problem stringlengths 10 7.47k | answer stringlengths 1 1.22k |
|---|---|
Variation 2 (Excerpt from the 2008 National Exam II, Science Question 21) Let the center of the ellipse be at the origin of the coordinate system, with $A(2,0), B(0,1)$ being two of its vertices. The line $y=k x(k>0)$ intersects $A B$ at point $D$, and intersects the ellipse at points $E$ and $F$. Find the maximum valu... | 2 \sqrt{2} |
781. A triangle moves in space so that its center of gravity remains stationary, and one of its vertices describes a circle, with the plane of the triangle remaining constantly perpendicular to the plane of the circle. What lines are described by the vertices of the triangle | Circles |
Which of the following propositions is correct? ( )
① If two intersecting lines in plane $\alpha$ are parallel to two intersecting lines in plane $\beta$, then plane $\alpha$ is parallel to plane $\beta$;
② If two planes each pass through two parallel lines, then these two planes are parallel to each other;
③ If ... | C |
2.183. $\frac{x \cdot|x-3|+x^{2}-9}{2 x^{3}-3 x^{2}-9 x}$. | \frac{3}{x(2x+3)},ifx\in(-\infty;-\frac{3}{2})\cup(-\frac{3}{2};0)\cup(0;3);\frac{1}{x},ifx\in(3;\infty) |
3.382. Find $\sin 2 \alpha$, if it is known that $2 \operatorname{tg}^{2} \alpha-7 \operatorname{tg} \alpha+3=0$ and the number $\alpha$ satisfies the inequalities: a) $\pi<\alpha<\frac{5 \pi}{4}$; b) $\frac{5 \pi}{4}<\alpha<\frac{3 \pi}{2}$. | \frac{4}{5};\frac{3}{5} |
What is the value of
\[\left(\left((2+1)^{-1}+1\right)^{-1}+1\right)^{-1}+1?\]
$\textbf{(A) } \frac58 \qquad \textbf{(B) }\frac{11}7 \qquad \textbf{(C) } \frac85 \qquad \textbf{(D) } \frac{18}{11} \qquad \textbf{(E) } \frac{15}8$ | \frac{11}7 |
\section*{Task 2 - 011112}
A steamboat travels from A to B in 3 hours and, with the same engine power, from B to A in \(4 \frac{1}{2}\) hours. How long does a vessel propelled only by the current take for the journey from A to B? | 18 |
10-1. There are several bowls on the table, each containing several grapes. Different bowls may contain different numbers of grapes. If 12 bowls are each added 8 more grapes, the average number of grapes in all the bowls will increase by 6. How many bowls are on the table | 16 |
If the terminal sides of angles $\alpha$ and $\beta$ are perpendicular, then the relationship between $\alpha$ and $\beta$ is ( )
A: $\beta = \alpha + 90^\circ$
B: $\beta = \alpha \pm 90^\circ$
C: $\beta = k \cdot 360^\circ + \alpha + 90^\circ$, $k \in \mathbb{Z}$
D: $\beta = k \cdot 360^\circ + \alpha \pm 90^\circ$, ... | D |
10. As shown in the figure, in square $\mathrm{ABCD}$, the area of isosceles right triangle $\mathrm{AEF}$ is $1 \mathrm{~cm}^2$, and the area of rectangle $\mathrm{EFGH}$ is $10 \mathrm{~cm}^2$. What is the area of square $\mathrm{ABCD}$ in $\mathrm{cm}^2$? | 24.5 |
6・69 Given $f(x)=x^{4}-4 x^{3}+(3+m) x^{2}-12 x+12$, where $m$ is a real parameter.
