problem stringlengths 10 7.47k | answer stringlengths 1 1.22k |
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Determine the number of multiplications required to calculate the value of the function $f(x) = 5x^6 - 7x^5 + 2x^4 - 8x^3 + 3x^2 - 9x + 1$ at $x = x$ using Horner's method, as opposed to 21 multiplications needed when using the direct method. | 6 |
1. The monotonic decreasing interval of the function $y=\log _{\frac{1}{5}}|x-2|$ is ( ).
A. $(-\infty, 2)$
B. $(-\infty,-2) \cup(2,+\infty)$
C. $(2,+\infty)$
D. $(0,2) \cup(2,+\infty)$ | C |
Given the function $f(x)=(x+1)\ln x-a(x-1)$.
$(1)$ When $a=4$, find the equation of the tangent line to the curve $y=f(x)$ at the point $(1,f(1))$;
$(2)$ If $f(x) > 0$ for $x\in(1,+\infty)$, find the range of values for $a$. | (-\infty,2] |
Given the function $f(x)=mx^{3}-3(m+1)x^{2}+(3m+6)x+1$, where $m < 0$
1. If the monotonically increasing interval of $f(x)$ is $(0,1)$, find the value of $m$.
2. When $x\in[-1,1]$, the slope of any tangent line to the graph of $y=f(x)$ is always greater than $3m$. Find the range of $m$. | (- \dfrac {4}{3},0) |
Let the constant $a\in \mathbb{R}$, and the sets $A=\left\{ x|\left( x-1 \right)\left( x-a \right)\geqslant 0 \right\}$, $B=\left\{ x|x\geqslant a-1 \right\}$. If $A\cup B=\mathbb{R}$, then the range of values for $a$ is
A: $(-\infty,2)$
B: $(-\infty,2]$
C: $(2,+\infty)$
D: $[2,+\infty)$ | \text{B} |
11.043. The base of a right parallelepiped is a rhombus with an area of $Q$. The areas of the diagonal sections are $S_{1}$ and $S_{2}$. Determine the volume and the lateral surface area of the parallelepiped. | \sqrt{\frac{S_{1}S_{2}Q}{2}};2\sqrt{S_{1}^{2}+S_{2}^{2}} |
Given the equation of a line is $3x - 4y + 2 = 0$.
(1) Find the equation of the line that passes through the point $(-2, 2)$ and is perpendicular to line $l$;
(2) Find the intersection of the lines $x - y - 1 = 0$ and $2x + y - 2 = 0$, and calculate the distance from this point to the given line. | 1 |
2. (5 points) The product of $1 \times 2 \times 3 \times 4 \times \cdots \times 2014$ ends with $\qquad$ zeros. | 501 |
15. (GBR 1) ${ }^{\mathrm{IMO} 6}$ Let $[x]$ denote the integer part of $x$, i.e., the greatest integer not exceeding $x$. If $n$ is a positive integer, express as a simple function of $n$ the sum
$$ \left[\frac{n+1}{2}\right]+\left[\frac{n+2}{4}\right]+\cdots+\left[\frac{n+2^{i}}{2^{i+1}}\right]+\cdots $$ | f(n)=n |
Task 2. Find the number of natural numbers not exceeding 2022 and not belonging to either the arithmetic progression $1,3,5, \ldots$ or the arithmetic progression $1,4,7, \ldots$ | 674 |
(14) Given the ellipse $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1(a>b>0)$ with an eccentricity of $\frac{1}{2}$, $F_{1}$ and $F_{2}$ are the left and right foci, respectively. A line passing through $F_{2}$ intersects the ellipse at points $A$ and $B$. If the maximum area of $\triangle F_{1} A B$ is 6, find the equatio... | \frac{x^{2}}{8}+\frac{y^{2}}{6}=1 |
Four, find the minimum perimeter of a convex pentagon on a Cartesian plane whose vertices are lattice points (i.e., the x and y coordinates of each vertex are integers).
untranslated text:
求平面直角坐标系中格点凸五边形 (即每个顶点的纵、横坐标都是整数的凸五边形)的周长的最小值.
translated text:
Find the minimum perimeter of a convex pentagon on a Cartesian ... | 2 + 3 \sqrt{2} |
5. Let $m$ be an integer, and the two roots of the equation $3 x^{2}+m x-2=0$ are both greater than $-\frac{9}{5}$ and less than $\frac{3}{7}$. Then $m=$ $\qquad$ .
