problem stringlengths 10 7.47k | answer stringlengths 1 1.22k |
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To promote her school's annual Kite Olympics, Genevieve makes a small kite and a large kite for a bulletin board display. The kites look like the one in the diagram. For her small kite Genevieve draws the kite on a one-inch grid. For the large kite she triples both the height and width of the entire grid.
[asy] for (in... | 189 |
Using the numbers $1$, $2$, $3$, and $4$ to form a four-digit number without repeating any digit, find the number of such four-digit numbers where exactly one even number is sandwiched between two odd numbers. | 8 |
Given $7A_6^x=20A_7^{x-1}$, where $x\in N_{+}$.
$(1)$ Find the value of $x$;
$(2)$ Find the value of $C_{20}^{20-x}+C_{17+x}^{x-1}$. | 1330 |
Standa and Jana received two three-digit numbers. Standa added a decimal point after the first digit in the first number, and after the second digit in the second number, then added the resulting decimal numbers and got the result 50.13. Jana added a decimal point after the second digit in the first number, and after t... | 765 |
3.375. Find $\operatorname{tg} \frac{x}{2}$, if it is known that $\sin x+\cos x=\frac{1}{5}$. | \operatorname{tg}\frac{x}{2}=2 |
Exercise 4. Find all functions $\mathrm{f}$ from $\mathbb{R}$ to $\mathbb{R}$ such that for all real numbers $x, y$,
$$
f(x) f(y)=f(x y)+x+y
$$ | f(x)=1+x |
2. Let $ABCD$ be a convex quadrilateral such that
$$
\measuredangle ABC=90^{\circ}, \measuredangle DAB=60^{\circ} \text{ and } \measuredangle BCD=120^{\circ}.
$$
Let the diagonals $AC$ and $BD$ intersect at point $M$, such that $\overline{BM}=1$ and $\overline{MD}=2$. Calculate the area of quadrilateral $ABCD$. | \frac{9}{2} |
Exercise 2. Anne drove her car for an integer (and non-zero) number of hours, and traveled an integer number of kilometers, at a speed of $55 \mathrm{~km} / \mathrm{h}$. At the beginning of the trip, the odometer read abc kilometers, where abc is a 3-digit number such that $a \geqslant 1$ and $a+b+c \leqslant 7$. At th... | 106 |
Given that $f(x)$ is an increasing function on $\mathbb{R}$, and for any $x \in \mathbb{R}$, it holds that $f\left[f(x)-3^x\right]=4$, find the value of $f(\log_3 9)$. | 10 |
Two dice are thrown. If at least one of the dice shows a 4 or a 5, the trial is considered successful. What is the expected number of successful trials, X, in 10 trials? | \frac{50}{9} |
7. Given the hyperbola $C: x^{2}-\frac{y}{3}=1$, with left and right foci $F_{1}$ and $F_{2}$, respectively. A line is drawn through point $F_{2}$ intersecting the right branch of the hyperbola $C$ at points $P$ and $Q$, such that $\angle F_{1} P Q=90^{\circ}$. Then the inradius of $\triangle F_{1} P Q$ is $\qquad$ | \sqrt{7}-1 |
Given an arithmetic progression $\{a_{n}\}$ with the sum of the first $n$ terms being $S_{n}=\left(x+2y+1\right)3^{n}+\left(x-y-4\right)$ (where $x \gt 0, y \gt 0$). The minimum value of $\frac{1}{x}+\frac{2}{y}$ is ______. | \frac{8}{3} |
13.2.1. Find $\bar{Y}$ and $\sigma_{y}$ for the sample
| $y_{j}$ | 0.5 | 1.3 | 2.1 | 2.9 | 3.7 |
| :---: | :---: | :---: | :---: | :---: | :---: |
| $m_{y_{j}}$ | 5 | 20 | 45 | 27 | 3 | | \bar{Y}=2.12,\sigma_{y}=0.71 |
Given $T_1 =2, T_{n+1}= T_{n}^{2} -T_n +1$ for $n>0.$ Prove:
(i) If $m \ne n,$ $T_m$ and $T_n$ have no common factor greater than $1.$
(ii) $\sum_{i=1}^{\infty} \frac{1}{T_i }=1.$ | \sum_{i=1}^{\infty} \frac{1}{T_i} = 1 |
Given a circle of radius $R$. Four circles of equal radii touch the given one externally, and each of these four circles touches two others. Find the radii of these four circles.
