problem stringlengths 37 4.98k | answer stringlengths 1 141 | subject stringclasses 8
values | level stringclasses 7
values |
|---|---|---|---|
2. (10 points) As shown in the figure, Han Mei has 2 pots of flowers on each side of her house. Each time, Han Mei moves one pot of flowers into her house according to the following rules: first choose the left or right side, then move the pot closest to the house on that side. To move all the flowers into the house, t... | 6 | Combinatorics | olympiads |
Example 1 The number of proper subsets of the set $\left\{x \left\lvert\,-1 \leqslant \log _{\frac{1}{x}} 10<-\frac{1}{2}\right., 1<\right.$ $x \in \mathbf{N}\}$ is $\qquad$
(1996, National High School Mathematics Competition) | 2^{90}-1 | Algebra | cn_contest |
Hugo, Evo, and Fidel are playing Dungeons and Dragons, which requires many twenty-sided dice. Attempting to slay Evo's [i]vicious hobgoblin +1 of viciousness,[/i] Hugo rolls $25$ $20$-sided dice, obtaining a sum of (alas!) only $70$. Trying to console him, Fidel notes that, given that sum, the product of the numbers wa... | 5 | Number Theory | aops_forum |
In the right-angled triangle $ABC$ at $C$, the angle bisector from vertex $B$ intersects side $AC$ at point $P$ and the circumcircle of the triangle at point $Q$. What are the angles of the triangle if $BP = 2PQ$? | 30,60,90 | Geometry | olympiads |
7
An engineer arrives at the train station at 8 o'clock in the morning every day. At exactly 8 o'clock, a car arrives at the station and takes the engineer to the factory. One day, the engineer arrived at the station at 7 o'clock and started walking towards the car. Meeting the car, he got in and arrived at the factor... | 50 | Algebra | olympiads |
2. Given numbers $x, y \in\left(0, \frac{\pi}{2}\right)$. Find the maximum value of the expression
$$
A=\frac{\sqrt{\cos x \cos y}}{\sqrt{\operatorname{ctg} x}+\sqrt{\operatorname{ctg} y}}
$$ | \frac{\sqrt{2}}{4} | Inequalities | olympiads |
# 4. Problem $4.1 *$
In the class, 6 students received a grade of 5, 7 received a grade of 4, and 1 received a grade of 3. The teacher told them to form pairs with different grades, where the student with the better grade would explain to the student with the worse grade where they made a mistake.
In how many ways co... | 5040 | Combinatorics | olympiads |
Example 1. Determine the character of the equilibrium point (0,0) of the system
\[
\left\{\begin{array}{l}
\frac{d x}{d t}=5 x-y \\
\frac{d y}{d t}=2 x+y
\end{array}\right.
\] | unstablenode | Calculus | olympiads |
On the plane are given $ k\plus{}n$ distinct lines , where $ k>1$ is integer and $ n$ is integer as well.Any three of these lines do not pass through the
same point . Among these lines exactly $ k$ are parallel and all the other $ n$ lines intersect each other.All $ k\plus{}n$ lines define on the plane a partitio... | k = 17 | Combinatorics | aops_forum |
(3) Let $n$ be a positive integer, $x=\left(1+\frac{1}{n}\right)^{n}, y=\left(1+\frac{1}{n}\right)^{n+1}$, then ( ).
(A) $x^{y}>y^{x}$
(B) $x^{y}=y^{x}$
(C) $x^{y}<y^{x}$
(D) Any of the above is possible | B | Algebra | olympiads |
[ Tangent circles [ Auxiliary area. The area helps to solve the task]
Two circles of radii $R$ and $r$ touch each other externally at point $A$. On the circle of radius $r$, a point $B$ diametrically opposite to point $A$ is taken, and a tangent $l$ is constructed at this point. Find the radius of the circle that is t... | \frac{r(R+r)}{R}orr+R | Geometry | olympiads |
You are playing a game called "Hovse."
Initially you have the number $0$ on a blackboard.
