problem stringlengths 37 4.98k | answer stringlengths 1 141 | subject stringclasses 8
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## Task B-1.3.
Marko spent 100 euros after receiving his salary. Five days later, he won $\frac{1}{4}$ of the remaining amount from his salary in a lottery and spent another 100 euros. After fifteen days, he received $\frac{1}{4}$ of the amount he had at that time and spent another 100 euros. In the end, he had 800 eu... | 2100 | Algebra | olympiads |
3. Let $O$ be the center of the circumcircle of triangle $ABC$, points $O$ and $B$ lie on opposite sides of line $AC$, $\angle AOC = 60^\circ$. Find the angle $AMC$, where $M$ is the center of the incircle of triangle $ABC$. | 165 | Geometry | olympiads |
There are $ n$ sets having $ 4$ elements each. The difference set of any two of the sets is equal to one of the $ n$ sets. $ n$ can be at most ? (A difference set of $A$ and $B$ is $ (A\setminus B)\cup(B\setminus A) $)
$\textbf{(A)}\ 3 \qquad\textbf{(B)}\ 5 \qquad\textbf{(C)}\ 7 \qquad\textbf{(D)}\ 15 \qquad\textb... | 7 | Combinatorics | aops_forum |
2. On 2016 cards, numbers from 1 to 2016 were written (each one once). Then $k$ cards were taken. What is the smallest $k$ such that among them there will be two cards with numbers, the difference of whose square roots is less than 1? | 45 | Number Theory | olympiads |
$2.9 \quad \frac{x^{2}+x-5}{x}+\frac{3 x}{x^{2}+x-5}+4=0$.
Solve the equation:
$2.9 \quad \frac{x^{2}+x-5}{x}+\frac{3 x}{x^{2}+x-5}+4=0$. | x_{1}=-5,x_{2}=1,x_{3,4}=-1\\sqrt{6} | Algebra | olympiads |
Find all real numbers a such that all solutions to the quadratic equation $ x^2 \minus{} ax \plus{} a \equal{} 0$ are integers. | a = 0, 4 | Number Theory | aops_forum |
7.3. Agent 007 wants to encrypt his number using two natural numbers m and n so that $0.07=\frac{1}{m}+\frac{1}{n}$. Can he do it? | 0.07=\frac{1}{50}+\frac{1}{20} | Number Theory | olympiads |
5. Find all positive integers $n$ such that $n^{4}-n^{3}+3 n^{2}+5$ is a perfect square. | 2 | Algebra | olympiads |
Consider the polynomial $ f(x) \equal{} ax^2 \plus{} bx \plus{} c$, with degree less than or equal to 2.
When $ f$ varies with subject to the constrain $ f(0) \equal{} 0,\ f(2) \equal{} 2$, find the minimum value of $ S\equal{}\int_0^2 |f'(x)|\ dx$. | 2 | Calculus | aops_forum |
How many positive whole numbers less than $100$ are divisible by $3$, but not by $2$? | 17 | Number Theory | aops_forum |
Example 19. A coin is tossed 5 times. The random variable $X$ is the number of times the head appears. Possible values of the variable $X: \cdot x_{0}=0$, $x_{1}=1, x_{2}=2, x_{3}=3, x_{4}=4, x_{5}=5$. Write the distribution law of the random variable $X$. | \begin{pmatrix}X&0&1&2&3&4&5\\\hlineP&\frac{1}{32}&\frac{5}{32}&\frac{10}{32}&\frac{10}{32}&\frac{5}{32}&\frac{1}{32}\end{pmatrix} | Combinatorics | olympiads |
3. There are three boxes $A$, $B$, and $C$, which contain 100, 50, and 80 balls of the same size, respectively. Each box contains some black balls. It is known that box $A$ has 15 black balls. If a box is randomly selected from the three boxes and then a ball is randomly drawn from this box, the probability of drawing ... | 22 | Combinatorics | olympiads |
5.12 Let real numbers $x_{1}, x_{2}, \cdots, x_{6}$ satisfy the conditions
$$
\left\{\begin{array}{l}
x_{1}^{2}+\cdots+x_{6}^{2}=6 \\
x_{1}+\cdots+x_{6}=0 .
