problem stringlengths 10 7.47k | answer stringlengths 1 1.22k |
|---|---|
On a bustling street, a middle-aged man is shouting "giving away money" while holding a small black cloth bag in his hand. Inside the bag, there are 3 yellow and 3 white ping-pong balls (which are identical in volume and texture). Next to him, there's a small blackboard stating:
Method of drawing balls: Randomly draw ... | 1920 |
Let $A=\{1,2,3,4\}$, $B=\{2,4,5,6\}$. Then $A\cup B$ is
A: $\{2,4\}$
B: $\{1,3,5,6\}$
C: $\{1,2,3,4,5,6\}$
D: $\left[1,6\right]$ | C |
A, B, and C, three students, participated in an exam. Both A and B scored 70 points, and C scored 65 points. Given the proposition p: If the passing score is below 70 points, then A, B, and C all fail. Among the following four propositions, the one that is the contrapositive of p is (γγ)
A: If the passing score is not ... | C |
Determine the equation of a line that satisfies the following conditions:
(1) It passes through point P(-1, 3) and is parallel to the line $x-2y+3=0$.
(2) For points A(1, 2) and B(3, 1), find the equation of the perpendicular bisector of line segment AB. | 4x - 2y - 5 = 0 |
Use the Horner's method to calculate the value of the polynomial $f(x) = 5x^5 + 4x^4 + 3x^3 + 2x^2 + x + 0.3$ when $x = 5$. The total number of additions and multiplications performed is ( )
A: 11
B: 10
C: 12
D: 13 | B |
Given vectors $|\vec{a}|= \sqrt{3}$, $|\vec{b}|=2$, and $\vec{a}\cdot(\vec{a}-\vec{b})=0$, find the magnitude of $\vec{a}-\vec{b}$. | 1 |
For non-zero vectors $\overrightarrow{a}$ and $\overrightarrow{b}$, "$\overrightarrow{a} + \overrightarrow{b} = \overrightarrow{0}$" is a (γγ) condition for "$\overrightarrow{a} \parallel \overrightarrow{b}$".
A: Sufficient but not necessary condition
B: Necessary but not sufficient condition
C: Sufficient and necessar... | \text{A} |
Problem 10.5. On the side $AB$ of the rectangle $ABCD$, a circle $\omega$ is constructed with $AB$ as its diameter. Let $P$ be the second intersection point of the segment $AC$ and the circle $\omega$. The tangent to $\omega$ at point $P$ intersects the segment $BC$ at point $K$ and passes through point $D$. Find $AD$,... | 24 |
Given $44_{(k)} = 36$, convert 67 into base $k$ to get (γγ)
A: $55_{(k)}$
B: $67_{(k)}$
C: $103_{(k)}$
D: $124_{(k)}$ | C |
6. (6 points) It is known that the work efficiency ratio of water pump A to water pump B is 3:4. If the two pumps work together to empty a water pool, it takes 15 hours to complete the task. Now, if pump B works for 9 hours and then is turned off, and pump A is then turned on, it will take $\qquad$ more hours to empty ... | 23 |
\begin{aligned} & \text { [The inscribed angle is half the central angle] } \\ & {[\quad \text { Symmetry helps solve the problem. }]}\end{aligned}
Point $X$ is located on the diameter $A B$ of a circle with radius $R$. Points $K$ and $N$ lie on the circle in the same half-plane relative to $A B$,
and $\angle K X A=\... | R |
3. In a square $ABCD$ with side length 2, $E$ is the midpoint of $AB$. Now, fold the figure along segments $EC$ and $ED$ so that segments $EA$ and $EB$ coincide (point $B$ overlaps with $A$), forming a tetrahedron $CDEA$. Then the volume of this tetrahedron is $\qquad$ | \frac{\sqrt{3}}{3} |
Example 2: Let $1 \leqslant k<n$, consider all positive integer sequences of length $n$, and find the total number of terms $T(n, k)$ in these sequences that are equal to $k$. (29th IMO Shortlist) | (n-k+3) \cdot 2^{n-k-2} |
Find all integers $k$ for which, there is a function $f: N \to Z$ that satisfies:
(i) $f(1995) = 1996$
(ii) $f(xy) = f(x) + f(y) + kf(m_{xy})$ for all natural numbers $x, y$,where$ m_{xy}$ denotes the greatest common divisor of the numbers $x, y$.