(2003, National Junior High School Mathematics League) | 4 |
Problem N2. Find all positive integers $n$ such that $36^{n}-6$ is a product of two or more consecutive positive integers.
| 1 |
2. Twenty numbers are arranged in a circle. It is known that the sum of any six consecutive numbers is 24. What is the number in the 12th position if the number in the 1st position is 1? | 7 |
Given $9^{m}=\dfrac{3}{2}$ and $3^{n}=\dfrac{1}{2}$, which of the following conclusions is correct?
A: $2m-n=1$
B: $2m-n=3$
C: $2m+n=3$
D: $2m=3n$ | A |
The 66th question: Find the smallest positive real number $\lambda$, such that for any three complex numbers $\mathrm{z}_{1}, \mathrm{z}_{2}, \mathrm{z}_{3} \in\{\mathrm{z} \in \mathrm{C} \| \mathrm{z} \mid<1\}$, if $\mathrm{z}_{1}+$ $z_{2}+z_{3}=0$, then $\left|z_{1} z_{2}+z_{2} z_{3}+z_{3} z_{1}\right|^{2}+\left|z_{1... | 1 |
In isosceles right triangle $ABC$, with $\angle A=90^{\circ}$, points $A$ and $B$ are on the right branch of hyperbola $D$. The segment $AB$ passes through the right focus $F$ of the hyperbola, and point $C$ is the left focus of the hyperbola $D$. Then, the value of $\dfrac{|AF|}{|FB|}=$
A: $\dfrac{2-\sqrt{2}}{2}$
B: $... | \text{B} |
Given that $y$ is directly proportional to $x+1$, when $x=1$, $y=4$, then when $x=2$, the value of $y$ is ______. | 6 |
The maximum value of the function $y = \sin x + \cos x$ is ______. | \sqrt{2} |
Rubaanov I.S.
A polynomial $P(x)$ of degree $n$ has $n$ distinct real roots. What is the maximum number of its coefficients that can be zero?
# | \frac{n}{2}forevenn,\frac{n+1}{2}foroddn |
Let $M=\{x|x>2\}$ and $N=\{x|1<x<3\}$. Which of the following statements is correct?
A: $M\cup N=M$
B: $M\cap N=\{x|2<x<3\}$
C: $M\cup N=\mathbb{R}$
D: $M\cap N=\{x|1<x<2\}$ | B |
Given that a normal vector of plane $\alpha$ is $\overrightarrow{n\_1}=(1,2,-2)$, and a normal vector of plane $\beta$ is $\overrightarrow{n\_2}=(-2,-4,k)$, find the value of $k$ if $\alpha$ is parallel to $\beta$. | 4 |
Given that $f(x)$ is an odd function defined on $\mathbb{R}$, and its graph is symmetric about the line $x = 1$.
(1) Find the value of $f(0)$;
(2) Prove that the function $f(x)$ is a periodic function;
(3) If $f(x) = x$ for $0 < x \leq 1$, find the analytic expression of $f(x)$ for $x \in [-1,1]$. | f(x) = x |
II If one side of the square $A B C D$ lies on the line $y=2 x-17$, and the other two vertices are on the parabola $y=x^{2}$. Then the minimum value of the area of the square is $\qquad$ . | 80 |
Example 9 Find the smallest positive integer $n(n>1)$, such that $\sqrt{\frac{1}{n} \sum_{i=1}^{n} i^{2}}$ is an integer.