If at any moment the number $x$ is written on the board, you can either:
$\bullet$ replace $x$ with $3x + 1$
$\bullet$ replace $x$ with $9x + 1$
$\bullet$ replace $x$ with $27x + 3$
$\bullet$ or replace $x$ with $\left \lfloor \f... | 127 | Combinatorics | aops_forum |
## Task $8 / 80$
We are looking for all natural numbers $n$ with the following properties:
1. It is $n=p_{1} \cdot p_{2}$; the product of two (proper) two-digit prime numbers $p_{1}$ and $p_{2}$.
2. For the cross sum $Q(n)$, it holds that $Q(n)=p_{1}$ with $p_{1}<p_{2}$.
3. The units digits of $p_{1}$ and $p_{2}$ are... | 629,1679 | Number Theory | olympiads |
Josefina the Quail dances by the marsh, using steps of double length — short steps measure $45 \mathrm{~cm}$, long steps $60 \mathrm{~cm}$. Over time, she has worn an oval path, which she dances around repeatedly during long nights. If she repeats three long steps forward and one short step back, then the ninetieth ste... | 162 | Number Theory | olympiads |
12.15. Solve the equation in integers
$$
x^{3}-2 y^{3}-4 z^{3}=0
$$ | 0 | Number Theory | olympiads |
A six place number is formed by repeating a three place number; for example, $256256$ or $678678$, etc. Any number of this form is always exactly divisible by:
$\textbf{(A)}\ 7 \text{ only} \qquad\textbf{(B)}\ 11 \text{ only} \qquad\textbf{(C)}\ 13 \text{ only} \qquad\textbf{(D)}\ 101 \qquad\textbf{(E)}\ 1001$ | 1001 | Number Theory | amc_aime |
The finite set $M$ of real numbers is such that among any three of its elements there are two whose sum is in $M$.
What is the maximum possible cardinality of $M$?
[hide=Remark about the other problems] Problem 2 is UK National Round 2022 P2, Problem 3 is UK National Round 2022 P4, Problem 4 is Balkan MO 2021 Shortlis... | 7 | Combinatorics | aops_forum |
2. If $k$ is a given real number, such that the following system of equations about $a, b$
$$
\left\{\begin{array}{l}
a+3 b-1=0, \\
a^{2}+b^{2}-4 a-6 b+13-k=0
\end{array}\right.
$$
has real solutions, then the range of values for $k$ is ( ).
(A) $k \geqslant 10$
(B) $k \geqslant 12$
(C) $k \geqslant 15$
(D) $k \in \ma... | A | Algebra | cn_contest |
Let's determine the sum of the fourth powers of two numbers, given that the sum of these numbers is 10 and their product is 4. | 8432 | Algebra | olympiads |
2. If real numbers $x, y$ satisfy $y^{2}=4 x$, then the range of $\frac{y}{x+1}$ is $\qquad$ . | [-1,1] | Algebra | cn_contest |
In the figure below, triangle $ABC$ and rectangle $PQRS$ have the same area and the same height 1. For each value of $x$ between 0 and 1, the trapezoid $ABED$ of height $x$ is drawn, and then the rectangle $PQNM$ of area equal to that of the trapezoid, as shown in the figure. Let $f$ be the function that associates eac... | f(x)=2x-x^2 | Geometry | olympiads |
11 If the function $f(x)=\sin (x+\theta)+\sqrt{3} \cos (x-\theta)$ is an even function, then $\theta=$ | \theta=k\pi-\frac{\pi}{6},k\in{Z} | Algebra | olympiads |
4. Suppose 40 objects are placed along a circle at equal distances. In how many ways can 3 objects be chosen from among them so that no two of the three chosen objects are adjacent nor diametrically opposite?