(1) Find all integer values of $m$ such that $f(x)-f(1-x)+4 x^{3}=0$ has at least one integer solution;
(2) Find all values of $m$ such that for every real number $x, f(x) \geqslant 0$. | [4,\infty) |
A positive integer $n$ between 10 and 99 , inclusive, is chosen at random. If every such integer is equally likely to be chosen, what is the probability that the sum of the digits of $n$ is a multiple of 7 ? | \frac{2}{15} |
N28 (25-6, Poland) Let $a, b, c, d$ be odd numbers, $0<a<b<c<d$, and $a d=$ $b c$. Prove that if
$$
a+d=2^{k}, b+c=2^{m},
$$
where $k, m$ are integers, then $a=1$. | 1 |
Exercise 20. Three lines $D_{1}, D_{2}, D_{3}$ are parallel to each other. The line $D_{2}$ is located between the other two. It is 13 centimeters from $D_{1}$ and 31 centimeters from $D_{3}$. Determine the area, expressed in square centimeters, of a square in which each of these three lines contains one vertex. | 1130 |
10) The manager of a restaurant with a capacity of 600 seats does not remember how many people he served during a large collective lunch. However, he remembers that if he tried to seat them at tables of 3, exactly one person would be left out, and the same thing happened when seating them at tables of 4, 5, or 6. Howev... | 301 |
Given the function $$y=\log_{a}(ax^{2}+x+a)$$ has its domain as $\mathbb{R}$, the range of values for $a$ is the set $M$; when its range is $\mathbb{R}$, the range of values for $a$ is the set $N$, then among the following expressions, the correct one is ( )
A: $M \supseteq N$
B: $M \cup N = \mathbb{R}$
C: $M \c... | C |
Example 3 Color each edge of a convex polyhedron red or yellow, and a dihedral angle between two faces of different colors is called a strange dihedral angle. The number of strange dihedral angles at a vertex $A$ is called the strangeness of that vertex, denoted as $S_{A}$. Prove that there exist two vertices $B$ and $... | S_{B}+S_{C}\leqslant4 |
Ten men sit side by side at a long table, all facing the same direction. Each of them is either a knight (and always tells the truth) or a knave (and always lies). Each of the people announces: "There are more knaves on my left than knights on my right." How many knaves are in the line? | 5 |
Among the following functions, the one that is increasing on the interval $(0,2)$ is ( ).
A: $y= \sqrt {2-x}$
B: $y= \dfrac {1}{x-2}$
C: $y=( \dfrac {1}{2})^{x-2}$
D: $y=\log _{ \frac {1}{2}}(2-x)$ | D |
5. The function $f(x)=x^{2}-2 a x+a$ has a minimum value on the interval $(-\infty, 1)$. Then the function $g(x)=\frac{f(x)}{x}$ on the interval $(1,+\infty)$ must $(\quad)$.
(A) have a minimum value
(B) have a maximum value
(C) be a decreasing function
(D) be an increasing function | D |
Given that the sum of the first $n$ terms of the sequence $\{a\_n\}$ is $S\_n=\frac{n(n+1)}{2}$.
(I) Find the general term formula of the sequence $\{a\_n\}$;
(II) Let $T\_n$ be the sum of the first $n$ terms of the sequence $\{b\_n\}$, where $b\_n=\frac{a_{n+1}}{2S\_n \cdot S_{n+1}}$, find $T\_n$;
(III) If there exist... | (-\infty, \frac{1}{12}] |
Positive integers m and n are both greater than 50, have a least common multiple equal to 480, and have a
greatest common divisor equal to 12. Find m + n.
| 156 |
Find all functions $f: \mathbf{Q}_{+} \rightarrow \mathbf{Q}_{+}$, such that
$$
f(x)+f(y)+2 x y f(x y)=\frac{f(x y)}{f(x+y)},
$$
where $\mathbf{Q}_{+}$ denotes the set of positive rational numbers.
(Li Shenghong) | f(x)=\frac{1}{x^{2}} |
Given the hyperbola $C$: $\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1$, its right latus rectum intersects with the two asymptotes at points $A$ and $B$, and its right focus is $F$. If $\triangle ABF$ is an equilateral triangle, then the eccentricity of the hyperbola $C$ is _______. | 2 |
(1) The imaginary part of the complex number $z= \frac{(i-1)^{2}+4}{i+1}$ is \_\_\_.
(2) The coefficient of $x^{2}$ in the expansion of $(2x- \frac{1}{2})^{6}$ is \_\_\_\_.
(3) The number of 5-digit numbers formed without repeating digits using the digits $12$, $3$, $4$, $5$, where the number of odd digits is \_\_\_\... | \frac{2 \sqrt{3}}{3} |
An opaque bag contains $1$ red ball and $2$ yellow balls, with no other differences between these balls. If one ball is randomly drawn from the bag, the probability of drawing a red ball is ____. | \frac{1}{3} |
8. Given the function
$$
f(x)=x^{3}-\log _{2}\left(\sqrt{x^{2}+1}-x\right) \text {. }
$$
For any real numbers $a, b (a+b \neq 0)$, the value of $\frac{f(a)+f(b)}{a^{3}+b^{3}}$ is ( ).
