problem stringlengths 10 7.47k | answer stringlengths 1 1.22k |
|---|---|
Which of the following statements about propositions is incorrect? ( )
A: The contrapositive of the proposition "If $x^2-3x+2=0$, then $x=1$" is "If $x \neq 1$, then $x^2-3x+2 \neq 0$".
B: If $p \lor q$ is a true proposition, then both $p$ and $q$ are true propositions.
C: "$x=2$" is a sufficient but not necessary cond... | B |
21. (7th grade) (Newton's problem) Grass grows at the same rate and density everywhere in the corner. It is known that 70 cows would eat it in 24 days, and 30 cows in 60 days. How many cows would eat all the grass in 96 days? (It is assumed that the cows eat the grass uniformly). | 20 |
Question: (1) In a region, there are 10,000 middle and elementary school students, including 2,000 high school students, 4,500 middle school students, and 3,500 elementary school students. Now, a sample of 200 is drawn from them; (2) From 1002 products produced by the same production line, 20 products are selected for ... | C |
[Pythagorean Theorem (direct and inverse) $]$ Law of Cosines $\quad]$
A circle inscribed in a right triangle with legs of 6 and 8 touches the hypotenuse at point $M$. Find the distance from point $M$ to the vertex of the right angle. | 2\sqrt{\frac{29}{5}} |
11. For the function $f(x)$, for any $x_{1}, x_{2}\left(x_{1} \neq x_{2}\right)$ in its domain, the following conclusions are drawn:
(1) $f\left(x_{1}+x_{2}\right)=f\left(x_{1}\right) \cdot f\left(x_{2}\right)$
(2) $f\left(x_{1} \cdot x_{2}\right)=f\left(x_{1}\right)+f\left(x_{2}\right)$
(3) $\frac{f\left(x_{1}\right)-... | (1),(3) |
If $m$ and $n$ are the roots of $x^2+mx+n=0 ,m \ne 0,n \ne 0$, then the sum of the roots is:
$\text{(A) } -\frac{1}{2}\quad \text{(B) } -1\quad \text{(C) } \frac{1}{2}\quad \text{(D) } 1\quad \text{(E) } \text{undetermined}$ | -1 |
A1. Raw silk before being processed into a rough thread. One meter of thread has a mass of approximately $0,13 \mathrm{mg}$. What is the mass of $1 \mathrm{~km}$ of thread?
(A) $0,0013 \mathrm{~g}$
(B) $0,013 \mathrm{~g}$
(C) $0,13 \mathrm{~g}$
(D) $1,3 \mathrm{~g}$
(E) $130 \mathrm{~g}$ | 0.13\mathrm{~} |
4. Students from two classes agreed to play football. In one of the classes, there were not enough players to form a team of 11 students, so they agreed to mix the students from both classes and then form two teams. The teacher noticed that in the first class, boys are 4/13 of the students, while in the second class, b... | 27 |
The equation of motion for a body is given by $$s = \frac{1}{4}t^4 - 3$$. Calculate the instantaneous velocity at t = 5. | 125 |
10,11
The edge of a regular tetrahedron is $4 \sqrt{6}$. Find the radius of the sphere that touches the lateral faces of the tetrahedron at points lying on the sides of the base. | 3 |
Given two planar vectors $\vec{a}$ and $\vec{b}$ satisfying $|\vec{a}|=|\vec{b}|=2$ and $(\vec{a} +2\vec{b})\cdot(\vec{a} - \vec{b})=-6$, then the angle between $\vec{a}$ and $\vec{b}$ is ( )
A: $\dfrac{\pi }{6}$
B: $\dfrac{\pi }{3}$
C: $\dfrac{2\pi }{3}$
D: $\dfrac{5\pi }{6}$ | C |
5. Uncle Li rents a piece of farmland, he pays the landowner 1000 yuan in cash and 2 bags of corn as rent. If the price of corn is 1 yuan per pound, the rent per acre is 180 yuan; if the price of corn is 1.5 yuan per pound, the rent per acre is 220 yuan. The area of the farmland Uncle Li rents is $\qquad$ acres. | 10 |
1. If positive real numbers $x, y$ satisfy $x^{3}+y^{3}+3 x y=1$, then the minimum value of $\left(x+\frac{1}{x}\right)^{3}+\left(y+\frac{1}{y}\right)^{3}$ is $\qquad$ . | \frac{125}{4} |
5. A bag contains $n$ red balls and 5 white balls. It costs 1 yuan to draw once, and each time two balls are drawn from the bag at once. If the two balls are of different colors, it is considered a win, and the prize is 2 yuan. Then when $n=$ $\qquad$, the expected value is maximized. | 4or5 |
Given an arithmetic sequence $\{a_n\}$, let $(S_n)$ denote the sum of its first $n$ terms. Given that $a_5=10$ and $S_7=56$.
