problem stringlengths 37 4.98k | answer stringlengths 1 141 | subject stringclasses 8
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|---|---|---|---|
Example 8 Given that positive integers $p, q$ are both prime numbers, and $7p+q, pq+11$ are also prime numbers. Find $p+q$.
保留源文本的换行和格式，直接输出翻译结果。 | 5 | Number Theory | cn_contest |
3. 15 children are playing hide-and-seek. Xiao Zhi is searching, and the other children are hiding. Now Xiao Zhi has already found 8 children, he still has $\qquad$ children left to find. | 6 | Logic and Puzzles | olympiads |
3. A student wrote a program for recoloring a pixel into one of 128 different colors. These colors he numbered with natural numbers from 1 to 128, and the primary colors received the following numbers: white color - number 1, red - 5, orange - 13, yellow - 21, green - 45, blue - 75, indigo - 87, violet - 91, black - 12... | 75 | Number Theory | olympiads |
Example 2 In the tetrahedron $A-B C D$, it is known that
$$
\begin{array}{l}
\angle A C B=\angle C B D \\
\angle A C D=\angle A D C=\angle B C D=\angle B D C=\theta,
\end{array}
$$
and $\cos \theta=\frac{\sqrt{10}}{10}$.
If the length of edge $A B$ is $6 \sqrt{2}$, then the volume of this pyramid is
$\qquad$ | 144 | Geometry | cn_contest |
30. (5 points)
On Lie Island, half of the people only lie on Wednesday, Friday, and Saturday, while the other half only lie on Tuesday, Thursday, and Sunday. One day, everyone on the island said, “I will tell the truth tomorrow.” So, this day is ( ).
A. Tuesday
B. Wednesday
C. Friday
D. Saturday
E. Sunday | C | Logic and Puzzles | olympiads |
Rectangle $P Q R S$ is divided into 60 identical squares, as shown. The length of the diagonal of each of these squares is 2. The length of $Q S$ is closest to
(A) 18
(B) 13
(C) 26
(D) 24
(E) 17
. Similarity criteria. In triangle $ABC$, a perpendicular line passing through the midpoint of side $AB$ intersects the extension of side $BC$ at point $M$, such that $MC: MB = 1: 5$. A perpendicular line passing through the midpoint of side $BC$ intersects side $AC$ at point $... | \angleA=\operatorname{arctg}2,\angleB=\operatorname{arctg}3,\angleC=45 | Geometry | olympiads |
In a class of 28 students, 4 prizes are to be distributed. In how many ways can this happen,
a) if the prizes are identical and a student can receive at most one prize?
b) if the prizes are identical and a student can receive multiple prizes?
c) if the prizes are different and a student can receive at most one prize... | 20475,31465,491400,614656 | Combinatorics | olympiads |
6. A box contains $m$ red balls, 9 white balls, and $n$ black balls, all of which are identical except for their color. If at least 17 balls must be drawn to ensure that there are 5 red balls, and at least 17 balls must be drawn to ensure that there are 8 balls of the same color. Then the value of the algebraic express... | D | Combinatorics | cn_contest |
For example, given $a_{1}=a_{2}=1, a_{n+1}=a_{n}+a_{n-1}(n \geqslant 2)$, find the general term formula of the sequence $\left\{a_{n}\right\}$. | a_{n}=\frac{1}{\sqrt{5}}[(\frac{1+\sqrt{5}}{2})^{n}-(\frac{1-\sqrt{5}}{2})^{n}] | Algebra | olympiads |
Let $n$ be a positive integer. Let $(a, b, c)$ be a random ordered triple of nonnegative integers such that $a + b + c = n$, chosen uniformly at random from among all such triples. Let $M_n$ be the expected value (average value) of the largest of $a$, $b$, and $c$. As $n$ approaches infinity, what value does $\frac{... | \frac{11}{18} | Combinatorics | aops_forum |
Example 5 Let the set $A=\{1,2,3,4,5,6\}$, and the mapping $f: A \rightarrow A$, such that its third composition $f \cdot f \cdot f$ is the identity mapping. How many such $f$ are there?