(2017, Singapore Mathematical Olympiad) | 337 |
We randomly select 3 points independently on the circumference of a circle. What is the probability that the triangle determined by them is acute? | \frac{1}{4} |
Given $$\sin\left( \frac {\pi}{4}+\alpha\right)= \frac {1}{2}$$, find the value of $$\frac {\sin\left( \frac {5\pi}{4}+\alpha\right)}{\cos\left( \frac {9\pi}{4}+\alpha \right)}\cdot \cos\left( \frac {7\pi}{4}-\alpha\right)$$. | - \frac {1}{2} |
A class needs to select one student from two students, A and B, to participate in a school knowledge competition. In the recent five selection tests, their scores are as shown in the table below:
| | Test 1 | Test 2 | Test 3 | Test 4 | Test 5 |
|---------|--------|--------|--------|--------|--------|
| A ... | \text{B} |
1. If $M=\left\{(x, y)|| \tan \pi y \mid+\sin ^{2} \pi x=0\right\}, N=\left\{(x, y) \mid x^{2}+y^{2} \leqslant 2\right\}$, then the number of elements in $M \cap N$ is
A. 4
B. 5
C. 8
D. 9 | 9 |
In trapezoid $ABCD$, the diagonals intersect at $E$, the area of $\triangle ABE$ is 72 and the area of $\triangle CDE$ is 50. What is the area of trapezoid $ABCD$? | 242 |
Let \\(F\\) be the focus of the parabola \\({{C}\_{1}}:{{y}^{2}}=2px(p > 0)\\), and point \\(A\\) be a common point of the parabola and one of the asymptotes of the hyperbola \\({{C}\_{2}}\\): \\(\\dfrac{{{x}^{2}}}{{{a}^{2}}}-\\dfrac{{{y}^{2}}}{{{b}^{2}}}=1(a > 0,b > 0)\\). If \\(AF\\) is perpendicular to the x-axis, t... | C |
We decompose a number into its prime factors; let $N=a^{\alpha} \times b^{\beta} \times c^{\gamma}$. First, we need to list all divisors of this number; second, determine the number of all divisors; third, prove that if these divisors are arranged in increasing order, the product of two divisors equidistant from the en... | 630^{12}=3909188328478827879681\cdot10^{12} |
The focal length of the hyperbola $\frac{x^2}{4} - \frac{y^2}{5} = 1$ is ( ).
A: $2$
B: $4$
C: $3$
D: $6$ | D |
Problem 5.3. The figure shows a plan of the road system of a certain city. In this city, there are 8 straight streets, and 11 intersections are named with Latin letters $A, B, C, \ldots, J, K$.
Three police officers need to be placed at some intersections so that at least one police officer is on each of the 8 streets... | B,G,H |
Problem 7. For what values of the parameter $a$ does the equation
$$
a \log _{5}^{2} x-(2 a+5) \log _{5} x+10=0
$$
have exactly two roots, one of which is five times larger than the other? | \frac{5}{3},5 |
Example 4. Find the integral $\int \sqrt{x} d x$. | \frac{2}{3}x\sqrt{x}+C |
Example 7. At two automatic machines, identical parts are manufactured. It is known that the productivity of the first machine is twice that of the second, and that the probability of producing a part of the highest quality on the first machine is 0.9, and on the second - 0.81. Parts manufactured during the shift on bo... | 0.87 |
In the mid-16th century, the English mathematician Robert Recorde first used "$=$" as an equal sign in his book "The Whetstone of Witte." Later, the English mathematician Thomas Harriot first used the symbols "$$," which were gradually accepted by the mathematical community. The introduction of inequality symbols had a... | C |