(I) Find the general term formula of the sequence $\{a_n\}$.
(II) If $\{b_n\} = \{a_n + (\sqrt{3})^{a_n}\}$, find the sum of the first $n$ terms of the sequence $\{b_n\}$, denoted as $(T_n)$. | T_n = n^2 + n + \frac{3^{n+1} - 3}{2} |
Two concentric circles have radii $1$ and $2$. Two points on the outer circle are chosen independently and uniformly at random. What is the probability that the chord joining the two points intersects the inner circle?
$\textbf{(A)}\ \frac{1}{6}\qquad \textbf{(B)}\ \frac{1}{4}\qquad \textbf{(C)}\ \frac{2-\sqrt{2}}{2}\q... | \frac{1}{3} |
7. On June 9, 1898, Britain forced the Qing government to sign a treaty leasing 975.1 square kilometers of Hong Kong land to Britain for 99 years. On July 1, 1997, it was known that it was a Tuesday. Therefore, June 9, 1898, was a ( ).
(A) Tuesday
(B) Wednesday
(C) Thursday
(D) Friday
(Note: In the Gregorian calendar, ... | C |
Problem 9.1. a) Draw all points in the plane with coordinates $(x ; y)$ such that
$$
(|3 x-y|-3)(|3 x+y|-3)=0
$$
b) Find all $x$ and $y$ for which
$$
\begin{array}{ccc}
(|3 x-y|-3)(|3 x+y|-3) & = & 0 \\
y-\{4 x\} & = & 0 \\
-1 \leq x \leq 1 & &
\end{array}
$$
(For a real number $x$ we denote the unique number in t... | \frac{5}{7},\frac{6}{7};\frac{6}{7},\frac{3}{7};1,0;-1,0 |
Given $a < b < c$, the two roots of the function $f(x) = (x - a)(x - b) + (x - b)(x - c) + (x - c)(x - a)$ are located in the intervals ( ).
A: $(a, b)$ and $(b, c)$
B: $(-\infty, a)$ and $(a, b)$
C: $(b, c)$ and $(c, +\infty)$
D: $(-\infty, a)$ and $(c, +\infty)$ | \text{A} |
4. Given that the circumradius of $\triangle A B C$ is 1, and $A B, B C, \frac{4}{3} C A$ form a geometric sequence in order, then the maximum value of $B C$ is $\qquad$ . | \frac{4\sqrt{2}}{3} |
\section*{Problem 3A - 261043A}
Determine all triples \(\left(x_{1}, x_{2}, x_{3}\right)\) of real numbers \(x_{1}, x_{2}, x_{3}\) that satisfy the following system of equations (1), (2), (3)!
\[
\begin{aligned}
x_{1}+x_{2}+x_{3} & =3 \\
x_{1}^{3}+x_{2}^{3}+x_{3}^{3} & =3 \\
x_{1} \cdot x_{2} \cdot x_{3} & =1
\end{al... | (1,1,1) |
7. Multiply $(F A C E)_{16}$ and $(B A D)_{16}$. | B 705736 |
Problem 6.6. Ksyusha runs twice as fast as she walks (both speeds are constant).