(1996 Japan Mathematical Olympiad Preliminary) | 81 | Combinatorics | olympiads |
## Problem 2
$100a + 10b + c$ is divisible by 17
$\Rightarrow 15a + 10b + c$ is divisible by 17 (1)
But $12a - 6b + c$ is divisible by 17
Adding the last two relations, we obtain that: $27a + 4b + 2c$ is divisible by 17 $2 p$.
Thus $5a + 2b + c$ is divisible by 17 $1 p$.
Subtracting the last relation from relatio... | 136,612,748 | Number Theory | olympiads |
Let $f^1(x)=x^3-3x$. Let $f^n(x)=f(f^{n-1}(x))$. Let $\mathcal{R}$ be the set of roots of $\tfrac{f^{2022}(x)}{x}$. If
\[\sum_{r\in\mathcal{R}}\frac{1}{r^2}=\frac{a^b-c}{d}\]
for positive integers $a,b,c,d$, where $b$ is as large as possible and $c$ and $d$ are relatively prime, find $a+b+c+d$. | 4060 | Algebra | aops_forum |
NT1 Find all the pairs positive integers $(x, y)$ such that
$$
\frac{1}{x}+\frac{1}{y}+\frac{1}{[x, y]}+\frac{1}{(x, y)}=\frac{1}{2}
$$
where $(x, y)$ is the greatest common divisor of $x, y$ and $[x, y]$ is the least common multiple of $x, y$.
| (8,8),(9,24),(24,9),(5,20),(20,5),(12,15),(15,12),(8,12),(12,8),(6,12),(12,6) | Number Theory | olympiads |
$2+$ [ Factorization ]
Find at least one integer solution to the equation $a^{2} b^{2}+a^{2}+b^{2}+1=2005$.
# | =2,b=20 | Number Theory | olympiads |
2. Let $n \geqslant 3, \omega=\cos \frac{2 \pi}{n}+\mathrm{i} \sin \frac{2 \pi}{n}$ be an $n$-th root of unity, and let $x_{i}(i=0,1, \cdots, n-1)$ be real numbers, with $x_{0} \geqslant x_{1} \geqslant \cdots \geqslant$ $x_{n-1}$. Find the necessary and sufficient conditions that $x_{0}, x_{1}, \cdots, x_{n-1}$ must s... | x_{0}=x_{1}=x_{2}=\cdots=x_{n-1} | Algebra | olympiads |
Lex has $\$ 2.65$. He has only dimes (worth $\$ 0.10$ each) and quarters (worth $\$ 0.25$ each). If Lex has more quarters than dimes, how many coins does he have in total?
(A) 12
(B) 13
(C) 16
(D) 19
(E) 22 | 13 | Algebra | olympiads |
It is known that the distance from the center of the circumscribed circle to the side $AB$ of triangle $ABC$ is equal to half the radius of this circle. Find the height of triangle $ABC$, dropped to the side $AB$, if it is less than $\sqrt{\frac{3}{2}}$, and the other two sides of the triangle are equal to 2 and 3. | 3\sqrt{\frac{3}{19}} | Geometry | olympiads |
4. Two players, A and B, are playing a game that is a best-of-seven series, where the first to win four games is declared the winner, and the series ends. If in each game, both players have a $\frac{1}{2}$ probability of winning, then the expected value of the number of games by which the winner exceeds the loser is $\... | \frac{35}{16} | Combinatorics | olympiads |
5. $[x]$ represents taking the integer part of the number $x$, for example $\left[\frac{15}{4}\right]=3$, etc. If $y=4\left(\frac{x+[u]}{4}-\left[\frac{x+[u]}{4}\right]\right)$, and when
$x=1,8,11,14$, $y=1$;
$x=2,5,12,15$, $y=2$;
$x=3,6,9,16$, $y=3$;
$x=4,7,10,13$, $y=0$.