## Task B-4.4.
Determine all polynomials $p$ with real coefficients for which the equality
$$
x \cdot p(x-1)=(x-2021) \cdot p(x)
$$
is satisfied for all real numbers $x$. | p(x)=\cdotx(x-1)(x-2)\cdots(x-2020),\in\mathbb{R} |
10. Place 2 different English books, 4 different German books, and 3 different Chinese books on the same shelf of a bookcase. If books of the same language are adjacent, there are $\qquad$ different ways to arrange them. | 1728 |
The polar equation of the line $\frac{\sqrt{3}}{3}x - y = 0$ (with the restriction $\rho \geqslant 0$) can be ( )
A: $\theta = \frac{\pi}{6}$
B: $\theta = \frac{7}{6} \pi$
C: $\theta = \frac{\pi}{6}$ and $\theta = \frac{7}{6} \pi$
D: $\theta = \frac{5}{6} \pi$ | C |
Example 4.27 Using the digits $1,2,3,4$ to form 6-digit numbers, each digit may not appear more than 2 times in the 6-digit number. How many different 6-digit numbers can be formed? | 1440 |
$2 \cdot 89$ Let $s$ be a list of positive integers (the list can contain the same number multiple times), which includes the number 68. The arithmetic mean of the numbers in $s$ is 56, but if 68 is removed, the arithmetic mean of the remaining numbers drops to 55. What is the largest number that could appear in $s$? | 649 |
Solve the system of equations
$$
\frac{1}{x}+\frac{1}{y}=\frac{1}{z} ; \quad \frac{1}{x+15}+\frac{1}{y-6}=\frac{1}{z} ; \quad \frac{1}{x+24}+\frac{1}{y-15}=\frac{1}{z}
$$ | x_1=-20,y_1=10,z_1=20;\quadx_2=-60,y_2=-30,z_2=-20 |
Jonathan has a magical coin machine which takes coins in amounts of $7, 8$, and $9$. If he puts in $7$ coins, he gets $3$ coins back; if he puts in $8$, he gets $11$ back; and if he puts in $9$, he gets $4$ back. The coin machine does not allow two entries of the same amount to happen consecutively. Starting with $15$ ... | 4 |
18. (IMO-28 Preliminary Question) Find 8 positive integers $n_{1}, n_{2}, \cdots, n_{8}$, such that they have the following property: for each integer $k, -1985 \leqslant k \leqslant 1985$, there are 8 integers $a_{1}, a_{2}, \cdots, a_{8}$, where each $a_{i}$ belongs to the set $\{-1,0,1\}$, such that $k=\sum_{i=1}^{8... | n_{i}=3^{i-1},i=1,2,\cdots,8 |
Given $|\overrightarrow{a}|=2$, $|\overrightarrow{b}|=3$, if $|\overrightarrow{a}+\overrightarrow{b}|=\sqrt{17}$, then $|\overrightarrow{a}-\overrightarrow{b}|=$____. | 3 |
The set $M=\{x|(x-1)(x-2)<0\}$, $N=\{x|x<a\}$. If $M \subseteq N$, then the range of the real number $a$ is ( )
A: $[2,+\infty)$
B: $(2,+\infty)$
C: $[1,+\infty)$
D: $(1,+\infty)$ | A |
There is a scoring method: using $60$ points as the base, $68$ points are recorded as $+8$ points. If a student scores $54$ points, it should be recorded as ______ points. | -6 |
120. Calculate $\left(\frac{9}{16}\right)^{-1 / 10}:\left(\frac{25}{36}\right)^{-3 / 2}-\left[\left(\frac{4}{3}\right)^{-1 / 2}\right]^{-2 / 5}\left(\frac{6}{5}\right)^{-3}$. | 0 |
9. If positive integers $m, n$ satisfy $\frac{(m+n)!}{n!}=5040$, then the value of $m!n$ is $\qquad$
Translate the above text into English, please retain the original text's line breaks and format, and output the translation result directly. | 144 |
1. It is known that in a set of five coins three are genuine (and have the same weight) while two coins are fakes, each of which has a different weight from a genuine coin. What is the smallest number of weighings on a scale with two cups that is needed to locate one genuine coin? | 2 |
In $\triangle ABC$, the lengths of the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively, and they satisfy the relationship $c\sin B = \sqrt{3}b\cos C$, $a^2 - c^2 = 2b^2$.
(I) Find the measure of angle $C$;
(II) If the area of $\triangle ABC$ is $21\sqrt{3}$, find the value of $b$. | 2\sqrt{7} |
Example 6. Given a right triangle $\triangle A B C$ with the lengths of the two legs $a, b$ being integers, and $a$ being a prime number. If the length of the hypotenuse $c$ is also an integer, prove that $2(a+b+1)$ is a perfect square.
---
The translation maintains the original text's line breaks and format. | (a+1)^2 |
Given the function $f(x)=|x+1|+|x-3|$.