\end{array}\right.
$$
Find the maximum possible value of $x_{1} x_{2} \cdots x_{6}$. | \frac{1}{2} | Algebra | olympiads |
1. (2002 National High School Competition Question) The line $\frac{x}{3}+\frac{y}{4}=1$ intersects the ellipse $\frac{x^{2}}{16}+\frac{y^{2}}{9}=1$ at points $A, B$. A point $P$ on the ellipse makes the area of $\triangle P A B$ equal to 3. How many such $P$ points are there?
A. 1
B. 2
C. 3
D. 4 | B | Geometry | olympiads |
Determine all triples of positive integers $(a, b, n)$ that satisfy the following equation: $a! + b! = 2^n$ | (1, 1, 1), (2, 2, 2), (3, 2, 3), (2, 3, 3) | Number Theory | aops_forum |
1. Given $P=\{x \mid x=3 k, k \in \mathbf{Z}\}, Q=\{x \mid x=3 k+1, k \in \mathbf{Z}\}, S$ $\{x \mid x=3 k-1, k \in \mathbf{Z}\}$. If $a \in P, b \in Q, c \in S$, then ( ).
A. $a+b-c \in P$
B. $a+b-c \in Q$
C. $a+b-c \in S$
D. $a+b-c \in P \cup Q$ | +b-\inS | Number Theory | olympiads |
A parabola passes through the point of intersection of the lines with equations $y=-x+3$ and $x-2 y-6=0$, as well as the the $x$-intercept of each line. If the parabola also passes through the point $(10, k)$, what is the value of $k$ ? | 14 | Algebra | olympiads |
There are 2016 customers who entered a shop on a particular day. Every customer entered the shop exactly once. (i.e. the customer entered the shop, stayed there for some time and then left the shop without returning back.)
Find the maximal $k$ such that the following holds:
There are $k$ customers such that either all ... | 45 | Combinatorics | olympiads_ref |
2. Let $\mathrm{i}$ be the imaginary unit, simplify $(\mathrm{i}+1)^{2016}+(\mathrm{i}-1)^{2016}=$ | 2^{1009} | Algebra | olympiads |
5. (20 points) In a right triangle $A B C\left(\angle C=90^{\circ}\right)$ with side $B C=a$, points $M$ and $N$ are the midpoints of sides $A B$ and $B C$, respectively. The bisector of $\angle A$ intersects line $M N$ at point $L$. Find the radius of the circumcircle of triangle $A C L$, if angle $\angle C B L=\alpha... | \frac{}{2\sin2\alpha} | Geometry | olympiads |
35. Let $f(x)=x^{2}+a x+b \cos x$, find all pairs of real numbers $(a, b)$, such that the equation $f(x)=0$ and $f(f(x))=0$ have the same and non-empty set of real solutions. | (,b)\mid0\leqslant<4,b=0 | Algebra | olympiads |
Example 4 As shown in Figure 4, in the acute triangle $\triangle ABC$, side $BC$ is the shortest side. Points $O, G, I, H$ are the circumcenter, centroid, incenter, and orthocenter of the triangle, respectively. When $OG = GI = IH$, then $AB : AC : BC =$ | 2: 2: 1 | Geometry | cn_contest |
The first three terms of a geometric sequence form a geometric sequence with a common ratio of 2, while the common ratios of the three geometric sequences form an arithmetic sequence with a common difference of 1. The sum of the second terms of the three geometric sequences is 24, and the sum of the first three terms o... | 1,2,4\ldots;2,6,18\ldots4,16,64\ldots\\\frac{192}{31},\frac{168}{31},\ldots\frac{384}{31},\frac{48}{31},\ldots;\frac{768}{31},\frac{864}{31}, | Algebra | olympiads |
8.3. In triangle $A B C$, side $A C$ is the largest. Points $M$ and $N$ on side $A C$ are such that $A M=A B$ and $C N=C B$. It is known that angle $N B M$ is three times smaller than angle $A B C$. Find $\angle A B C$. | 108 | Geometry | olympiads |
Let $ n $ be a natural number. How many numbers of the form $ \pm 1\pm 2\pm 3\pm\cdots\pm n $ are there? | \frac{n(n+1)}{2} + 1 | Combinatorics | aops_forum |
8. Solve the system $\left\{\begin{array}{l}\log _{4} x-\log _{2} y=0 \\ x^{2}-5 y^{2}+4=0 .\end{array}\right.$ | {1;1},{4;2} | Algebra | olympiads |
## Task $1 / 86$
We are looking for the smallest natural number $n$, which is the product of 3 prime factors $p_{1} ; p_{2} ; p_{3}$, and it holds that: $p_{3}=55 \cdot p_{1} \cdot p_{2}+1$ and $p_{3}>p_{2}>p_{1}$. | 1986 | Number Theory | olympiads |
A straight quadrilateral pyramid has a square base; each of the side edges is equal to the diagonal of the base. What is the surface area and volume of the pyramid if the diagonal of the base is $d$? | F=\frac{^{2}}{2}(1+\sqrt{7}),K=\frac{^{3}}{12}\sqrt{3} | Geometry | olympiads |
1. Riana has been asked to erase digits from the number 12323314 to obtain a number which reads the same from left to right as it does from right to left. What is the smallest number of digits Riana needs to erase?