Then the expression for $u$ is
(A) $\frac{x+2}... | D | Algebra | olympiads |
1. If $f(x)=\sqrt{x+27}+\sqrt{13-x}+\sqrt{x}$, then the maximum value of $f(x)$ is $\qquad$ | 11 | Algebra | cn_contest |
Example 11 Let real numbers $a, b$ satisfy $0<a<1,0<b<1$, and $ab=\frac{1}{36}$, find the minimum value of $u=\frac{1}{1-a}+\frac{1}{1-b}$. (Example 4 from [2]) | \frac{12}{5} | Algebra | inequalities |
3. Given the function $f(n)=\frac{20^{n}+3^{n}}{n!}, n \in \mathbf{N}$. Then, the value of $n$ that maximizes $f(n)$ is, $n=$ | 19 | Algebra | cn_contest |
1. A mathematician and a physicist started running on a running track towards the finish line at the same time. After finishing, the mathematician said: "If I had run twice as fast, I would have beaten the physicist by 12 seconds." And after finishing, the physicist said: "If I had run twice as fast, I would have beate... | 16 | Algebra | olympiads |
$17 \cdot 154$ Let $P_{1}, P_{2}, P_{3}$ be the equilateral triangles constructed on the sides $AB$, $BC$, $CA$ of the right triangle $\triangle ABC$ (with $C$ as the right angle). Then the $(\quad)$ of $P_{1}$ is the sum of the ( ) of $P_{2}$ and $P_{3}$.
(A) area, area.
(B) perimeter, perimeter.
(C) sum of interior a... | A | Geometry | olympiads |
3. Find the smallest positive number $\alpha$, such that there exists a positive number $\beta$, for which the inequality
$$
\sqrt{1+x}+\sqrt{1-x} \leqslant 2-\frac{x^{\alpha}}{\beta}
$$
holds for $0 \leqslant x \leqslant 1$. | \alpha=2,\beta=4 | Inequalities | olympiads |
1. The sequence $\left\{a_{n}\right\}$ has nine terms, $a_{1}=a_{9}=1$, and for each $i \in\{1,2, \cdots, 8\}$, we have $\frac{a_{i+1}}{a_{i}} \in\left\{2,1,-\frac{1}{2}\right\}$. The number of such sequences is $\qquad$ .
(2013, National High School Mathematics League Competition) | 491 | Combinatorics | olympiads |
5. If the distances from the center of the ellipse to the focus, the endpoint of the major axis, the endpoint of the minor axis, and the directrix are all positive integers, then the minimum value of the sum of these four distances is $\qquad$ .
| 61 | Geometry | cn_contest |
Let $ABCD$ be a trapezium in which $AB \parallel CD$ and $AB = 3CD$. Let $E$ be then midpoint of the diagonal $BD$. If $[ABCD] = n \times [CDE]$, what is the value of $n$?
(Here $[t]$ denotes the area of the geometrical figure$ t$.) | 8 | Geometry | aops_forum |
8. (3 points) Fill in the circles in the figure with $10, 15, 20, 30, 40$, and $60$ so that the product of the 3 numbers at the vertices of the three small triangles $A, B, C$ are all equal. What is the maximum equal product? $\qquad$ . | 18000 | Logic and Puzzles | olympiads |
5. As shown in Figure 1, a circle centered at point $M$ on the hyperbola $\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1(a, b>0)$ is tangent to the $x$-axis at one of the foci $F$ of the hyperbola, and intersects the $y$-axis at points $P$ and $Q$. If $\triangle M P Q$ is an equilateral triangle, then the eccentricity of th... | \sqrt{3} | Geometry | olympiads |
5. If $n \leqslant 2011$, then the number of positive integers $n$ such that $1+17 n$ is a perfect square is ( ) .
(A) 20
(B) 22
(C) 24
(D) 26 | A | Number Theory | cn_contest |
Find the 21st term of the sequence that starts like this:
$$
1 ; 2+3 ; 4+5+6 ; 7+8+9+10 ; 11+12+13+14+15 ; \ldots
$$ | 4641 | Number Theory | olympiads |
Consider the domain $D$ expressed by the following system inequality:
\[x^2+(y-1)^2\leq 1,\ x\geq \frac{\sqrt{2}}{3}.\]
Suppose that the line $l$ passes through the origin and the common part of $l$ and $D$ is line segment.
