problem stringlengths 37 4.98k | answer stringlengths 1 141 | subject stringclasses 8
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|---|---|---|---|
4. Let's say a subset $\mathrm{P}$ of the set $\mathrm{M}=\{1,2,3, \ldots, 42\}$ is halfish if it contains 21 elements and each of the 42 numbers in the sets $\mathrm{P}$ and $\mathrm{Q}=\{7 x ; x \in \mathrm{P}\}$ gives a different remainder when divided by 43. Determine the number of halfish subsets of the set M.
(J... | 128 | Combinatorics | olympiads |
3. The length of a rectangle is three times its width. If the length is decreased by 5 and the width is increased by 5 , the rectangle becomes a square. Determine the length of the original rectangle. | 15 | Algebra | olympiads |
Problem 6. Each face of a cube $6 \times 6 \times 6$ is divided into $1 \times 1$ cells. The cube is covered with $2 \times 2$ squares such that each square covers exactly four cells, no squares overlap, and each cell is covered by the same number of squares. What is the maximum value that this identical number can tak... | 3 | Combinatorics | olympiads |
【Example 3】Let $n$ be a positive integer. A particle starts at the origin $A(0,0)$ and ends at $B(n, n)$, making non-decreasing moves. The entire path lies below the "diagonal" and touches it at most. Try to find the number of such paths. | \frac{C_{2n}^{n}}{n+1} | Combinatorics | olympiads |
Rectangle $A B C D$ is made up of six squares. The areas of two of the squares are shown. The perimeter of rectangle $A B C D$, in centimetres, is
(A) 50
(B) 44
(C) 46
(D) 52
(E) 48
 | 48 | Geometry | olympiads |
\section*{Task 1 - 081211}
At the European Championships for women's rowing in August 1966, the GDR, as the most successful country, received 37 points, and the USSR received 36.5 points. Both countries received exactly one of the three medals awarded in each of the 5 disciplines: Single Sculls, Double Sculls, "Coxed ... | 3 | Logic and Puzzles | olympiads |
Given the real number $s$. Solve the inequality
$$
\log _{\frac{1}{s}} \log _{s} x>\log _{s} \log _{s} x
$$ | 1<x<for>1;0<x<for0<<1 | Inequalities | olympiads |
Three. (50 points) Given a positive integer $n$, let $n$ real numbers $a_{1}, a_{2}, \cdots, a_{n}$ satisfy the following $n$ equations:
$$
\sum_{i=1}^{n} \frac{a_{i}}{i+j}=\frac{4}{2 j+1}(j=1,2,3, \cdots, n) .
$$
Determine the value of the sum $S=\sum_{i=1}^{n} \frac{a_{i}}{2 i+1}$ (expressed in the simplest form inv... | S=1-\frac{1}{(2 n+1)^{2}} | Algebra | cn_contest |
8. Let $n$ be a given positive integer, and the sum $\sum_{1 \leqslant i<j \leqslant n}\left|x_{i}-x_{j}\right|=\left|x_{1}-x_{2}\right|+\left|x_{1}-x_{3}\right|+\cdots+$ $\left|x_{1}-x_{n}\right|+\left|x_{2}-x_{3}\right|+\left|x_{2}-x_{4}\right|+\cdots+\left|x_{2}-x_{n}\right|+\left|x_{n-2}-x_{n}\right|+\left|x_{n-2}-... | \left[\frac{n^{2}}{4}\right] | Combinatorics | inequalities |
Eight congruent equilateral triangles, each of a different color, are used to construct a regular octahedron. How many distinguishable ways are there to construct the octahedron? (Two colored octahedrons are distinguishable if neither can be rotated to look just like the other.)
[asy]import three;
import math;
size(180... | 1680 | Combinatorics | aops_forum |
82. A vanguard unit of the PLA sets off from location $A$ via location $B$ to reach location $C$. Three hours later, the unit is urgently dispatched to send a communicator by car to make contact. It is known that the communicator's car speed is 20 kilometers per hour faster than the vanguard unit's speed. The vanguard ... | 240 | Algebra | olympiads |
1. Calculate: $123456789 \times 8+9=$ | 987654321 | Algebra | olympiads |
12. If $\vec{a}+2 \vec{b}=\{-3,1\}, 2 \vec{a}-\vec{b}=\{2,4\}$, then $\vec{a}+\vec{b}=$ | {-\frac{7}{5},\frac{7}{5}} | Algebra | olympiads |
1. An object at rest in space is suspended by three ropes. It is known that the magnitudes of the forces in the three ropes are $1 \mathrm{~N}, 2 \mathrm{~N}, 3 \mathrm{~N}$, and the angle between any two ropes is $60^{\circ}$. The magnitude of the gravitational force acting on the object is $\qquad$ $\mathrm{N}$.