# | \frac{R}{\sqrt{2}-1} |
Determine the correct option(s) among the following derivatives:
\[
\begin{align*}
&(1)\left(\frac{\cos x}{x}\right)' = -\frac{\sin x}{x^2} \\
&(2)\left[(x^2 + x + 1)e^x\right]' = (2x + 1)e^x \\
&(3)\left(\frac{2x}{x^2 + 1}\right)' = \frac{2 - 2x^2}{(x^2 + 1)^2} \\
&(4)\left(e^{3x + 1}\right)' = 3e^{3x + 1}
\end{align*... | \text{B} |
## Problem II - 5
The natural numbers 22, 23, and 24 have the following property: the exponents of the prime factors in their factorization are all odd:
$$
22=2^{1} \cdot 11^{1} ; \quad 23=23^{1} ; \quad 24=2^{3} \cdot 3^{1}
$$
What is the largest number of consecutive natural numbers that can have this property? Ju... | 7 |
3A. The altitudes of the acute-angled triangle $ABC$ intersect at point $O$, such that $\overline{OC} = \overline{AB}$. Determine the angle $ACB$? | 45 |
I4.1 In triangle $A B C, \angle B=90^{\circ}, B C=7$ and $A B=24$. If $r$ is the radius of the inscribed circle, find $r$. | 3 |
Given an arithmetic sequence $\{a_n\}$ with the general term formula $a_n = 2n - 1$, insert one 2 between $a_1$ and $a_2$, two 2's between $a_2$ and $a_3$, and so on, inserting $n$ 2's between $a_n$ and $a_{n+1}$ to form a new sequence $\{b_n\}$. If $a_{10} = b_k$, find the value of $k$.
Options:
A: 45
B: 50
C: 5... | C |
(1) In the set $A=\{1,2,3, \cdots, 2011\}$, the number of elements whose last digit is 1 is $\qquad$ . | 202 |
Given that $a$, $b$, and $c$ are the sides opposite to angles $A$, $B$, and $C$ in $\triangle ABC$, and $\overrightarrow{AB}\cdot \overrightarrow{AC}=4$, and $ac\sin B=8\sin A$. Find:
$(1)$ the value of angle $A$;
$(2)$ the range of values for $\sin A\sin B\sin C$. | (0, \frac{3\sqrt{3}}{8}] |
Find the value of $x$ in the equation $\log _{3}(1+2\cdot 3^{x})=x+1$. | x = 0 |
8. Define the sequence $\left\{a_{n}\right\}: a_{n}=n^{3}+4, n \in \mathbf{N}_{+}$, let $d_{n}=\left(a_{n}, a_{n+1}\right)$, i.e., $d_{n}$ is the greatest common divisor of $a_{n}$ and $a_{n+1}$, then the maximum value of $d_{n}$ is $\qquad$. | 433 |
Example 8 $n$ teams are to hold a home-and-away double round-robin match (each pair of teams plays two matches, each having one home match), and each team can play multiple away matches within a week (from Sunday to Saturday). However, if a team has a home match in a certain week, no away matches for that team can be s... | 6 |
If positive numbers $a$ and $b$ satisfy $ab = a + b + 3$, then the range of values for $ab$ is \_\_\_\_\_\_. | [9, +\infty) |
Example 8. Find the integral $\int \cos \frac{x}{4} d x$. | 4\sin\frac{x}{4}+C |
3. For the system of equations $\left\{\begin{array}{l}x-y=2, \\ m x+y=3\end{array}\right.$, if the solution corresponds to a point in the first quadrant of the Cartesian plane, then the range of values for $m$ is | -1<m<\frac{3}{2} |
Example 3 Given that $a, b, c$ are three distinct real numbers. Try to solve the system of equations about $x, y, z$
$$
\left\{\begin{array}{l}
\frac{x}{a^{3}}-\frac{y}{a^{2}}+\frac{z}{a}=1, \\
\frac{x}{b^{3}}-\frac{y}{b^{2}}+\frac{z}{b}=1, \\
\frac{x}{c^{3}}-\frac{y}{c^{2}}+\frac{z}{c}=1 .