Find the maximum length of the line segment $L$, then when $L$ is maximized, find the value of ... | \cos \theta = \frac{\sqrt{3}}{3} | Geometry | aops_forum |
1. Solve the equation
$$
(\sqrt{3-\sqrt{8}})^{x}+(\sqrt{3+\sqrt{8}})^{x}=6
$$ | x_1=2,x_2=-2 | Algebra | olympiads |
A square and equilateral triangle have the same perimeter. If the triangle has area $16\sqrt3$, what is the area of the square?
[i]Proposed by Evan Chen[/i] | 36 | Geometry | aops_forum |
13. Given that $(m-2)$ is a positive integer and it is also a factor of $3 m^{2}-2 m+10$. Find the sum of all such values of $m$. | 51 | Algebra | olympiads |
Example 3. Solve the equation: $\sqrt[5]{171-x}+\sqrt[5]{104+x}=5$.
| x_1=139, x_2=-72 | Algebra | cn_contest |
When boarding the plane, a queue of p passengers forms, each with a ticket for one of $\mathrm{n}$ seats. The first in line is a crazy old lady. She rushes into the cabin and sits in a random seat (possibly even her own). Next, passengers take their seats in order, and if their seat is already occupied, they sit in a r... | \frac{1}{2} | Combinatorics | olympiads |
33. One of the interior angles of a triangle is equal to $61^{\circ}$, and one of its exterior angles is equal to one of its interior angles. Then, the smallest interior angle of the triangle is $\qquad$ degrees. | 29 | Geometry | olympiads |
Task 3. On the board, all such natural numbers from 3 to 223 inclusive are written, which when divided by 4 leave a remainder of 3. Every minute, Borya erases any two of the written numbers and instead writes their sum, decreased by 2. In the end, only one number remains on the board. What can it be? | 6218 | Number Theory | olympiads |
## Task Condition
Find the second-order derivative $y_{x x}^{\prime \prime}$ of the function given parametrically.
$$
\left\{\begin{array}{l}
x=\ln t \\
y=\operatorname{arctg} t
\end{array}\right.
$$ | \frac{\cdot(1-^2)}{(1+^2)^2} | Calculus | olympiads |
2. The square root of a two-digit number is expressed as an infinite decimal fraction, the first four digits of which (including the integer part) are the same. Find this number without using tables. | 79 | Number Theory | olympiads |
12. (10 points) As shown in the figure, in triangle $A B C$, $A F=2 B F, C E=3 A E, C D=2 B D$, connect $C F$ intersecting $D E$ at point $P$, find the value of $\frac{\mathrm{EP}}{\mathrm{DP}}$. | \frac{9}{4} | Geometry | olympiads |
Solve the following system of equations:
$$
\begin{gathered}
\log _{x}(x+y)+\log _{y}(x+y)=4 \\
(x-1)(y-1)=1 .
\end{gathered}
$$ | 2 | Algebra | olympiads |
Example 9 Let $a, b, c$ be positive integers, and the quadratic equation $a x^{2}+b x+c=0$ has two real roots whose absolute values are both less than $\frac{1}{3}$. Find the minimum value of $a+b+c$.
(2005, National High School Mathematics League, Fujian Province Preliminary | 25 | Algebra | cn_contest |
Let $E$ be the midpoint of side $[AB]$ of square $ABCD$. Let the circle through $B$ with center $A$ and segment $[EC]$ meet at $F$. What is $|EF|/|FC|$?