(Gol... | 5 | Algebra | olympiads |
3B. Given a parallelogram $A B C D$. The bisector of $\angle D A B$ intersects side $D C$ at point $L$, and diagonal $B D$ at point $K$, such that $\overline{D K}: \overline{K B}=3: 4$. Calculate the length of segment $L C$, if the perimeter of the parallelogram is 28. | 2 | Geometry | olympiads |
2. Given the function $g(x)=\frac{4 \sin ^{4} x+7 \cos ^{2} x}{4 \cos ^{4} x+\sin ^{2} x}$. Find:
a) the roots of the equation $g(x)=4$;
b) the maximum and minimum values of the function $g(x)$. | )\\frac{\pi}{3}+k\pi,k\in\mathrm{Z},\frac{\pi}{2}+k\pi,k\in\mathrm{Z};b)g_{\}=\frac{7}{4},g_{\max}=\frac{63}{15} | Algebra | olympiads |
5. [5 points] On a plane with a given rectangular Cartesian coordinate system, a square is drawn with vertices at points $(0 ; 0),(0 ; 65),(65 ; 65)$ and ( $65 ; 0)$. Find the number of ways to choose two grid nodes inside this square (not including its boundary) such that at least one of these nodes lies on one of the... | 500032 | Combinatorics | olympiads |
2. Karl found some valuable stones in a cave: each 5-pound stone is worth 14 yuan, each 4-pound stone is worth 11 yuan, and each 1-pound stone is worth 2 yuan. It is known that there are at least 20 of each type of stone, and Karl can take out a total of 18 pounds of stones from the cave. Then the maximum value of the ... | C | Number Theory | cn_contest |
## 223. Math Puzzle $12 / 83$
In a grandfather clock, the seconds pendulum completes one oscillation in only 0.9999 seconds. As a result, the clock runs fast.
By how much does the error increase within one week? | 60.48\mathrm{~}\approx1 | Logic and Puzzles | olympiads |
Example 2 Find four consecutive integers, each of which is divisible by $2^{2}, 3^{2}, 5^{2}$, and $7^{2}$ respectively. | 29348,29349,29350,29351 | Number Theory | number_theory |
# 3. Clone 1
On an island, there live knights who always tell the truth, and liars who always lie. Before a friendly match, 30 islanders gathered in T-shirts with numbers on them—arbitrary natural numbers. Each of them said: “I have a T-shirt with an odd number.” After that, they exchanged T-shirts, and each said: “I ... | 15 | Logic and Puzzles | olympiads |
11.3. Find all solutions of the system of equations in real numbers:
$$
\left\{\begin{array}{l}
x^{5}=y^{3}+2 z \\
y^{5}=z^{3}+2 x \\
z^{5}=x^{3}+2 y
\end{array}\right.
$$ | (0,0,0),\(\sqrt{2},\sqrt{2},\sqrt{2}) | Algebra | olympiads |
Problem 9.6. On a line, two red points and several blue points are marked. It turned out that one of the red points is contained in exactly 56 segments with blue endpoints, and the other - in 50 segments with blue endpoints. How many blue points are marked? | 15 | Combinatorics | olympiads |
【Question 6】Person A and Person B stand facing each other 30 meters apart, playing "Rock, Paper, Scissors". The winner moves forward 3 meters, the loser moves back 2 meters, and in the event of a tie, both move forward 1 meter. After 15 rounds, Person A is 17 meters from the starting point, and Person B is 2 meters fro... | 7 | Logic and Puzzles | olympiads |
[ Systems of nonlinear algebraic equations ] Higher degree equations (miscellaneous) $\quad]$
Solve the system of equations:
$x^{3}-y=6$,
$y^{3}-z=6$,
$z^{3}-x=6$. | (2,2,2) | Algebra | olympiads |
12.19 Given the function $y=x^{4}-6 x^{2}+1$. Find the maximum and minimum values of its derivative on the interval $[-1,3]$.