\end{array}\right.
$$ | x=abc, y=ab+bc+ca, z=a+b+c |
A certain factory packages a certain product in boxes, with each box containing $100$ items. Each box of products must be inspected before delivery to customers. If any defective items are found during inspection, they are replaced with qualified items. During inspection, 10 items are randomly selected from the box for... | \frac{3}{10} |
Given an ellipse C: $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1$ ($a > b > 0$) with two foci $F_1(-1, 0)$ and $F_2(1, 0)$, and the ellipse C passes through point $P\left(\frac{4}{3}, \frac{1}{3}\right)$. The equation of ellipse C is __________. | \frac{x^{2}}{2} + y^{2} = 1 |
Given the function $f(x)=|2x-1|+|x+1|$,
$(1)$ Solve the inequality $f(x) < 4$;
$(2)$ If there exists a real number $x_{0}$ such that the inequality $f(x_{0}) < |m+t|+|t-m|$ holds for any real number $t$, find the range of values for $m$. | (-\infty,- \frac{3}{4})\cup( \frac{3}{4},+\infty) |
141. How can we express with a single inequality sign that the signs of two real numbers $a$ and $b$ are the same? that they are different? | >0;<0 |
Estimate the value of $\sqrt{10}+2$ is in which of the following ranges?
A: between $2$ and $3$
B: between $3$ and $4$
C: between $4$ and $5$
D: between $5$ and $6$ | D |
Beatriz adores odd numbers. How many numbers between 0 and 1000 can she write using only odd digits? | 155 |
Given $m\in \mathbb{R}$, if the line $l_{1}$: $mx+y+1=0$ is parallel to the line $l_{2}$: $9x+my+2m+3=0$, then $m=$____. | m = 3 |
Definition: For a sequence $\{a_n\}$, the product of its first $n$ terms $T_n = a_1 \cdot a_2 \cdot \ldots \cdot a_n$, and the sequence $a_n = 2^{9-n}$, then among the following equations, the correct one is ( )
A: $T_1 = T_{19}$
B: $T_3 = T_{17}$
C: $T_5 = T_{12}$
D: $T_8 = T_{11}$ | C |
4. (5 points) Xiao Pang and Xiao Ya have a total of 66 science books and 92 story books. Among them, Xiao Pang's science books are twice as many as Xiao Ya's, and Xiao Ya's story books are three times as many as Xiao Pang's. How many more books does Xiao Ya have than Xiao Pang? $\qquad$ books. | 24 |
A rectangular solid with length, width, and height of 1, 2, and 3, respectively, is circumscribed by a sphere. The volume of this sphere is ( )
A: $14\pi$
B: $28\pi$
C: $\dfrac{7\sqrt{14}}{3}\pi$
D: $\dfrac{14\sqrt{14}}{3}\pi$ | C |
106. Easy Division. By dividing the number 8101265822784 by 8, you will see that the answer can be obtained simply by moving the 8 from the beginning to the end of the number!
Could you find a number starting with 7 that can be divided by 7 in such a simple way? | 7101449275362318840579 |
2.225. $\left(x^{2}-6 x+1+\left(\frac{\frac{x-3}{1+3 x}-\frac{x-5}{1+5 x}}{1+\frac{(x-5)(x-3)}{(1+5 x)(1+3 x)}}\right)^{-1}\right)^{1 / 2}$. | |x-3| |
Example 1 Find the minimum value of the function with real variables $x$ and $y$
$$
u(x, y)=x^{2}+\frac{81}{x^{2}}-2 x y+\frac{18}{x} \sqrt{2-y^{2}}
$$
(2nd "Hope Cup" National Mathematics Invitational Competition) | 6 |
In a box, there are $6$ products, including $4$ good ones and $2$ defective ones. Two products are randomly selected. (List all the elementary events)$(1)$ Find the probability of selecting exactly one defective product.$(2)$ Find the probability of selecting two good products.$(3)$ Find the probability of selecting a ... | \frac{3}{5} |
The graph of the quadratic function $y=ax^{2}-4ax+c\left(a \lt 0\right)$ passes through four points $A(-2$,$y_{1})$, $B(0$,$y_{2})$, $C(3$,$y_{3})$, $D(5$,$y_{4})$. Which of the following statements must be correct?