Clarification: $N = \{1,2,3,...\}$ and $Z = \{...-2,-1,0,1,2,...\}$ . | k = -1 |
18. Solve the equation $\sqrt{x^{2}-4}+2 \sqrt{x^{2}-1}=\sqrt{7} x$. | \frac{2}{3}\sqrt{21} |
Given the function $f(x)=\cos (2x- \frac {\pi}{3})+\sin ^{2}x-\cos ^{2}x+ \sqrt {2}$.
$(1)$ Find the smallest positive period of the function $f(x)$ and the intervals of monotonic increase;
$(2)$ If there exists $t\in\left[ \frac {\pi}{12}, \frac {\pi}{3}\right]$ satisfying $\left[f(t)\right]^{2}-2 \sqrt {2}f(t)-m > 0$... | (-\infty,-1) |
8. (15 points) A primary school is conducting a height statistics survey. There are 99 students whose height does not exceed $130 \mathrm{~cm}$, with an average height of $122 \mathrm{~cm}$. There are 72 students whose height is not less than $160 \mathrm{~cm}$, with an average height of $163 \mathrm{~cm}$. The average... | 621 |
31. A middle school has decided to organize a study tour for some classes. There are 16 teachers and 284 students. There are two types of large buses available: Bus A can carry 30 people and costs 300 yuan to rent; Bus B can carry 42 people and costs 400 yuan to rent. The school plans for the total rental cost of the s... | 3,2900 |
Given the set $M=\{-1, 1\}$, the number of sets $N$ that satisfy $N \subseteq M$ is ( )
A: 1
B: 2
C: 3
D: 4 | D |
6. Find all real-coefficient polynomials $P$ such that for all non-zero real numbers $x, y, z$ satisfying $2xyz = x + y + z$, we have
$$
\begin{array}{l}
\frac{P(x)}{yz} + \frac{P(y)}{zx} + \frac{P(z)}{xy} \\
= P(x-y) + P(y-z) + P(z-x).
\end{array}
$$ | P(x)=(x^2+3) |
The Goldbach Conjecture states that "every even integer greater than 2 can be expressed as the sum of two prime numbers," such as 8=3+5. What is the probability that the sum of two different numbers randomly selected from the prime numbers not exceeding 13 is even, expressed as a fraction? | \frac{2}{3} |
13. If the computer executes the following program, what value will be printed by statement 5? ( ).
1. Initial values $x=3, S=0$.
2. $x=x+2$.
3. $S=S+x$.
4. If $S \geqslant 10000$, then proceed to 5; otherwise, continue execution.
5. Print x.
6. Stop.
(A)19. (b)21. (C)23. (D) 199 .
(E) 201 . | E |
Find the equation of the line passing through points A(2, -1) and B(5, 1) in two-point form, and then convert it into point-slope form, slope-intercept form, and intercept form. | \frac{x}{\frac{7}{2}} + \frac{y}{-\frac{7}{3}} = 1 |
In the positive arithmetic sequence $\{a_{n}\}$, if $a_{3}=2$, then the range of the common difference $d$ is ______. | [0,1) |
Begunni A.
What can be the product of several different prime numbers if it is divisible by each of them, decreased by 1?
Find all possible values of this product.
# | 6,42,1806 |
1.3. Replace the ellipsis with such a natural number $p$ so that the answer to the following question is unique: how many lines are drawn on the plane, if it is known that they intersect at ... different points? | 2 |
Given the parabola $C$: $y^{2}=4x$ with the intersection point $F$, the line $y=x-1$ intersects $C$ at points $A$ and $B$, and intersects the asymptotes of the hyperbola $E$: $\dfrac {x^{2}}{a^{2}}- \dfrac {y^{2}}{b^{2}}=2$ ($a > 0,b > 0$) at points $M$ and $N$. If the midpoint of segment $AB$ is the same as that of $M... | C |
9.1. A finite set of distinct real numbers $X$ is called good if each number in $X$ can be represented as the sum of two other distinct numbers in $X$. What is the minimum number of elements that a good set $X$ can contain?
