problem stringlengths 10 7.47k | answer stringlengths 1 1.22k |
|---|---|
After the line $y=2x$ is translated $3$ units downwards, the resulting line passes through the point $\left(m+2,-5\right)$. Find the value of $m$. | -3 |
9.4. Find the minimum value of the expression $(\sqrt{2(1+\cos 2 x)}-\sqrt{3-\sqrt{2}} \sin x+1) \cdot(3+2 \sqrt{7-\sqrt{2}} \cos y-\cos 2 y)$. If the answer is not an integer, round it to the nearest integer. | -9 |
1. If $\frac{100}{x}>\sqrt{1994}-25>\frac{125}{x+2}$, then the positive integer $x$ is ( ).
(A) 4
(B) 5
(C) 6
(D) 7 | B |
392. A rectangular plate is immersed in water, positioned vertically. Its horizontal side is 1 m, the vertical side is $2 \mathrm{m}$. The top side is at a depth of 0.5 m. Determine the force of water pressure on the plate. | 29430 |
27. [15] Let $\left\{a_{i}\right\}_{i \geq 0}$ be a sequence of real numbers defined by
$$
a_{n+1}=a_{n}^{2}-\frac{1}{2^{2020 \cdot 2^{n}-1}}
$$
for $n \geq 0$. Determine the largest value for $a_{0}$ such that $\left\{a_{i}\right\}_{i \geq 0}$ is bounded. | 1+\frac{1}{2^{2020}} |
The conjugate complex number of $z=\frac{1}{1+i}$ (where $i$ is the imaginary unit) corresponds to a point in the complex plane located in quadrant ( ).
A: 1
B: 2
C: 3
D: 4 | A |
Problem 4. A square area was paved with square tiles (all tiles are the same). A total of 20 tiles adjoin the four sides of the area. How many tiles were used in total?
 | 36 |
Example 4.2.5. Let $n \in \mathbb{N}$. Find the minimum value of the following expression
$$f(x)=|1+x|+|2+x|+\ldots+|n+x|, \quad(x \in \mathbb{R})$$ | \left\{\begin{array}{l}
m(m+1) \text { if } n=2 m(m \in \mathbb{N}) \\
(m+1)^{2} \text { if } n=2 m+1(m \in \mathbb{N})
\end{array}\right.} |
Given vectors $\overrightarrow {m}=(\cos\alpha- \frac { \sqrt {2}}{3}, -1)$, $\overrightarrow {n}=(\sin\alpha, 1)$, and $\overrightarrow {m}$ is collinear with $\overrightarrow {n}$, and $\alpha\in[-\pi,0]$.
(Ⅰ) Find the value of $\sin\alpha+\cos\alpha$;
(Ⅱ) Find the value of $\frac {\sin2\alpha}{\sin\alpha-\cos\al... | \frac {7}{12} |
The graph of the function $f(x)=\sin (2x- \frac {\pi}{6})$ is translated to the right by $\frac {\pi}{12}$ units. One of the axes of symmetry of the resulting graph is __( )__.
A: $x= \frac {\pi}{4}$
B: $x= \frac {3\pi}{8}$
C: $x= \frac {5\pi}{12}$
D: $x= \frac {7\pi}{24}$ | C |
Initially Alex, Betty, and Charlie had a total of $444$ peanuts. Charlie had the most peanuts, and Alex had the least. The three numbers of peanuts that each person had formed a geometric progression. Alex eats $5$ of his peanuts, Betty eats $9$ of her peanuts, and Charlie eats $25$ of his peanuts. Now the three number... | 108 |
Given that the center of the ellipse $C$ is at the origin, its right focus is at $(\sqrt{3}, 0)$, and its eccentricity is $\frac{\sqrt{3}}{2}$.
(I) Find the equation of the ellipse $C$.