A: If $y_{1}y_{2} \gt 0$, then $y_{3}y_{4} \gt 0$
B: If $y_{2}y_{4} \lt 0$, then $y_{1}y_{3} \lt 0$
C... | \text{B} |
17. In the equation "中环杯是 + 最棒的 $=2013$", different Chinese characters represent different digits. What are the possible values of "中 + 环 + 杯 + 是 + 最 + 棒 + 的"? (If there are multiple solutions, please list them all.)
翻译结果：
17. In the equation "Zhong Huan Bei Shi + Zui Bang De $=2013$", different Chinese characters rep... | 24,33 |
Find all four-digit natural numbers $\overline{xyzw}$ with the property that their sum plus the sum of their digits equals $2003$. | 1978 |
20. (2002 Shanghai Spring College Entrance Examination) Triangular prism $O A B-O A_{1} B_{1}$, plane $O B B_{1} O_{1} \perp O A B, \angle O_{1} O B=$ $60^{\circ}, \angle A O B=90^{\circ}$, and $O B=O O_{1}=2, O A=\sqrt{3}$. Find:
(1) The size of the dihedral angle $O_{1}-A B-O$;
(2) The distance $d$ between the skew l... | \sqrt{3} |
In acute triangle $\triangle ABC$, the sides opposite angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. It is known that $2c\sin A = \sqrt{3}a$.
$(1)$ Find the measure of angle $C$.
$(2)$ If $b=2$ and $c=\sqrt{7}$, find the area of $\triangle ABC$. | S = \frac{3\sqrt{3}}{2} |
## Task 2 - 090922
Given is a cube with edge length $a_{1}$ and volume $V_{1}$, and a regular tetrahedron with edge length $a_{2}$ and volume $V_{2}$. For the edge lengths, it holds that $a_{1}: a_{2}=1: \sqrt{2}$.
Calculate the ratio $V_{1}: V_{2}$! | 3:1 |
9. (16 points) Given the parabola $y=x^{2}$ and the ellipse $(x-3)^{2}+4 y^{2}=1$, the focus of the parabola is $F$, and $A$ is a point on the ellipse. The line $F A$ intersects the parabola at points $B$ and $C$. Tangents to the parabola are drawn through points $B$ and $C$. Find the locus of the intersection of the t... | -\frac{1}{4}\quad(\frac{-3-\sqrt{33}}{64}\leqslantx\leqslant\frac{-3+\sqrt{33}}{64}) |
To protect the ecological environment, a mountainous area in our city has started to implement the policy of returning farmland to forest since 2005. It is known that at the end of 2004, the forest coverage area of this mountainous area was $a$ acres.
(1) Assuming the annual natural growth rate of forest coverage are... | 7.2\% |
21-12 For a convex $n$-sided polygon, if circles are constructed with each side as the diameter, the $n$-sided polygon must be completely covered by these $n$ circles. Then the maximum value of $n$ is
(A) 3.
(B) 4.
(C) 5.
(D) Greater than 5.