| 7720 | Combinatorics | olympiads |
Example 4. Find the orthogonal trajectories of the family of lines $y=k x$. | x^{2}+y^{2}=C(C\geqslant0) | Calculus | olympiads |
3. On the table, there are candies of three types: caramels, toffees, and lollipops. It is known that there are 8 fewer caramels than all the other candies, and there are 14 fewer toffees than all the other candies. How many lollipops are on the table? Be sure to explain your answer. | 11 | Algebra | olympiads |
Four disks with disjoint interiors are mutually tangent. Three of them are equal in size and the fourth one is smaller. Find the ratio of the radius of the smaller disk to one of the larger disks. | \frac{2\sqrt{3} - 3}{3} | Geometry | aops_forum |
Calcule:
a) $1678^{2}-1677^{2}$
b) $1001^{2}+1000^{2}$
c) $19999^{2}$
d) $2001^{2}+2002^{2}+2003^{2}$ | 3355,2002001,399960001,12024014 | Algebra | olympiads |
Question 89: If planar vectors $\vec{a} 、 \vec{b} 、 \vec{c}$ satisfy $\vec{a} \cdot(\vec{a}+\vec{c})=0,|\vec{a}+\vec{b}-2 \vec{c}|=2$, try to find the maximum value of $\vec{a} \cdot \vec{b}$. | \frac{1}{3} | Algebra | olympiads |
Solve the following equation:
$$
\frac{3+2 x}{1+2 x}-\frac{5+2 x}{7+2 x}=1-\frac{4 x^{2}-2}{7+16 x+4 x^{2}}
$$ | \frac{7}{8} | Algebra | olympiads |
22. Hogwarts School of Witchcraft and Wizardry is holding a magic competition, with 246 people signing up to participate. The school has a total of 255 boys, and the number of boys participating in the competition is 11 more than the number of girls not participating. The total number of students at the magic school is... | 490 | Algebra | olympiads |
23. What is the maximum area that a triangle with sides \(a, b, c\) can have, given the following constraints:
\[
0 \leqslant a \leqslant 1 \leqslant b \leqslant 2 \leqslant c \leqslant 3 \text { ? }
\] | 1 | Geometry | olympiads |
(a) Three lines $l,m,n$ in space pass through point $S$. A plane perpendicular to $m$ intersects $l,m,n $ at $A,B,C$ respectively. Suppose that $\angle ASB = \angle BSC = 45^o$ and $\angle ABC = 90^o$. Compute $\angle ASC$.
(b) Furthermore, if a plane perpendicular to $l$ intersects $l,m,n$ at $P,Q,R$ respectively an... | PQ = 1, QR = \sqrt{2}, PR = \sqrt{3} | Geometry | aops_forum |
5. In trapezoid $A B C D$ with bases $A D$ and $B C$, the angle $B C D$ is known to be $120^{\circ}$. A circle with radius 1 is inscribed in this angle, passing through points $A, B$ and $D$. Find the area of triangle $A B D$. Answer: $\frac{\sqrt{3}}{4}$. | \frac{\sqrt{3}}{4} | Geometry | olympiads |
[
Find the equation of the center of the Kiepert hyperbola: a) in trilinear coordinates; b) in barycentric coordinates.
# | ((b^{2}-^{2})^{2}:(^{2}-^{2})^{2}:(^{2}-b^{2})^{2}) | Geometry | olympiads |
Find all sets of integers $n\geq 2$ and positive integers $(a_1, a_2, \dots, a_n)$ that satisfy all of the following conditions:
[list]
[*] $a_1 < a_2 < \cdots < a_n$
[*] $a_n$ is a prime number.
[*] For any integer $k$ between $1$ and $n$, $a_k$ divides $a_1+a_2+\cdots+a_n$.
[/list] | (1, 2, 3) | Number Theory | aops_forum |
14. (20 points) Given sets $A$ and $B$ are both sets of positive integers, and $|A|=20,|B|=16$. Set $A$ satisfies the following condition: if $a, b, m, n \in A$, and $a+b=$ $m+n$, then $\{a, b\}=\{m, n\}$. Define
$$
A+B=\{a+b \mid a \in A, b \in B\} \text {. }
$$
Determine the minimum value of $|A+B|$. | 200 | Combinatorics | cn_contest |
4・155 To find the minimum value of $n$ for which the following system of equations
$$\left\{\begin{array}{l}
\sin x_{1}+\sin x_{2}+\cdots+\sin x_{n}=0 \\
\sin x_{1}+2 \sin x_{2}+\cdots+n \sin x_{n}=100
\end{array}\right.$$
has a solution. | 20 | Algebra | inequalities |
Which of the following numbers is largest?
