problem stringlengths 37 4.98k | answer stringlengths 1 141 | subject stringclasses 8
values | level stringclasses 7
values |
|---|---|---|---|
7-1. Petya has stickers. If he gives 5 stickers to each of his friends, he will have 8 stickers left. If he wants to give 6 stickers to each of his friends, he will be short of 11 stickers. How many friends does Petya have? | 19 | Algebra | olympiads |
Example 12 Let $n$ be a fixed integer, $n \geqslant 2$.
(1) Determine the smallest constant $c$ such that the inequality
$$
\sum_{1 \leqslant i<j \leqslant n} x_{i} x_{j}\left(x_{i}^{2}+x_{j}^{2}\right) \leqslant c \cdot\left(\sum_{i=1}^{n} x_{i}\right)^{4}
$$
holds for all non-negative real numbers $x_{1}, x_{2}, \cd... | \frac{1}{8} | Inequalities | olympiads |
Solve the following equation on the set of real numbers:
$$
x_{1}^{2}+x_{2}^{2}+x_{3}^{2}+\cdots+x_{n}^{2}-x_{n+1}=\sqrt{x_{1}+x_{2}+\cdots+x_{n}-x_{n+1}}-\frac{n+1}{4}
$$ | x_{1}=\cdots=x_{n}==\frac{1}{2},\quadx_{n+1}=\frac{2n-1}{4} | Algebra | olympiads |
7.281. $\left\{\begin{array}{l}y \cdot x^{\log _{y} x}=x^{2.5} \\ \log _{3} y \cdot \log _{y}(y-2 x)=1 .\end{array}\right.$ | (3;9) | Algebra | olympiads |
Consider a circle of radius 1. Describe a regular $n$-sided polygon around it and inscribe a regular $n$-sided polygon in it. Denote their perimeters by $P_{n}$ (for the circumscribed) and $p_{n}$ (for the inscribed).
a) Find $P_{4}, p_{4}, P_{6}$, and $p_{6}$.
b) Prove that the following recurrence relations hold: $... | P_{96}\approx6.285429,p_{96}\approx6.282064 | Geometry | olympiads |
[The Pigeonhole Principle (continued).]
In a photo studio, 20 birds flew in - 8 sparrows, 7 wagtails, and 5 woodpeckers. Each time the photographer clicks the camera shutter, one of the birds flies away (permanently). How many shots can the photographer take to be sure: he will have at least four birds of one species ... | 7 | Combinatorics | olympiads |
The number 519 is formed using the digits 5,1 and 9 . The three digits of this number are rearranged to form the largest possible and then the smallest possible three digit numbers. What is the difference between these largest and smallest numbers?
(A) 332
(B) 432
(C) 792
(D) 756
(E) 720
## Part B: Each correct answer... | 792 | Number Theory | olympiads |
Problem 7.1. In the picture, nine small squares are drawn, with arrows on eight of them. The numbers 1 and 9 are already placed. Replace the letters in the remaining squares with numbers from 2 to 8 so that the arrows from the square with the number 1 point in the direction of the square with the number 2 (the number 2... | In\\A\there\is\the\\6,\in\B-2,\in\C-4,\in\D-5,\in\E-3,\in\F-8,\in\G-7 | Logic and Puzzles | olympiads |
5. Let $A B C$ and $P Q R$ be two triangles. If $\cos A=\sin P, \cos B=\sin Q$ and $\cos C=\sin R$, what is the largest angle (in degrees) among the six interior angles of the two triangles?