(China Fujian Province Fuzhou City High School Mathematics Competition, 1993) | 4 |
Problem 9.1. In a store, there are 20 items, the costs of which are different natural numbers from 1 to 20 rubles. The store has decided to have a promotion: when buying any 5 items, one of them is given as a gift, and the customer chooses which item to receive for free. Vlad wants to buy all 20 items in this store, pa... | 136 |
4. Solve the system $\left\{\begin{array}{c}x^{2}+y^{2}+2 x+6 y=-5 \\ x^{2}+z^{2}+2 x-4 z=8 \\ y^{2}+z^{2}+6 y-4 z=-3\end{array}\right.$. | (1;-2;-1),(1;-2;5),(1;-4;-1),(1;-4;5),(-3;-2;-1),(-3;-2;5),(-3;-4;-1),(-3;-4;5) |
Given the sequence 1, 0, 1, 0, ..., which of the following options cannot be its general term formula?
A: $$\frac {1}{2}[1+(-1)^{n+1}]$$
B: $$\sin^{2} \frac {n\pi}{2}$$
C: $$\frac {1}{2}[1+(-1)^{n}]$$
D: $$\frac {1-\cos n\pi}{2}$$ | \text{C} |
## Task B-1.2.
Matko collects stickers with portraits of athletes that are sold in packs of 5 stickers, and one pack costs 7 kuna. He gave $30 \%$ of his stickers to his younger brother, then his mother bought Matko 4 more packs of stickers. After that, Matko gave his brother a quarter of his current number of sticker... | 82 |
(6) Given $O$ is the circumcenter of $\triangle A B C$, $A B=2 a, A C=\frac{2}{a}, \angle B A C$
$120^{\circ}$. If $\overrightarrow{A O}=\alpha \overrightarrow{A B}+\beta \overrightarrow{A C}$, then the minimum value of $\alpha+\beta$ is $\qquad$. | 2 |
To obtain the graph of the function $y=\sin \left(3x- \frac {\pi}{4}\right)$, one only needs to shift the graph of the function $y=\cos 3x$ by
A: Moving it to the right by $\frac {\pi}{4}$ units
B: Moving it to the left by $\frac {\pi}{4}$ units
C: Moving it to the right by $\frac {3\pi}{4}$ units
D: Moving it to the l... | \text{A} |
An urn contains marbles of four colors: red, white, blue, and green. When four marbles are drawn without replacement, the following events are equally likely:
(a) the selection of four red marbles;
(b) the selection of one white and three red marbles;
(c) the selection of one white, one blue, and two red marbles; an... | 21 |
5. In an acute-angled triangle $\mathrm{ABC}$, a line $s$ passing through vertex A and the median from vertex B divide the height dropped from vertex C into three equal parts. In what ratio does the line $s$ divide the side BC? | 1:4 |
$17 \cdot 121$ In $\triangle A B C$, $\angle B A C=120^{\circ}$, point $P$ is a point inside $\triangle A B C$, then
(A) $P A+P B+P C < A B+A C$.
(C) $P A+P B+P C=A B+A C$.
(D) The size relationship between $P A+P B+P C$ and $A B+A C$ depends on the position of point $P$. (China Guangzhou, Wuhan, Fuzhou, etc. five citi... | D |
Example 16: Three piles of chess pieces, A, B, and C, are moved as follows: First, take chess pieces from pile A and place them into piles B and C according to the number of pieces in piles B and C; then, take chess pieces from pile B and place them into piles A and C according to the number of pieces in piles A and C;... | 13,7,4 |
In the Cartesian coordinate system, if point $P_{1}(a,-5)$ and $P_{2}(3,b)$ are symmetric with respect to the $y$-axis, then $a+b=\_\_\_\_\_\_.$ | -8 |
Given the function $f(x)=2\sin(\pi-x)\cos(x)-2\sqrt{3}\cos^2(x)$.
$(1)$ Find the maximum value of $f(x)$.