On Tuesday, when she left home for school, she first walked, and then, when she realized she was running late, she started running. The distance she walked was twice the distance she ran. In the end, it took her exactly 30 minutes to get ... | 24 |
A=15, A=-A+5, the final value of A is ____. | -10 |
4. As shown in Figure 2, in the right triangle $\triangle ABC$, $\angle ABC = 90^{\circ}$, $AB = \frac{1}{2} AC$. Take a point $P$ on the line $AB$ or $BC$ such that $\triangle PAC$ is an isosceles triangle. Then, the number of points $P$ that satisfy the condition is $(\quad)$.
(A) 4
(B) 6
(C) 7
(D) 8 | B |
7. Let $\theta_{1}, \theta_{2}$ be acute angles, and $\frac{\sin ^{2020} \theta_{1}}{\cos ^{2018} \theta_{2}}+\frac{\cos ^{2020} \theta_{1}}{\sin ^{2018} \theta_{2}}=1$, then $\theta_{1}+\theta_{2}=\triangle$, | \theta_{1}+\theta_{2}=\frac{\pi}{2} |
VII.5. In the square $ABCD$, a point $P$ is chosen on side $BC$ and a point $S$ on side $CD$, such that $\triangle APB = \triangle APS$. Determine the measure of angle $PAS$.
---
(Translation provided as requested, maintaining the original formatting and structure.) | 45 |
[ Numerical inequalities. Comparing numbers.]
What is greater, $2^{700}$ or $5^{300}$? | 2^{700} |
34. Consider an equilateral triangle $A B C$, where $A B=B C=C A=2011$. Let $P$ be a point inside $\triangle A B C$. Draw line segments passing through $P$ such that $D E\|B C, F G\| C A$ and $H I \| A B$. Suppose $D E: F G: H I=8: 7: 10$. Find $D E+F G+H I$. | 4022 |
1. If $A, B$ are two interior angles of an acute $\triangle A B C$, then the complex number
$$
z=(\cos B-\sin A)+i(\sin B-\cos A)
$$
in the complex plane corresponds to a point located in
(A) the first quadrant;
(B) the second quadrant;
(C) the third quadrant;
(D) the fourth quadrant. | B |
Given sets $A=\{y|y=x^{2}-2x-3,x\in R\}$ and $B=\{y|y=-x^{2}+2x+13,x\in R\}$, find $A\cap B$. | [-4,14] |
8.3. In trapezoid $A B C D$ with base $A D$, it is known that $A B=B C, A C=C D$ and $B C+C D=A D$. Find the angles of the trapezoid. | \angleA=72,\angleB=108,\angleC=144,\angleD=36 |
Example 6: There are three types of goods, A, B, and C. If you buy 3 pieces of A, 7 pieces of B, and 1 piece of C, it costs 3.15 yuan; if you buy 4 pieces of A, 10 pieces of B, and 1 piece of C, it costs 4.20 yuan. Now, if you buy 1 piece each of A, B, and C, it will cost $\qquad$ yuan.
(1985. National Junior High Scho... | 1.05 |
The sum of all positive divisors of 120 that are composite numbers but not odd equals to ____. | 334 |
2. Let $A B C$ be a triangle such that $A B=7$, and let the angle bisector of $\angle B A C$ intersect line $B C$ at $D$. If there exist points $E$ and $F$ on sides $A C$ and $B C$, respectively, such that lines $A D$ and $E F$ are parallel and divide triangle $A B C$ into three parts of equal area, determine the numbe... | 13 |
3. In the sequence $\left\{a_{n}\right\}$, for $1 \leqslant n \leqslant 5$, we have $a_{n}=n^{2}$; and for all positive integers $n$, we have $a_{n+5}+a_{n+1}=a_{n+4}+a_{n}$. Then $a_{2023}=$ $\qquad$ . | 17 |
Determine the values of $m$ and $n$ if the domain and range of the function $f(x) = -x^{2} + 2x$ are given by the intervals $[m, n]$ and $[3m, 3n]$ respectively. Find the sum $m + n$. | -1 |
In triangle $\triangle ABC$, given $A(1,1)$ and $B(3,-2)$.