(1 mark)
設 $A B C$ 和 $P Q R$ 為三角形。若 $\cos A=\sin P 、 \cos B=\sin Q$ 且 $\cos C=\sin R$,則兩個三角形六個內角中最大的一個（以「度」為單位）是多少？ | 135 | Geometry | olympiads |
11. Giovanni draws a regular 9-gon with a pencil and connects each of its vertices to the center, drawing a total of 18 segments and thus obtaining nine triangles. He then traces some of the drawn segments with a pen, ensuring that each of the nine triangles has exactly one side traced with a pen. In how many ways can ... | 76 | Combinatorics | olympiads |
17. A moving point moves on the integer points in the first quadrant of the Cartesian coordinate system (including the integer points on the $x$-axis and $y$-axis of the first quadrant), with the movement rules being $(m, n) \rightarrow(m+1, n+1)$ or $(m, n) \rightarrow$ $(m+1, n-1)$. If the moving point starts from th... | 9 | Combinatorics | cn_contest |
6. The range of the function $f(x)=\sqrt{3 x-6}+\sqrt{3-x}$ is
$\qquad$ | [1,2] | Algebra | cn_contest |
12.126. Find the angle in the axial section of a cone if a sphere with its center at the vertex of the cone, touching its base, divides the volume of the cone in half. | 2\arccos\frac{1+\sqrt{17}}{8} | Geometry | olympiads |
13. $(x, y)=\left(x_{1}, y_{1}\right),\left(x_{2}, y_{2}\right)\left(x_{1}<x_{2}\right)$ are two integer solutions of the equation $x^{2}-y^{2}-2 x+6 y-8=0$. In the Cartesian coordinate system, point $A\left(x_{1}, y_{1}\right)$ and point $B\left(x_{2}, y_{2}\right)$ are symmetric about $P(1,3)$, and point $C(5,-1)$. I... | (-1,1) | Algebra | cn_contest |
How many five-digit numbers can be written with these digits:
(1) $0,1,2$;
(2) $1,1,0,2,3$. | 48 | Combinatorics | olympiads |
Dividing the numbers 1200, 1640, and 1960 by the same (integer) number, we got remainders of 3, 2, and 7, respectively. What could the divisor have been? | 9,21,63 | Number Theory | olympiads |
Example 3. Find $\int \ln ^{2} x d x$. | x\ln^{2}x-2x\lnx+2x+C | Calculus | olympiads |
1. (10 points) The little rabbit and the little turtle travel from location $A$ to the forest amusement park. The little rabbit starts at 9:00 AM, jumping 40 meters per minute, and after every 3 minutes of jumping, it plays on the spot for 2 minutes. The little turtle starts at 6:40 AM, crawling 10 meters per minute wi... | 2370 | Logic and Puzzles | olympiads |
rainbow is the name of a bird. this bird has $n$ colors and it's colors in two consecutive days are not equal. there doesn't exist $4$ days in this bird's life like $i,j,k,l$ such that $i<j<k<l$ and the bird has the same color in days $i$ and $k$ and the same color in days $j$ and $l$ different from the colors it has i... | 2n - 1 | Combinatorics | aops_forum |
## [ [Mean proportionals in a right triangle]
[ Median to the hypotenuse ]
In a right triangle $A B C$, angle $A C B$ is a right angle. Let $E$ be the point of intersection of the angle bisector of angle $A B C$ with side $A C$. Point $D$ is the midpoint of side $A B$, and $O$ is the point of intersection of segments... | \frac{8b^{2}\sqrt{11}}{5} | Geometry | olympiads |
281. Find the six-digit number $\overline{x y 342 z}$, divisible by 396. | 453420413424 | Number Theory | olympiads |
4. Given point $P(x, y)$ satisfies $(x-4 \cos \theta)^{2}+$ $(y-4 \sin \theta)^{2}=4(\theta \in \mathbf{R})$. Then the area of the region where point $P(x, y)$ is located is ( ).
(A) $36 \pi$
(B) $32 \pi$
(C) $20 \pi$
(D) $16 \pi$ | B | Geometry | cn_contest |
5. Find all positive integers $n$ such that the sum $1+2+3+\cdots+n$ is a three-digit number composed of the same digit. | 36 | Number Theory | cn_contest |
30-1 Let $p, q$ be natural numbers, condition A: $p^{3}-q^{3}$ is even; condition B: $p+q$ is even. Then
(A) A is a sufficient but not necessary condition for B.
(B) A is a necessary but not sufficient condition for B.
(C) A is a necessary and sufficient condition for B.
(D) A is neither a sufficient condition nor a ne... | C | Number Theory | olympiads |
In the base ten number system the number $526$ means $5 \times 10^2+2 \times 10 + 6$.
In the Land of Mathesis, however, numbers are written in the base $r$.
Jones purchases an automobile there for $440$ monetary units (abbreviated m.u).
He gives the salesman a $1000$ m.u bill, and receives, in change, $340$ m.u. The... | 8 | Number Theory | amc_aime |
4. In a week of 7 days, it rained for 5 days, the number of ways it could have rained for exactly 3 consecutive days is $\qquad$.
将上面的文本翻译成英文，请保留源文本的换行和格式，直接输出翻译结果。
Note: The last sentence is a repetition of the instruction and should not be included in the translation output. Here is the correct translation:
4. In... | 9 | Combinatorics | olympiads |
5. Let the set of all permutations $X=(x_{1}, x_{2}, \cdots, x_{9})$ of $1,2, \cdots, 9$ be $A$. For any $X \in A$, let
\[
\begin{array}{l}
f(X)=x_{1}+2 x_{2}+\cdots+9 x_{9}, \\
M=\{f(X) \mid X \in A\} .
\end{array}
\]
Find $|M|$ (where $|M|$ denotes the number of elements in the set $M$).