$(2)$ In acute triangle $\triangle ABC$, where $a$, $b$, $c$ are the sides opposite to angles $A$, $B$, $C$ respectively, if $f(A)=0$, $b=4$, and $c=3$, find the length of the altitude $AD$ from side $BC$. | \frac{6\sqrt{39}}{13} |
Given $2 \leqslant \int_{1}^{2} (k+1)dx \leqslant 4$, find the range of the real number $k$. | [1,3] |
A certain unit plans to build a batch of simple houses (the house type is rectangular, with a height of 2.5 meters). The front and back walls use 2.5-meter high color steel plates, and the two sides use 2.5-meter high composite steel plates. The prices of both types of steel plates are calculated by length (i.e., the h... | \frac{20}{3} |
(7) Let $\left(1+x-x^{2}\right)^{10}=a_{0}+a_{1} x+a_{2} x^{2}+\cdots+a_{20} x^{20}$, then $a_{0}+$ $a_{1}+2 a_{2}+3 a_{3}+\cdots+20 a_{20}=$ $\qquad$ | -9 |
## Task B-4.2.
Students wrote posts for Instagram in pairs to invite people to the "Math Evening." The pair that collects the most likes expects a reward. Maja collected 3731, Matko 2754, but they claim that together they have 6705, which makes other pairs suspicious of cheating.
If we allow the possibility that the ... | 8 |
In tetrahedron OABC, a point M in space satisfies $\overrightarrow{OM}=\frac{1}{2}\overrightarrow{OA}+\frac{1}{6}\overrightarrow{OB}+λ\overrightarrow{OC}$. If $\overrightarrow{MA}$, $\overrightarrow{MB}$, and $\overrightarrow{MC}$ are coplanar, then the value of λ is:
A: $\frac{1}{2}$
B: $\frac{1}{3}$
C: $\frac{5}{1... | \text{B} |
The sequence $\{a_n\}$ satisfies $a_1=1$, $na_{n+1}=(n+1)a_n+n(n+1)$ for $n \in \mathbb{N}^*$.
(1) Prove that the sequence $\left\{ \dfrac{a_n}{n} \right\}$ is an arithmetic sequence;
(2) Let $T_n = a_1 - a_2 + a_3 - a_4 + \ldots + (-1)^{n+1}a_n$, find the expression for $T_{2n}$. | T_{2n} = -2n^2 - n |
When the input is $x=5$, the output value of $y$ from the following program is $\_\_\_\_\_\_\_\_$.
$INPUT x$
$IF x < 0 THEN$
$y=(x+1)\*(x+1)$
$ELSE$
$y=(x-1)\*(x-1)$
$END IF$
$PRINT y$
$END$ | y = 16 |
Definition: The function f(x) is defined as the largest integer not exceeding x, denoted as [x]. For example, f(1.6) = [1.6] = 1, f(-2.4) = [-2.4] = -3. Determine the value of f(lg81). | 1 |
I5.3 It is known that $b^{16}-1$ has four distinct prime factors, determine the largest one, denoted by $c$ | 257 |
1. Consider the vectors $\vec{a}=(3, x), \vec{b}=(\sqrt{x-2}, 4)$, then $\vec{a} \cdot \vec{b} \leq|\vec{a}||\vec{b}|$, which is equivalent to
$$
3 \sqrt{x-2}+4 x \leq \sqrt{\left(9+x^{2}\right)(x+14)}
$$
Equality is possible if and only if the vectors are collinear
$$
\frac{\sqrt{x-2}}{3}=\frac{4}{x}>0 \Leftrightar... | 6 |
If $f(x) = 3x + \sin x$, then find the range of $m$ for which the inequality $f(2m-1) + f(3-m) > 0$ holds. | m > -2 |
Let $z=\frac{1+3i}{1+i}$, then $|\overline{z}|$ is ______. | \sqrt{5} |
Let ellipse M be defined by the equation $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ ($a > b > 0$) with an eccentricity of $\frac{\sqrt{2}}{2}$. Let point A be $(a, 0)$, point B be $(0, -b)$, and the distance from the origin O to the line AB be $\frac{2\sqrt{3}}{3}$.
(I) Find the equation of ellipse M.