(II) If the line $l: y = kx + \sqrt{2}$ always intersects the ellipse $C$ at two distinct points $A$ and $B$, and $\overrightarrow{O... | k \in \left(-\frac{\sqrt{3}}{3}, -\frac{1}{2}\right) \cup \left(\frac{1}{2}, \frac{\sqrt{3}}{3}\right) |
Given the complex number $z=4+3i$, where $i$ is the imaginary unit, find the modulus of $z$ and the value of $\frac{1+i}{z}$. | \frac{7+i}{25} |
Let \\(P(x,y)\\) be a point on the curve \\(C\\): \\( \sqrt { \frac {x^{2}}{25}}+ \sqrt { \frac {y^{2}}{9}}=1\\), with \\(F_{1}(-4,0)\\) and \\(F_{2}(4,0)\\). Find the maximum value of \\(|PF_{1}|+|PF_{2}|\\). | 10 |
Let $a$, $b$, $c$, $d$, $e$, $f$ and $g$ be seven distinct positive integers not bigger than $7$. Find all primes which can be expressed as $abcd+efg$ | 179 |
Ancient Chinese mathematical work "Zengshan Suanfa Tongzong" records the "measuring rod with rope" problem: "One rod and one rope, the rope is longer than the rod by a 'tuo'. When the rope is folded back to measure the rod, it is shorter than the rod by a 'tuo'." The gist of it is: there is a rod and a rope, and when u... | \text{A} |
7. $A_{1}(i=1,2,3,4)$ are subsets of the set $S=\{1,2, \cdots, 2005\}$, $F$ is the set of all ordered quadruples $\left(A_{1}, A_{2}, A_{3}, A_{1}\right)$, then the value of $\sum_{1}\left|A_{1} \cup A_{2} \cup A_{3} \cup A_{4}\right|$ is $\qquad$ . | 2^{8016}\times2005\times15 |
Convert the number 89 to base-5. | 324_5 |
In a certain group, each member has a probability $p$ of using mobile payment, and the payment methods are independent of each other. Let $X$ be the number of people out of 10 members in the group who use mobile payment, with a variance of $DX=2.4$ and $P(X=4) < P(X=6)$. Determine the value of $p$. | p = 0.6 |
Example 3. Find the unit vector and direction cosines of the vector $\vec{a}=\{6,-2,-3\}$. | \cos\alpha=\frac{6}{7},\cos\beta=-\frac{2}{7},\cos\gamma=-\frac{3}{7} |
If set $A=\{x \,|\, |x|=x\}$, and $B=\{x \,|\, x^2+x \geq 0\}$, then $A \cap B=$ ? | [0, +\infty) |
Let $\triangle A B C$ be a triangle such that the height relative to side $B C$ is not less than side $B C$ and the height relative to side $A B$ is not less than side $A B$. Determine the measures of the angles of this triangle. | 45,45,90 |
Given a sequence ${a_n}$ that satisfies ${a_{n+1}} + {a_n} = (n+1)\cos \frac{n\pi}{2}$ $(n\geqslant 2, n\in \mathbb{N^*})$, and ${S_n}$ is the sum of the first $n$ terms of the sequence. If ${S_{2017}} + m = 1012$ and ${a_1}m > 0$, find the minimum value of $\frac{1}{{a_1}} + \frac{1}{m}$. | 1 |
From the numbers \\(1, 2, 3, 4, 5\\), two different numbers are randomly selected. Let event \\(A\\) be "the sum of the two numbers is even", and event \\(B\\) be "both numbers are even". Then, \\(P(B|A)=\\) . | \dfrac{1}{4} |
9. Given the ellipse
$$
\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1(a>b>0)
$$
with an eccentricity of $\frac{\sqrt{3}}{2}$, a line with a slope of 1 passing through the point $M(b, 0)$ intersects the ellipse at points $A$ and $B$. Let $O$ be the origin. If
$$
\overrightarrow{O A} \cdot \overrightarrow{O B}=\frac{32}{5} ... | \frac{x^{2}}{16}+\frac{y^{2}}{4}=1 |
1. Given is $\triangle A B C$, over whose side $A C$ a circle $k$ is constructed as if over a diameter. The circle $k$ passes through the midpoint of side $B C$, and intersects side $A B$ at point $D$ such that $A D=\frac{3}{2} D B$. If $A C=60$, calculate the area of $\triangle A B C$. | 1440 |
Proizvolov V.V.