| y_{\text{}}^{\}=-8,y_{\text{max}}^{\}=72 | Calculus | olympiads |
Blinkov A.d:
The square of the sum of the digits of the number $A$ is equal to the sum of the digits of the number $A^{2}$. Find all such two-digit numbers $A$. | 10,11,12,13,20,21,22,30,31 | Number Theory | olympiads |
7. (10 points) The teacher is doing calculation practice with Jiajia, Fangfang, and Mingming. The teacher first gives each of them a number, and then asks them to each pick 3 cards with numbers on them. Jiajia picks 3, 6, 7, Fangfang picks 4, 5, 6, and Mingming picks 4, 5, 8. The teacher then asks them to each multiply... | 7 | Algebra | olympiads |
Exercise 6. A set of $\mathrm{n}$ cells in an $\mathrm{n} \times \mathrm{n}$ grid is said to be distributed if it never includes two cells in the same row or column. In how many ways can one color some (possibly none) cells of an $\mathrm{n} \times \mathrm{n}$ grid so that all distributed sets contain the same number o... | 2^{n+1}-2 | Combinatorics | olympiads |
4. A. As shown in Figure 2, the area of $\triangle A B C$ is 24, point $D$ is on line segment $A C$, and point $F$ is on the extension of line segment $B C$, with $B C=4 C F$. If quadrilateral $D C F E$ is a parallelogram, then the area of the shaded part in the figure is ( ).
(A) 3
(B) 4
(C) 6
(D) 8 | C | Geometry | cn_contest |
1. For what values of $a, b$, and $c$ do the lines $y=a x+b, y=b x+c, y=c x+a$ pass through the point $(1 ; 3)$? | =b==1.5 | Algebra | olympiads |
5. In an arbitrary triangular pyramid $A B C D$, a section is made by a plane intersecting the edges $A B, D C$, and $D B$ at points $M, N, P$ respectively. Point $M$ divides edge $A B$ in the ratio $A M: M B=1: 3$. Point $N$ divides edge $D C$ in the ratio $D N: N C=4: 3$. Point $P$ divides edge $D B$ in the ratio $D ... | AQ:QC=4:27 | Geometry | olympiads |
Find all functions $f: \mathbb{Q} \rightarrow \mathbb{Q}$ satisfying $f(0)=0$ and
$$
\forall x, y \in \mathbb{Q}, \quad f(f(x)+f(y))=x+y
$$
The following problem serves to introduce a classic and very useful trick... | f(x)=xf(x)=-x | Algebra | olympiads |
12・19 If $a \pm b i(b \neq 0)$ are the complex roots of the equation $x^{3}+q x+r=0$, where $a$, $b, q$ and $r$ are real numbers. Then $q$ expressed in terms of $a, b$ is
(A) $a^{2}+b^{2}$.
(B) $2 a^{2}-b^{2}$.
(C) $b^{2}-a^{2}$.
(D) $b^{2}-2 a^{2}$.
(E) $b^{2}-3 a^{2}$.
(23rd American High School Mathematics Examinati... | b^{2}-3a^{2} | Algebra | olympiads |
Example 13 (CMO-18 Problem) Find all triples of positive integers $(a, m, n)$ satisfying $a \geqslant 2, m \geqslant 2$ such that $a^{n}+203$ is divisible by $a^{m}+1$.
Find all triples of positive integers $(a, m, n)$ satisfying $a \geqslant 2, m \geqslant 2$ such that $a^{n}+203$ is divisible by $a^{m}+1$. | \begin{pmatrix}(2,2,4k+1),(2,3,6k+2),(2,4,8k+8),(2,6,12k+9),(3,2,4k+3),\\(4,2,4k+4),(5,2,4k+1),(8,2,4k+3),(10,2,4k+2),(203, | Number Theory | olympiads |
7) How many positive numbers $n$ are there such that $n+30>n^{2}$ ?