(A) always greater than zero
(B) always equal to zero
(C) always less than zero
(D) the sign is uncertain | A |
## Aufgabe 24/75
Aus einer Tabelle der Fakultäten will jemand den Wert für 20! entnehmen. Dabei stellt er fest, dass zwei Ziffern unleserlich sind:
$$
20!=2 \bullet \bullet 2902008176640000
$$
Wie kann man die unleserlichen Ziffern ermitteln, ohne das Produkt auszurechnen?
| 2432902008176640000 |
2. Let $a$ be a real number, and the two parabolas
$$
y=x^{2}+x+a \text { and } x=4 y^{2}+3 y+a
$$
have four intersection points.
(1) Find the range of values for $a$;
(2) Prove that these four intersection points are concyclic, and find the coordinates of the center of the circle. | (-\frac{3}{8},\frac{1}{8}) |
If $x$ and $y$ are positive integers such that $(x-4)(x-10)=2^y$, then Find maximum value of $x+y$ | 16 |
27.11. (Belgium, 77). Three shooters $A, B, C$ decided to duel simultaneously. They positioned themselves at the vertices of an equilateral triangle and agreed on the following: the first shot is taken by $A$, the second by $B$, the third by $C$, and so on in a circle; if one of the shooters is eliminated, the duel con... | 3 |
The expression $(\sqrt{100}+\sqrt{9}) \times(\sqrt{100}-\sqrt{9})$ is equal to
(A) 91
(B) 19
(C) 9991
(D) 9919
(E) 10991 | 91 |
In the sequence $\{a_n\}$, $a_1=1$, $a_{n+1}=a_n^2-1$ ($n\ge1$), then the sum of $a_1+a_2+a_3+a_4+a_5$ equals to ( )
A: $-1$
B: $1$
C: $0$
D: $2$ | \text{A} |
Line segment $\overline{AB}$ is a diameter of a circle with $AB=24$. Point $C$, not equal to $A$ or $B$, lies on the circle. As point $C$ moves around the circle, the centroid (center of mass) of $\triangle{ABC}$ traces out a closed curve missing two points. To the nearest positive integer, what is the area of the r... | 50 |
5. The exam consists of $N \geqslant 3000$ questions. Each of the 31 students has learned exactly 3000 of them, and every question is known by at least 29 students. Before the exam, the teacher openly laid out all the question cards in a circle. He asked the students to point to one of the questions and explained that ... | 3100 |
C2. An expression like
$$
x=1+\frac{1}{2+\frac{1}{3+\frac{1}{4}}}
$$
is called a continued fraction.
(a) Write $x$ given above as a reduced fraction of the form $\frac{a}{b}$ where $a$ and $b$ are positive integers.
(b) Write $\frac{355}{113}$ as a continued fraction in the form $a+\frac{1}{b+\frac{1}{c}}$, where $a, b... | p=4,q=17 |
Given the function $f(x)=xe^{x}-a( \frac{1}{2} x^{2}+x)(a\in R)$.
(1) If $a=0$, find the equation of the tangent line to the curve $y=f(x)$ at the point $(1,f(1))$;
(2) If $x\in (-2,0)$, $f(x)\leqslant 0$ always holds, find the range of values for the real number $a$;
(3) When $a > 0$, discuss the monotonicity of th... | (-\infty, \frac{2}{e}] |
2. Given that $A, B, C, D$ are four non-coplanar points on a sphere, and $A B=B C=A D=1, B D=A C=\sqrt{2}, B C \perp$ $A D$, then the volume of the sphere is $\qquad$ . | \frac{\sqrt{3}}{2}\pi |
14.1.2 ** Find all pairs of positive integers $(a, b)$, such that: $a^{2}+b+1$ is a power of a prime, $a^{2}+b+1$ divides $b^{2}-a^{3}-1$, but does not divide $(a+b-1)^{2}$. | =2^{},b=2^{2}-1,=2,3,4,\cdots |
If the front view of a horizontally placed cylinder is similar to its lateral development view, then the ratio of the lateral area to the total area of this cylinder is ( )
A: $\dfrac{\sqrt{\pi}}{\sqrt{\pi}+1}$
B: $\dfrac{2\sqrt{\pi}}{2\sqrt{\pi}+1}$
C: $\dfrac{2}{2\sqrt{\pi}+1}$
D: $\dfrac{1}{\sqrt{\pi}+1}$ | B |
10. For the non-empty subsets of the set $A=\{1,2, \cdots, 10\}$, subsets whose elements' sum is a multiple of 10 are called good subsets. Therefore, the number of good subsets of $A$ is $\qquad$. | 103 |
A square garden has an area of 9 square metres. The perimeter of the garden is $N$ metres. What is the value of $N$ ?