$\text{(A)}\ 9.12344 \qquad \text{(B)}\ 9.123\overline{4} \qquad \text{(C)}\ 9.12\overline{34} \qquad \text{(D)}\ 9.1\overline{234} \qquad \text{(E)}\ 9.\overline{1234}$ | B | Number Theory | amc_aime |
5. Let the three roots of the cubic equation $x^{3}+p x+1=0$ correspond to points in the complex plane that form an equilateral triangle. Then $p=$ , and the area of this equilateral triangle is $\qquad$ . | p=0, S=\frac{3 \sqrt{3}}{4} | Algebra | cn_contest |
Solve the following equation in the set of integers:
$$
x^{3}+y^{3}=8^{30}
$$ | (2^{30},0)(0,2^{30}) | Number Theory | olympiads |
4. Let $A=\{1,2,3, \cdots, 1997\}$, for any 999-element subset $X$ of $A$, if there exist $x, y \in X$, such that $x<y$ and $x \mid y$, then $X$ is called a good set. Find the largest natural number $a(a \in A)$, such that any 999-element subset containing $a$ is a good set.
(《Mathematics in Middle School》1999 Issue 1 ... | 665 | Combinatorics | olympiads |
2. Find the smallest positive real number $k$ such that for any 4 distinct real numbers $a, b, c, d$ not less than $k$, there exists a permutation $p, q, r, s$ of $a, b, c, d$ such that the equation $\left(x^{2}+p x+q\right)\left(x^{2}+r x+s\right)=0$ has 4 distinct real roots. (Feng Zhigang) | 4 | Algebra | cn_contest |
One, (20 points) Given real numbers $x$, $y$, and $a$ satisfy
$$
x+y=x^{3}+y^{3}=x^{5}+y^{5}=a \text {. }
$$
Find all possible values of $a$. | \pm 2, \pm 1, 0 | Algebra | cn_contest |
Example 12 (2008 National High School Joint Competition Hubei Province Preliminary Test Question) Let the sequence $\{f(n)\}$ satisfy: $f(1)=1$, $f(2)=2, \frac{f(n+2)}{f(n)}=\frac{f^{2}(n+1)+1}{f^{2}(n)+1} \quad(n \geqslant 1)$.
(1) Find the recurrence relation between $f(n+1)$ and $f(n)$, i.e., $f(n+1)=g[f(n)]$;
(2) P... | 63<f(2008)<78 | Algebra | olympiads |
9. Given that 854 is a factor of the five-digit number $x$, and the last two digits of $x$ are 72, then the sum of all five-digit numbers $x$ that meet the conditions is $\qquad$ . | 73444 | Number Theory | olympiads |
36. Given $x=\sqrt{\frac{a-\sqrt{a^{2}-4}}{2 a}}(\mathrm{a}>0)$, then $\frac{x}{\sqrt{1-x^{2}}}+\frac{\sqrt{1-x^{2}}}{x}=$ $\qquad$ | a | Algebra | olympiads |
The Tournament of Towns is held once per year. This time the year of its autumn round is divisible by the number of the tournament: $2021\div 43 = 47$. How many times more will the humanity witness such a wonderful event?