$
\textbf{(A)}\ 2
\qquad\textbf{(B)}\ \dfrac{3}{2}
\qquad\textbf{(C)}\ \sqrt{5}-1
\qquad\textbf{(D)}\ 3
\qquad\textbf{(E)}\ \sqrt{3}
$ | \frac{3}{2} | Geometry | aops_forum |
9. (16 points) Given the function
$$
f(x)=a \cos x+b \cos 2 x+c \cos 3 x,
$$
and $f(x) \geqslant-1$ always holds. Find the maximum value of $a-b+c$. | 1 | Inequalities | olympiads |
9. Let set $A$ consist entirely of positive integers, and for any $x, y \in A (x \neq y)$, we have $|x-y| \geqslant \frac{1}{25} x y$. How many numbers can $A$ contain at most? | 9 | Number Theory | olympiads |
Problem 4. Simplify the expression
$$
M=\frac{2}{\sqrt{4-3 \sqrt[4]{5}+2 \sqrt[4]{25}-\sqrt[4]{125}}}
$$ | \sqrt[4]{5}+1 | Algebra | olympiads |
A number is said to be a palindrome if reading from right to left is the same as reading from left to right. For example, the numbers 23432 and 18781 are palindromes. How many 4-digit palindrome numbers are divisible by 9? | 10 | Number Theory | olympiads |
Example 2 Given that $f(x)$ is an $n(>0)$-degree polynomial of $x$, and for any real number $x$, it satisfies $8 f\left(x^{3}\right)-x^{6} f(2 x)-$ $2 f\left(x^{2}\right)+12=0$
(1). Find $f(x)$. | f(x)=x^{3}-2 | Algebra | olympiads |
## Task 5 - 250735
In a study group, Rainer presents his classmates with the following task: He takes three cards, each with a two-digit number, and holds them in his hand so that no one else can see the numbers, and the six digits of the three cards read as a six-digit number when placed side by side.
He does this (... | 30 | Number Theory | olympiads |
Example 11. In a box, there are identical items manufactured by two machines: $40 \%$ of the items are made by the first machine, the rest - by the second. Defects in the production of the first machine constitute $3 \%$, of the second $-2 \%$. Find the probability that a randomly selected item will be defective. | 0.024 | Algebra | olympiads |
In the right-angled triangle $\mathrm{ABC}$, the angle at vertex $B$ is $30^{\circ}$. The center of the square constructed outward on the hypotenuse $\mathrm{ABC}$ is $D$. What is the measure of the angle $A D B$? | 60 | Geometry | olympiads |
6. Given the arithmetic sequence $\left\{a_{n}\right\}$, and $S_{5}=28, S_{10}=36$, then $S_{15}$ equals ( ).
A. 80
B. 40
C. 24
D. -48 | 24 | Algebra | olympiads |
[Pairings and groupings; bijections ] [ Products and factorials $\quad$]
A natural number $A$ has exactly 100 different divisors (including 1 and $A$). Find their product. | A^{50} | Number Theory | olympiads |
32. Given $3 a-2|b|=5, 4|a|-6 a=3 b$, then $a^{2}+b^{2}=$
Translate the above text into English, please retain the original text's line breaks and format, and output the translation result directly. | 13 | Algebra | olympiads |
8. Six people, all of different weights, are trying to build a human pyramid: that is, they get into the formation
$$
\begin{array}{c}
\text { A } \\
\text { B C } \\
\text { D E F }
\end{array}
$$
We say that someone not in the bottom row is "supported by" each of the two closest people beneath her or him. How many d... | 16 | Combinatorics | olympiads |
4. If the two real roots of $x^{2}+a x+b=0$ are $\alpha$, $\beta(\alpha \beta \neq 0)$, then the two real roots of $b x^{2}+a x+1=0$ | \frac{1}{\alpha}, \frac{1}{\beta} | Algebra | cn_contest |
Problem A