$\begin{array}{llll}\text { (A) infinite } & \text { (B) } 30 & \text { (C) } 6 & \text { (D) } 5\end{array}$
(E) none of the above answers is correct. | 5 | Inequalities | olympiads |
1. In a chess tournament, 10 players participated. Each player played one game with each of the other players.
a) How many games did each individual player play?
b) How many games of chess were played in total at the tournament?
Explain your answers. | 45 | Combinatorics | olympiads |
35. A ball $P$ starts oscillating back and forth from point $A$ 8 times. If moving to the right is defined as positive and moving to the left as negative, and the 8 movements are recorded (in millimeters) as $:+12,-10,+9,-6,+8.5,-6,+8,-7$.
(1) Find the distance in millimeters from point $A$ to where the ball $P$ stops:... | 8.5,1.33 | Algebra | olympiads |
In triangle $A B C$, the bisector $C D$ of the right angle $A C B$ is drawn; $D M$ and $D N$ are the altitudes of triangles $A D C$ and $B D C$, respectively.
Find $A C$, given that $A M=4, B N=9$.
# | 10 | Geometry | olympiads |
6. $\underbrace{2 \times 2 \times \ldots \times 2}_{20 \uparrow 2}-1$
The unit digit of the result is $\qquad$ | 5 | Number Theory | olympiads |
Let $p>3$ be a prime and let $a_{1}, a_{2}, \ldots, a_{\frac{p-1}{2}}$ be a permutation of $1,2, \ldots, \frac{p-1}{2}$. For which $p$ is it always possible to determine the sequence $a_{1}, a_{2}, \ldots, a_{\frac{p-1}{2}}$ if for all $i, j \in\left\{1,2, \ldots, \frac{p-1}{2}\right\}$ with $i \neq j$ the residue of $... | For all primes p>5 | Number Theory | olympiads_ref |
Putnam 1997 Problem A5 Is the number of ordered 10-tuples of positive integers (a 1 , a 2 , ... , a 10 ) such that 1/a 1 + 1/a 2 + ... + 1/a 10 = 1 even or odd? Solution | odd | Combinatorics | olympiads |
22. Let $S$ be the set of all non-zero real-valued functions $f$ defined on the set of all real numbers such that
$$
\mathrm{f}\left(x^{2}+y f(z)\right)=x \mathrm{f}(x)+z \mathrm{f}(y)
$$
for all real numbers $x, y$ and $z$. Find the maximum value of $\mathrm{f}(12345)$, where $\mathrm{f} \in S$. | 12345 | Algebra | olympiads |
Find all function $f:\mathbb{R}\rightarrow \mathbb{R}$ such that for any three real number $a,b,c$ , if $ a + f(b) + f(f(c)) = 0$ :
$$ f(a)^3 + bf(b)^2 + c^2f(c) = 3abc $$. | f(x) = 0 | Other | aops_forum |
9. (40 points) On an island, there live only 50 knights, who always tell the truth, and 15 commoners, who can either tell the truth or lie. A scatterbrained professor, who came to the island to give a lecture, forgot what color hat he was wearing. How many of the local residents should the professor ask about the color... | 31 | Logic and Puzzles | olympiads |
3. 1 Simplify:
$$
\log _{a}\left[\left(\frac{m^{4} n^{-4}}{m^{-1} n}\right)^{-3} \div\left(\frac{m^{-2} n^{2}}{m n^{-1}}\right)^{5}\right]
$$
Here $m$, $n$, and $a$ are all positive numbers, $a \neq 1$. | 0 | Algebra | olympiads |
\section*{Problem 2 - 161242}
Given a natural number \(n \geq 1\).
Determine the number of different ways to distribute \(2 n\) red, \(2 n\) green, and \(2 n\) black balls into two containers \(Q_{1}\) and \(Q_{2}\) such that each container contains \(3 n\) balls.
Hint:
Two distribution possibilities are considered... | 3n^{2}+3n+1 | Combinatorics | olympiads |
Let $ABCDEF$ be a regular hexagon with side length $a$. At point $A$, the perpendicular $AS$, with length $2a\sqrt{3}$, is erected on the hexagon's plane. The points $M, N, P, Q,$ and $R$ are the projections of point $A$ on the lines $SB, SC, SD, SE,$ and $SF$, respectively.
