problem stringlengths 10 7.47k | answer stringlengths 1 1.22k |
|---|---|
7.230. $\log _{x+1}(x-0.5)=\log _{x-0.5}(x+1)$. | 1 |
The medians of the sides $AC$ and $BC$ in the triangle $ABC$ are perpendicular to each other. Prove that $\frac12 <\frac{|AC|}{|BC|}<2$. | \frac{1}{2} < \frac{|AC|}{|BC|} < 2 |
Example 4.2.11 Use $1 \times 2$ dominoes to cover a $4 \times 4$ chessboard, how many covering schemes are there. | 36 |
Problem 4. Tamarа walked down the street and greeted Sasha, who was standing on the sidewalk. After walking $30 \mathrm{~m}$, Sasha started chasing her. After how many steps will Sasha catch up with Tamarа, if the length of Sasha's step is $85 \mathrm{~cm}$, and the length of Tamarа's step is $75 \mathrm{~cm}$? Explain... | 300 |
Given $9^x - 12 \cdot 3^x + 27 \leq 0$, find the range of the function $y = \log_2^2 x - \log_2 x + 2$. | \left[\frac{7}{4}, 2\right] |
Example 5. Convert $1-\frac{1}{4} \sin ^{2} 2 \alpha-\sin ^{2} \beta-\cos ^{4} \alpha$ into a product form. | \sin (\alpha+\beta) \sin (\alpha-\beta) |
Construct a right-angled triangle iron frame with an area of $1m^{2}$, using iron pipes of the following four lengths, which is the most reasonable (sufficient and least wasteful) option?
A: $3.5m$
B: $4.8m$
C: $5m$
D: $5.2m$ | \text{C} |
Determine the largest positive integer $n$ which cannot be written as the sum of three numbers bigger than $1$ which are pairwise coprime. | 17 |
3. [15] Let $p(x)=a_{n} x^{n}+a_{n-1} x^{n-1}+\ldots+a_{0}$, where each $a_{i}$ is either 1 or -1 . Let $r$ be a root of $p$. If $|r|>\frac{15}{8}$, what is the minimum possible value of $n$ ? | 4 |
If $f(x)=3-2x$, then the solution set for $|f(x+1)+2| \leqslant 3$ is __________. | [0, 3] |
Given that $p$ and $q$ are simple propositions, the statement "the proposition $p \vee q$ is true" is a ( ) condition for the statement "the proposition $p \land q$ is false".
A: Sufficient but not necessary condition
B: Necessary but not sufficient condition
C: Necessary and sufficient condition
D: Neither sufficient ... | D |
Example 16 (Mathematical Olympiad Problems from "Intermediate Mathematics" Issue 4, 2004) Try to find the maximum number of elements in a set $S$ that satisfies the following conditions:
(1) Each element in $S$ is a positive integer not exceeding 100;
(2) For any two different elements $a$ and $b$ in $S$, there exists ... | 50 |
The period of the function $y = \cos^2 x - \sin^2 x$ is . | \pi |
8. Given the ellipse $\frac{x^{2}}{4}+\frac{y^{2}}{3}=1$ with left and right foci $F_{1}, F_{2}$, a line $l$ passing through the right focus intersects the ellipse at points $P, Q$. Then the maximum value of the area of the incircle of $\triangle F_{1} P Q$ is $\qquad$ | \frac{9\pi}{16} |
Given a point P(1, -2) on the terminal side of angle $\alpha$, then $$\frac {\sin\alpha+2\cos\alpha}{\sin\alpha -\cos\alpha }$$ equals \_\_\_\_\_\_. | 0 |
1. Find the number of points in the $x O y$ plane having natural coordinates $(x, y)$ and lying on the parabola $y=-\frac{x^{2}}{4}+5 x+39$. | 12 |
16. Find the sum: $1+3+5+\ldots+1999$. | 1000000 |
2. As shown in Figure 1, given that $G$ is the centroid of $\triangle A B O$. If $P Q$ passes through point $G$, and
$$
\begin{array}{l}
O A=a, O B=b, \\
O P=m a, O Q=n b,
\end{array}
$$
then $\frac{1}{m}+\frac{1}{n}=$ $\qquad$ | 3 |
Example 8. Given $1 x=(3+2 \sqrt{2})^{-1}$, $y=(3-2 \sqrt{2})^{-1}$, find the value of $(x+1)^{-1}+(y+1)^{-1}$. | 1 |
In the arithmetic sequence $\{a_n\}$, if $a_3=2$, then the sum of the first 5 terms of this sequence is ( )
A: 10
B: 16
C: 20
D: 32 | A |
2. The number of positive integer solutions to the system of equations $\left\{\begin{array}{l}x y+y z=63, \\ x z+y z=23\end{array}\right.$ is ( ).