Let $a$ and $b$ be positive whole numbers such that $\frac{4.5}{11}<\frac{a}{b}<\frac{5}{11}$. Find the fraction $\frac{a}{b}$ for which the sum $a+b$ is as small as possible. Justify your answer. | \frac{3}{7} | Number Theory | olympiads |
Denote by $\mathbb{R}^{+}$ the set of all positive real numbers. Find all functions $f: \mathbb{R}^{+} \rightarrow \mathbb{R}^{+}$ such that
$$
x f\left(x^{2}\right) f(f(y))+f(y f(x))=f(x y)\left(f\left(f\left(x^{2}\right)\right)+f\left(f\left(y^{2}\right)\right)\right)
$$
for all positive real numbers $x$ and $y$. | f(y) = \frac{1}{y} | Algebra | olympiads_ref |
Solve the following equation:
$\cos x + \sqrt{3} \sin x = \sqrt{2}$. | \frac{\pi}{12}+2k\pior\frac{7\pi}{12}+2\pi | Algebra | olympiads |
### 9.303 Find integer values of $x$ that satisfy the inequality
$$
\log _{0.3}(\sqrt{x+5}-x+1)>0
$$ | 3 | Inequalities | olympiads |
3. (2 points) A poor student wrote the following incorrect formulas for the sine and cosine of a sum: $\sin (\alpha+\beta)=$ $\sin \alpha+\sin \beta$ and $\cos (\alpha+\beta)=\cos \alpha+\cos \beta$. In his defense, he said that for some $\alpha$ and $\beta$ his formulas are still correct. Find all such pairs $(\alpha,... | \alpha=\\frac{\pi}{3}+2\pi,\in\mathbb{Z} | Algebra | olympiads |
1. (3 points) $98 \times 196+2 \times 196+198=$ | 19798 | Algebra | olympiads |
## Task 5
All 20 Young Pioneers of Class 1a, who are participating in the ABC Action "Sniffing Nose," meet at the school in the afternoon.
16 Young Pioneers are already there, how many are still missing? | 4 | Other | olympiads |
1. $[\mathbf{1 0}]$ How many ways are there to place pawns on an $8 \times 8$ chessboard, so that there is at most 1 pawn in each horizontal row? Express your answer in the form $p_{1}^{e_{1}} \cdot p_{2}^{e_{2}} \cdots$, where the $p_{i}$ are distinct primes and the $e_{i}$ are positive integers. | 3^{16} | Combinatorics | olympiads |
Example 3 Let $X=\{1,2, \cdots, 100\}$, for any non-empty subset $M$ of $X$, the sum of the maximum and minimum numbers in $M$ is called the characteristic of $M$, denoted as $m(M)$, find the average of the characteristics of all non-empty subsets of $X$. | 101 | Combinatorics | olympiads |
12. Given that the center of ellipse $C$ is at the origin, the foci are on the $x$-axis, the eccentricity is $\frac{\sqrt{3}}{2}$, and the area of the triangle formed by any three vertices of ellipse $C$ is $\frac{1}{2}$.
(1) Find the equation of ellipse $C$;
(2) If a line $l$ passing through $P(\lambda, 0)$ intersects... | \lambda\in(-1,-\frac{1}{3})\cup(\frac{1}{3},1) | Geometry | olympiads |
【Question 13】
City A and City B are 55 kilometers apart. Xiao Wang starts from City A to City B, first riding a bicycle for 25 kilometers, then switching to a bus, which travels at twice the speed. Upon arriving in City B, he finds that the time spent cycling is 1 hour more than the time spent on the bus. Xiao Wang's... | 10 | Algebra | olympiads |
13. The tens digit of a two-digit number is three more than the units digit. When this two-digit number is divided by the sum of its digits, the answer is 7 remainder 3 . What is the sum of the digits of the two-digit number?
A 5
B 7
C 9
D 11
E 13 | 7 | Algebra | olympiads |
If point $M(x,y)$ lies on the line with equation $y=x+2$ and $1<y<3$, calculate the value of
$A=\sqrt{y^2-8x}+\sqrt{y^2+2x+5}$ | 5 | Algebra | aops_forum |
9. Find all values of $a$ for which the system of equations $2 y-2=a(x-1), \quad \frac{2 x}{|y|+y}=\sqrt{x}$ has at least one solution, and solve it for each $a$.