[list=a]
[*]Prove that the points $M, N, P,... | \theta = 30^\circ | Geometry | aops_forum |
Problem 9.6. Find all pairs of natural prime numbers $p$, $q$, that satisfy the equation
$$
3 p^{4}+5 q^{4}+15=13 p^{2} q^{2}
$$ | (2,3) | Number Theory | olympiads |
2. If the complex number $z=\frac{\sqrt{3}}{2}+\frac{1}{2} \mathrm{i}$, then $z^{2016}=(\quad)$.
(A) -1
(B) $-\mathrm{i}$
(C) $\mathrm{i}$
(D) 1 | D | Algebra | cn_contest |
Irrational Equations
[ Variable Substitution (etc.).
Let $a$ be a given real number, $n$ be a natural number, $n>1$.
Find all such $x$ that the sum of the $n$-th roots of the numbers $x^{n}-a^{n}$ and $2 a^{n}-x^{n}$ equals the number $a$.
# | x_{1}=\sqrt[n]{2},x_{2}= | Algebra | olympiads |
1. A classroom has 30 students and 30 desks arranged in 5 rows of 6 . If the class has 15 boys and 15 girls, in how many ways can the students be placed in the chairs such that no boy is sitting in front of, behind, or next to another boy, and no girl is sitting in front of, behind, or next to another girl? | 2\cdot15!^{2} | Combinatorics | olympiads |
Two circles touch each other at $C$; draw a common tangent to the two circles, which touches the circles at points $A$ and $B$. Calculate the sides of the triangle $ABC$, if the radii of the circles ($R$ and $r$) are given. | AB=2\sqrt{Rr},\BC=2R\sqrt{\frac{r}{R+r}},\AC=2r\sqrt{\frac{R}{R+r}} | Geometry | olympiads |
## Task A-2.3. (8 points)
In triangle $A B C$, the angles $\varangle C A B=35^{\circ} \text{ and } \varangle A B C=60^{\circ}$ are known. If $t$ is the tangent to the circumcircle of this triangle at vertex $C$, and $p$ is the line parallel to line $A B$ through vertex $C$, determine the angle between lines $p$ and $t... | 25 | Geometry | olympiads |
13) Knowing that the equilateral triangle in the figure has a side length of 3 and that the arc of the circle is tangent to two sides, what is the area of the gray figure?
(A) $\sqrt{3}-\frac{\pi}{6}$
(B) $\pi-\sqrt{3}$
(C) $2 \sqrt{3}-\pi$
(D) $\frac{\sqrt{3}+\pi}{6}$
(E) $3 \sqrt{3}-\pi$
$ defined in $-1<x<1$ satisfies the following properties (i) , (ii), (iii).
(i) $f'(x)$ is continuous.
(ii) When $-1<x<0,\ f'(x)<0,\ f'(0)=0$, when $0<x<1,\ f'(x)>0$.
(iii) $f(0)=-1$
Let $F(x)=\int_0^x \sqrt{1+\{f'(t)\}^2}dt\ (-1<x<1)$. If $F(\sin \theta)=c\theta\ (c :\text{constant})$ ... | f(x) = -\sqrt{1 - x^2} | Calculus | aops_forum |
14. As shown in Figure 6, it is known that quadrilateral $ABCD$ is inscribed in a circle $\odot O$ with a diameter of 3, diagonal $AC$ is the diameter, the intersection point of diagonals $AC$ and $BD$ is $P$, $AB=BD$, and $PC=0.6$. Find the perimeter of quadrilateral $ABCD$. | 1+2\sqrt{2}+\sqrt{3}+\sqrt{6} | Geometry | cn_contest |
Find a four-digit number that, when divided by 131, gives a remainder of 112, and when divided by 132, gives a remainder of 98.
# | 1946 | Number Theory | olympiads |
The lengths of the three sides of a triangle are $7, x+4$ and $2 x+1$. The perimeter of the triangle is 36 . What is the length of the longest side of the triangle?