## | 12 |
14) How many angles greater than $90^{\circ}$ can a quadrilateral (non-intersecting) have?
(A) It always has at least one
(B) it has at most one
(C) it has at most two
(D) it has at most three
(E) it can have four. | D |
Given the function $f(x)=\sin (ωx+φ)$, where $ω > 0$ and $|φ| < \dfrac {π}{2}$.
(1) If $\cos \dfrac {π}{4}\cos φ-\sin \dfrac {3π}{4}\sin φ=0$, find the value of $φ$;
(2) Under the condition of (1), if the distance between the adjacent symmetry axes of the function $f(x)$ graph is $\dfrac {π}{3}$, find the analytical ex... | \dfrac {π}{12} |
9. Using 6 different digits $a, b, c, d, e, f$ to form six three-digit numbers $\overline{a b c}, \overline{b c d}, \overline{c d e}, \overline{d e f}, \overline{e f a}, \overline{f a b}$. If $\frac{\overline{a b c}+\overline{b c d}+\overline{c d e}+\overline{d e f}+\overline{e f a}+\overline{f a b}}{3}$ is a perfect s... | 256789 |
5. Several teams are divided into 4 groups, with at least two teams in each group. Each group conducts a round-robin tournament (each pair of teams in the group plays one match), with a total of 66 matches played. Question: How many teams are there in total? (List all possible numbers of participating teams) | 21,22,23,24,25 |
The function $y=\cos^2\left(x+ \frac{\pi}{4}\right)-\sin^2\left(x+ \frac{\pi}{4}\right)$ is ( )
A: An odd function with a period of $\pi$
B: An even function with a period of $\pi$
C: An odd function with a period of $2\pi$
D: An even function with a period of $2\pi$ | \text{A} |
There are 5 high school graduates applying for universities, with 3 universities to choose from. Each person can only fill in one preference. The number of possible application schemes is ( )
A: 15
B: 8
C: $3^5$
D: $5^3$ | \text{C} |
30. [15] The function $f: \mathbb{Z}^{2} \rightarrow \mathbb{Z}$ satisfies
- $f(x, 0)=f(0, y)=0$, and
- $f(x, y)=f(x-1, y)+f(x, y-1)+x+y$
for all nonnegative integers $x$ and $y$. Find $f(6,12)$. | 77500 |
Find the solution set for the inequality $x^2 < x$. | (0, 1) |
Given a complex number $z$ satisfies $(1+i)z=1-3i$ (where $i$ is the imaginary unit),
(1) Find the imaginary part of the complex number $z$;
(2) If the complex number $(1+ai)z$ is a pure imaginary number, find the value of the real number $a$;
(3) If the conjugate of the complex number $z$ is denoted as $\overset... | \frac { \sqrt {5}}{2} |
3. 1. 12 ** Given the sequence $a_{k}=2^{k}, k=1,2, \cdots, n$. Find the sum of all possible products $a_{i} a_{j}(1 \leqslant$ $i \leqslant j \leqslant n)$. | \frac{4}{3}(2^{n}-1)(2^{n+1}-1) |
Given \\(f(x)= \begin{cases} 2^{x}, & x\leqslant 0 \\ \log_{\frac{1}{2}}x, & x > 0\end{cases}\\), then \\(f[f(4)]=\\) \_\_\_\_\_\_. | \frac{1}{4} |
21st CanMO 1989 Problem 3 Tranform a number by taking the sum of its digits. Start with 1989 1989 and make four transformations. What is the result? | 9 |
Let $n$ be a positive integer. Determine the smallest positive integer $k$ with the following property: it is possible to mark $k$ cells on a $2 n \times 2 n$ board so that there exists a unique partition of the board into $1 \times 2$ and $2 \times 1$ dominoes, none of which contains two marked cells. | 2n |
Given $a \lt b$, which of the following expressions may not hold true?