$(1)$ If the line $l$ passes through point $M(2,0)$ and the distances from points $A$ and $B$ to $l$ are equal, find the equation of line $l$.
$(2)$ If the angle bisector of $\angle C$ lies on the line with equation $x+y-3=0$, find the equation of line $BC$... | 4x + y - 10 = 0 |
$17 \cdot 6$ In triangle $ABC$, $\angle C=\theta, \angle B=2 \theta$, where $0^{\circ}<\theta<60^{\circ}$. A circle with center $A$ and radius $AB$ intersects $AC$ at $D$, and intersects $BC$ (if necessary, extend $BC$) at $B, E$ (E may coincide with B). Then $EC=AD$ is true if
(A) no value of $\theta$ is suitable.
(B)... | E |
1.2. Gavriila found out that the front tires of the car last for 24000 km, while the rear tires last for 36000 km. Therefore, he decided to swap them at some point to maximize the distance the car can travel. Find this distance (in km). | 28800 |
The "Rhind Mathematical Papyrus" is one of the oldest mathematical texts in the world. It contains the following problem: Divide 100 loaves of bread among 5 people so that each person receives an amount forming an arithmetic sequence, and the sum of the larger three portions is $\dfrac{1}{7}$ of the sum of the smaller ... | \dfrac{115}{3} |
A4 The areas of two similar triangles are in the ratio 4 : 9. One of the sides of the smaller triangle measures $6 \mathrm{~cm}$. The corresponding side of the larger triangle measures:
(A) $9 \mathrm{~cm}$
(B) $13.5 \mathrm{~cm}$
(C) $4 \mathrm{~cm}$
(D) $12 \mathrm{~cm}$
(E) none of the above | 9\mathrm{~} |
3. In the tetrahedron $D-ABC$, it is known that $AB=$ $BC=2, AB \perp BC, BC \perp CD, DA \perp AB$, $\angle CDA=60^{\circ}$. Then the volume of the tetrahedron $D-ABC$ is $\qquad$ . | \frac{4}{3} |
Given the function $f(x)= \sqrt {2}\sin \omega x\cos \omega x+ \sqrt {2}\cos ^{2}\omega x- \dfrac { \sqrt {2}}{2}(\omega > 0)$, if $x= \dfrac {\pi}{4}$ is an axis of symmetry for the function $f(x)$, then the value of the real number $\omega$ can be $(\quad)$.
A: $1$
B: $\dfrac {1}{2}$
C: $\dfrac {1}{4}$
D: $\dfrac {1}... | B |
Given vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ satisfy $(\overrightarrow{a}+\overrightarrow{b})\cdot(\overrightarrow{a}-2\overrightarrow{b})=-6$, and $|\overrightarrow{a}|=1$, $|\overrightarrow{b}|=2$.
$(1)$ Find $\overrightarrow{a}\cdot\overrightarrow{b}$;
$(2)$ If the angle between $\overrightarrow{a... | \theta = \frac{2\pi}{3} |
Each of the $39$ students in the eighth grade at Lincoln Middle School has one dog or one cat or both a dog and a cat. Twenty students have a dog and $26$ students have a cat. How many students have both a dog and a cat?