(1) Find the range of real number $a$ such that the inequality $f(x) < a$ has a solution for $x$.
(2) If the solution set of the inequality $f(x) < a$ is $(b,\frac{7}{2})$, find the value of $a+b$. | 3.5 |
3. (8 points) Xiaopang used all of this month's salary to buy a stock. On the first day, the stock price increased by $\frac{1}{10}$, on the second day it decreased by $\frac{1}{11}$, on the third day it increased by $\frac{1}{12}$, and on the fourth day it decreased by $\frac{1}{13}$. At this point, the value of his s... | 5000 |
For each $a$, $1<a<2$, the graphs of functions $y=1-|x-1|$ and $y=|2x-a|$ determine a figure. Prove that the area of this figure is less than $\frac13$. | \frac{1}{3} |
5. Given that $f(x)$ is a function defined on $\mathbf{R}$, $f(1)=1$ and for any $x \in \mathbf{R}$, we have $f(x+5) \geqslant f(x)+5$, $f(x+1) \leqslant f(x)+1$. If $g(x)=f(x)+1-x$, then $g(2002)=$ $\qquad$ . | 1 |
35. If $S=\frac{1}{1+1^{2}+1^{4}}+\frac{2}{1+2^{2}+2^{4}}+\frac{3}{1+3^{2}+3^{4}}+\ldots \ldots \ldots+\frac{200}{1+200^{2}+200^{4}}$, find the value of $80402 \times S$. | 40200 |
5. Let $x$ be the length of the smaller segment. On the upper side of the quadrilateral, which has a length of 1, there are 3 small segments and one large segment. Therefore, the length of the large segment is $(1-3x)$. On the lower side of the quadrilateral, which has a length of 2, there are 3 large segments and one ... | 5:1 |
Let's determine all real-valued functions $f$ defined on the positive integers such that for any positive integer $n$, the following holds:
$$
\frac{p}{f(1)+f(2)+\ldots+f(n)}=\frac{p+1}{f(n)}-\frac{p+1}{f(n+1)}
$$
where $p$ is a fixed positive number. | f(n)=\binom{p+n-1}{p} |
The number represented by $\square$ so that $\frac{1}{2}+\frac{1}{4}=\frac{\square}{12}$ is
(A) 3
(B) 12
(C) 9
(D) 6
(E) 15
Part B: Each correct answer is worth 6. | 9 |
Given that point P(x,y) is in the second quadrant, and $|x| = 2$, $y$ is the square root of 1, then the coordinates of point P are \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_. | (-2, 1) |
Given the set $M=\{x|1\leqslant x\leqslant 10$,$x\in N^{*}\}$, for its non-empty subset $A$, each element $k$ in $A$ is multiplied by $\left(-1\right)^{k}$ and then summed. For example, if $A=\{1,4,7\}$, the sum can be calculated as $\left(-1\right)^{-1}\times 1+\left(-1\right)^{4}\times 4+\left(-1\right)^{7}\times 7=-... | 2560 |
Exercise 12. A circle with diameter $2 n-1$ is drawn at the center of a chessboard $2 n \times 2 n$. How many squares are crossed by an arc of the circle? | 8n-4 |
Example 2.61. Calculate the surface area formed by the rotation of the arc of the circle $x^{2}+y^{2}=16(y>0)$ over the segment $-1 \geqslant x \geqslant 1$ around the $O X$ axis. | 16\pi |
Solve the following equation:
$$
|2 x-3|=\frac{6 x+17}{11}
$$ | x_1=\frac{25}{8},x_2=\frac{4}{7} |
Given a complex number $z$ satisfies $\frac {1+i}{1-i} \cdot z = 3+4i$, then the conjugate of $z$ is ( )
A: $4+3i$
B: $-4+3i$
C: $-4-3i$
D: $4-3i$ | A |
6. In a regular tetrahedron $ABCD$, the line segment connecting the midpoint of edge $AD$ and the centroid of face $BCD$ is $MN$, and the line segment connecting the midpoint of edge $CD$ and the centroid of face $ABC$ is $PQ$. Then the angle formed by $MN$ and $PQ$ is $\qquad$ | \arccos \frac{1}{18} |
Given vectors $\overrightarrow{a} = (x, y)$, $\overrightarrow{b} = (-1, 2)$, and $\overrightarrow{a} + \overrightarrow{b} = (1, 3)$, find the magnitude of $\overrightarrow{a} - 2\overrightarrow{b}$. | 5 |