A: $a-1 \lt b-1$
B: $-2a \gt -2b$
C: $2a+1 \lt 2b+1$
D: $m^{2}a \gt m^{2}b$ | D |
1. The product of four numbers - the roots of the equations $x^{2}+2 b x+c=0$ and $x^{2}+2 c x+b=0$, is equal to 1. It is known that the numbers $b$ and $c$ are positive. Find them. | b==1 |
Given $a=0.5^{\frac{1}{2}}$, $b=0.5^{\frac{1}{3}}$, and $c=0.5^{\frac{1}{4}}$, the relationship among $a$, $b$, and $c$ is ( )
A: $a>b>c$
B: $ab>c$ | B |
5. If one of the legs of a right triangle is 12, and the other two sides are integers, then the number of right triangles that meet this condition is ( ).
(A) 1
(B) 6
(C) 4
(D) infinitely many | C |
7. The inverse function of $y=f(x)$ is $y=f^{-1}(x), y=f(x-1)$ passes through the point $(3,3)$. Then the graph of the function $y=f^{-1}(x+2)$ must pass through the point | (1,2) |
In the Cartesian coordinate system, if point $P(m,m-n)$ is symmetric to point $Q(2,3)$ with respect to the origin, then point $M(m,n)$ is in the
A: First quadrant
B: Second quadrant
C: Third quadrant
D: Fourth quadrant | B |
To obtain the graph of the function $y=\sin 2x$, one only needs to shift the graph of $y=\sin (2x+ \frac {\pi}{4})$ by
A: shifting it left by $\frac {\pi}{8}$ units
B: shifting it right by $\frac {\pi}{8}$ units
C: shifting it left by $\frac {\pi}{4}$ units
D: shifting it right by $\frac {\pi}{4}$ units | B |
Example 6. (1MO-27 Preliminary Question) Let $A, B, C$ be three points on the edge of a circular pool, with $B$ due west of $C$, and $A B C$ forming an equilateral triangle with side lengths of 86 meters. A swimmer starts from $A$ and swims directly to $B$. After swimming $x$ meters, he reaches point $\boldsymbol{E}$, ... | 12 |
A woman from a village started at 10:31 and arrived at village $B$ at 13:43, traveling at a constant speed. On the same day, a man started from $B$ at 9:13 on the same route and arrived at $A$ at 11:53, also at a constant speed. They met at a bridge along the way, and after passing each other, the woman left the bridge... | 11:13 |
In a soccer match, based on the attacking and defending situation on the field, the goalkeeper will move back and forth in front of the goal. If we take the goal line as the reference, moving forward is recorded as a positive number, and moving backward is recorded as a negative number. The movement record of a goalkee... | 3 |
Determine the constant term in the expansion of the binomial $(4^{x} - 2^{-x})^{6}$ when $x\in R$. | 15 |
6. (10 points) The sum of 13 different natural numbers is 996, and the sum of the digits of these numbers are all equal. Find these 13 numbers.
The text above is translated into English, preserving the original text's line breaks and format. | 6,15,24,33,42,51,60,105,114,123,132,141,150 |
F19 (43-5, India) Find all functions $f$ from the set of real numbers $\mathbf{R}$ to $\mathbf{R}$ such that for all $x, y, z, t \in \mathbf{R}$, we have
$$
(f(x)+f(z))(f(y)+f(t))=f(x y-z t)+f(x t+y z) .
$$ | f(x)=0,f(x)=\frac{1}{2},f(x)=x^2 |
Task B-2.4. (20 points) Two circles with diameters of $6 \mathrm{~cm}$ and $18 \mathrm{~cm}$ touch each other externally. Calculate the area of the figure bounded by the circles and their common external tangent. | 36\sqrt{3}-\frac{33\pi}{2} |
Given an ellipse with foci on the $x$-axis and a minor axis length of 4, the eccentricity is $\frac{\sqrt{5}}{5}$.