(A) 7
(B) 12
(C) 17
(D) 15
(E) 16 | 17 | Algebra | olympiads |
3. [4 points] Solve the inequality $\left(\sqrt{x^{3}-10 x+7}+1\right) \cdot\left|x^{3}-18 x+28\right| \leqslant 0$. | -1+\sqrt{15} | Inequalities | olympiads |
141. Find the greatest common divisor of all six-digit numbers composed of the digits $1,2,3,4,5,6$ (without repetition). | 3 | Number Theory | olympiads |
In a regular quadrilateral pyramid, two identical spheres of radius $r$ are placed, with their centers located on the axis of symmetry of the pyramid. One of the spheres touches all the lateral faces of the pyramid, while the second touches the base of the pyramid and the first sphere. Find the height of the pyramid fo... | (6+2\sqrt{3})r | Geometry | olympiads |
A $21 \mathrm{~m}$ long, $5 \mathrm{~m}$ wide, and $5 \mathrm{~m}$ high barn has a spider sitting on the centerline of one of its square walls, $1 / 2$ meter from the ceiling. On the centerline of the opposite wall, $1 / 2$ meter from the floor, a fly has been caught in a web. What is the shortest path along which the ... | 26 | Geometry | olympiads |
10. Katherine and James are jogging in the same direction around a pond. They start at the same time and from the same place and each jogs at a constant speed. Katherine, the faster jogger, takes 3 minutes to complete one lap and first overtakes James 8 minutes after starting. How many seconds does it take James to com... | 288 | Algebra | olympiads |
3. Given the sequence $\left\{x_{n}\right\}$ satisfies
$$
x_{1}=2,(n+1) x_{n+1}=x_{n}+n \text {. }
$$
Then the general term of the sequence $\left\{x_{n}\right\}$ is $x_{n}=$ $\qquad$ . | x_{n}=1+\frac{1}{n!} | Algebra | cn_contest |
Example 5 Given that the function $y=f(x)$ defined on $\mathbf{R}$ satisfies $f(x)-f(2-$ $x)=0$ for any $x \in \mathbf{R}$, is its graph an axis-symmetric figure? If so, write down the equation of the axis of symmetry. | 1 | Algebra | olympiads |
7. Find all pairs of integers ( $x, y$ ) for which the equation $x^{2}+y^{2}=x+y+2$ holds.
ANSWER: $(-1,0) ;(-1,1) ;(0,-1) ;(0,2) ;(1,-1),(1,2),(2,0) ;(2,1)$ | (-1,0);(-1,1);(0,-1);(0,2);(1,-1),(1,2),(2,0);(2,1) | Algebra | olympiads |
3. Given the equation about $x$: $x^{3}+(1-a) x^{2}-2 a x+a^{2}$ $=0$ has only one real root. Then the range of the real number $a$ is $\qquad$. | a<-\frac{1}{4} | Algebra | cn_contest |
10.2. Petya runs down from the fourth floor to the first floor 2 seconds faster than his mother rides the elevator. Mother rides the elevator from the fourth floor to the first floor 2 seconds faster than Petya runs down from the fifth floor to the first floor. How many seconds does it take Petya to run down from the f... | 12 | Algebra | olympiads |
3. In $\triangle A B C$, it is known that $D$ is a point on side $B C$, $\frac{B D}{D C}=\frac{1}{3}$, $E$ is the midpoint of side $A C$, $A D$ intersects $B E$ at point $O$, and $C O$ intersects $A B$ at point $F$. Find the ratio of the area of quadrilateral $B D O F$ to the area of $\triangle A B C$ | \frac{1}{10} | Geometry | cn_contest |
6. In the permutation $a_{1}, a_{2}, a_{3}, a_{4}, a_{5}$ of $1,2,3,4,5$, the number of permutations that satisfy the conditions $a_{1}a_{3}, a_{3}a_{5}$ is ( ).
(A) 8
(B) 10
(C) 14
(D) 16 | D | Combinatorics | cn_contest |
[
Diameter, main properties
On the leg $B C$ of the right triangle $A B C$ as a diameter, a circle is constructed, intersecting the hypotenuse at point $D$ such that $A D: B D=1: 3$. The height dropped from the vertex $C$ of the right angle to the hypotenuse is 3. Find the leg $B C$.