A 1
B 2
C 3
D 4
E 5 | 3 | Logic and Puzzles | olympiads |
A robot is placed on the grid shown. The robot starts on square 25 , initially facing square 32 . The robot (i) moves 2 squares forward in the direction that it is facing, (ii) rotates clockwise $90^{\circ}$, and (iii) moves 1 square forward in the new direction. Thus, the robot moves to square 39, then turns to face s... | 16 | Logic and Puzzles | olympiads |
2. 10 people go to the bookstore to buy books, it is known that: (1) each person bought three types of books; (2) any two people have at least one book in common.
How many people at most could have bought the book that was purchased by the fewest people? | 5 | Combinatorics | olympiads |
Example 3 (2000 Hebei Provincial Competition Question) Given $a, b \in \mathbf{R}^{+}, m, n \in \mathbf{R}, m^{2} n^{2}>a^{2} m^{2}+b^{2} n^{2}$, let $M=\sqrt{m^{2}+n^{2}}, N=a+b$, then the size relationship between $M$ and $N$ is ( ).
A. $M>N$
B. $M<N$
C. $M=N$
D. The size relationship between $M$ and $N$ cannot be de... | M>N | Inequalities | olympiads |
2. Find the non-negative real solutions of the system of equations:
$$
\left\{\begin{array}{l}
x+y+z=3 x y, \\
x^{2}+y^{2}+z^{2}=3 x z, \\
x^{3}+y^{3}+z^{3}=3 y z
\end{array}\right.
$$ | (x,y,z)=(0,0,0)or(1,1,1) | Algebra | olympiads |
3. Today is Sunday, 1 day ago was Saturday, $\cdots \cdots,\left(5^{5}\right)^{5}$ days ago was ( ).
A. Monday
B. Tuesday
C. Wednesday
D. Thursday
E. Friday
F. Saturday
G. Sunday | Tuesday | Number Theory | olympiads |
Izmeystiev I.V.
The network of bus routes in the suburb of Amsterdam is organized in such a way that:
a) each route has exactly three stops;
b) any two routes either have no common stops at all or have only one common stop. What is the maximum number of routes that can be in this suburb if there are a total of 9 sto... | 12 | Combinatorics | olympiads |
Find the sum of all possible sums $a + b$ where $a$ and $b$ are nonnegative integers such that $4^a + 2^b + 5$ is a perfect square. | 9 | Number Theory | aops_forum |
2. Due to the flood, water flooded the basement room, so we turned on a water pump that pumps out 180 liters of water per minute. The room is 3.7 m long, 2.3 m wide, and the water level in the room was 1.2 m.
a) Write how the amount of water in the room changed during pumping, as a function of time.
b) Write how long... | 56 | Algebra | olympiads |
II. (20 points) As shown in Figure 4, in circle $\odot O$, chord $AB$ divides the circle into two segments with areas in the ratio $1:3$. Find the size of the central angle $\angle AOB$ (require precision to $1'$ or 0.001 radians).
| 2.31 | Geometry | cn_contest |
Let $n \in \mathbb{N}^{*}$. Find the number of $n$-digit numbers whose digits are among $\{2,3,7,9\}$ and which are divisible by 3. | \frac{4^{n}+2}{3} | Number Theory | olympiads |
(1) Given $\theta \in\left[\frac{5 \pi}{4}, \frac{3 \pi}{2}\right]$, then $\sqrt{1-\sin 2 \theta}-\sqrt{1+\sin 2 \theta}$ can be simplified to ( ).