# | 6 |
All the roots of the polynomial $z^6-10z^5+Az^4+Bz^3+Cz^2+Dz+16$ are positive integers, possibly repeated. What is the value of $B$?
$\textbf{(A) }{-}88 \qquad \textbf{(B) }{-}80 \qquad \textbf{(C) }{-}64 \qquad \textbf{(D) }{-}41\qquad \textbf{(E) }{-}40$ | -88 |
22nd CanMO 1990 Problem 1 A competition is played amongst n > 1 players over d days. Each day one player gets a score of 1, another a score of 2, and so on up to n. At the end of the competition each player has a total score of 26. Find all possible values for (n, d). | (n,)=(3,13),(12,4),(25,2) |
Sampling methods include ( )
A: Random sampling, systematic sampling, and stratified sampling
B: Random number method, drawing lots, and stratified sampling method
C: Simple random sampling, stratified sampling, and systematic sampling
D: Systematic sampling, stratified sampling, and random number method | C |
Given that the points (1,2) and (1,1) lie on opposite sides of the line $3x-y+m=0$, the range of values for the real number $m$ is __________. | -2 < m < -1 |
Given that $i$ is the imaginary unit, if $(-1-2i)z=1-i$ then the point represented by $\overline{z}$ in the complex plane lies in ( )
A: First quadrant
B: Second quadrant
C: Third quadrant
D: Fourth quadrant | D |
How many numbers between 1 and 601 are multiples of 3 or multiples of 4?
(a) 100
(b) 150
(c) 250
(d) 300
(e) 430 | 300 |
Given the inequality $(x+y)\left( \frac {1}{x}+ \frac {a}{y}\right)\geqslant 9$ holds for any positive real numbers $x$, $y$, then the minimum value of the positive real number $a$ is (γ).
A: $2$
B: $4$
C: $6$
D: $8$ | B |
3. A number from 0 to 1 inclusive was written on the board. Each of the twenty-five students in the class approached the board, erased the number $x$ written there at the moment, and wrote $\frac{1}{2} \cos \pi x$ instead.
For which initial value of the number will the final result be the maximum?
If there are multip... | 0;1 |
Give the power function $y=(m^{2}-m-1)x^{m^{2}-2m- \frac {1}{3}}$, if it is a decreasing function when $x \in (0, +\infty)$, then the analytic expression of this power function is \_\_\_\_\_\_ . | y = x^{- \frac {1}{3}} |
Let $X \sim B(n, P)$, $E(X)=15$, and $D(X)=11.25$. Find the value of $n$.
A: $45$
B: $50$
C: $55$
D: $60$ | \text{D} |
$$
\begin{array}{l}
a^{2}+b^{2}+c^{2}+d^{2}+(1-a)^{2}+(a-b)^{2}+ \\
(b-c)^{2}+(c-d)^{2}+(d-1)^{2}
\end{array}
$$
When the value of the above algebraic expression is minimized, what is the value of $a+b+c+d$? Prove your conclusion. | \frac{6}{5} |
Given $a+b=4$ and $ab=1$, find:
(1) The value of $a^2+b^2$;
(2) The value of $(a-b)^2$. | 12 |
Let $p$ be a polynomial with integer coefficients such that $p(15)=6$, $p(22)=1196$, and $p(35)=26$. Find an integer $n$ such that $p(n)=n+82$. | 28 |
2. Let complex numbers $z, w$ satisfy $|z|=3,(z+\bar{w})(\bar{z}-w)=7+4 \mathrm{i}$, where $\mathrm{i}$ is the imaginary unit, $\bar{z}, \bar{w}$ represent the conjugates of $z, w$ respectively, then the modulus of $(z+2 \bar{w})(\bar{z}-2 w)$ is $\qquad$ | \sqrt{65} |
Given the function $f(x) = 3x + 1$, the value of $\lim_{\Delta x \to 0} \frac{f(1-\Delta x) - f(1)}{\Delta x}$ is (γγ)
A: $-\frac{1}{3}$
B: $\frac{1}{3}$
C: $\frac{2}{3}$
D: 0 | \text{A} |
For any positive integer $n$, let $D_n$ denote the set of all positive divisors of $n$, and let $f_i(n)$ denote the size of the set
$$F_i(n) = \{a \in D_n | a \equiv i \pmod{4} \}$$
where $i = 1, 2$.