# | 6 | Geometry | olympiads |
11. Let $a, b$ be real numbers. Then the minimum value of $a^{2}+a b+b^{2}-$ $a-2 b$ is $\qquad$. | -1 | Algebra | cn_contest |
13.355. A car, having traveled a distance from $A$ to $B$, equal to 300 km, turned back and after 1 hour 12 minutes from leaving $B$, increased its speed by 16 km/h. As a result, it spent 48 minutes less on the return trip than on the trip from $A$ to $B$. Find the original speed of the car. | 60 | Algebra | olympiads |
The base of an isosceles trapezoid $AB = a$ and the side parallel to it $DC = b$. Draw a circle that is tangent to the sides of the trapezoid and connect the points of tangency $M$ and $N$ on the sides $AD$ and $BC$. What is the length of $MN$? | \frac{2ab}{+b} | Geometry | olympiads |
27. N8 (IRN) Let $p$ be a prime number and let $A$ be a set of positive integers that satisfies the following conditions: (i) the set of prime divisors of the elements in $A$ consists of $p-1$ elements; (ii) for any nonempty subset of $A$, the product of its elements is not a perfect $p$ th power. What is the largest p... | (p-1)^2 | Number Theory | olympiads_ref |
3. The seller has a balance scale. Help the seller come up with a set of 4 weights that can be used to weigh any whole number of kilograms from 1 to 12 on these scales. No more than two weights can be used for each weighing; weights can be placed on different pans of the scale. | 1,2,5,10 | Logic and Puzzles | olympiads |
4. If $x_{1}, x_{2}$ are two distinct real roots of the equation $x^{2}+2 x-k=0$, then $x_{1}^{2}+x_{2}^{2}-2$ is ( ).
(A) Positive
(B) Zero
(C) Negative
(D) Not greater than zero
Translate the text above into English, please keep the original text's line breaks and format, and output the translation result directly. | A | Logic and Puzzles | cn_contest |
5. [4] Compute
$$
\lim _{h \rightarrow 0} \frac{\sin \left(\frac{\pi}{3}+4 h\right)-4 \sin \left(\frac{\pi}{3}+3 h\right)+6 \sin \left(\frac{\pi}{3}+2 h\right)-4 \sin \left(\frac{\pi}{3}+h\right)+\sin \left(\frac{\pi}{3}\right)}{h^{4}}
$$ | \frac{\sqrt{3}}{2} | Calculus | olympiads |
Given an $n \times n \times n$ grid of unit cubes, a cube is [i]good[/i] if it is a sub-cube of the grid and has side length at least two. If a good cube contains another good cube and their faces do not intersect, the first good cube is said to [i]properly[/i] contain the second. What is the size of the largest possib... | (n-1)^3 + (n-2)^3 | Combinatorics | aops_forum |
4・241 Given $2 \lg (x-2 y)=\lg x+\lg y$, try to find $x: y$.
Translate the above text into English, please retain the original text's line breaks and format, and output the translation result directly. | \frac{x}{y}=4 | Logic and Puzzles | olympiads |
7. (1) Does there exist a cube that can be expressed as the sum of two squares?
(2) Does there exist a fourth power that can be expressed as the sum of two squares? | 15^{2}+20^{2}=5^{4} | Number Theory | olympiads |
Example 7 The function $f(x)$ is defined on the set of real numbers, and for all real numbers $x$ it satisfies the equations: $f(2+x)=f(2-x)$ and $f(x+7)=f(7-x)$. Suppose $x=0$ is a root of $f(x)=0$, and let $N$ denote the number of roots of $f(x)=0$ in the interval $[-1000,1000]$. Find the minimum value of $N$. | 401 | Algebra | cn_contest |
$19 \cdot 30$ If a convex polygon has exactly three obtuse interior angles, the maximum number of sides this polygon can have is
(A) 4 .
(B) 5 .
(C) 6 .
(D) 7 .
(E) 8 .
(36th American High School Mathematics Examination, 1985) | 6 | Geometry | olympiads |
248. The equation of the hyperbola $y=\frac{1-3 x}{2 x-1}$ can be transformed using a parallel translation of the coordinate axes to the form $X Y=m$. Plot this hyperbola. | XY=-0.25 | Algebra | olympiads |
8. find all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ such that for all real $x, y$ holds
$$
f(f(x)-f(y))=(x-y)^{2} f(x+y)
$$
## Solution | f(x)=0,\quadf(x)=x^{2},\quadf(x)=-x^{2} | Algebra | olympiads |
17. Carla rolls, all together, 4 dice, with faces numbered from 1 to 6. What is the probability that the product of the 4 numbers rolled is 24?