(A) 1
(B) 2
(C) 3
(D) 4 | B |
Example 7 Find the smallest positive period of the function $f(x)=\sin 2 x+2 \cos 2 x+3 \tan 4 x$. | \pi |
One. (20 points) Arrange natural numbers in a triangular number table as shown in Figure 4, following the principle of increasing from top to bottom and from left to right (each row has one more number than the previous row). Let $a_{i j}\left(i, j \in \mathbf{N}_{+}\right)$ be the number located in the $j$-th position... | \frac{5}{2}-\frac{1}{n}-\frac{1}{n+1} |
The fraction $\frac{1}{x+1}$ is meaningful when ______. | x \neq -1 |
The function $f(x)=2\sin ωx (ω > 0)$ is monotonically increasing on the interval $[0, \frac{π}{3}]$, and the maximum value on this interval is $\sqrt{2}$. Determine the value of $ω$. | \frac{3}{4} |
3. As shown in Figure 1, quadrilateral DEFG is inscribed in $\triangle A B C$, and it is known that the areas of $\triangle A D E$, $\triangle E F C$, and $\triangle D B G$ are $1$, $3$, and $1$, respectively. Therefore, the area of $\square D E F G$ is ( ).
(A) $2 \sqrt{3}$
(B) 2
(C) 3
(D) 4 | D |
The area of the figure enclosed by the curve \\(y = \dfrac{1}{x}\\) and the lines \\(y = x\\), \\(y = 2\\) is \\(({ })\\)
A: \\(3 + \ln 2\\)
B: \\(3 - \ln 2\\)
C: \\(\dfrac{3}{2} + \ln 2\\)
D: \\(\dfrac{3}{2} - \ln 2\\) | D |
Six points $A, B, C, D, E, F$ are chosen on a circle anticlockwise. None of $AB, CD, EF$ is a diameter. Extended
$AB$ and $DC$ meet at $Z, CD$ and $FE$ at $X, EF$ and $BA$ at $Y. AC$ and $BF$ meets at $P, CE$ and $BD$ at $Q$ and $AE$ and $DF$ at $R.$ If $O$ is the point of intersection of $YQ$ and $ZR,$ find the $\ang... | 90^\circ |
Calculate $3 \times 37$ then $27 \times 37$, then give a criterion for divisibility by 37. | \sum_{k=0}^{3}(n_{3k}+10n_{3k+1}-11n_{3k+2}) |
4. [4] Circle $\omega$ has radius 5 and is centered at $O$. Point $A$ lies outside $\omega$ such that $O A=13$. The two tangents to $\omega$ passing through $A$ are drawn, and points $B$ and $C$ are chosen on them (one on each tangent), such that line $B C$ is tangent to $\omega$ and $\omega$ lies outside triangle $A B... | 17 |
1. Find all pairs of prime numbers $p, q$ for which there exists a natural number $a$ such that
$$
\frac{p q}{p+q}=\frac{a^{2}+1}{a+1}
$$ | p=q=2p=q=5 |
Given $\sin \alpha =\frac{3}{5}$, and $\alpha$ is an angle in the second quadrant.