(II) Suppose line ... | S_{max} = \frac{10\sqrt{2}}{9} |
5. Find all positive integers $n$ such that the sum $1+2+3+\cdots+n$ is a three-digit number composed of the same digit. | 36 |
Given that the function $f(x) = \cos(\omega x - \frac{\omega \pi}{6})$ ($\omega > 0$) has the smallest positive period of $\pi$, the graph of the function $f(x)$ can be obtained by ( )
A: Shifting the graph of the function $g(x) = \cos(2x)$ to the left by $\frac{\pi}{3}$ units
B: Shifting the graph of the function $... | \text{D} |
Define a function $f(x)$ with domain $\mathbb{R}$ such that for any $x$, $f(x) = f(4-x)$. If $f(x)$ is monotonically increasing for $x \in [2, +\infty)$, then for $2 < a < 4$, we have ( )
A: $f(2^a) < f(2) < f(\log_2a)$
B: $f(2) < f(2^a) < f(\log_2a)$
C: $f(2) < f(\log_2a) < f(2^a)$
D: $f(\log_2a) < f(2^a) < f(2)$ | \text{C} |
## Problem Statement
Calculate the volume of the tetrahedron with vertices at points \( A_{1}, A_{2}, A_{3}, A_{4} \) and its height dropped from vertex \( A_{4} \) to the face \( A_{1} A_{2} A_{3} \).
\( A_{1}(1 ; -1 ; 2) \)
\( A_{2}(2 ; 1 ; 2) \)
\( A_{3}(1 ; 1 ; 4) \)
\( A_{4}(6 ; -3 ; 8) \) | 3\sqrt{6} |
1. Let the arithmetic sequence $\left\{a_{n}\right\}$ satisfy $3 a_{8}=5 a_{13}$ and $a_{1}>0, S_{n}$ be the sum of its first $n$ terms. Then the largest $S_{n}(n \in N)$ is ( ).
(A) $S_{10}$
(B) $S_{11}$
(C) $S_{20}$
(D) $S_{21}$ | C |
14. Asymmetric coin. (From 9th grade, 2 points) Billy Bones has two coins - a gold one and a silver one. One of them is symmetric, and the other is not. It is unknown which coin is asymmetric, but it is known that the asymmetric coin lands heads with a probability of \( p = 0.6 \).
Billy Bones tossed the gold coin, an... | \frac{5}{9} |
The range of the function $f(x)=x^{2}+2x$, where $x\in\[-2,1\]$ is $(\quad\quad)$.
A: $[-1,3]$
B: $[4,8]$
C: $[1,3]$
D: $[2,3]$ | \text{A} |
5. Given that the area of $\triangle A B C$ is 4, the side lengths are $B C=a, C A=b, A B=c(c>b)$, and $A D$ is the angle bisector of $\angle A$. $C^{\prime}$ is the symmetric point of $C$ with respect to the line $A D$. If $\triangle C^{\prime} B D$ is similar to $\triangle A B C$, find the minimum perimeter of $\tria... | 4\sqrt{2}+4 |
10. For the letter arrangement abcd, there are ( ) different rearrangements such that no two letters that were adjacent in the original arrangement are adjacent in the new arrangement (for example, ab and ba should not appear in the new arrangement).
(A) 0
(B) 1
(C) 2
(D) 3
(E) 4 | C |
In triangle $\mathrm{ABC}$, the following relationship holds between the sides:
$$
\left(a^{2}+b^{2}+c^{2}\right)^{2}=4 b^{2}\left(a^{2}+c^{2}\right)+3 a^{2} c^{2}
$$
What is the measure of angle $\beta$? | \beta=30 |
Using the analytical method, also known as the method of seeking cause from the effect, prove the following: Given that $a > b > c$ and $a + b + c = 0$, demonstrate that $\sqrt{b^2 - ac} 0$
B: $a - c > 0$
C: $(a - b)(a - c) > 0$
D: $(a - b)(a - c) < 0$ | C |
Let the general term of the sequence $\{a_n\}$ be $a_n=n\cos \frac {n\pi}{2}$. If the sum of the first $n$ terms is $S_n$, then $S_{2012}=$ ______. | 1006 |
Given a sequence 1, 1, 2, 1, 2, 4, 1, 2, 4, 8, 1, 2, 4, 8, 16, ... where the first term is 2^0, the next two terms are 2^0, 2^1, the next three terms are 2^0, 2^1, 2^2, and so on. What is the sum of the first 50 terms of this sequence?