Determine the smallest positive integer $m$ such that $2f_1(m) - f_2(m) = 2017$. | 2 \cdot 5^{2016} |
The equation of the directrix of the parabola $x^{2}=ay$ is $y=1$. Then, the value of $a$ is (γ).
A: $- \dfrac {1}{2}$
B: $-2$
C: $- \dfrac {1}{4}$
D: $-4$ | D |
$(1)\sqrt{\frac{1}{3}}+\sqrt{27}Γ\sqrt{9}$οΌ$(2)\sqrt{32}-3\sqrt{\frac{1}{2}}+\sqrt{\frac{1}{8}}$. | \frac{11\sqrt{2}}{4} |
If a certain commodity is sold at 80% of its marked price, a profit of 20% can be made. If it is sold at the original marked price, the profit made will be (γγ)
A: 25%
B: 40%
C: 50%
D: 66.7% | C |
Example 1 Calculate $\left(\frac{137}{227}\right)$. | -1 |
The condition "$m<$" means "the quadratic equation $x^2 + x + m = 0$ has real solutions". This is a:
A: Sufficient but not necessary condition
B: Sufficient and necessary condition
C: Necessary but not sufficient condition
D: Neither sufficient nor necessary condition | \text{A} |
14. Asymmetric coin. (From 9th grade, 2 points) Billy Bones has two coins - a gold one and a silver one. One of them is symmetric, and the other is not. It is unknown which coin is asymmetric, but it is known that the asymmetric coin lands heads with a probability of \( p = 0.6 \).
Billy Bones tossed the gold coin, an... | \frac{5}{9} |
The domain of the function $f(x) = \arcsin(\log_{m}(nx))$ is a closed interval of length $\frac{1}{2013}$ , where $m$ and $n$ are positive integers and $m>1$. Find the remainder when the smallest possible sum $m+n$ is divided by 1000. | 371 |
5. Let $x, y, z$ be non-negative real numbers, and satisfy $\left\{\begin{array}{l}x+y-z=4 \\ x-y+3 z=6\end{array}\right.$, then the minimum value of $4 x+3 y+2 z$ is
Translate the above text into English, please retain the original text's line breaks and format, and output the translation result directly. | 19 |
Given that the graph of the power function $y=f(x)$ passes through the point $(\sqrt{\mathbf{2}}, 2\sqrt{\mathbf{2}})$, find $f(2)=\_\_\_\_\_\_.$ | 8 |
Shift the graph of the function $y=\sin 2x$ to the right by $\frac{\pi}{12}$ units, and then stretch the x-coordinates of each point on the graph by a factor of 2 (keeping the y-coordinates unchanged). The equation of the resulting graph is \_\_\_\_\_\_. | y=\sin\left(x- \frac{\pi}{6}\right) |
222. If $\ a^{p} x^{q}$ is one of the terms in the expansion of $(a+x)^{n}$, what condition do $p$ and $q$ satisfy? What is the coefficient $M$? | \frac{n(n-1)(n-2)\ldots(n-p+1)}{1\cdot2\cdot3\ldotsp} |
14. Taran thought of a whole number and then multiplied it by either 5 or 6 . Krishna added 5 or 6 to Taran's answer. Finally Eshan subtracted either 5 or 6 from Krishna's answer. The final result was 73 . What number did Taran choose?
A 10
B 11
C 12
D 13
E 14 | 12 |
In the Cartesian coordinate system $xOy$, the equation of curve $C$ is: $(x-1)^{2}+y^{2}=1$. Establish a polar coordinate system with $O$ as the pole and the non-negative half-axis of $x$ as the polar axis.