[i]Alexey Zaslavsky[/i] | 4 | Number Theory | aops_forum |
2. The four-digit number $\overline{a a b b}$ is a perfect square. Then $\overline{a a b b}=(\quad)$.
(A) 7744
(B) 6655
(C) 8833
(D) 4477 | A | Number Theory | cn_contest |
12. Let $S=\{1,2,3, \cdots, 100\}$, find the smallest positive integer $n$, such that every $n$-element subset of $S$ contains 4 pairwise coprime numbers. | 75 | Number Theory | olympiads |
2. What is the remainder when $3^{2020}$ is divided by 73 ? | 8 | Number Theory | olympiads |
5. Let $F_{1}$ and $F_{2}$ be
the left and right foci of the hyperbola $C: \frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1(a>0, b>0)$,
and let the line $l$ passing through $F_{2}$ intersect the right branch of the hyperbola $C$ at points $A$ and $B$, and
$$
\overrightarrow{A F_{1}} \cdot \overrightarrow{A F_{2}}=0, \overrig... | \frac{\sqrt{17}}{3} | Geometry | olympiads |
1. Come up with five different natural numbers whose product is 1000. | 1,2,4,5,25 | Number Theory | olympiads |
1. Tom and Geck are ordered to write a big slogan for the 2020 Moscow Mathematical Olympiad (MMO). The slogan is written on rectangular wooden boards, each 5 cm wide, and then joined together horizontally. Tom writes MMO, and Geck writes 2020. Each letter and each digit is 9 cm wide. Can Geck manage to use fewer boards... | 8 | Logic and Puzzles | olympiads |
5. If the function $f(x)=\frac{a+\sin x}{2+\cos x}+b \tan x$ has a sum of its maximum and minimum values equal to 4, then $a+b=$ $\qquad$ | 3 | Algebra | cn_contest |
5. If $\sqrt{3-a}-\sqrt{a+1}>\frac{1}{2}$ always holds, then the range of values for $a$ is . $\qquad$ | \left[-1,1-\frac{\sqrt{31}}{8}\right) | Inequalities | cn_contest |
2. (Average) For the upcoming semester, 100 math majors can take up to two out of five math electives. Suppose 22 will not take any math elective in the coming semester. Also,
- 7 will take Algebraic Number Theory and Galois Theory
- 12 will take Galois Theory and Hyperbolic Geometry
- 3 will take Hyperbolic Geometry a... | 17 | Combinatorics | olympiads |
Consider a standard twelve-hour clock whose hour and minute hands move continuously. Let $m$ be an integer, with $1 \leq m \leq 720$. At precisely $m$ minutes after 12:00, the angle made by the hour hand and minute hand is exactly $1^\circ$.
Determine all possible values of $m$. | 458 | Geometry | aops_forum |
【Question 1】
Given $128 \div x+75 \div x+57 \div x=6.5$, then $x=$ $\qquad$.
untranslated part:
```
【第 1 题】
已知 $128 \div x+75 \div x+57 \div x=6.5$, 那么 $x=$ $\qquad$.
```
translated part:
```
【Question 1】
Given $128 \div x+75 \div x+57 \div x=6.5$, then $x=$ $\qquad$.
``` | 40 | Algebra | olympiads |
11.5. Let $M$ be some set of pairs of natural numbers $(i, j), 1 \leq i<j \leq n$ for a fixed $n \geq 2$. If a pair $(i, j)$ belongs to $M$, then no pair $(j, k)$ belongs to it. What is the largest set of pairs that can be in the set $M$? | \frac{n^{2}}{4}forevenn,\frac{n^{2}-1}{4}foroddn | Combinatorics | olympiads |
10. If real numbers $b, c$ satisfy $b^{2}+c^{2}=1$, and
$$
f(x)=a x+b \sin x+c \cos x
$$
has two perpendicular tangent lines on its graph, then the range of values for $a$ is $\qquad$. | \{0\} | Calculus | cn_contest |
9. (3 points) Using the digits $0, 1, 2, 3, 4$, the number of even numbers that can be formed without repeating any digit is $\qquad$. | 163 | Combinatorics | olympiads |
How many integers $x$ satisfy
$$
-5 < x-1 \leq 5 ?
$$
(a) 8
(b) 9
(c) 10
(d) 11
(e) 12 | 10 | Inequalities | olympiads |
In triangle $ABC$, let $I, O, H$ be the incenter, circumcenter and orthocenter, respectively. Suppose that $AI = 11$ and $AO = AH = 13$. Find $OH$.
[i]Proposed by Kevin You[/i] | 10 | Geometry | aops_forum |
Starting with the "1" in the centre, the spiral of consecutive integers continues, as shown. What is the sum of the number that appears directly above 2007 and the number that appears directly below 2007 ?