$\textbf{(A)}\ 7\qquad \textbf{(B)}\ 13\qquad \textbf{(C)}\ 19\qquad \textbf{(D)}\ 39\qquad \textbf... | 7 |
Let $\overline{AB}$ be a diameter of circle $\omega$. Extend $\overline{AB}$ through $A$ to $C$. Point $T$ lies on $\omega$ so that line $CT$ is tangent to $\omega$. Point $P$ is the foot of the perpendicular from $A$ to line $CT$. Suppose $\overline{AB} = 18$, and let $m$ denote the maximum possible length of segment ... | 432 |
# Task 8. (16 points)
Let all companies in the country have a certain rank, which is a natural number. When two companies of ranks $m$ and $n$ merge, a new company of rank $(m+n)$ is formed. The profit of the resulting company will be $m \cdot n$ more than the sum of the profits of the companies forming it. The profit... | 63 |
1. Calculate: $45 \times 1234321-3333 \times 9999=$ | 22217778 |
Walter has exactly one penny, one nickel, one dime and one quarter in his pocket. What percent of one dollar is in his pocket?
$\text{(A)}\ 4\% \qquad \text{(B)}\ 25\% \qquad \text{(C)}\ 40\% \qquad \text{(D)}\ 41\% \qquad \text{(E)}\ 59\%$ | D |
14.54 If there is a positive integer $m$, such that $m$! ends with exactly $n$ zeros, then the positive integer $n$ is called a "factorial tail". How many positive integers less than 1992 are not "factorial tails"?
(10th American Mathematical Invitational, 1992) | 396 |
35. Find the total number of positive four-digit integers $N$ satisfying both of the following properties:
(i) $N$ is divisible by 7 , and
(ii) when the first and last digits of $N$ are interchanged, the resulting positive integer is also divisible by 7. (Note that the resulting integer need not be a four-digit number.... | 210 |
XV OM - I - Problem 11
In triangle $ ABC $, angle $ A $ is $ 20^\circ $, $ AB = AC $. On sides $ AB $ and $ AC $, points $ D $ and $ E $ are chosen such that $ \measuredangle DCB = 60^\circ $ and $ \measuredangle EBC = 50^\circ $. Calculate the angle $ EDC $. | 30 |
3. Find the rational number - the value of the expression
$$
2 \cos ^{6}(5 \pi / 16)+2 \sin ^{6}(11 \pi / 16)+3 \sqrt{2} / 8
$$ | \frac{5}{4} |
Given a set of data $3$, $6$, $9$, $a$, $5$, $7$, and the unique mode is $6$, then the average of this set of data is ______. | 6 |
Given $f(x) = \begin{cases} \ln x, & x > 1 \\ 2x + \int_{0}^{m} 3t^{2} \, dt, & x \leq 1 \end{cases}$, and $f(f(e)) = 10$, the value of $m$ is ( )
A: 2
B: -1
C: 1
D: -2 | A |
12. A circle touches the sides of triangle $P Q R$ at the points $S, T$ and $U$ as shown. Also $\angle P Q R=\alpha^{\circ}, \angle P R Q=\beta^{\circ}$ and $\angle T S U=\gamma^{\circ}$.
Which of the following gives $\gamma$ in terms of $\alpha$ and $\beta$ ?
A $\frac{1}{2}(\alpha+\beta)$
B $180-\frac{1}{2}(\alpha+\be... | \gamma=\frac{1}{2}(\alpha+\beta) |
A box contains three cards, each marked with the numbers $1$, $2$, and $3$. Apart from the numbers marked on them, the three cards are identical. Draw one card at a time randomly with replacement for $3$ times, and denote the numbers on the drawn cards as $a$, $b$, and $c$ in order.
(Ⅰ) Find the probability that the ... | \dfrac {8}{9} |
[ Arithmetic. Mental arithmetic, etc.] $[$ Work problems $]$
Three diggers dug three holes in two hours. How many holes will six diggers dig in five hours?
# | 15 |
Which of the following statements is correct?
A: A quadrilateral with equal diagonals is a parallelogram.
B: A quadrilateral with perpendicular diagonals is a rhombus.
C: A quadrilateral with all right angles is a rectangle.