Given fifty different natural numbers, twenty-five of which do not exceed 50, and the rest are greater than 50 but do not exceed 100. Moreover, no two of them differ exactly by 50. Find the sum of these numbers. | 2525 |
2. Every hour, Martina reads an equal number of pages of a book. First, Martina calculated how many pages she should read every hour to finish the book exactly by the given deadline. Then she calculated that she would finish reading the entire book 2.5 hours before the deadline if she read two more pages per hour than ... | 360 |
The management department of a seafood market plans to build a wholesale greenhouse with an area of 2400m2, containing a total of 80 storefronts of type A and type B. The average area of each type A storefront is 28m2 with a monthly rent of 400 yuan; each type B storefront has an average area of 20m2 with a monthly ren... | 40 |
Let $n$ be a positive integer and consider an arrangement of $2n$ blocks in a straight line, where $n$ of them are red and the rest blue. A swap refers to choosing two consecutive blocks and then swapping their positions. Let $A$ be the minimum number of swaps needed to make the first $n$ blocks all red and $B$ be the ... | n^2 |
Let $f(n)$ be the smallest prime which divides $n^4+1$. What is the remainder when the sum $f(1)+f(2)+\cdots+f(2014)$ is divided by $8$?
$
\textbf{(A)}\ 1
\qquad\textbf{(B)}\ 3
\qquad\textbf{(C)}\ 5
\qquad\textbf{(D)}\ 7
\qquad\textbf{(E)}\ \text{None of the preceding}
$ | 5 |
If the line $y=kx$ and the circle $(x-1)^2+y^2=1$ have two intersection points that are symmetric with respect to the line $x-y+b=0$, then the values of $k$ and $b$ are ( )
A: $k=-1$, $b=1$
B: $k=-1$, $b=-1$
C: $k=1$, $b=1$
D: $k=1$, $b=-1$ | \text{B} |
1. Given real numbers $a$, $b$, $c$ satisfy $a^{2}+2 b=7$, $b^{2}-2 c=-1$, $c^{2}-6 a=-17$. Then the value of $a+b+c$ is ( ).
(A) 2
(B) 3
(C) 4
(D) 5 | B |
Three tickets to Jingyuetan Park and two tickets to Changying Century City are placed into five identical envelopes. Xiao Ming randomly draws one envelope. The probability that the envelope contains exactly a ticket to Jingyuetan Park is ____. | \frac{3}{5} |
Given the function $f(x) = \sqrt{3} (\sin^2(x) - \cos^2(x)) + 2\sin(x)\cos(x)$.
(1) Find the smallest positive period of $f(x)$;
(2) If $x \in [0, \frac{\pi}{3}]$, find the range of $f(x)$. | [-\sqrt{3}, \sqrt{3}] |
6. $100 n$ positive numbers $x_{1}, x_{2}, \cdots, x_{n}$ sum up to 1. Let $S$ be the largest of the following numbers:
$$\frac{x_{1}}{1+x_{1}}, \frac{x_{2}}{1+x_{1}+x_{2}}, \cdots, \frac{x_{n}}{1+x_{1}+x_{2}+\cdots+x_{n}} .$$
Find the smallest possible value of $S$. For what values of $x_{1}, x_{2}, \cdots, x_{n}$ is... | 1-2^{-\frac{1}{n}} |
An arc of a circle measures $300^{\circ}$ and its length is $2 \mathrm{~km}$. What is the nearest integer to the measure of the radius in meters?