(A) $2 \sin \theta$
(B) $-2 \sin \theta$
(C) $-2 \cos \theta$
(D) $2 \cos \theta$ | 2\cos\theta | Algebra | olympiads |
14, 43 students, each carrying a different amount of money ranging from 8 cents to 5 yuan. Each student spent all their money on picture cards. There are only two types of picture cards, 3 cents each and 5 cents each, and each student tried to buy as many 5-cent cards as possible. How many 3-cent cards did they buy in ... | 84 | Number Theory | olympiads |
## Task Condition
Calculate the area of the parallelogram constructed on vectors $a_{\text {and }} b$.
\[
\begin{aligned}
& a=4 p-q \\
& b=p+2 q \\
& |p|=5 \\
& |q|=4 \\
& (\widehat{p, q})=\frac{\pi}{4}
\end{aligned}
\] | 90\sqrt{2} | Algebra | olympiads |
6. The line segment connecting two points on a sphere is called a chord of the sphere.
For a sphere with a radius of 4, the lengths of two chords \( AB \) and \( CD \) are \( 2 \sqrt{7} \) and \( 4 \sqrt{3} \) respectively. \( M \) and \( N \) are the midpoints of \( AB \) and \( CD \) respectively, and the endpoints ... | A | Geometry | cn_contest |
Example 7 (1992 National High School League Question) Let the sequence $a_{1}, a_{2}, \cdots, a_{n}, \cdots$ satisfy $a_{1}=a_{2}=1, a_{3}=2$, and for any positive integer $n$, $a_{n} a_{n+1} a_{n+2} \neq 1$. Also, $a_{n} a_{n+1} a_{n+2} a_{n+3}=a_{n}+a_{n-1}+a_{n+2}+a_{n+3}$, then the value of $a_{1}+a_{2}+\cdots+a_{1... | 200 | Algebra | olympiads |
1. (8 points) Calculate: $80 \times 37 + 47 \times 63=$ | 5921 | Algebra | olympiads |
## Task 1 - 180611
In a district of Leipzig, 260 large apartments were renovated.
For one tenth of these apartments, each apartment has $55 \mathrm{~m}^{2}$ of living space; for one quarter of the 260 apartments, each apartment has $67 \mathrm{~m}^{2}$ of living space; each of the other 260 apartments has $80 \mathrm... | 19305\mathrm{~}^{2} | Algebra | olympiads |
Find all pairs of non-zero natural numbers $(k, n)$ for which
$$
1!+2!+\cdots+k!=1+2+\cdots+n
$$ | (1,1),(2,2),(5,17) | Number Theory | olympiads_ref |
3B. Determine all complex numbers $z$ for which
$$
z \bar{z}+1=-i(z-\bar{z})
$$ | i | Algebra | olympiads |
5. A sequence of digits consists of the first 222 natural numbers written in a row. In this sequence, we cross out the digits that are in odd positions. After that, we again cross out the digits that are in (new) odd positions. We repeat this procedure until only one digit remains. Which digit will it be?
## Ministry ... | 0 | Number Theory | olympiads |
A die is a cube with its faces numbered 1 through 6 . One red die and one blue die are rolled. The sum of the numbers on the top two faces is determined. What is the probability that this sum is a perfect square? | \frac{7}{36} | Combinatorics | olympiads |
3. On a chessboard, $8 \times 8$, there are 63 coins of 5 denari each and one coin of 10 denari, placed such that there is exactly one coin in each square. There are enough 5, 10, and 20 denari coins available. The following exchanges of three coins on the board with other three coins are possible:
$$
\begin{array}{ll... | no | Logic and Puzzles | olympiads |
6. In the chicken and rabbit cage, there are a total of 40 heads, the number of rabbit feet is 8 less than 10 times the number of chicken feet, so the number of rabbits is $\qquad$. | 33 | Algebra | olympiads |
24. Given real numbers $a$, $b$, $c$, the polynomial
$$
g(x)=x^{3}+a x^{2}+x+10
$$
has three distinct roots, and these three roots are also roots of the polynomial
$$
f(x)=x^{4}+x^{3}+b x^{2}+100 x+c
$$
Then the value of $f(1)$ is $(\quad)$.