A: 1044
B: 1024
C: 1045
D: 1025 | A |
Given the function $f(x)= \begin{cases} 3^{x}, & (x > 0) \\ -x^{ \frac {1}{3}}, & (x\leqslant 0)\end{cases}$, find the value of $f(\log _{3}4)$. | 4 |
21. Five participants of the competition became its prize winners, scoring 20, 19, and 18 points and taking first, second, and third places, respectively. How many participants won each prize place if together they scored 94 points? | 1,2,2 |
The coefficient of the \\(k+1\\)th term in the expansion of \\((2x+5y)^{2016}\\) is
A: \\( C_{2016}^{k}2^{2016-k}5^{k-1}\\)
B: \\( C_{2016}^{k-1}2^{2017-k}5^{k-1}\\)
C: \\( C_{2016}^{k-1}\\)
D: \\( C_{2016}^{k}2^{2016-k}5^{k}\\) | D |
5. Answer: 1) $\alpha_{1}=\operatorname{arctg} \frac{24}{7}+2 \pi k, k \in Z \quad$ 2) $\alpha_{2}=\pi+2 \pi k, k \in Z$
$$
\text { 3) } \alpha_{3}=\operatorname{arctg} \frac{4+3 \sqrt{24}}{4 \sqrt{24}-3}+2 \pi k, k \in Z \quad \text { 4) } \alpha_{4}=\operatorname{arctg} \frac{3 \sqrt{24}-4}{4 \sqrt{24}+3}+2 \pi k, k... | \alpha_{1}=\operatorname{arctg}\frac{24}{7}+2\pik,k\inZ\quad\alpha_{2}=\pi+2\pik,k\inZ\quad\alpha_{3}=\operatorname{arctg}\frac{4+3\sqrt{24}}{4\sqrt{24}-3}+2\pik,k\inZ\ |
Which of the following statements is correct?
A: When a fair coin is tossed $10$ times, it will definitely land heads up $5$ times.
B: "The sun rises from the west tomorrow" is an impossible event.
C: "A basketball player shoots once from the free-throw line and scores" is a certain event.
D: "$a$ is a real number,... | \text{B} |
In $\triangle ABC$, $D$ is the midpoint of $AB$. A line through $D$ parallel to $BC$ intersects $AC$ at $E$. If $DE = 4$, then $BC = \boxed{8}$. | 8 |
4. Given the sequences $\left\{a_{n}\right\}$ and $\left\{b_{n}\right\}$ satisfy:
$$
b_{n}=\left\{\begin{array}{ll}
a_{\frac{n+1}{2}}, & n \text { is odd; } \\
\sqrt{a_{n+1}}, & n \text { is even. }
\end{array}\right.
$$
If $\left\{b_{n}\right\}$ is a geometric sequence, and $a_{2}+b_{2}=108$, then the general term fo... | a_{n}=9^{n} |
$4 \cdot 2 a, b$ are real numbers, among the following five propositions, which one is correct?
(A) If $a>b$, then $a^{2}>b^{2}$.
(B) If $|a|>b$, then $a^{2}>b^{2}$.
(C) If $a>|b|$, then $a^{2}>b^{2}$.
(D) If $a^{2}>b^{2}$, then $a>b$.
(E) If $a \neq|b|$, then $a^{2} \neq b^{2}$.
(China Beijing Junior High School Mathe... | C |
Let $\{a_n\}$ be an arithmetic sequence with a positive common difference. If $a_1+a_2+a_3=15$ and $a_1a_2a_3=45$, then $a_{2009}+a_{2010}+a_{2011}$ equals to which of the following options?
A: 21111
B: 22111
C: 23111
D: 24111 | \text{D} |
$14 \cdot 36$ In the sequence $\left[\frac{1^{2}}{1980}\right],\left[\frac{2^{2}}{1980}\right],\left[\frac{3^{2}}{1980}\right], \cdots,\left[\frac{1980^{2}}{1980}\right]$, how many different numbers are there?
(6th All-Russian Mathematical Olympiad, 1980) | 1486 |
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