$(1)$ Find the polar equation of curve $C$;
$(2)$ The polar equation of line $l_{1}$ is $2\rho\sin(\theta+\fr... | 4 |
$$
\begin{array}{l}
16\left(\frac{1}{5}-\frac{1}{3} \times \frac{1}{5^{3}}+\frac{1}{5} \times \frac{1}{5^{5}}-\frac{1}{7} \times \frac{1}{5^{7}}+ \\
\frac{1}{9} \times \frac{1}{5^{9}}-\frac{1}{11} \times \frac{1}{5^{11}}\right)-4\left(\frac{1}{239}-\frac{1}{3} \times \frac{1}{239^{3}}\right) \\
=
\end{array}
$$ | 3.14159265 |
The password for a bank savings card consists of 6 digits. When someone tries to withdraw money from an ATM, they forget the last digit of the password but remember that the last digit is an even number. The probability that they will enter the correct password in no more than two attempts is (γγ)
A: $$\frac {1}{5}$$
... | \text{B} |
1. As shown in Figure 1, label the six vertices of a regular hexagon with the numbers $0,1,2,3,4,5$ in sequence. An ant starts from vertex 0 and crawls counterclockwise, moving 1 edge on the first move, 2 edges on the second move, $\cdots \cdots$ and $2^{n-1}$ edges on the $n$-th move. After 2005 moves, the value at th... | 1 |
2. Xiao Pang and his father played a dart game together, each of them threw 5 times, father scored 48 points, Xiao Pang's score was 8 points less than half of his father's, Xiao Pang scored $\qquad$ points. | 16 |
Let the universal set $U=\mathbb{R}$, $A=\{x|0<x<2\}$, $B=\{x||x|\leq1\}$. Then, the set $(C_UA) \cap B$ is ( )
A: $\{x|-1\leq x<0\}$
B: $\{x|-1\leq x\leq1\}$
C: $\{x|1\leq x\leq2\}$
D: $\{x|-1\leq x\leq0\}$ | \text{D} |
## problem statement
Let $k$ be the coefficient of similarity transformation with the center at the origin. Is it true that point $A$ belongs to the image of plane $a$?
$A(-2 ; 4 ; 1)$
$a: 3 x+y+2 z+2=0$
$k=3$ | 6\neq0 |
6. (10 points) The area of a sector of a circle is 100. For what value of the radius of the circle will the perimeter of this sector be minimal? If the answer is not an integer, round to the tenths.
Answer: 10 | 10 |
In the sequence $\{a\_n\}$, $a\_1=1$, and the point $(a\_n,a\_{n+1})$ is on the line $y=2x$. What is the value of $a\_4$?
A: $7$
B: $8$
C: $9$
D: $16$ | B |
Consider the following propositions:
1. If we pair each real number $a$ with $ai$, then there is a one-to-one correspondence between the set of real numbers and the set of pure imaginary numbers.
2. The necessary and sufficient condition for two complex numbers to be conjugates of each other is that their product is a... | C |
11) Fabio finds an old combination lock; to open the lock, one must align three digits in the correct order, each of which can vary from 0 to 9. Fabio does not remember the correct combination, but he is sure that the sum of the three digits is 10. How many attempts will he have to make, at most, to find the correct co... | 63 |
Given vectors $\overrightarrow {a}=(m,2)$, $\overrightarrow {b}=(-1,n)$, (where $n>0$) and $\overrightarrow {a}\cdot \overrightarrow {b}=0$, if point P$(m,n)$ lies on the circle $x^2+y^2=5$, then the magnitude of $2\overrightarrow {a}+ \overrightarrow {b}$ equals \_\_\_\_\_\_. | \sqrt {34} |
Randomly select a number $x$ in the interval $[0,\pi]$, the probability of the event "$\sin x \geqslant \cos x$" occurring is ( ).