(A) 4014
(B) 4016
(C) 4018
(D) 4020
(E) 4022
| 17 | 16 | 15 | 14 | 13 |
| :---: | :---: | :---: | :---: | :---: |... | 4022 | Number Theory | olympiads |
36. Given $\frac{x}{1 \times 2}+\frac{x}{2 \times 3}+\frac{x}{3 \times 4}+\ldots+\frac{x}{999 \times 1000}=999$. Then $x=$ | 1000 | Algebra | olympiads |
2. As shown in Figure 1, in the obtuse triangle $\triangle ABC$, $BC=1, \angle A = 30^{\circ}, D$ is the midpoint of side $BC$, and $G$ is the centroid of $\triangle ABC$. If $B$ and $C$ are fixed points, when point $A$ moves, the range of the length of segment $GD$ is ( ).
(A) $0 < GD \leqslant \frac{\sqrt{13}}{6}$
(B... | B | Geometry | cn_contest |
Example 8 Find all positive integers $a, b$ such that
$$4^{a}+4 a^{2}+4=b^{2}$$ | (2,6) \text{ or } (4,18) | Number Theory | number_theory |
6. Three circles with radii $1,2,3$ touch each other externally. Find the radius of the circle passing through the three points of tangency of these circles. | 1 | Geometry | olympiads |
(2) Given the cubic function $f(x)=a x^{3}+b x^{2}+c x+d,(a, b, c, d \in \mathbf{R})$, proposition $p: y=f(x)$ is a monotonic function on $\mathbf{R}$;
proposition $q: y=f(x)$ intersects the $x$-axis at exactly one point. Then $p$ is ( ) of $q$.
(A) a sufficient but not necessary condition
(B) a necessary but not suffi... | A | Algebra | olympiads |
10. If $0<a, b, c<1$ satisfy the condition $ab+bc+ca=1$, then the minimum value of $\frac{1}{1-a}+\frac{1}{1-b}+\frac{1}{1-c}$ is $\qquad$ | \frac{3(3+\sqrt{3})}{2} | Inequalities | cn_contest |
8. If $p$ is a prime number, and $p+3$ divides $5p$, then the last digit of $p^{2009}$ is $\qquad$ . | 2 | Number Theory | cn_contest |
Let $\omega$ be the unit circle centered at the origin of $R^2$. Determine the largest possible value for the radius of the circle inscribed to the triangle $OAP$ where $ P$ lies the circle and $A$ is the projection of $P$ on the axis $OX$. | \frac{\sqrt{2} - 1}{2} | Geometry | aops_forum |
Example 5 Given the ellipse $C: \frac{x^{2}}{24}+\frac{y^{2}}{16}=1$, the line $l: \frac{x}{12}+\frac{y}{8}=1$, and point $P$ on $l$, the ray $O P$ intersects the ellipse at point $R$. Point $Q$ is on $O P$ such that $|O Q| \cdot|O P|=|O R|^{2}$. Find the equation of the trajectory of point $Q$ as point $P$ moves along... | \frac{(x-1)^{2}}{\frac{5}{2}}+\frac{(y-1)^{2}}{\frac{5}{3}}=1 | Geometry | olympiads |
$7 \cdot 26$ In $\triangle A B C$, $B D, C E$ are the altitudes on sides $A C, A B$, respectively, then $\frac{D E}{B C}$ equals
(A) $\frac{A E}{A B}$.
(B) $\sin A$.
(C) $\cos A$.
(D) $|\cos A|$.
("Zu Chongzhi Cup" Junior High School Mathematics Invitational Competition, 1988) | D | Geometry | olympiads |
10.296. Find the area of a right-angled triangle if the radii $R$ and $r$ of the circumscribed and inscribed circles are given. | r(2R+r) | Geometry | olympiads |
15. The figure $A B C D E F$ is a regular hexagon. Evaluate the quotient
$$
\frac{\text { Area of hexagon } A B C D E F}{\text { Area of triangle } A C D} \text {. }
$$ | 3 | Geometry | olympiads |
Folklore
Find all pairs of natural numbers $(a, b)$ for which the equality $\operatorname{LCM}(a, b)-\operatorname{GCD}(a, b)={ }^{a b} / 5$ holds.