D: A parallelogram with a pair of adjacent sides equal is a square. | C |
Calculate the result of $\left(-a\right)^{3}\cdot a^{2}$ is ______. | -a^{5} |
10. If $0<x \leqslant 1, a=\left(\frac{\sin x}{x}\right)^{2}, b=\frac{\sin x}{x}$, $c=\frac{\sin x^{2}}{x^{2}}$. Then the size relationship of $a, b, c$ is $\qquad$ . | a<b \leqslant c |
6. $\frac{2 \cos 10^{\circ}-\sin 20^{\circ}}{\cos 20^{\circ}}$ The value is $\qquad$ | \sqrt{3} |
Let the function be $f(x)=mx- \dfrac{m}{x}-2\ln x$.
$(1)$ When $m=1$ and $x > 1$, prove that: $f(x) > 0$;
$(2)$ If for all $x\in[1, \sqrt{3}]$, $f(x) < 2$ holds, find the range of the real number $m$. | (-\infty, \sqrt{3}(1+\ln \sqrt{3})) |
A circular disk is divided by $2n$ equally spaced radii($n>0$) and one secant line. The maximum number of non-overlapping
areas into which the disk can be divided is
$\textbf{(A) }2n+1\qquad \textbf{(B) }2n+2\qquad \textbf{(C) }3n-1\qquad \textbf{(D) }3n\qquad \textbf{(E) }3n+1$ | 3n+1 |
Find the angles of a triangle if it is known that all squares inscribed in it are equal (each of the squares is inscribed such that two of its vertices lie on one side of the triangle, and the other vertices on the other two sides of the triangle).
# | 60 |
1. Determine the largest natural number such that any two adjacent digits written in the same order form a two-digit number divisible by 23. | 46923 |
Task B-4.2. If $z+z^{-1}=2 \cos \frac{\alpha}{2012}$, determine $\alpha$ for which $z^{2012}+z^{-2012}=1$ | \alpha=\\frac{\pi}{3}+2k\pi,\quadk\in\mathbb{Z} |
Two cars start at the same time from city A heading towards city B. One of them travels at a constant speed of $60 \mathrm{~km} / \mathrm{h}$ and the other at a constant speed of $70 \mathrm{~km} / \mathrm{h}$. If the faster car makes this trip in 15 minutes less than the other car, what is the distance between the two... | 105\mathrm{~} |
Simplify first, then evaluate: $\left(3x+2y\right)\left(3x-2y\right)-5x\left(x-y\right)-\left(2x-y\right)^{2}$, where $x=-\frac{1}{3}$ and $y=-2$. | -14 |
5. In how many different ways can four children, Ante, Bruno, Cvijeta, and Dunja, distribute four identical pencils among themselves?
## Ministry of Science and Education of the Republic of Croatia Agency for Education and Education Croatian Mathematical Society
## COUNTY COMPETITION IN MATHEMATICS March 24, 2022. | 35 |
Define the function $f(x)$ on $\mathbb{R}$ as follows:
$$
f(x) = \begin{cases}
\log_{2}(8-x) & \text{if } x \leq 0 \\
f(x-1) & \text{if } x > 0
\end{cases}
$$
Then, $f(3) =$ （ ）
A: 3
B: 2
C: $\log_{2}9$
D: $\log_{2}7$ | A |
Given lines $l\_1$: $A\_1x+B\_1y=1$ and $l\_2$: $A\_2x+B\_2y=1$ intersect at point $P(2,3)$, derive the equation of the line passing through points $P\_1(A\_1,B\_1)$ and $P\_2(A\_2,B\_2)$. | 2x+3y-1=0 |
47. Find the limit of the function:
1) $f(x)=x^{3}-5 x^{2}+2 x+4$ as $x \rightarrow-3$;
2) $\varphi(t)=t \sqrt{t^{2}-20}-\lg \left(t+\sqrt{t^{2}-20}\right)$ as $t \rightarrow 6$. | 23 |
3. $\mathrm{ABCD}$ is a trapezoid with bases $\mathrm{AD}=6$ and $\mathrm{BC}=10$. It turns out that the midpoints of all four sides of the trapezoid lie on the same circle. Find its radius.