A: $\dfrac{1}{4}$
B: $\dfrac{1}{2}$
C: $\dfrac{3}{4}$
D: $1$ | C |
8. Solve the system $\left\{\begin{array}{l}x \log _{2} 3+y=\log _{2} 18 ; \\ 5^{x}=25^{y} .\end{array}\right.$
 | {2;1} |
14. What is the smallest number of triangular pyramids (tetrahedrons) into which a cube can be divided? | 5 |
Example 4 In $\triangle A B C$, $\angle A B C=50^{\circ}$, $\angle A C B=30^{\circ}$, $Q$ is a point inside the triangle, $\angle Q B A=$ $\angle Q C A=20^{\circ}$. Find the degree measure of $\angle Q A B$. | 20^{\circ} |
Randomly select a number $k$ in the interval $[-1,1]$, and find the probability that the line $y=k(x+2)$ intersects with the circle $x^{2}+y^{2}=1$. The options are:
A: $\frac{1}{2}$
B: $\frac{1}{3}$
C: $\frac{\sqrt{3}}{2}$
D: $\frac{\sqrt{3}}{3}$ | D |
Given vectors $\overrightarrow{a}=(x,x-1), \overrightarrow{b}=(1,2)$, and $\overrightarrow{a}$ is parallel to $\overrightarrow{b}$, find $\overrightarrow{a} \cdot \overrightarrow{b} = $____. | \overrightarrow{a} \cdot \overrightarrow{b} = -5 |
In the arithmetic sequence $\{a_n\}$, it is known that $a_6+a_9+a_{13}+a_{16}=20$. Then, $S_{21}$ equals (γγ)
A: 100
B: 105
C: 200
D: 0 | \text{B} |
19.15 If the number of sides of a convex polygon increases from 3 to $n$, then the sum of the exterior angles of the polygon
(A) increases.
(B) decreases.
(C) remains unchanged.
(D) cannot be predicted.
(E) becomes $(n-3)$ straight angles.
(1st American High School Mathematics Examination, 1950) | C |
One, (20 points) If the two sides of a right triangle, $x, y$, are both prime numbers, and make the algebraic expressions $\frac{2 x-1}{y}$ and $\frac{2 y+3}{x}$ both positive integers, find the inradius $r$ of this right triangle. | 2 |
A certain unit has launched a staff walking activity. If the staff walks $5000$ steps a day, they are considered to have met the standard. If Xia walked $6200$ steps, it is recorded as $+1200$ steps. If Chen walked $4800$ steps, it is recorded as ____ steps. | -200 |
Xiaopang arranged the 50 integers $1, 2, 3, 4, ..., 49, 50$ in ascending order without any gaps, and then inserted a β+β between each pair of adjacent numbers, resulting in an addition expression: $1+2+3+4+5+6+7+8+9+1+0+1+1+\cdots+4+9+5+0$. Please calculate the sum of this addition expression, the result is $\qquad$ _ | 330 |
17 Five boys and two girls stand in a row, with boy A must stand in the middle, and the girls must be adjacent, then the number of different ways to arrange them is
A. $C_{5}^{1} P_{4}^{4} P_{2}^{2}$
B. $C_{4}^{1} P_{4}^{4} P_{2}^{2}$
C. $P_{6}^{6}$
D. $P_{5}^{5}$ | C_{4}^{1}P_{4}^{4}P_{2}^{2} |
4. A problem from the collection LILAVATI, by the Indian mathematician Bhaskara:
For the same price, I bought eight of these rubies, then ten emeralds, and finally a hundred pearls, which you now wear in your earrings. If I add together one of each of these precious stones, their price is only three coins less than ha... | 2,\,20,\,25 |
In the book "Algorithms Statute", there is a problem: "A mute comes to buy meat and finds it difficult to express the amount of money. One pound (16 ounces) less thirty cents, eight ounces more eighteen cents, try to ask those who can calculate, how much meat is it in total?" The meaning is that a mute comes to buy mea... | 6 |
6. For a positive integer, if it reads the same forward and backward, it is called a "palindrome" (such as $1, 11, 121$ are palindromes, while 110 is not a palindrome). If all palindromes are arranged in ascending order:
$1,2, \cdots, 9,11,22, \cdots$.
Then the 2012th palindrome is ( ).
(A) 1011101
(B) 1013101
(C) 1021... | B |
The isoelectric point of glycine is the pH at which it has zero charge. Its charge is $-\frac13$ at pH $3.55$, while its charge is $\frac12$ at pH $9.6$. Charge increases linearly with pH. What is the isoelectric point of glycine? | 5.97 |
7. Three parallel lines touch a sphere with center at point O at points K, L, and M. It is known that the radius of the sphere is $5 \sqrt{2}$, the area of triangle OKL is 7, and the area of triangle KLM is greater than 50. Find the angle KML.