(a) 157
(b) 284
(c) 382
(d) 628
(e) 764 | 382 |
Let the function $f(x)$ defined on $\mathbb{R}$ satisfy $f(-x) + f(x) = x^2$. If $f'(x) < x$ for $x < 0$, then the solution set for the inequality $f(x) + \frac{1}{2} \geq f(1-x) + x$ is _________. | x \in (-\infty, \frac{1}{2}] |
Which is the largest positive integer that is 19 times larger than the sum of its digits? | 399 |
10.2 In a company of $n$ people, 100000 ruble coins need to be distributed equally. How many different values of $n$ exist for which such a distribution is possible? | 36 |
15. As shown in Figure 4, in tetrahedron $ABCD$, $AB \perp BC$, $BC \perp CD$, $CD \perp AB$.
(1) Identify the side face that is perpendicular to face $BCD$, and prove it;
(2) If $AB = BC = 1$,
$CD = x$, the dihedral angle $C-AD$
$-B$ has a plane angle $\alpha$, $\sin \alpha$
$= f(x)$, find the expression for $f(x)$ an... | f(x)=\frac{1}{\sqrt{2}} \sqrt{1+\frac{1}{1+x^{2}}}\left(x \in \mathbf{R}_{+}\right), \left(\frac{\pi}{4}, \frac{\pi}{2}\right) |
Given a moving point P on the circle C: $x^2 + y^2 - 2x + 2y = 0$, find the product of the maximum and minimum distances from point P to the line $x - y + 1 = 0$. | \frac{5}{2} |
Given an arithmetic-geometric sequence $\{a\_n\}$ with a sum of the first $n$ terms denoted as $S\_n$ and a common ratio of $\frac{3}{2}$.
(1) If $S\_4 = \frac{65}{24}$, find $a\_1$;
(2) If $a\_1=2$, $c\_n = \frac{1}{2}a\_n + bn$, and $c\_2$, $c\_4$, $c\_5$ form an arithmetic sequence, find $b$. | -\frac{3}{16} |
Given $\overrightarrow{a}=(3,2)$ and $\overrightarrow{b}=(6,y)$, if $\overrightarrow{a} \parallel \overrightarrow{b}$, then $y$ equals ( ).
A: $-9$
B: $-4$
C: $4$
D: $9$ | C |
On a Cartesian coordinate plane, points $(1, 2)$ and $(7, 4)$ are opposite vertices of a square. What is the area of the square? | 20 |
Given vectors $\overrightarrow{a}=(\sin x, \sqrt {3}\sin x)$ and $\overrightarrow{b}=(\sin x,\cos x)$, where $x\in[0, \dfrac {\pi}{2}]$.
(I) Find the value of $x$ when $|\overrightarrow{a}|=|\overrightarrow{b}|$;
(II) Define a function $f(x)=\overrightarrow{a}\cdot\overrightarrow{b}$. By shifting the graph of $f(x)$ to... | \dfrac {\pi}{6} |
If the function $f(x) = (2m-1)x^3$ is a power function, then $m=$ . | 1 |
Given that the sum of the coefficients of all odd terms in the binomial expansion of $(\sqrt{x} - \sqrt[3]{x})^n$ is 512, find all the rational terms (terms with an integer exponent) in the expansion. | x^5, \binom{10}{4} x^4 |
Given the partial correspondence between the independent variable $x$ and the function $y$ in the quadratic function $y=ax^{2}+bx+c$ as shown in the table:
| $x$ | $\ldots $ | $0$ | $1$ | $2$ | $3$ | $\ldots $ |
|-----|-----------|-----|-----|-----|-----|-----------|
| $y$ | $\ldots $ | $-1$| $2$ | $3$ | $2$ | $\ldots... | D |
II. (Full marks for this question:
20 points)
Through a point $P$ outside $\odot O$
draw a tangent $P_{N}$ to $\odot O$,
with the point of tangency being $N$. Let $M$ be the midpoint of $P N$. Then the circle passing through $P$ and $M$
intersects $\odot O$ at $A$,
$B$, and the extension of $B A$ intersects $P N$ at ... | P M=3 M Q |
12. In the arithmetic sequence $\left\{a_{n}\right\}$, it is known that
$$
a_{20} \simeq \frac{1}{a}, a_{201}=\frac{1}{b}, a_{2012}=\frac{1}{c} \text {. }
$$
Then $1992 a c-1811 b c-181 a b=$ | 0 |
Let the function $f(x) = \sin(\omega x + \varphi)$. Consider the condition P: "$f(0) = 0$" and the condition Q: "$f(x)$ is an odd function". Then, P is the ( ) condition for Q.