(A) -9009
(B) -8008
(C) -7007
(D) -6006
(E) -5005 | C | Algebra | cn_contest |
In the diagram, $D$ is on side $A C$ of $\triangle A B C$ so that $B D$ is perpendicular to $A C$. If $A B=29$, $A C=69$, and $B D=20$, what is the length of $B C$ ?
## | 52 | Geometry | olympiads |
6. If we pick (uniformly) a random square of area 1 with sides parallel to the $x-$ and $y$-axes that lies entirely within the 5 -by- 5 square bounded by the lines $x=0, x=5, y=$ $0, y=5$ (the corners of the square need not have integer coordinates), what is the probability that the point $(x, y)=(4.5,0.5)$ lies within... | \frac{1}{64} | Geometry | olympiads |
Let $ z = \frac{1}{2}(\sqrt{2} + i\sqrt{2}) $. The sum $$ \sum_{k = 0}^{13} \dfrac{1}{1 - ze^{k \cdot \frac{i\pi}{7}}} $$
can be written in the form $ a - bi $. Find $ a + b $. | 7 - 7i | Calculus | aops_forum |
26. Find the minimum value of the expression $\left(a^{2}+x^{2}\right) / x$, where $a>0$ is a constant, and $x>0$ is a variable. | 2a | Algebra | olympiads |
8. The real numbers $x, y$ and $z$ are a solution $(x, y, z)$ of the equation $\left(x^{2}-9\right)^{2}+\left(y^{2}-4\right)^{2}+\left(z^{2}-1\right)^{2}=0$. How many different possible values are there for $x+y+z ?$ | 7 | Algebra | olympiads |
11.4 Another participant in the meaningless activity contest brought with him $N$ unit squares, from which he immediately, in front of the amazed jury, formed a rectangle with sides differing by 9. Not stopping there, the participant then formed a large square from these same $N$ squares, but this time 6 squares were l... | 10,22,70,682 | Number Theory | olympiads |
In parallelogram $ABCD$, $AC=10$ and $BD=28$. The points $K$ and $L$ in the plane of $ABCD$ move in such a way that $AK=BD$ and $BL=AC$. Let $M$ and $N$ be the midpoints of $CK$ and $DL$, respectively. What is the maximum walue of $\cot^2 (\tfrac{\angle BMD}{2})+\tan^2(\tfrac{\angle ANC}{2})$ ? | 2 | Geometry | aops_forum |
A sequence of positive integers $a_{1}, a_{2}, \ldots$ is such that for each $m$ and $n$ the following holds: if $m$ is a divisor of $n$ and $m<n$, then $a_{m}$ is a divisor of $a_{n}$ and $a_{m}<a_{n}$. Find the least possible value of $a_{2000}$. | 128 | Number Theory | olympiads |
5. (1990 American High School Mathematics Examination) First, select $a$ from $\{1,2,3, \cdots, 99,100\}$, then select $b$ from the same set. The probability that the last digit of $3^{a}+7^{b}$ is 8 is ( ).