# | {4,20} | Number Theory | olympiads |
4. Given a circle (c) with center $O$ and radius $R$. The point $O_{1}$, which lies on the circle (c), is the center of another circle with radius $\frac{R}{2}$.
If $A$ and $B$ are the intersection points of these two circles, determine the approximate value of $\measuredangle A O B$ (in radians), and then the volume ... | \frac{13}{192}\piR^{3} | Geometry | olympiads |
Find all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ such that for all $x, y \in \mathbb{R}$,
$$
f(x-f(y))=1-x-y
$$ | x\mapsto\frac{1}{2}-x | Algebra | olympiads |
At Typico High School, $60\%$ of the students like dancing, and the rest dislike it. Of those who like dancing, $80\%$ say that they like it, and the rest say that they dislike it. Of those who dislike dancing, $90\%$ say that they dislike it, and the rest say that they like it. What fraction of students who say they d... | \textbf{(D)}25 | Logic and Puzzles | amc_aime |
Example 1 Try to determine all triples $(p, q, n)$ that simultaneously satisfy
$$
\begin{array}{l}
q^{n+2} \equiv 3^{n+2}\left(\bmod p^{n}\right), \\
p^{n+2} \equiv 3^{n+2}\left(\bmod q^{n}\right)
\end{array}
$$
where $p, q$ are odd primes, and $n$ is an integer greater than 1. ${ }^{[1]}$ | (3,3,n)(n=2,3,\cdots) | Number Theory | olympiads |
# 6. Variant 1.
The vertices of the triangle have coordinates $A(1 ; 3.5), B(13.5 ; 3.5), C(11 ; 16)$. Consider horizontal lines given by the equations $y=n$, where $n$ is an integer. Find the sum of the lengths of the segments cut off on these lines by the sides of the triangle. | 78 | Geometry | olympiads |
A subset $S$ of the set $M=\{1,2,.....,p-1\}$,where $p$ is a prime number of the kind
$12n+11$,is [i]essential[/i],if the product ${\Pi}_s$ of all elements of the subset
is not less than the product $\bar{{\Pi}_s}$ of all other elements of the set.The
[b]difference[/b] $\bigtriangleup_s=\Pi_s-\bar{{\Pi}_s}$ is call... | 2 | Number Theory | aops_forum |
Determine all non negative integers $k$ such that there is a function $f : \mathbb{N} \to \mathbb{N}$ that satisfies
\[ f^n(n) = n + k \]
for all $n \in \mathbb{N}$ | k = 0 | Logic and Puzzles | aops_forum |
21. The diagram shows a shaded semicircle of diameter 4 , from which a smaller semicircle has been removed. The two semicircles touch at exactly three points. What fraction of the larger semicircle is shaded?