$(1)$ Find the standard equation of the ellipse;
$(2)$ If line $l$ passes through the left focus of the ellipse and intersects the ellipse at points $M$ and $N$, with $\left| MN \right|=\dfrac{16}{9}\... | y = x + 1 \quad \text{or} \quad y = -x - 1 |
Example 1 Find all non-square numbers $n \in \mathbf{N}_{+}$, such that $[\sqrt{n}]^{3} \mid n^{2}$. (1)
(3rd Northern Mathematical Olympiad Invitational Competition) | n=2,3,8,24 |
What is the least positive integer $m$ such that the following is true?
[i]Given $\it m$ integers between $\it1$ and $\it{2023},$ inclusive, there must exist two of them $\it a, b$ such that $1 < \frac ab \le 2.$ [/i] \[\mathrm a. ~ 10\qquad \mathrm b.~11\qquad \mathrm c. ~12 \qquad \mathrm d. ~13 \qquad \mathrm e. ~... | 12 |
For example, in the sequence $x_{1}, x_{2}, \cdots, x_{n}, \cdots$, the sum of any three consecutive terms is 20, and $x_{1}=9, x_{12}=7$. Find the value of $x_{2000}$. | 4 |
18. A uniformly mass-distributed cubic die is randomly rolled repeatedly until an odd number appears. Then, the probability that each even number appears at least once before an odd number appears is ( ).
(A) $\frac{1}{120}$
(B) $\frac{1}{32}$
(C) $\frac{1}{20}$
(D) $\frac{3}{20}$
(E) $\frac{1}{6}$ | \frac{1}{20} |
The value of $1.000+0.101+0.011+0.001$ is
(A) 1.112
(B) 1.113
(C) 1.111
(D) 1.1111
(E) 1.101 | 1.113 |
4. Let $f(x)$ be a polynomial function of degree 2016 whose 2016 zeroes have a sum of $S$. Find the sum of the 2016 zeroes of $f(2 x-3)$ in terms of $S$. | \frac{1}{2}S+3024 |
The intercept of the tangent line of the graph of the function $f\left(x\right)=x^{3}-2x^{2}+3x+1$ at $x=1$ on the $x$-axis is ( ).
A: $1$
B: $\frac{1}{2}$
C: $-\frac{1}{2}$
D: $0$ | C |
Find the smallest positive integer $n$ such that there are at least three distinct ordered pairs $(x,y)$ of positive integers such that \[x^2-y^2=n.\] | 45 |
4. Point $B$ lies on segment $A C$ of length 5. Find the distance between the midpoints of segments $A B$ and $B C$.
---
Note: The translation preserves the original formatting and structure of the text. | 2.5 |
Find the roots of the equation $1-x+\frac{x(x-1)}{2!}+\frac{x(x-1)(x-2)}{3!}+\ldots+(-1)^{n} \frac{x(x-1) \ldots(x-n+1)}{n!}=0$. | 1,2,\ldots,n |
Simplify first and then evaluate $(\frac{x^2}{x+1}-x+1)÷\frac{x^2-1}{x^2+2x+1}$. Choose a value for $x$ that you like, substitute it into the expression, and calculate the result. | 1 |
8,9
What is the maximum number of numbers that can be chosen from the set $1,2, \ldots, 1963$, so that the sum of no two numbers is divisible by their difference? | 655 |
When $x=$, the value of the fraction $\frac{1-x^2}{x-1}$ is $0$. | -1 |
Let points $F_1(-3,0)$ and $F_2(3,0)$. If the locus of point $M$ satisfying the condition $|MF_1|+|MF_2|=2m+1$ is an ellipse, and the locus of point $N$ satisfying the condition $\left| |NF_1|-|NF_2| \right| = 2m-1$ is a hyperbola, then the range of the real number $m$ is __________. | \left( \dfrac{5}{2}, \dfrac{7}{2} \right) |
(Finding all functions from $\mathbb{R}-\{0,1\}$ to $\mathbb{R}$ such that
$$
f(x)+f\left(\frac{1}{1-x}\right)=x
$$ | f(x)=\frac{1}{2}(x+1-\frac{1}{x}-\frac{1}{1-x}) |
\section*{Problem 3 - 151033}
When printing a math problem, instead of \(\left(1+a^{2} x^{2}\right): x^{2}=b\) (with given numbers \(a, b\)), the equation \(\left(1+a^{2} x^{2}\right) \cdot x^{2}=b\) (with the same numbers \(a, b\)) was printed by mistake.