A: Necessary and sufficient condition
B: Sufficient but not necessary condition
C: Necessary but not sufficient condition
D: Neither sufficient... | \text{A} |
3. Determine all pairs $(a, b)$ of real numbers for which the quadratic trinomials $P(x)=$ $=x^{2}+a x+b$ and $Q(x)=x^{2}+b x+a$ have the following property: each of the equations
$$
a P(x)+b Q(x)=0 \quad \text{and} \quad a Q(x)+b P(x)=0
$$
is a quadratic equation with a double root.
(Jaroslav Švrček) | (4,4) |
The length of the interval of solutions of the inequality $a \le 2x + 3 \le b$ is $10$. What is $b - a$?
$\mathrm{(A)}\ 6 \qquad \mathrm{(B)}\ 10 \qquad \mathrm{(C)}\ 15 \qquad \mathrm{(D)}\ 20 \qquad \mathrm{(E)}\ 30$ | 20 |
If proposition $p$ is true, and proposition $q$ is false, then among the following, the true proposition is ( )
A: $p \land q$
B: $p \land (\lnot q)$
C: $(\lnot p) \lor q$
D: $(\lnot p) \land (\lnot q)$ | \text{B} |
2. On a line, several points were marked, including points $A$ and $B$. All possible segments with endpoints at the marked points are considered. Vasya calculated that point $A$ is inside 50 of these segments, and point $B$ is inside 56 segments. How many points were marked? (The endpoints of a segment are not consider... | 16 |
Let $a$, $b$, and $c$ be the lengths of the sides of triangle $ABC$. Prove that $$a^2 + b^2 + c^2 < 2(ab + bc + ac).$$ | a^2 + b^2 + c^2 < 2(ab + bc + ac) |
3. From a right prism with a square base of side length $L_{1}$ and height $H$, we extract a frustum of a pyramid, not necessarily a right one, with square bases of side lengths $L_{1}$ (for the lower base) and $L_{2}$ (for the upper base), and height $H$. The two resulting pieces are shown in the following image.
 x+a^{2}+a+3+\frac{1}{a} \quad(a \neq 0)$ intersects the x-axis at two points. In which direction do the branches of this parabola point - upwards or downwards? | downwards |
\section*{Task 2 - V11122}
The Octavia Touring Sport Car from Škoda Automobile Works Prague reaches a speed of \(80 \frac{\mathrm{km}}{\mathrm{h}}\) in 14 seconds after starting.
a) How many kilometers has it covered in this time (assuming uniform acceleration)? b) In what time, starting from the moment of the start,... | 52\mathrm{~} |
2. For a positive integer $n$, let $1 \times 2 \times \cdots \times n=n$!. If $M=1!\times 2!\times \cdots \times 10!$, then the number of perfect cubes among the positive divisors of $M$ is ( ).