$(1)$ Find the value of $\tan \alpha$;
$(2)$ Find the value of $\frac{{2\sin\alpha+3\cos\alpha}}{{\cos\alpha-\sin\alpha}}$. | \frac{6}{7} |
The solution to the equation $\log_{2}(x+1)^{2}+\log_{4}(x+1)=5$ is. | 3 |
Pedrinho is playing with three triangular pieces with sides $(5,8,10),(5,10,12)$ and $(5,8,12)$ as shown in the drawing below. He can join two pieces by exactly gluing the sides of the same length. For example, he can join the side 10 of the first piece with the side 10 of the second, but he cannot join the side 10 of ... | 49 |
13.135. A team of mechanics can complete a certain task of processing parts 15 hours faster than a team of apprentices. If the team of apprentices works for 18 hours on this task, and then the team of mechanics continues working on the task for 6 hours, only 0.6 of the entire task will be completed. How much time does ... | 45 |
Let $x \in \mathbb{R}$. A necessary but not sufficient condition for $2x > 2$ is ( )
A: $x > -1$
B: $x 3$
D: $x < 3$ | \text{A} |
5. What remainder do we get when the number $2002^{2001}$ is divided by 2003? Justify your answer.
Mathematical Competition for High School Students in Slovenia
## Optional Competition
March 28, 2003
## Problems for 3rd Year Students | 2002 |
Given the sequence $\{a\_n\}$ satisfying: $a\_1=1$, $a_{n+1}= \frac{a\_n}{2a\_n+3}(n∈N^{})$, find $a\_4$. | \frac{1}{53} |
On $x-y$ plane, let $C: y=2006x^{3}-12070102x^{2}+\cdots.$
Find the area of the region surrounded by the tangent line of $C$ at $x=2006$ and the curve $C.$ | \frac{1003}{6} |
Given an ellipse $(C)$: $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (a > b > 0)$, with an eccentricity of $\frac{\sqrt{3}}{2}$, and two foci $F\_1$, $F\_2$. A line passing through $F\_1$ intersects the ellipse $(C)$ at points $M$ and $N$, and the perimeter of $\triangle F\_2MN$ is $8$.
(1) Find the equation of the ellipse ... | 2 |
12. If $\sec \alpha \sqrt{1+\tan ^{2} \alpha}+\tan \alpha \sqrt{\csc ^{2} \alpha-1}=\tan ^{2} \alpha$, determine the sign of the product $\sin (\cos \alpha) \cdot \cos (\sin \alpha)$. | positive |
In a soccer league with 6 teams $(P, Q, R, S, T, W)$, each team must eventually play each other team exactly once. So far, $P$ has played one match, $Q$ has played 2 matches, $R$ has played 3 matches, $S$ has played 4 matches, and $T$ has played 5 matches. How many matches has $W$ played so far?
(A) 1
(B) 2
(C) 3
(D) 4... | 3 |
Given an arithmetic sequence with a first term of $1$, an even number of terms, the sum of its odd terms is $341$, and the sum of its even terms is $682$. What is the number of terms in this sequence?
( )
A: $4$
B: $6$
C: $8$
D: $10$ | \text{D} |
4. A horse stands at the corner of a chessboard, a white square. With each jump, the horse can move either two squares horizontally and one vertically or two vertically and one horizontally (like a knight moves). The horse earns two carrots every time it lands on a black square, but it must pay a carrot in rent to rabb... | 0 |
19. The expression $\frac{7 n+12}{2 n+3}$ takes integer values for certain integer values of $n$.
What is the sum of all such integer values of the expression?
A 4
B 8
C 10
D 12
E 14 | 14 |
2. Let $m$ be a given positive integer, then the largest positive integer $n$ such that $m^{2}+n$ divides $n^{2}+m$ is $\qquad$ . | m^{4}-m^{2}+m |
Given the following four propositions, the correct one(s) is (are) ( )
① If $\cos(A-B)\cos(B-C)\cos(C-A)=1$, then $\triangle ABC$ is an equilateral triangle;
② If $\sin A=\cos B$, then $\triangle ABC$ is a right triangle;
③ If $\cos A\cos B\cos C<0$, then $\triangle ABC$ is an obtuse triangle;
④ If $\sin 2A=\s... | C |
5. Given real numbers $a, b$ satisfy $a>b>0, c$ satisfies that a triangle can be formed with $a, b, c$ as its side lengths. If the largest angle among the three interior angles of the triangle is to have the minimum value, then $c$ should be equal to ( ).