13. Given that the angle between vector $\boldsymbol{a}$ and $\boldsymbol{b}$ is $120^{\circ}$, and $|a|=2,|b|=5$. Then $(2 a-b) \cdot a=$ $\qquad$ | 13 |
There is a bag containing 5 white balls and 3 yellow balls of the same size. If two balls are taken out at random, given that the first ball drawn is a yellow one, the probability of drawing a yellow ball the second time is __________. | \frac{2}{7} |
4. In the Cartesian coordinate system $x O y$, the ellipse $C: \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1(a>b>0)$ has left and right foci $F_{1}$ and $F_{2}$, respectively. The chords $S T$ and $U V$ of the ellipse $C$ are parallel to the $x$-axis and $y$-axis, respectively, and intersect at point $P$. It is known that ... | \sqrt{15} |
Task B-1.4. A cattleman has secured food for his 24 cows for 18 weeks. How many cows would he have to sell after 8 weeks so that he would have enough food for another 12 weeks? | 4 |
One morning at school, consecutive integers starting from 1 were written on the board up to a certain number. The troublemaker erased one of the numbers carefully. The whole incident would have been forgotten if someone hadn't noted that the arithmetic mean of the remaining numbers was $\frac{45}{4}$. Let's try to figu... | 6 |
In triangle $ABC$, the sides opposite to angles $A$, $B$, and $C$ are denoted as $a$, $b$, and $c$ respectively. Given that $a=2$, $b=3$, and $\angle C=2\angle A$.
$(I)$ Find the value of $c$;
$(II)$ Calculate the area of $\triangle ABC$. | \frac{3\sqrt{15}}{4} |
Given $(x-1)(x+1)^9 = a + a_1x + a_2x^2 + \ldots + a_9x^9 + a_{10}x^{10}$, find the value of $a_2 + a_4 + a_6 + a_8 + a_{10}$.
A: $-1$
B: $0$
C: $1$
D: $2$ | \text{C} |
Task B-3.5. A barrel has the shape of a right cylinder with its axis in a horizontal position, and the radius of the base is $2 \mathrm{dm}$, and the height is $6 \mathrm{dm}$. The barrel is submerged in water to half the radius of the base. Calculate the volume of the submerged part of the barrel. | 8\pi-6\sqrt{3}\mathrm{~}^{3} |
Let the set $A=\{x|x\geqslant -1\}$, and $B=\{x|y=\ln (x-2)\}$. Then, the intersection of $A$ and the complement of $B$ in $\mathbb{R}$, denoted as $A\cap \complement_{\mathbb{R}}B$, is
A: $[-1,2)$
B: $[2,+\infty)$
C: $[-1,2]$
D: $[-1,+\infty)$ | C |
Find all solutions of $2^n + 7 = x^2$ in which n and x are both integers . Prove that there are no other solutions. | (n, x) = (1, 3) |
2. Determine the remainder of the division of the number $3^{100}$ by 13. | 3 |
146. $\int(2 x+3)^{4} d x$.
Translate the text above into English, keeping the original text's line breaks and format, and output the translation result directly.
146. $\int(2 x+3)^{4} d x$. | 0.1(2x+3)^{5}+C |
Problem 9.1. Find all real numbers $a$ for which the equation $x^{2}+a x+3 a^{2}-$ $7 a-19=0$ has real roots $x_{1}$ and $x_{2}$ such that
$$
\frac{1}{x_{1}-2}+\frac{1}{x_{2}-2}=-\frac{2 a}{13}
$$
Peter Boyvalenkov | 4 |
Given a triangle $\triangle ABC$ with sides $a$, $b$, and $c$ opposite to the angles $A$, $B$, and $C$ respectively, and given that $c=\sqrt{2}$, $b=\sqrt{6}$, and $B=120^\circ$, find the area of $\triangle ABC$. | \frac{\sqrt{3}}{2} |
It is known that the point $(1,2)$ lies on the graph of the function $f(x) = a^x$ (where $a>0$ and $a\neq 1$). The sum of the first $n$ terms of the sequence $\{a_n\}$ is given by $S_n = f(n) - 1$.