(A) 468
(B) 684
(C) 846
(D) 648 | 468 |
Given the function $f(x)=\ln x+\dfrac{1}{\ln x}$, determine the correct conclusion from the options below:
( )
A: If ${{x}\_{1}},{{x}\_{2}}({{x}\_{1}} 0$ and $x\ne 1, f(x)\geqslant 2$
D: There exists ${{x}\_{0}} > 0$ such that $f(x)$ is an increasing function on $({{x}\_{0}},+\infty )$ | D |
In triangle ABC, the angles opposite to sides a, b, and c are A, B, and C, respectively. They satisfy the condition: cosA(b - 2c) = a(2cosC - $\sqrt{3}$sinB), and c = 2b, $|$$\overrightarrow{AB}$$ + 2$$\overrightarrow{AC}$$|$ = 6. Find the value of a. | 3 |
9.1. Find the smallest integer $x$ that satisfies the inequality $\frac{100}{|x|}>x^{2}+1$. | -4 |
Problem 4. Joseph solved 6 problems correctly, Darko solved 5 problems incorrectly, and Petre had the same number of correct and incorrect solutions. How many problems did all three solve correctly together, if Joseph solved twice as many problems correctly as Darko? | 13 |
Task B-2.4. Dundo and Maro were competing by giving each other unusual math problems. Thus, Maro was given the task to calculate the exact value of the cosine of an angle of $36^{\circ}$. He was not allowed to use any tools except paper and a pencil for writing and calculating.
It's impossible, Maro thought, until Dun... | \frac{1+\sqrt{5}}{4} |
Example 5. As shown in the figure, in $\triangle A B C$, $D, E$ are points on $B C, C A$ respectively, $B E, A D$ intersect at $H$, and $\frac{B D}{D C}=\frac{1}{3}, \frac{C E}{E A}=\frac{1}{2}, S_{\triangle A B C}=1$. Find the area of $\triangle A H E$. | \frac{4}{9} |
Solve the following equation:
$$
\sin x + \cos x = \sqrt{\frac{3}{2}}.
$$ | x_{1}=\frac{\pi}{12}\2k\pi,\quadx_{2}=\frac{5\pi}{12}\2k\pi |
Let $F_1(-4,0)$ and $F_2(4,0)$ be fixed points, and let the moving point $M$ satisfy $|MF_1|+|MF_2|=8$. Then, the trajectory of the moving point $M$ is ( )
A: Ellipse
B: Straight line
C: Circle
D: Line segment | \text{D} |
Given that $a$ and $b$ are two different lines, and $\alpha$ and $\beta$ are two different planes, a sufficient condition for $a \perp b$ is ( )
A: $a \perp \alpha$, $b \parallel \beta$, $\alpha \perp \beta$
B: $a \parallel \alpha$, $b \perp \beta$, $\alpha \parallel \beta$
C: $a \perp \alpha$, $b \perp \beta$, $\alpha... | \text{B} |
If $f(x)$ is an odd function defined on $\mathbb{R}$ and $f(x) = \log(x+1)$ when $x > 0$, then $f(x) = (\ \ \ )$ when $x < 0$.
A: $\log(1-x)$
B: $-\log(x+1)$
C: $-\log(1-x)$
D: None of the above | \text{C} |
Problem 130. Given an angle of $54^{\circ}$. Divide it into three equal parts using a compass and a straightedge. | 18 |
Example 1-5 The number of $n$-ary Boolean functions $f\left(x_{1}, x_{2}, \cdots, x_{n}\right)$. | 2^{2^{n}} |
If the point $\left(m-4,1-2m\right)$ is in the third quadrant, then the range of $m$ is ______. | \frac{1}{2} < m < 4 |
1. A social club has $2 k+1$ members, each of whom is proficient in the same $k$ languages, and any two members can only communicate in one language. If there do not exist three members who can communicate pairwise in the same language, let $A$ be the number of subsets of three members, where each subset's three member... | \frac{2 k(k-2)(2 k+1)}{3} |
Given that the sum of the first $n$ terms of the sequence $\{a_{n}\}$ is $S_{n}$, $2S_n=n(a_1+a_n)$, where $n \in \mathbb{N}^*$.
$(1)$ Prove that the sequence $\{a_{n}\}$ is an arithmetic sequence.