If there are multiple correct answers, list them in any order separated by a semicolon. | 4 |
3. A bridge is built across a river. One quarter of the bridge is over the left bank of the river and one third of the bridge is over the right bank. The river is $120 \mathrm{~m}$ wide. How long is the bridge?
A $150 \mathrm{~m}$
B $190 \mathrm{~m}$
C $240 \mathrm{~m}$
D $288 \mathrm{~m}$
E $324 \mathrm{~m}$ | 288\mathrm{~} |
At a party there are only single women and married men with their wives. The probability that a randomly selected woman is single is $\frac25$. What fraction of the people in the room are married men?
$\textbf{(A)}\ \frac13\qquad \textbf{(B)}\ \frac38\qquad \textbf{(C)}\ \frac25\qquad \textbf{(D)}\ \frac{5}{12}\qquad \... | \textbf{(B)}\\frac{3}{8} |
Given $a=\sqrt{17}-1$, then the value of $a^{5}+2a^{4}-17a^{3}-a^{2}+18a-17$ is ______. | -1 |
3. As shown in Figure 1, in $\triangle A B C$, $O$ is the midpoint of side $B C$, and a line through $O$ intersects lines $A B$ and $A C$ at two distinct points $M$ and $N$ respectively. If
$$
\begin{array}{l}
\overrightarrow{A B}=m \overrightarrow{A M}, \\
\overrightarrow{A C}=n \overrightarrow{A N},
\end{array}
$$
t... | 2 |
Given the function $$f(x)= \begin{cases} (3-a)^{x}, & x\leq 2 \\ \log_{a}(x-1)+3, & x>2 \end{cases}$$ is a monotonically increasing function on its domain, then the range of values for $a$ is ( )
A: $(3- \sqrt {3}, 2)$
B: $( \sqrt {5}-1, \sqrt {3})$
C: $(1, \sqrt {3})$
D: $(1, 3- \sqrt {3})$ | A |
I1.1 Find $a$, if $a=\log _{\frac{1}{4}} \frac{1}{2}$. | \frac{1}{2} |
The numbers $p, q, r$, and $t$ satisfy $p<q<r<t$. When these numbers are paired, each pair has a different sum and the four largest sums are 19,22,25, and 28 . What is the the sum of the possible values for $p$ ? | \frac{17}{2} |
Given vectors $\overrightarrow{a} = (\cos \alpha, \sin \alpha)$ and $\overrightarrow{b} = (\cos \beta, \sin \beta)$, with $|\overrightarrow{a} - \overrightarrow{b}| = \frac{2\sqrt{5}}{5}$.
(Ⅰ) Find the value of $\cos (\alpha - \beta)$;
(Ⅱ) If $0 < \alpha < \frac{\pi}{2}$ and $-\frac{\pi}{2} < \beta < 0$, and $\sin \b... | \frac{33}{65} |
15. Intersecting Cylinders. The axes of symmetry of two straight circular cylinders, each with a diameter of 2 cm, intersect at a right angle. What is the volume of the common part of these cylinders? | \frac{16}{3}\mathrm{~}^{3} |
We define the polynomial $$P (x) = 2014x^{2013} + 2013x^{2012} +... + 4x^3 + 3x^2 + 2x.$$ Find the largest prime divisor of $P (2)$. | 61 |
3. In a regular quadrilateral pyramid $P-A B C D$, it is known that $A_{1} 、 C_{1}$ are the midpoints of $P A 、 P C$ respectively. Then
$$
\frac{V_{\text {=floor } A_{1}}-B C_{1} D}{V_{\text {regular quadrilateral pyramid } P-A B C D}}=
$$
$\qquad$ | \frac{1}{4} |
5. Among A, B, and C, one is a priest, one is a liar, and one is a gambler. The priest never lies, the liar always lies, and the gambler sometimes tells the truth and sometimes lies. A says: “I am the priest.” B says: “I am the liar.” C says: “I am the gambler.” Then, among the three, $\qquad$ is the priest. | A |
What is the maximum value of the function $$\frac{1}{\left|x+1\right|+\left|x+2\right|+\left|x-3\right|}?$$
$\text{(A) }\frac{1}{3}\qquad\text{(B) }\frac{1}{4}\qquad\text{(C) }\frac{1}{5}\qquad\text{(D) }\frac{1}{6}\qquad\text{(E) }\frac{1}{7}$ | \frac{1}{5} |
3. Solve the inequality $\frac{4 x^{4}+1}{4 \sqrt{2}} \leq x \sqrt{x^{4}-\frac{1}{4}}$. | \sqrt{\frac{1+\sqrt{2}}{2}} |
If the three lines $ax+2y+8=0$, $4x+3y-10=0$, and $2x-y=0$ intersect at one point, then the value of the real number $a$ is ______. | -12 |