(1) Find the general formula for the $n$-th term of the sequence $\{a_n\}$;
(2) Let $b_n = \log_a (a_{n+1})$, find the sum... | T_n = (n-1)\cdot2^n + 1 |
Simplify: $\sqrt {2+\cos 20 ^{\circ} -\sin ^{2}10 ^{\circ} }=$ \_\_\_\_\_\_ . | \sqrt {3}\cos 10^{\circ} |
G6.2 If $(p, q),(5,3)$ and $(1,-1)$ are collinear, find $q$. | 5 |
5. (2005 Slovenia Mathematical Olympiad) Find the smallest prime $p$ such that $p^{3}+2 p^{2}+p$ has exactly 42 divisors. | 23 |
12.87 Determine all integer solutions of the indeterminate equation
$$
x^{3}+x^{2} y+x y^{2}+y^{3}=8\left(x^{2}+x y+y^{2}+1\right)
$$
(Luxembourg and four other countries' International Mathematical Competition, 1980) | (x,y)=(8,2)\text{or}(2,8) |
In triangle $ABC$, $AB = AC$. The distances from the two trisection points of the side $AB$ to the vertex $C$ are 17 cm and 20 cm, respectively. What could be the distance from vertex $A$ to the trisection points of the base? | x_1=\sqrt{585}\approx24.2 |
Solve the following equations using appropriate methods:$(1)x\left(x-5\right)=3x-15$;$(2)2y^{2}-9y+5=0$. | y_1=\frac{9+\sqrt{41}}{4}, y_2=\frac{9-\sqrt{41}}{4} |
Let the function $y=f(x)$ have an inverse function $y=f^{-1}(x)$, and the graph of the function $y=x-f(x)$ passes through the point $(1,2)$. Then, the graph of the function $y=f^{-1}(x)-x$ must pass through the point $(\_,\_)$. | (-1,2) |
381. The numbers $p$ and $p+15$ are prime. Find all such $p$. | 2 |
24. To mail a letter in the U.S.A. it costs 20 cents for the first ounce and 18 cents for each additional ounce or fraction thereof. Find a formula involving the greatest integer function for the cost of mailing a letter. Could it possibly cost $\$ 1.08$ or $\$ 1.28$ to mail a letter? | 20+18[\mathrm{x}-1], \$ 1.08 \text{ no, } \$ 1.28 \text{ yes} |
Problem 9.6. The teacher wrote a number on the board. Sasha decided to divide it by 102 with a remainder, and Masha - by 103. It turned out that the quotient obtained by Sasha and the remainder obtained by Masha add up to 20. What remainder did Sasha get? List all possible options. | 20 |
Evaluate the integral \\(\int_{-4}^{4}{\sqrt{16-x^{2}}}dx\). | 8\pi |
Given that $a$, $b$, $c$, and $d$ are all positive numbers, and $a^2+b^2=4$, $cd=1$, find the minimum value of $(a^2c^2+b^2d^2)(b^2c^2+a^2d^2)$. | 16 |
In a table tennis single round-robin tournament with 15 athletes, each athlete plays against every other athlete once. If Athlete 1 wins $x_{1}$ matches, Athlete 2 wins $x_{2}$ matches, ‥ and Athlete 15 wins $x_{15}$ matches, find the value of $x_{1}+x_{2}+\cdots+x_{15}$. | 105 |
2. A person is chasing a car stopped at a traffic light at a uniform speed of $6 \mathrm{~m}$ /s. When he is $25 \mathrm{~m}$ away from the car, the traffic light changes from red to green, and the car starts to move with a uniform acceleration of $1 \mathrm{~m} / \mathrm{s}^{2}$. Then ( ).
(A) the person can catch up ... | D |
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