(A) $5 / 162$
(B) $7 / 324$
(C) $1 / 36$
(D) $13 / 324$
(E) $5 / 144$ | \frac{13}{324} | Combinatorics | olympiads |
After Sally takes $20$ shots, she has made $55\%$ of her shots. After she takes $5$ more shots, she raises her percentage to $56\%$. How many of the last $5$ shots did she make?
$\textbf{(A)}\ 1\qquad\textbf{(B)}\ 2\qquad\textbf{(C)}\ 3\qquad\textbf{(D)}\ 4\qquad\textbf{(E)}\ 5$ | 3 | Algebra | amc_aime |
70. Given $a: b: c=2: 3: 4, a b+b c+c a=13$, then $b c=$ | 6 | Algebra | olympiads |
Tokarev S.i.
The set of five-digit numbers $\{N_1, \dots, N_k\}$ is such that any five-digit number, all of whose digits are in increasing order, coincides in at least one digit with at least one of the numbers $N_1, \dots, N_k$. Find the smallest possible value of $k$. | 1 | Combinatorics | olympiads |
All faces of a tetrahedron are right-angled triangles. It is known that three of its edges have the same length $s$. Find the volume of the tetrahedron. | \frac{s^3}{6} | Geometry | olympiads_ref |
## problem statement
Find the point of intersection of the line and the plane.
$\frac{x+2}{1}=\frac{y-2}{0}=\frac{z+3}{0}$
$2 x-3 y-5 z-7=0$ | (-1,2,-3) | Algebra | olympiads |
2. A three-digit number $X$ was written with three different digits $A B C$. Four schoolchildren made the following statements. Petya: “The largest digit in the number $X$ is $B$”. Vasya: “$C=8$”. Tolya: “The largest digit is $C$”. Dima: “$C$ is the arithmetic mean of $A$ and $B$”. Find the number $X$, given that exact... | 798 | Logic and Puzzles | olympiads |
a) Let $n$ be a positive integer. Prove that $ n\sqrt {x-n^2}\leq \frac {x}{2}$ , for $x\geq n^2$.
b) Find real $x,y,z$ such that: $ 2\sqrt {x-1} +4\sqrt {y-4} + 6\sqrt {z-9} = x+y+z$ | (x, y, z) = (2, 8, 18) | Inequalities | aops_forum |
For the digits of the decimal number $\overline{a b c d}$, it holds that $a>b>c>d$. These same digits, in some order, are also the digits of the difference $\overline{a b c d}-\overline{d c b a}$. Which is this four-digit number? | 7641 | Number Theory | olympiads |
On the table are written three natural numbers $a, b, c$, for which the following holds:
- the greatest common divisor of numbers $a, b$ is 15,
- the greatest common divisor of numbers $b, c$ is 6,
- the product of numbers $b, c$ is 1800,
- the least common multiple of numbers $a, b$ is 3150.
What are these numbers?
... | 315,150,12 | Number Theory | olympiads |
6. Find all pairs of natural numbers $n$ and $k$ for which $(n+1)^{k}=$ $n!+1$. (As usual, $n!$ denotes the product of all natural numbers not exceeding $n$. For example, $4!=1 \cdot 2 \cdot 3 \cdot 4$.) | n=k=1,n=2k=1,n=4k=2 | Number Theory | olympiads |
6. A sequence is recursively defined as: $t_{1}=1$, for $n>1$, when $n$ is even, $t_{n}=1+t_{\frac{1}{2} ;}$ when $n$ is odd, $t_{n}=\frac{1}{t_{n-1}}$. If it is known that $t_{n}=\frac{19}{87}$, then the sum of the digits of $n$ is
A. 15
B. 17
C. 19
D. 21
E. 23 | 15 | Algebra | olympiads |
4. Let the function be
$$
\begin{array}{l}
f(x)=\sqrt{10-6 \cos x}+\sqrt{\frac{17}{8}-\frac{3 \sqrt{2}}{2} \sin x}+ \\
\sqrt{19-2 \sqrt{2} \cos x-8 \sin x} \text {. } \\
\end{array}
$$
For all real numbers $x$, the minimum value of $f(x)$ is
$\qquad$ . | \frac{21 \sqrt{2}}{4}-1 | Algebra | cn_contest |
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