(A) $a$
(B) $\frac{a+b}{2}$
(C) $\sqrt{a b}$
(D) $b$ | A |
Among the following expressions, the one whose minimum value is $4$ is $(\quad)$.
A: $y=\dfrac{x}{2}+\dfrac{8}{x}$
B: $y=\sin x+\dfrac{4}{\sin x}$ $(0 < x < \pi)$
C: $y=\mathrm{e}^{x}+4\mathrm{e}^{-x}$
D: $y=\sqrt{x^{2}+1}+\dfrac{2}{\sqrt{x^{2}+1}}$ | \text{C} |
Problem 5. Ana wants to buy several pencils. If she buys 6 pencils, she will have 7 denars left, but if she buys 10 pencils, she will be short of 5 denars. How much does each pencil cost, and how many denars did Ana have? | 25 |
Determine all functions from $\mathbb{R}$ to $\mathbb{R}$ satisfying:
$$
\forall(x, y) \in \mathbb{R}^{2}, f(f(x)+9 y)=f(y)+9 x+24 y
$$ | f(x)=3x |
10.134. In an isosceles trapezoid, one base is equal to $40 \mathrm{~cm}$, and the other is 24 cm. The diagonals of this trapezoid are perpendicular to each other. Find its area. | 1024\mathrm{~}^{2} |
A set of data has a mean of 2.8 and a variance of 3.6. If each data point in this set is multiplied by 2 and then increased by 60, the mean and variance of the new data set are respectively ( )
A: 62.8, 3.6
B: 62.8, 14.4
C: 65.6, 3.6
D: 65.6, 14.4 | \text{D} |
4. Given an isosceles right triangle \(ABC\) with hypotenuse \(AB\). Point \(M\) is the midpoint of side \(BC\). On the smaller arc \(AC\) of the circumcircle of triangle \(ABC\), point \(K\) is chosen. Point \(H\) is the foot of the perpendicular dropped from \(K\) to line \(AB\). Find the angle \(\angle CAK\), given ... | 22.5 |
# Problem №2 (15 points)
An empty glass jar is three times lighter than the same jar filled to the brim with milk. If it is known that the volume occupied by the glass is ten times less than the volume occupied by the milk, then compare the densities of glass and milk. The density of a material is defined as the ratio... | \rho_{M}=0.2\rho_{CT} |
Given $a > b$, the quadratic inequality $ax^{2}+2x+b\geqslant 0$ holds for all real numbers $x$, and there exists $x_{0}\in\mathbb{R}$, such that $ax_{0}^{2}+2x_{0}+b=0$, then the minimum value of $\dfrac{a^{2}+b^{2}}{a-b}$ is $\boxed{2\sqrt{2}}$. | 2\sqrt{2} |
If angle $θ$ is in the fourth quadrant, then $\frac {π}{2}+θ$ is in which quadrant?
A: First quadrant
B: Second quadrant
C: Third quadrant
D: Fourth quadrant | A |
Let $x \in \mathbb{R}$, and let $i$ be the imaginary unit. Then, "$x = -3$" is the condition for "the complex number $z = (x^2 + 2x - 3) + (x - 1)i$ to be purely imaginary" is:
A: A sufficient but not necessary condition
B: A necessary but not sufficient condition
C: A necessary and sufficient condition
D: Neither a su... | C |
The solution set of the inequality $\left| \frac{x-2}{x} \right| > \frac{x-2}{x}$ is $(\quad)$
A: $(0,2)$
B: $(-\infty,0)$
C: $(2,+\infty)$
D: $(-\infty,0)\cup(0,+\infty)$ | A |
Toss a fair $1$ yuan coin $10$ times, with $9$ times landing heads and $1$ time landing tails. If the coin is tossed for the $11$th time, the probability of landing heads is ( )
A: $\frac{1}{10}$
B: $\frac{1}{2}$
C: $\frac{9}{10}$
D: Unable to determine | B |
For any $x \in \mathbb{R}$, the graph of the function $y=(k^2-k-2)x^2-(k-2)x-1$ is always below the x-axis. Find the range of the real number $k$. | \left(-\frac{2}{5}, 2\right] |
A bank employee is filling an empty cash machine with bundles of $\$ 5.00, \$ 10.00$ and $\$ 20.00$ bills. Each bundle has 100 bills in it and the machine holds 10 bundles of each type. What amount of money is required to fill the machine?
