $c y + b z = a$; $a z + c x = b$; $b x + a y = c$. Find the minimum value of the function $f(x, y, z) = \frac{x^2}{1 + x} + \frac{y^2}{1 + y} + \frac{z^2}{1 + z}$. | \frac{1}{2} |
21,023 | 4. If the inequality
$$
\left|a x^{2}+b x+a\right| \leqslant x
$$
holds for all $x \in[1,2]$, then the maximum value of $3 a+b$ is
$\qquad$ . | 3 |
22,054 | Rectangle \(ABCD\) has area 2016. Point \(Z\) is inside the rectangle and point \(H\) is on \(AB\) so that \(ZH\) is perpendicular to \(AB\). If \(ZH : CB = 4 : 7\), what is the area of pentagon \(ADCZB\)? | 1440 |
3,387 | On the island of Unfortune, there live knights who always tell the truth and liars who always lie. One day, 2022 natives gathered at a round table, and each of them made the statement:
"Next to me sit a knight and a liar!"
It is known that three knights made a mistake (i.e., unintentionally lied). What is the maximum number of knights that could have been at the table? | 1349 |
5,106 | A geometric sequence with common ratio $r \neq 0$ is a sequence of numbers $s_{0}, s_{1}, s_{2}$, $\ldots$ that satisfies for any index $k, s_{k+1}=s_{k} \times r$. Determine an expression for $s_{n}$ in terms of $s_{0}, r$, and $n$. | s_0 \cdot r^n |
20,128 | 13.438 By mixing $2 \mathrm{~cm}^{3}$ of three substances, 16 g of the mixture was obtained. It is known that $4 \mathrm{r}$ of the second substance occupies a volume that is $0.5 \mathrm{~cm}^{3}$ larger than $4 \mathrm{r}$ of the third substance. Find the density of the third substance, given that the mass of the second substance in the mixture is twice the mass of the first. | 4 |
18,764 | Find all values of the parameter \( b \) for which there exists a value of \( a \) such that the system
$$
\left\{\begin{array}{l}
x^{2}+y^{2}+2 b(b-x+y)=4 \\
y=\frac{9}{(x+a)^{2}+1}
\end{array}\right.
$$
has at least one solution \((x, y)\). | [-11, 2) |
26,170 | 9. (20 points) Inside an acute triangle $A B C$, a point $M$ is marked. The lines $A M, B M$, $C M$ intersect the sides of the triangle at points $A_{1}, B_{1}$ and $C_{1}$ respectively. It is known that $M A_{1}=M B_{1}=M C_{1}=3$ and $A M+B M+C M=43$. Find $A M \cdot B M \cdot C M$. | 441 |
53,033 | Van has equal-sized balls, of which 2 are red, 2 are white, and 2 are blue. Among the monochromatic balls, one is made of gold, and the other is made of silver. What is the minimum number of weighings needed to select the three gold balls using a two-pan balance? (Balls made of the same material have the same weight.) | 2 |
54,658 | What is the remainder in the following division:
$$
\left(x^{1001}-1\right):\left(x^{4}+x^{3}+2 x^{2}+x+1\right)
$$
and also when the divisor is the following polynomial:
$$
x^{8}+x^{6}+2 x^{4}+x^{2}+1
$$
(the remainder being the polynomial that results after determining the last non-negative power term of the quotient). | -2x^{7}-x^{5}-2x^{3}-1 |
25,003 | Real numbers \(a, b, c\) and a positive number \(\lambda\) such that \(f(x)=x^{3}+a x^{2}+b x+c\) has three real roots \(x_{1}, x_{2}, x_{3}\), satisfying:
(1) \(x_{2}-x_{1}=\lambda\);
(2) \(x_{3}>\frac{1}{2}\left(x_{1}+x_{2}\right)\).
Find the maximum value of \(\frac{2 a^{3}+27 c-9 a b}{\lambda^{3}}\). | \dfrac{3\sqrt{3}}{2} |
12,793 | Let \(a, b, n, c \in \mathbb{N}^{*}\) such that \((a + bc)(b + ac) = 19^n\). Show that \(n\) is even. | n \text{ is even} |
57,602 | 14.A. Choose $n$ numbers from $1,2, \cdots, 9$. Among them, there must be some numbers (at least one, or possibly all) whose sum is divisible by 10. Find the minimum value of $n$. | 5 |
7,860 | Equilateral triangles $BCM$ and $CDN$ are constructed on the sides $BC$ and $CD$ of parallelogram $ABCD$ outside (inside) of it. Prove that triangle $AMN$ is equilateral. | \triangle AMN \text{ is equilateral} |
22,693 | Extend 1 Let non-negative real numbers $a, b, c, x, y, z$ satisfy $a+b+c=x+y+z=1$.
Find the minimum value of $\left(a-x^{2}\right)\left(b-y^{2}\right)\left(c-z^{2}\right)$. | -\dfrac{1}{4} |
4,136 | In the sequence \(\left\{x_{n}\right\}\), \(x_{1} > 3\), and \(x_{n}=\frac{3 x_{n-1}^{2}-x_{n-1}}{4\left(x_{n-1}-1\right)} \ \text{for} \ n=2,3, \cdots\). Prove that \(3 < x_{n+1} < x_{n} \ \text{for} \ n=1,2, \cdots\). | 3 < x_{n+1} < x_n \text{ for all } n \geq 1 |
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