1. Let the decimal part of $\frac{2}{\sqrt{5}-1}$ be denoted as $\alpha$. Then the value of $\alpha(\alpha+1)$ is ( ).
(A) 1
(B) a positive rational number less than 1
(C) a positive irrational number less than 1
(D) a positive irrational number greater than 1 | A |
Let $x>0$. Show that $x+\frac{1}{x} \geqslant 2$. Find the cases of equality. | 1 |
[Mathematical logic (miscellaneous).] [ Arithmetic. Mental arithmetic, etc. ]
Rabbits are sawing a log. They made 10 cuts. How many chunks did they get?
# | 11 |
48th Putnam 1987 Problem A2 An infinite sequence of decimal digits is obtained by writing the positive integers in order: 123456789101112131415161718192021 ... . Define f(n) = m if the 10 n th digit forms part of an m-digit number. For example, f(1) = 2, because the 10th digit is part of 10, and f(2) = 2, because the 1... | 1984 |
Example 4. As shown in the figure, $C$ is a point on the diameter $A B$, $C E$ is perpendicular to the tangent $D E$, and $D$ is the point of tangency. Prove: $A B \cdot C E$ $=A C \cdot B C+C D^{2}$.
Analysis: To make the equation useful, we need to substitute terms containing the common factor $C D$. Extend $D C$ to... | A B \cdot C E=A C \cdot B C+C D^{2} |
B3. Complete the text so that the expression $10-(1.08+3 \cdot 0.52+5 \cdot 0.29)$ correctly represents the calculation of the given problem.
Špela bought three yogurts, five cherries, and one kilogram of flour at the store.
She paid with a $\qquad$ euro bill.
Each yogurt costs $\qquad$ euros.
Each cherry costs $\q... | 5.91 |
5. Let quadrilateral $ABCD$ be a cyclic quadrilateral, and now four relations are given.
(1) $\sin A=\sin C$;
(2) $\sin A+\sin C=0$;
(3) $\cos B+\cos D=0$;
(4) $\cos B=\cos D$.
The number of relations that always hold is ).
(A) 1
(B) 2
(C) 3
(D) 4 | B |
5. In how many ways can a rectangular board of size $2 \times 13$ be covered with rectangular tiles of size $1 \times 2?$ (The tiles are laid so that they do not overlap and fit entirely on the board.) | 377 |
Problem 2. Gosha entered a natural number into the calculator. Then he performed the following operation, consisting of two actions, three times: first, he extracted the square root, and then took the integer part of the obtained number. In the end, he got the number 1. What is the largest number that Gosha could have ... | 255 |
4. If a sequence of numbers $a_{1}, a_{2}, \cdots$ satisfies that for any positive integer $n$ there is $a_{1}+a_{2}+\cdots+a_{n}=n^{3}$, then the value of $\frac{1}{a_{2}-1}+\frac{1}{a_{3}-1}+\cdots+\frac{1}{a_{100}-1}$ is ( ).
(A) $\frac{33}{100}$
(B) $\frac{11}{100}$
(C) $\frac{11}{99}$
(D) $\frac{33}{101}$ | A |
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