(A) $\$ 30000$
(B) $\$ 25000$
(C) $\$ 35000$
(D) $\$ 40000$
(E) $\$ 45000$
Part... | 35000 |
If the graph of the inverse proportion function $y=\frac{k}{x}(k≠0)$ passes through the point $\left(3,-2\right)$, then the graph of this inverse proportion function is in which quadrant?
A: First and second quadrants
B: First and third quadrants
C: Second and third quadrants
D: Second and fourth quadrants | D |
2. In the final stage of a professional bowling tournament, the top five players compete as follows: First, the fifth-place player competes with the fourth-place player, the loser gets fifth place, the winner competes with the third-place player; the loser gets third place, the winner competes with the first-place play... | 16 |
If the equation $x^2 - my^2 + 2x + 2y = 0$ represents two lines, find the value of $m$. | m = 1 |
The number of white balls and red balls in a jar is in the ratio of $3: 2$.
If there are 9 white balls, how many red balls are there?
(A) 5
(B) 8
(C) 2
(D) 6
(E) 3 | 6 |
10.415 A line parallel to the bases of a given rectangular trapezoid divides it into two trapezoids, into each of which a circle can be inscribed. Find the bases of the original trapezoid, if its lateral sides are equal to $c$ and $d$, where $c<d$. | \frac{(\sqrt{+}\\sqrt{-})^2}{4} |
p1. Determine what is the smallest positive integer by which the number $2010$ must be multiplied so that it is a perfect square.
p2. Consider a chessboard of $8\times 8$. An [i]Alba trajectory[/i] is defined as any movement that contains $ 8$ white squares, one by one, that touch in a vertex. For example, the white ... | 2010 |
The sequence $\{2^{n}-1\}$ forms a set $A_{n}=\{1,3,7,…,2^{n}-1\}$ with its first $n$ terms. From the set $A_{n}$, any $k(k=1,2,…,n)$ numbers are chosen, and the sum of all possible products of these $k$ numbers is denoted as $T_{k}$ (if only one number is chosen, the product is defined as the number itself). Let $S_{n... | 2^{\frac {n(n+1)}{2}}-1 |
Example 5: A total of 240 students participating in a large group performance stand in a row facing the coach, and they report numbers in sequence from left to right as 1, 2, 3, 4, 5, ... The coach asks all students to remember their reported numbers and perform the following actions: First, all students who reported n... | 118 |
Given the function $f(x)=4x^{3}-3x^{2}-18x+27$, $x\in R$.
$(1)$ Find the monotonic intervals and extreme values of $f(x)$;
$(2)$ Find the maximum and minimum values of $f(x)$ on the interval $[0,3]$. | \frac{27}{4} |
$-(a^{3})^{4}=\_\_\_\_\_\_.$ | -a^{12} |
\section*{Problem 4 - 061034}
Determine all real numbers \(k\) for which the equation
\[
x^{2}+x+3=k\left(x^{2}+5\right)
\]
is a quadratic equation in \(x\) that
a) has a double root!
b) has two distinct real roots! | \frac{1}{2}<k<\frac{11}{10} |
Given the curve $C$: $x^2 + y^2 - 2x - 4y + m = 0$, where $O$ is the coordinate origin.
(1) For what value of $m$ does the curve $C$ represent a circle?