A. $\frac{1}{16}$
B. $\frac{1}{8}$
C. $\frac{3}{16}$
D. $\frac{1}{5}$
E. $\frac{1}{4}$ | \frac{3}{16} | Number Theory | olympiads |
In the Egyptian triangle, whose sides are 3, 4, and 5 units long, inscribe a rectangle whose vertices lie on the sides of the triangle and the ratio of its sides is $1: 3$. Determine the lengths of the sides of the rectangle. | \frac{12}{13},\frac{36}{13},\frac{4}{5},\frac{12}{5},\frac{60}{61},\frac{180}{61},\frac{20}{29},\frac{60}{29} | Geometry | olympiads |
Let $a_1,a_2,\dots,a_{2017}$ be reals satisfied $a_1=a_{2017}$, $|a_i+a_{i+2}-2a_{i+1}|\le 1$ for all $i=1,2,\dots,2015$. Find the maximum value of $\max_{1\le i<j\le 2017}|a_i-a_j|$. | 508032 | Inequalities | aops_forum |
5. In the country of Lemonia, coins of denominations $3^{n}, 3^{n-1} \cdot 4, 3^{n-2} \cdot 4^{2}, 3^{n-3} \cdot 4^{3}, \ldots, 3 \cdot 4^{n-1}, 4^{n}$ piastres are in circulation, where $n-$ is a natural number. A resident of the country went to the bank without any cash on hand. What is the largest amount that the ba... | 2\cdot4^{n+1}-3^{n+2} | Number Theory | olympiads |
The percent that $M$ is greater than $N$ is:
$(\mathrm{A})\ \frac{100(M-N)}{M} \qquad (\mathrm{B})\ \frac{100(M-N)}{N} \qquad (\mathrm{C})\ \frac{M-N}{N} \qquad (\mathrm{D})\ \frac{M-N}{N} \qquad (\mathrm{E})\ \frac{100(M+N)}{N}$ | \frac{100(M-N)}{N} | Algebra | amc_aime |
Find the value of $\alpha+2 \beta$ angle, if $\operatorname{tg} \alpha=\frac{1}{7}, \operatorname{tg} \beta=\frac{1}{3}$? | \alpha+2\beta=45 | Algebra | olympiads |
5. Let the complex numbers $z_{1}, z_{2}$ correspond to points $A, B$ in the complex plane, respectively, and $\left|z_{1}\right|=4, 4 z_{1}^{2}-2 z_{1} z_{2}+z_{2}^{2}=0$. $O$ is the origin. Then the area of $\triangle O A B$ is
(A) $8 \sqrt{3}$;
(B) $4 \sqrt{3}$;
(C) $6 \sqrt{3}$;
(D) $12 \sqrt{3}$. | 8\sqrt{3} | Algebra | olympiads |
2. As shown in Figure 2, person A at point $A$ on the shore notices person B in distress at point $B$ in the water. Point $B$ is 30 meters away from the shore at $C$, and $\angle BAC=15^{\circ}$. Person A runs on the shore at a speed that is $\sqrt{2}$ times their swimming speed in the water. Given that A's swimming sp... | 15 \sqrt{2}+5 \sqrt{6} | Geometry | cn_contest |
9・36 Find the smallest real number $c$ such that for any positive sequence $\left\{x_{n}\right\}$, if $x_{1}+x_{2}+\cdots+x_{n} \leqslant x_{n+1}, n=1,2,3, \cdots$, then
$$\sqrt{x_{1}}+\sqrt{x_{2}}+\cdots+\sqrt{x_{n}} \leqslant c \sqrt{x_{1}+x_{2}+\cdots+x_{n}}, n=1,2,3, \cdots$$ | \sqrt{2}+1 | Inequalities | inequalities |
Let $S$ be the set $\{1, 2, ..., 10\}$. Let $A$ be a subset of $S$.
We arrange the elements of $A$ in increasing order, that is, $A = \{a_1, a_2, ...., a_k\}$ with $a_1 < a_2 < ... < a_k$.
Define [i]WSUM [/i] for this subset as $3(a_1 + a_3 +..) + 2(a_2 + a_4 +...)$ where the first term contains the odd numbered terms... | 2^{n-3} (5n^2 + 5n + 2) | Combinatorics | aops_forum |
(2) The points $(x, y)$ in the plane that satisfy the constraints $\left\{\begin{array}{l}x \geqslant 2, \\ x+y \leqslant 0, \\ x-y-10 \leqslant 0\end{array}\right.$ form a region $D$. The region $D$ is symmetric to region $E$ with respect to the line $y=2 x$. Then, the distance between the closest points in regions $D... | \frac{12\sqrt{5}}{5} | Geometry | olympiads |
Example 1 Find $a, b, c$, such that they satisfy the inequality $a^{2}+b^{2}+c^{2}+3<a b+3 b+2 c(a, b, c \in \mathbf{Z})$. | =1,b=2,=1 | Inequalities | olympiads |
10. If for all positive numbers $x, y$, the inequality $\sqrt{x}+\sqrt{y}$ $\leqslant a \sqrt{x+y}$ holds, then the minimum value of $a$ is $\qquad$ | \sqrt{2} | Inequalities | cn_contest |
Below the standard representation of a positive integer $n$ is the representation understood by $n$ in the decimal system, where the first digit is different from $0$. Everyone positive integer n is now assigned a number $f(n)$ by using the standard representation of $n$ last digit is placed before the first.