$(2)$ If the sequence $\{a_{n}\}$ satisfying the condition in $(1)$ has $S_{3}=6$ and $S_{5}=15$, let $b_n=\frac{a_n}{(n+1... | T_n < 1 |
1. What is the last digit of $11^{6}+14^{6}+16^{8}$? | 3 |
3. As shown in Figure 1, given a square $A B C D$ with a side length of $5, E$ and $F$ are two points outside the square, $B E=D F=4, A E=C F=$ 3. Then $E F=$ . $\qquad$ | 7 \sqrt{2} |
1. There are 70 apples in two boxes. Taking 6 apples from the first box and putting them into the second box makes the number of apples in both boxes equal. Originally, there were $\qquad$ apples in the first box. | 41 |
A triangle has sides $a, b, c$, and its area is $\frac{(a+b+c)(a+b-c)}{4}$. What is the measure of the largest angle of the triangle? | 90 |
4. In a cube $A B C D-A_{1} B_{1} C_{1} D_{1}$ with edge length 1, points $E$, $F$, and $G$ are the midpoints of edges $A A_{1}$, $C_{1} D_{1}$, and $B C$ respectively. Then the volume of the tetrahedron $B_{1} E F G$ is $\qquad$ | \frac{3}{16} |
5. Given 9 points in space, where no four points are coplanar, a line segment is drawn between every pair of points. These line segments can be colored blue or red, or left uncolored. Find the minimum value of $n$ such that if any $n$ of these line segments are arbitrarily colored red or blue, the set of these $n$ line... | 33 |
Exercise 14. Let $p$ be a prime number and $a_{k} \cdots a_{0}$ its representation in base 10. We define
$$
Q_{p}(x)=a_{k} x^{k}+a_{k-1} x^{k-1}+\cdots+a_{1} x+a_{0}
$$
Show that $\mathrm{Q}_{p}$ has no integer roots, except for 4 values of $p$ which we will determine. | 11,13,17,19 |
For each positive integer $n$ we consider the function $f_{n}:[0,n]\rightarrow{\mathbb{R}}$ defined by $f_{n}(x)=\arctan{\left(\left\lfloor x\right\rfloor \right)} $, where $\left\lfloor x\right\rfloor $ denotes the floor of the real number $x$. Prove that $f_{n}$ is a Riemann Integrable function and find $\underset{n... | \frac{\pi}{2} |
A group of 6 students decided to make [i]study groups[/i] and [i]service activity groups[/i] according to the following principle:
Each group must have exactly 3 members. For any pair of students, there are same number of study groups and service activity groups that both of the students are members.
Supposing there ... | 8 |
1. Mrs. Pekmezac loves fruit. In her garden, she harvested $5 \frac{3}{4} \mathrm{~kg}$ of blueberries, $4 \frac{1}{2} \mathrm{~kg}$ of cherries, $2 \frac{19}{20} \mathrm{~kg}$ of raspberries, and $1 \frac{4}{5} \mathrm{~kg}$ of blackberries. She decided to store the fruit in the refrigerator, in bags of 75 dag. Fruit ... | 20 |
Given the function $f(x) = x^{3} + \frac{5}{2}x^{2} + ax + b$ (where $a, b$ are constants), with its graph being the curve $C$.
(1) When $a = -2$, find the monotonic decreasing interval of the function $f(x)$;
(2) Let the derivative of the function $f(x)$ be $f'(x)$. If there exists a unique real number $x_{0}$ such ... | \lambda = 4 |
In triangle $\triangle ABC$, let $a$, $b$, and $c$ be the lengths of the sides opposite to angles $A$, $B$, and $C$ respectively. Then, "$a≤\frac{1}{2}({b+c})$" is a ( ) condition for angle $A$ to be acute.