(2) If the curve $C$ intersects with the line $x + 2y - 3 = 0$ at points $M$ and $N$, and $OM \perp ON$, find the value of $m$. | m = \frac{12}{5} |
Given a sequence $\{a_n\}$ whose sum of the first $n$ terms is $S_n$, and it satisfies $S_n + \frac{1}{2}a_n = 1$ for $n \in \mathbb{N}^*$,
$(1)$ Find the general formula for the sequence $\{a_n\}$;
$(2)$ Let $b_n = \log_{3}(1-S_{n+1})$ for $n \in \mathbb{N}^*$, find the value of the positive integer $n$ that satisfi... | 100 |
Sharygin I.F.
The angle at the vertex $A$ of the isosceles triangle $ABC (AB = AC)$ is $20^{\circ}$. On the side $AB$, we lay off the segment $AD$, equal to $BC$. Find the angle $BCD$. | 70 |
Among the following sets of three line segments, the one that cannot form a triangle is:
A: $2cm$, $3cm$, $4cm$
B: $3cm$, $6cm$, $6cm$
C: $2cm$, $2cm$, $6cm$
D: $5cm$, $6cm$, $7cm$ | C |
8. As shown in Figure 3, given a regular tetrahedron $O-ABC$ with three lateral edges $OA, OB, OC$ mutually perpendicular, and each of length $2, E, F$ are the midpoints of edges $AB, AC$ respectively, $H$ is the midpoint of line segment $EF$, and a plane is constructed through $EF$ intersecting the lateral edges $OA, ... | \sqrt{5} |
Given the function $f(x) = \begin{cases} x^2 + 5x + 2, & x \leqslant a \\ x + 2, & x > a \end{cases}$, the function $g(x) = f(x) - 2x$ has exactly three distinct zeros. Determine the range of values for $z = 2^a$.
A: $\left[\frac{1}{2}, 2\right)$
B: $[1, 4]$
C: $\left[\frac{1}{4}, 4\right)$
D: $\left[\frac{1}{2}, 4\ri... | D |
I1.1 Let $A=15 \times \tan 44^{\circ} \times \tan 45^{\circ} \times \tan 46^{\circ}$, find the value of $A$. | 15 |
2. Find the value of $\sum_{k=3}^{1000}\left\lfloor\log _{2} k\right\rfloor$. | 7986 |
Among the following 4 inequalities: (1) $\int_{0}^{1} \sqrt{x} \, dx < \int_{0}^{1} 3x \, dx$; (2) $\int_{0}^{\frac{\pi}{4}} \sin x \, dx < \int_{0}^{\frac{\pi}{4}} \cos x \, dx$;
(3) $\int_{0}^{1} e^{-x} \, dx < \int_{0}^{1} e^{-x^2} \, dx$; (4) $\int_{0}^{2} \sin x \, dx < \int_{0}^{2} x \, dx$. The number of in... | \text{D} |
Solve the equation: $\frac{x}{x+1}=\frac{3}{x}+1$. | x = -\frac{3}{4} |
A point $A(2,1)$ is taken inside the circle $M: (x-1)^{2}+y^{2}=4$ and a chord $BC$ passing through $A$ intersects the circle at points $B$ and $C$. Tangents $PB$ and $PC$ are drawn through points $B$ and $C$ respectively. The two tangents intersect at point $P$. The equation of the locus of point $P$ is:
A: $y-5=0$
... | \text{C} |
Using the rounding method, approximate $0.6457$ to the nearest hundredth. Which of the following is correct?
A: $0.6$
B: $0.65$
C: $0.646$
D: $0.645$ | B |
3. (7 points) Given an arithmetic progression. The sum of its first 10 terms is 60, and the sum of its first 20 terms is 320. What can the 15th term of this progression be? | 25 |
Given that the polar coordinate equation of curve $C\_1$ is $ρ \cos \left( \left. θ- \frac{π}{3} \right. \right)=-1$, and the polar coordinate equation of curve $C\_2$ is $ρ=2 \sqrt{2}\cos \left( \left. θ- \frac{π}{4} \right. \right)$. With the pole as the coordinate origin and the polar axis as the positive half of th... | \frac{3+ \sqrt{3}+2 \sqrt{2}}{2} |
Given the function, $f(x)=a\ln x-ax-3(a\in\mathbb{R})$.