Examples:... | 105263157894736842 | Number Theory | aops_forum |
1. Given $1 \leqslant a_{1} \leqslant a_{2} \leqslant a_{3} \leqslant a_{4} \leqslant a_{5} \leqslant a_{6} \leqslant$ 64. Then the minimum value of $Q=\frac{a_{1}}{a_{2}}+\frac{a_{3}}{a_{4}}+\frac{a_{5}}{a_{6}}$ is $\qquad$ . | \frac{3}{4} | Inequalities | cn_contest |
B1. Draw the set of points $(x, y)$ in a rectangular coordinate system in the plane that satisfy the condition $(x \leq 4) \wedge(y \geq-1) \wedge(y \leq x)$. Calculate the area of the shape that the set of points represents. | \frac{25}{2} | Geometry | olympiads |
6. The ratio of the sides of a triangle, which is inscribed in a circle of radius $2 \sqrt{3}$, is $3: 5: 7$. Find the area of the triangle.
(1 mark)
6. A triangle is inscribed in a circle with a radius of $2 \sqrt{3}$. The ratio of its side lengths is $3: 5: 7$. Find the area of the triangle. | \frac{135}{49}\sqrt{3} | Geometry | olympiads |
3. Let $a, b, c$ be the three sides of a right triangle, with $c$ being the hypotenuse. The maximum value of $k$ such that the inequality $a^{2}(b+c)+b^{2}(c+a)$ $+c^{2}(a+b) \geqslant k a b c$ holds for all right triangles is $\qquad$. | 2+3\sqrt{2} | Inequalities | cn_contest |
2. In a $11 \times 11$ square grid, we sequentially wrote the numbers $1,2, \ldots, 121$ from left to right and from top to bottom. Using a $3 \times 3$ square tile, we covered exactly nine cells in all possible ways. In how many cases was the sum of the nine covered numbers a perfect square of an integer? | 6 | Number Theory | olympiads |
2. A paper punch can be placed at any point on a plane. When it works, it can punch out points that are at an irrational distance from it. What is the minimum number of paper punches needed to punch out all points on the plane? | 3 | Number Theory | cn_contest |
The number obtained from the last two nonzero digits of $90!$ is equal to $n$. What is $n$?
$\textbf{(A)}\ 12 \qquad \textbf{(B)}\ 32 \qquad \textbf{(C)}\ 48 \qquad \textbf{(D)}\ 52 \qquad \textbf{(E)}\ 68$ | 12 | Number Theory | amc_aime |
96. Xi Xi remembers that the English word "hello" is composed of three different letters $h, e, o$ and two identical letters $l$, but she doesn't remember the order, so the number of possible incorrect spellings Xi Xi might have is $\qquad$ kinds. | 59 | Combinatorics | olympiads |
Task 1. Determine all pairs of prime numbers whose difference of squares is equal to 120. | (31,29),(17,13),(13,7) | Number Theory | olympiads |
In parallelogram $A B C D$, the bisector of angle $B A D$ intersects side $C D$ at point $M$, and $\frac{D M}{M C}=2$. It is known that angle $C A M$ is equal to $\alpha$. Find angle $BAD$.
# | 2\operatorname{arctg}(5\operatorname{tg}\alpha) | Geometry | olympiads |
12. Teacher Li brought a stack of art paper, which was just enough to be evenly distributed among 24 students. Later, 8 more students came, so each person received 2 fewer sheets than originally. How many sheets of art paper did Teacher Li bring in total? | 192 | Algebra | olympiads |
At the theater children get in for half price. The price for $5$ adult tickets and $4$ child tickets is $$24.50$. How much would $8$ adult tickets and $6$ child tickets cost?