Despite this, the resulting equation had the same non-empty s... | {1,-1} |
A cube with sides of length 3cm is painted red
and then cut into 3 x 3 x 3 = 27 cubes with sides of length 1cm.
If a denotes the number of small cubes (of 1cm x 1cm x 1cm) that
are not painted at all, b the number painted on one sides, c the
number painted on two sides, and d the number painted on three
sides, determin... | -9 |
Given sets $A=\{x|x^{2}-4x-5=0\}$ and $B=\{x|x^{2}-1=0\}$, find $A\cup B$. | \{-1,1,5\} |
1) The sequence $\{2 n-1\}\left(n \in \mathbf{N}^{*}\right)$ is grouped in the original order with the requirement that the $n$-th group has $2^{n}$ terms. Then 2009 is in the ( ) group.
(A) 7
(B) 8
(C) 9
(D) 10 | 9 |
Let's determine the area of the right-angled triangle, given that $\alpha=38^{\circ} 40^{\prime}$ and the angle bisector of $\alpha$ is $f_{a}=7.8 \mathrm{~cm}$. | 21.67\mathrm{~}^{2} |
A unit radius circle is surrounded in the plane by $n$ equal radius circles, such that each of them externally touches the unit circle and the two adjacent circles in the "chain". Determine the radius of the circles as a function of $n$. Also, calculate the radii to four decimal places for the first four possible value... | \frac{\sin\frac{\pi}{n}}{1-\sin\frac{\pi}{n}} |
Given that the sequence $\{a_n\}$ is an arithmetic sequence with the sum of the first $n$ terms denoted as $S_n (n \in \mathbb{N}^*)$, and $\{b_n\}$ is a geometric sequence with the first term being 2 and a common ratio greater than 0. It is known that $b_2 + b_3 = 12$, $b_3 = a_4 - 2a_1$, and $S_{11} = 11b_4$.
1. Fin... | T_n = (3n - 4) \cdot 2^{n+2} + 16 |
5. The unit square in the first quadrant of the coordinate plane ( $0 \leq x, y \leq 1$ ) is divided into squares with side length $2 \cdot 10^{-4}$. How many nodes of this partition (inside the unit square) lie on the parabola $y=x^{2}$? | 49 |
The equation of the circle with endpoints A(-3, -1) and B(5, 5) as the diameter is ______. | (x - 1)^2 + (y - 2)^2 = 25 |
G1.3 Let $n$ be the product 3659893456789325678 and 342973489379256 . Determine the number of digits of $n$. | 34 |
Which of the following options is a quadratic equation in one variable $x$?
A: $x^{2}+3x-2=x^{2}$
B: $3x^{2}+2x+4=0$
C: $ax^{2}+bx+c=0$
D: $x^2+\frac{1}{x}=1$ | B |
Which of the following square roots can be combined with $\sqrt{3}$?
A: $\sqrt{18}$
B: $\sqrt{48}$
C: $\sqrt{54}$
D: $\sqrt{63}$ | B |
There is more than one integer greater than $1$ which, when divided by any integer $k$ such that $2 \le k \le 11$, has a remainder of $1$.
What is the difference between the two smallest such integers?
$\textbf{(A) }2310\qquad \textbf{(B) }2311\qquad \textbf{(C) }27,720\qquad \textbf{(D) }27,721\qquad \textbf{(E) }\t... | 27720 |
6. Let $n$ be the answer to this problem. $a$ and $b$ are positive integers satisfying
$$
\begin{array}{l}
3 a+5 b \equiv 19 \quad(\bmod n+1) \\
4 a+2 b \equiv 25 \quad(\bmod n+1)
\end{array}
$$
Find $2 a+6 b$.
Proposed by: David Vulakh | 96 |
Three robots, A, B, and C, are racing over 110 meters hurdles. When A reaches the finish line, B is 1 meter behind, and C is 2 meters behind; when B reaches the finish line, C is still 1.01 meters behind. After the race, C protests, claiming, "The track is inaccurate, the results should be invalidated." How many meters... | \text{A} |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.