Answer. $\angle \mathrm{KML}=\arccos \frac{1}{5 \sqrt{2}}$. | \arccos\frac{1}{5\sqrt{2}} |
2. Due to the price reduction of $20 \%$, for 240 denars, one can buy 1 meter more fabric than what could be bought for 270 denars before the reduction. Determine the price of the fabric before the reduction. | 30 |
8. The maximum volume of a cone inscribed in a sphere with radius 1 is . $\qquad$ | \frac{32 \pi}{81} |
Given the function $f(x)=\frac{3^x}{3^x+1}-a$ is an odd function.
(β
) Find the value of the real number $a$;
(β
‘) Determine the monotonicity of the function on $\mathbb{R}$ and prove it using the definition of function monotonicity;
(β
’) Find the range of the function $f(x)$. | \left(- \frac{1}{2}, \frac{1}{2}\right) |
$37 \cdot 6$ points $(x, y)$ where both $x, y$ are rational numbers are called rational points. In a circle with radius $r$ and center $(a, b)$, if $a$ is a rational number and $b$ is an irrational number, the number of rational points on this circle is
(A) at most one.
(B) at most two.
(C) at most three.
(D) infinitel... | B |
A circle is inscribed in a sector that is one sixth of a circle of radius 6 . (That is, the circle is tangent to both segments and the arc forming the sector.) Find, with proof, the radius of the small circle.
A: Parallel
B: Intersecting
C: Within the plane
D: Parallel or within the plane | \text{D} |
5. In the box, there are 100 cubes, all of whose sides are painted red, green, or yellow. Among them, 75 cubes have at least one red side, 80 cubes have at least one green side, and 85 cubes have at least one yellow side. What is the smallest possible number of cubes that have sides of all three colors? | 40 |
4.86 It is known that $\sin \alpha - \cos \alpha = n$. Find $\sin ^{3} \alpha - \cos ^{3} \alpha$. | \frac{3n-n^{3}}{2} |
4. Let $A, B$ be the vertices of the major axis of the ellipse $\Gamma$, $E, F$ be the two foci of $\Gamma$, $|A B|=4$, $|A F|=2+\sqrt{3}$, and $P$ be a point on $\Gamma$ such that $|P E| \cdot|P F|=2$. Then the area of $\triangle P E F$ is $\qquad$ . | 1 |
3. In space, there are four different planes. The set of possible numbers of intersection lines formed by these four planes is ( ).
(A) $\{1,2,3,4,5,6\}$
(B) $\{0,1,2,3,4,5,6\}$
(C) $\{0,1,3,4,5,6\}$
(D) $\{0,1,2,3,5,6\}$ | C |
# Task 8.2
A team consisting of Petya, Vasya, and a one-person scooter is participating in a race. The distance is divided into segments of equal length, the number of which is 42, with a checkpoint at the beginning of each. Petya runs a segment in 9 minutes, Vasya in 11 minutes, and on the scooter, either of them can... | 18 |
Yura put $2001$ coins of $1$, $2$ or $3$ kopeykas in a row. It turned out that between any two $1$-kopeyka coins there is at least one coin; between any two $2$-kopeykas coins there are at least two coins; and between any two $3$-kopeykas coins there are at least $3$ coins. How many $3$-koyepkas coins could Yura put? | 501 |
2. When $50$, $72$, and $157$ are divided by the positive integer $a$, the sum of their remainders is 27. Then $a$ is ( ).
(A) 9
(B) 15
(C) 21
(D) 35 | C |
6. A checker can move in one direction along a strip divided into cells, moving either to the adjacent cell or skipping one cell in a single move. In how many ways can it move 10 cells? 11 cells? | 89 |
4. Given equation A: $x^{2}+p_{1} x+q_{1}=0$, equation B: $x^{2}+p_{2} x+q_{2}=0$, where $p_{1} γ p_{2} γ q_{1} γ q_{2}$ are all real numbers, and satisfy $p_{1} p_{2}=2\left(q_{1}+q_{2}\right)$. Then $(\quad)$.
(A) Both A and B must have real roots
(B) Neither A nor B has real roots
(C) At least one of A and B has rea... | C |
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