# | \begin{aligned}&\in(-\infty;0]\cup{1},x_{1}=0,y_{1}=1-\frac{}{2};x_{2}=1,y_{2}=1;\\&\in(0;1)\cup(1;2),x_{1}=0,y_{1}=1-\frac{}{2};x_{2}= | Algebra | olympiads |
5. Let $x \neq y$, and the sequences $x, a_{1}, a_{2}, a_{3}, y$ and $b_{1}, x, b_{2}, b_{3}, y, b_{4}$ are both arithmetic sequences, $\frac{b_{4}-b_{3}}{a_{2}-a_{1}}=$ $\qquad$ . | \frac{8}{3} | Algebra | olympiads |
Example 8 As shown in Figure 8-5, in $\triangle A B C$, $B M: B C=1: 3, B N: B A=3: 5, A M$ intersects $C N$ at point $P$. If $\overrightarrow{B C}=\vec{a}, \overrightarrow{B A}=\vec{b}$, express $\overrightarrow{B P}$ in terms of $\vec{a}, \vec{b}$. | \overrightarrow{BP}=\frac{1}{6}\vec{}+\frac{1}{2}\vec{b} | Geometry | olympiads |
Example 7. Let $n=1990$. Find the value of $\frac{1}{2^{n}}\left(1-3 C_{n}^{2}+3^{2} C_{n}^{4}-3^{3} C_{n}^{6}+\cdots+3^{994} C_{n}^{1988}-3^{995} C_{n}^{1990}\right)$. (1990, National High School Competition) | -\frac{1}{2} | Combinatorics | cn_contest |
Example 1-3 The number of 0,1 symbol strings of length $n$ is $2^{n}$.
. | 2^{n} | Combinatorics | olympiads |
1. It is known that Rochelle made 8 hamburgers with 3 pounds of meat. She still needs to prepare 24 hamburgers for the neighborhood picnic. Then she needs ( ) pounds of meat.
(A) 6
(B) $6 \frac{2}{3}$
(C) $7 \frac{1}{2}$
(D) 8
(E) 9 | E | Algebra | cn_contest |
3. Frane has a total of $108 \mathrm{kn}$ in his piggy bank in $5 \mathrm{kn}$, $2 \mathrm{kn}$, and $1 \mathrm{kn}$ coins. The value of the $5 \mathrm{kn}$ and $2 \mathrm{kn}$ coins is the same. The number of $1 \mathrm{kn}$ coins is equal to the number of $5 \mathrm{kn}$ and $2 \mathrm{kn}$ coins combined. How many c... | 8 | Algebra | olympiads |
Find all functions $f: \mathbb N \cup \{0\} \to \mathbb N\cup \{0\}$ such that $f(1)>0$ and
\[f(m^2+3n^2)=(f(m))^2 + 3(f(n))^2 \quad \forall m,n \in \mathbb N\cup \{0\}.\] | f(n) = n | Other | aops_forum |
$A, B, C$ and $D$ need to divide a certain amount of money in such a way that the ratio of their shares is $3: 4: 5: 6$. However, the distribution occurred in the ratio of $3: 4: 6: 7$, as a result, $C$ and $D$ together received 1400 crowns more. What was the total amount to be divided? | 36000 | Algebra | olympiads |
6.156. $20\left(\frac{x-2}{x+1}\right)^{2}-5\left(\frac{x+2}{x-1}\right)^{2}+48 \frac{x^{2}-4}{x^{2}-1}=0$. | x_{1}=\frac{2}{3},x_{2}=3 | Algebra | olympiads |
## Task B-3.5.
Grandpa Ivo, a retired mathematician, turned every game with his grandson into a math problem. So, in response to the question of how to make a paper dragon, he gave a very unusual answer. The dragon has the shape of a kite, whose diagonals are determined by the vectors
$\overrightarrow{A C}=(5 a+4) \v... | B(-\frac{3}{5},-\frac{4}{5}),C(5,0) | Geometry | olympiads |
Example 9 Let $M=\{1,2,3, \cdots, 40\}$, find the smallest positive integer $n$, such that $M$ can be divided into $n$ pairwise disjoint subsets and for any 3 numbers $a, b, c$ (not necessarily distinct) taken from the same subset, $a \neq b+c$.
| 4 | Combinatorics | olympiads |
11 ・11 Let $(x+a)^{4}=x^{4}+a_{1} x^{3}+a_{2} x^{2}+a_{3} x+a_{4}$. If $a_{1}+a_{2}+a_{3}=64$, then $\alpha$ equals
(A) -4 .