$\textbf{(A) }$35\qquad \textbf{(B) }$38.50\qquad \textbf{(C) }$40\qquad \textbf{(D) }$42\qquad \textbf{(E) }$42.50$ | 38.50 | Algebra | amc_aime |
4. In the coordinate plane, a point whose both coordinates are integers is called an integer point. For any natural number $n$, connect the origin $O$ with the point $A_{n}(n, n+3)$, and let $f(n)$ denote the number of integer points on the line segment $O A_{n}$, excluding the endpoints. Then
$$
f(1)+f(2)+\cdots+f(199... | 1326 | Number Theory | olympiads |
10.2. There are 2004 small boxes on the table, each containing 1 ball. It is known that some of the balls are white, and there are an even number of white balls. You are allowed to point to any 2 boxes and ask: "Do they contain at least 1 white ball?" How many times do you need to ask, at a minimum, to determine a box ... | 2003 | Combinatorics | cn_contest |
11 Given the radius of a sphere is 2, and the planes of three great circles on the sphere are mutually perpendicular, then the volume of the octahedron formed by the intersection points of the three great circles is ( ).
(A) $\frac{16}{3}$
(B) $\frac{32}{3}$
(C) $\frac{64}{3}$
(D) 32 | \frac{32}{3} | Geometry | olympiads |
The symbol $\odot$ represents a special operation with numbers; some examples are $2 \odot 4=10,3 \odot 8=27,4 \odot 27=112$ and $5 \odot 1=10$. What is the value of $4 \odot(8 \odot 7)$?
(a) 19
(b) 39
(c) 120
(d) 240
(e) 260 | 260 | Algebra | olympiads |
1. In the set of real numbers, solve the equation
$$
\log _{1997}\left(\sqrt{1+x^{2}}+x\right)=\log _{1996}\left(\sqrt{1+x^{2}}-x\right)
$$ | 0 | Algebra | olympiads |
In $\square A B C D$, $C E \perp A B, C F \perp A D$, with feet of the perpendiculars at $E$ and $F$. Let $E F$ intersect the diagonal $B D$ at point $P$. If $A B: A D=2: 3$, find $P F: P E$. | 4:9 | Geometry | cn_contest |
Problem 1. a) Find the last digit of the number $x=2^{0}+2^{1}+2^{2}+\ldots+2^{2016}$ b) Let $n=1+3+5+\cdots+2015$. Show that $n$ is a perfect square. | 1008^2 | Number Theory | olympiads |
12. Let $f(x)+g(x)=\sqrt{\frac{1+\cos 2 x}{1-\sin x}}\left(x \in\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)\right)$, and $f(x)$ is an odd function, $g(x)$ is an even function, then $[f(x)]^{2}-[g(x)]^{2}=$ $\qquad$ . | -2\cosx | Algebra | olympiads |
2. 31 cars started simultaneously from one point on a circular track: the first car at a speed of 61 km/h, the second at 62 km/h, and so on (the 31st at 91 km/h). The track is narrow, and if one car overtakes another by a full lap, they crash into each other, both fly off the track, and are eliminated from the race. In... | 76 | Logic and Puzzles | olympiads |
4. In a circle with a radius of 1, there is an inscribed polygon. If all its sides are greater than 1 but less than $\sqrt{2}$, then the number of sides of this polygon must be
(A) 7 ;
(B) 6 ;
(C) 5 ;
(D) 4 . | 5 | Geometry | olympiads |
Maria buys computer disks at a price of $4$ for $$5$ and sells them at a price of $3$ for $$5$. How many computer disks must she sell in order to make a profit of $$100$?
$\text{(A)}\ 100 \qquad \text{(B)}\ 120 \qquad \text{(C)}\ 200 \qquad \text{(D)}\ 240 \qquad \text{(E)}\ 1200$ | 240 | Algebra | amc_aime |
2.60. Pentagon $A B C D E$ is inscribed in a circle. The distances from point $E$ to the lines $A B, B C$ and $C D$ are $a, b$ and $c$ respectively. Find the distance from point $E$ to the line $A D$. | \frac{ac}{b} | Geometry | olympiads |
11. There are 25 children in the class. Two are chosen at random for duty. The probability that both duty students will be boys is $\frac{3}{25}$. How many girls are in the class? | 16 | Combinatorics | olympiads |
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