A $\frac{2}{\pi}$
B $\frac{1}{2}$
C $\frac{\sqrt{2}}{3}$
D $\frac{\sqrt{2}}{2}$
E $\frac{3}{4 \pi}$ | \frac{1}{2} | Geometry | olympiads |
30. If a die is tossed 500 times at random, how many times is the probability of getting a 1 (the face with 1 dot facing up) the greatest? | 83 | Combinatorics | olympiads |
2・109 Let $S=\left\{A=\left(a_{1}, \cdots, a_{8}\right) \mid a_{i}=0\right.$ or $\left.1, i=1,2, \cdots, 8\right\}$. For two elements $A=\left(a_{1}, \cdots, a_{8}\right)$ and $B=\left(b_{1}, \cdots, b_{8}\right)$ in $S$, let
$$
d(A, B)=\sum_{i=1}^{8}\left|a_{i}-b_{i}\right|,
$$
and call it the distance between $A$ an... | 4 | Combinatorics | olympiads |
[The ratio of the areas of triangles with a common base or common height] Complex [The ratio of the areas of triangles with a common angle
In triangle $ABC$, a line is drawn from vertex $A$, intersecting side $BC$ at point $D$, which lies between points $B$ and $C$, and $\frac{CD}{BC}=\alpha\left(\alpha<\frac{1}{2}\ri... | 4(1-\alpha) | Geometry | olympiads |
\section*{Problem 5 - 041245}
Determine all digit triples \((x, y, z)\) with \(x, y, z \neq 0\), such that
\[
\sqrt{(x x x \ldots x)-(y y y \ldots y)}=(z z z \ldots z)
\]
\((x x x \ldots x): 2n\) digits; \((y y y \ldots y): n\) digits; \((z z z \ldots z): n\) digits
is satisfied for at least two different positive ... | (1,2,3)(4,8,6) | Number Theory | olympiads |
6. Given that $A$ and $B$ are two subsets of the set $\{1,2, \cdots, 100\}$, satisfying that $A$ and $B$ have the same number of elements, and $A \cap B$ is an empty set. If $n \in A$, then $2n+2 \in B$, the maximum number of elements in the set $A \cup B$ is ( ).
(A) 62
(B) 66
(C) 68
(D) 74 | B | Combinatorics | cn_contest |
A store received a shipment of walnuts and peanuts in $1 \mathrm{~kg}$ packages. The invoice only stated that the value of the shipment was $1978 \mathrm{Ft}$, and its weight was $55 \mathrm{~kg}$. The delivery people remembered the following:
- the walnuts were more expensive;
- the price per kilogram is a two-digit ... | 43\mathrm{Ft} | Number Theory | olympiads |
## Task B-1.5.
For real numbers $a, b, c$, which are not equal to zero, it holds that $a+b+c=0$. What is
$$
\frac{a^{2}}{b c}+\frac{b^{2}}{a c}+\frac{c^{2}}{a b} ?
$$ | 3 | Algebra | olympiads |
## Task Condition
Derive the equation of the normal to the given curve at the point with abscissa \( x_{0} \).
\[
y=2 x^{2}-3 x+1, x_{\bar{u}}=1
\] | -x+1 | Calculus | olympiads |
(7) As shown in the figure, the side length of square $A B C D$ is $3, E$ is the midpoint of $D C, A E$ intersects $B D$ at $F$, then the value of $\overrightarrow{F D} \cdot \overrightarrow{D E}$ is $\qquad$ . | -\frac{3}{2} | Geometry | olympiads |
7. The solution set of the inequality $\sqrt{2 x+5}>x+1$ is | -\frac{5}{2}\leqslantx<2 | Inequalities | olympiads |
How many ways are there to rearrange the letters of CCAMB such that at least one C comes before the A?
[i]2019 CCA Math Bonanza Individual Round #5[/i] | 40 | Combinatorics | aops_forum |
Oldjuk meg az
$$
x^{4}-14 x^{3}+71 x^{2}-154 x+120=0
$$
egyenletet, ha tudjuk azt, hogy két gyökének összege 5, a másik két gyökének szorzata 20.
| x_{1}=2,x_{2}=3,x_{3}=4,x_{4}=5 | Algebra | olympiads |
5. Find all functions $f: \mathbf{R} \rightarrow \mathbf{R}$, such that for all $x, y \in \mathbf{R}$, we have
$$
f(1+x y)-f(x+y)=f(x) f(y),
$$
and $f(-1) \neq 0$. | f(x)=x-1 | Algebra | cn_contest |
## Problem Statement
Calculate the definite integral:
$$
\int_{0}^{3} \frac{d x}{\left(9+x^{2}\right)^{3 / 2}}
$$ | \frac{\sqrt{2}}{18} | Calculus | olympiads |
Example 4 If $3 x-y-1=0$, find
$$
z=\left|\sqrt{x^{2}+y^{2}-8 x-2 y+17}-\sqrt{x^{2}+y^{2}-8 y+16}\right|
$$
the maximum value. | \sqrt{5} | Algebra | cn_contest |
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