(B) 4 .
(C) -2 .
(D) 2 .
(China Shaanxi Province Junior High School Mathematics Competition, 1997) | 2 | Algebra | olympiads |
10.206. What integers represent the sides of an isosceles triangle if the radius of the inscribed circle is $3 / 2 \mathrm{~cm}$, and the radius of the circumscribed circle is $25 / 8$ cm | 5 | Geometry | olympiads |
Question 1 In $\triangle A B C$, $A B=A C, \angle C A B$ and $\angle A B C$'s internal angle bisectors intersect the sides $B C$ and $C A$ at points $D$ and $E$ respectively. Let $K$ be the incenter of $\triangle A D C$. If $\angle B E K=45^{\circ}$, find all possible values of $\angle C A B$ ${ }^{[1,2]}$. | 60^{\circ} \text{ or } 90^{\circ} | Geometry | cn_contest |
3. A right-angled triangle has integer lengths of sides. Its perimeter is the square of a natural number. We also know that one of its legs has a length equal to the square of a prime number. Determine all possible values of this length.
(Patrik Bak) | 9 | Number Theory | olympiads |
12. For a residential building, the construction investment is 250 yuan per square meter, considering a lifespan of 50 years, and an annual interest rate of $5 \%$, then the monthly rent per square meter should be $\qquad$ yuan to recover the entire investment. | 1.14 | Algebra | olympiads |
23. If $x$ is positive, find the minimum value of $\frac{\sqrt{x^{4}+x^{2}+2 x+1}+\sqrt{x^{4}-2 x^{3}+5 x^{2}-4 x+1}}{x}$. If $x$ is positive, find the minimum value of $\frac{\sqrt{x^{4}+x^{2}+2 x+1}+\sqrt{x^{4}-2 x^{3}+5 x^{2}-4 x+1}}{x}$. | \sqrt{10} | Algebra | olympiads |
51st Putnam 1990 Problem A6 How many ordered pairs (A, B) of subsets of {1, 2, ... , 10} can we find such that each element of A is larger than |B| and each element of B is larger than |A|. Solution | 17711 | Combinatorics | olympiads |
B4. The infinite sequence of numbers
$$
0,1,2,2,1,-1,-2,-1,1,3, \ldots
$$
satisfies the following rule. For each quadruple of consecutive numbers $\ldots, a, b, c, d, \ldots$ in the sequence, it always holds that $d$ is equal to $c$ minus the smallest of the two numbers $a$ and $b$. Thus, the ninth number in the sequ... | 2187 | Algebra | olympiads |
2.1. Find the integer part of the number $a+\frac{9}{b}$, where $a$ and $b-$ are respectively the integer and fractional part of the number $\sqrt{76-42 \sqrt{3}}$. | 12 | Algebra | olympiads |
The equation with integer coefficients $x^{4}+a x^{3}+b x^{2}+c x+d=0$ has four positive roots, counting multiplicities.
Find the smallest possible value of the coefficient $b$ under these conditions. | 6 | Algebra | olympiads |
10.3. Laura has 2010 lamps connected with 2010 buttons in front of her. For each button, she wants to know the corresponding lamp. In order to do this, she observes which lamps are lit when Richard presses a selection of buttons. (Not pressing anything is also a possible selection.) Richard always presses the buttons ... | 11 | Combinatorics | olympiads |
A triangle's vertices are: $A(0 ; 0), B(4 ; 1), C(4 ; 0)$. Let's write the equation of the line that passes through the point $D(0 ;-1)$ and bisects the area of the triangle $ABC$. | \frac{1}{2}x-1 | Geometry | olympiads |
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