A: Sufficient but not necessary condition
B: Necessary but not sufficient condition
C: Necessary and suffici... | \text{A} |
Given that points $A$ and $B$ are any two points symmetric about the origin on the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ where $a>0$ and $b>0$, and point $P$ lies on the hyperbola (distinct from points $A$ and $B$), if the product of the slopes of lines $PA$ and $PB$ is $\frac{5c-4a}{2a}$, then the eccentricity... | B |
7. In the tetrahedron $SABC$, $\triangle ABC$ is an equilateral triangle. The projection of point $A$ onto the plane $SBC$ is $H$, which is the orthocenter of $\triangle SBC$. The dihedral angle $H-AB-C$ has a plane angle of $30^{\circ}$, and $SA=2\sqrt{3}$. What is the volume of the tetrahedron $SABC$? | \frac{9}{4}\sqrt{3} |
3rd Centromerican 2001 Problem A3 Find all squares which have only two non-zero digits, one of them 3. | 36,3600,360000, |
Given the function $$f(x)= \frac {x^{2}}{1+x^{2}}$$, find the value of $$f(1)+f(2)+f(3)+…+f(2018)+f(1)+f( \frac {1}{2})+f( \frac {1}{3})+…f( \frac {1}{2018})$$.
A: 2017
B: 1009
C: 2018
D: 4036 | C |
Let $U=\{1,2,4,6,8\}$, $A=\{1,2,4\}$, $B=\{2,4,6\}$. Which of the following conclusions is correct?
A: $A\subseteq B$
B: $B\subseteq A$
C: $A\cap B=\{2\}$
D: $A\cap (\complement _{U}B)=\{1\}$ | D |
Given the function $f(x)=\ln x-ax$.
$(1)$ When $a= \frac {2}{e}$, find the equation of the tangent line to the function $f(x)$ at $x=e$;
$(2)$ If the solution set of the inequality $\ln x-ax > 0$ contains exactly one integer, find the range of the real number $a$. | \left[ \frac {1}{2}\ln 2, \frac {1}{3}\ln 3\right) |
Given that $x_{0}$ is a root of the equation $x^{3}e^{x-4}+2\ln x-4=0$, then the value of ${e^{\frac{{4-{x_0}}}{2}}}+2\ln{x_0}$ is:
A: $3$
B: $4$
C: $5$
D: $6$ | B |
Given two lines $l_1$ and $l_2$, if the product of their slopes is -1, then statement $q$: $l_1 \perp l_2$. The statement $p$ is a ( ) condition for $q$.
A: Sufficient but not necessary condition
B: Necessary but not sufficient condition
C: Both sufficient and necessary condition
D: Neither sufficient nor necessary con... | \text{A} |
Distribute four balls, labeled 1, 2, 3, and 4, randomly to four people: A, B, C, and D, with each person receiving one ball. The event "Person A gets ball number 1" and the event "Person B gets ball number 1" are ( )
A: Mutually exclusive but not complementary events
B: Complementary events
C: Independent events
D: Non... | \text{A} |
Given $a=\log _{5}2$, $b=\sin 55^{\circ}$, $c=0.5^{0.6}$, then which of the following is true?
A: $c \gt b \gt a$
B: $a \gt c \gt b$
C: $b \gt c \gt a$
D: $b \gt a \gt c$ | C |
Consider an $n \times n$ grid. A $T$-tetromino is a set of four cells of the grid in a "T" shape. Find the values of $n$ such that the $n \times n$ grid can be covered by $T$-tetrominoes without overlapping pieces. | 4\midn |
In triangle $ABC$, let the lengths of sides opposite to angles $A$, $B$, and $C$ be $a$, $b$, and $c$ respectively. Given vectors $\vec{m} = (\sin C, \sin B \cos A)$ and $\vec{n} = (b, 2c)$, and that $\vec{m} \cdot \vec{n} = 0$,
(1) Find the magnitude of $\angle A$;
(2) If $a = 2\sqrt{3}$ and $\sin B + \sin C = 1$, fin... | \sqrt{3} |
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