(1) When $a=1$, find the intervals of monotonicity for the function $f(x)$.
(2) If the tangent line to the graph of $y=f(x)$ at the point $(2,f(2))$ has an angle of inclination of $45^{\circ}$, for what range of values of $m$ does the function $g(x)=x^{3}+x^{2... | - \frac{37}{3} < m < -9 |
Let $ a$ and $ b$ be positive real numbers with $ a\ge b$. Let $ \rho$ be the maximum possible value of $ \frac{a}{b}$ for which the system of equations
\[ a^2\plus{}y^2\equal{}b^2\plus{}x^2\equal{}(a\minus{}x)^2\plus{}(b\minus{}y)^2\]has a solution in $ (x,y)$ satisfying $ 0\le x<a$ and $ 0\le y<b$. Then $ \rho^2$ ca... | 7 |
The imaginary part of the complex number $z= \frac {3+i}{1+i}$ is ( )
A: $-i$
B: $i$
C: $-1$
D: $1$ | C |
To implement the "double reduction" policy, Teacher Liu divided the $25$ students in the class into several groups for mutual learning. Each group can only consist of $2$ or $3$ students. How many possible grouping schemes are there?
A: $4$ schemes
B: $3$ schemes
C: $2$ schemes
D: $1$ scheme | \text{A} |
Let set $A=\{1,2,\ldots,n\}$, where $n\geq4$, $n\in\mathbb{N}^{*}$. If $X\subseteq A$, and $2\leq\text{Card}(X)\leq n-2$ (where $\text{Card}(X)$ represents the number of elements in set $X$), let $a_X$ represent the sum of the largest and smallest numbers in $X$, then
(1) When $n=5$, the number of sets $X$ is $\boxed... | n+1 |
19. Within 60, there are four distinct prime numbers arranged in ascending order to form an arithmetic sequence (the difference between adjacent numbers is equal), there are $\qquad$ possible cases. | 7 |
17. Find the digits $a$ and $b$ in the number $\overline{42 a 4 b}$, given that this number is divisible by 72. | =8,b=0or=0,b=8 |
A certain school is a traditional school for table tennis sports. In order to further promote the development of this sport, the school is planning to purchase several pairs of straight paddles and horizontal paddles from a sports equipment store. For every pair of paddles purchased, they must also buy 10 table tennis ... | 10000 \text{ yuan} |
Let \\(f(x)=3^{x}+3x-8\\). When using the bisection method to find an approximate solution of the equation \\(3^{x}+3x-8=0\\) within \\(x∈(1,2)\\), it was found that \\(f(1) 0\\), and \\(f(1.25) < 0\\). Therefore, the root of the equation falls within the interval
A: \\((1,1.25)\\)
B: \\((1.25,1.5)\\)
C: \\((1.5,2)\\)... | \text{B} |
7. (10 points) The Spring Festival is approaching. Starting from January 17, 2011 (Monday), workers in the factory gradually return home for the New Year to reunite with their families. If the number of workers leaving the factory each day is the same, by January 31, there are still 121 workers left in the factory. Dur... | 120 |
## Problem Statement
Find the cosine of the angle between vectors $\overrightarrow{A B}$ and $\overrightarrow{A C}$.
$A(5 ; 3 ;-1), B(5 ; 2 ; 0), C(6 ; 4 ;-1)$ | -\frac{1}{2} |
2. At a cactus lovers' meeting, 80 cactus enthusiasts presented their collections, each consisting of cacti of different species. It turned out that no species of cactus is present in all collections at once, but any 15 people have cacti of the same species. What is the smallest total number of cactus species that can ... | 16 |
3. (17 points) In triangle $A B C \quad A B=4, B C=6$, angle $A B C$ is $30^{\circ}, B D-$ is the bisector of triangle $A B C$. Find the area of triangle $A B D$. | 2.4 |
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