(Xiong Bin) | 121 | Combinatorics | cn_contest |
10. (12 points) 1 kilogram of soybeans can be made into 3 kilograms of tofu, and 1 kilogram of soybean oil requires 6 kilograms of soybeans. Tofu sells for 3 yuan per kilogram, and soybean oil sells for 15 yuan per kilogram. A batch of soybeans weighs 460 kilograms, and after being made into tofu or soybean oil and sol... | 360 | Algebra | olympiads |
## Task B-2.3.
The equation $y=x^{2}+b x+c$ defines a set of parabolas whose roots are consecutive integers. To which set of points do the vertices of these parabolas belong? Determine its equation. | \frac{-1}{4} | Algebra | olympiads |
3. For $a, b \in \mathbf{R}$, let
$$
\max \{a, b\}=\left\{\begin{array}{ll}
a, & a \geqslant b ; \\
b, & a<b,
\end{array}\right.
$$
The function $f(x)=\max \left\{2^{-x},-|x-1|+2\right\}$. Then the equation $f(x)=a$ has three roots, the range of the real number $a$ is $\qquad$ | (1,2) | Algebra | olympiads |
19. A group of 40 boys and 28 girls stand hand in hand in a circle facing inwards. Exactly 18 of the boys give their right hand to a girl. How many boys give their left hand to a girl?
A 12
B 14
C 18
D 20
E 22 | 18 | Combinatorics | olympiads |
Let's calculate the sum of the following series:
$$
\frac{1}{1}+\frac{2}{2}+\frac{3}{4}+\frac{4}{8}+\frac{5}{16}+\ldots+\frac{n}{2^{n-1}}
$$
What does the sum approach as the number of terms grows beyond any bound? | 4 | Algebra | olympiads |
A class collects 50 dollars to buy flowers for a classmate who is in the hospital. Roses cost 3 dollars each, and carnations cost 2 dollars each. No other flowers are to be used. How many different bouquets could be purchased for exactly 50 dollars?
$\mathrm{(A)}\ 1 \qquad \mathrm{(B)}\ 7 \qquad \mathrm{(C)}\ 9 \qquad ... | 9 | Combinatorics | amc_aime |
After Clive assembled and wound his clock (see problem $\underline{32798}$), setting it by his grandfather's, it started running backward. How many times a day will it show the correct time? | 4 | Logic and Puzzles | olympiads |
1. A true-false test has ten questions. If you answer five questions "true" and five "false," your score is guaranteed to be at least four. How many answer keys are there for which this is true? | 22 | Combinatorics | olympiads |
## Task B-2.5.
In an equilateral triangle $ABC$, the midpoint of side $\overline{AB}$ is point $P$. Points $G$ and $H$ are between points $A$ and $P$, and points $I$ and $J$ are between points $P$ and $B$. Points $D, E$, and $F$ divide the length of $\overline{AC}$ into four equal parts. Parallels to side $\overline{A... | 10:7 | Geometry | olympiads |
[ Sum of angles in a triangle. Theorem about the exterior angle.]
$B K$ is the bisector of triangle $A B C$. It is known that $\angle A K B: \angle C K B=4: 5$. Find the difference between angles $A$ and $C$ of triangle $A B C$.
# | 20 | Geometry | olympiads |
. Determine the units digit of $1789^{1789}$.
. | 9 | Number Theory | olympiads |
## Task A-4.5. (4 points)
A rectangular path with a width of $1.5 \mathrm{~m}$ and a length of $20 \mathrm{~m}$ needs to be tiled with identical tiles in the shape of an isosceles right triangle with legs of length $50 \mathrm{~cm}$, such that the legs are parallel to the sides of the rectangle. Determine the number o... | 2^{120} | Combinatorics | olympiads |
5. The sequence $1,2,1,2,2,1,2,2,2,1,2,2,2,2,1,2, \cdots$ has 2s separating each pair of 1s, with the $n$-th pair of 1s separated by $n$ 2s. What is the sum of the first 1234 terms of the sequence?
A. 1996
B. 2419
C. 2429
D. 2439 | 2419 | Number Theory | olympiads |
2. Through the right focus of the hyperbola $x^{2}-\frac{y^{2}}{2}=1$, a line $l$ intersects the hyperbola at points $A$ and $B$. If a real number $\lambda$ makes $|A B|=\lambda$ such that there are exactly 3 lines $l$, then $\lambda=$
(Proposed by the Problem Committee) | 4 | Geometry | cn_contest |
3. (2003 Thailand Mathematical Olympiad) Define in the set of rational numbers: $f(0)=0, f(1)=1$,
$$
f(x)=\left\{\begin{array}{l}
\frac{f(2 x)}{4}, 0<x<\frac{1}{2} ; \\
\frac{3}{4}+\frac{f(2 x-1)}{4}, \frac{1}{2} \leqslant x<1 .
\end{array}\right.
$$
If $x$ is represented in binary, such as $x=\left(0 . b_{1} b_{2} b_... | f(x)=00b_{1}b_{1}b_{2}b_{2}b_{3}b_{3}\cdots | Number Theory | olympiads |
7. (10 points) As shown in the figure, the length $AB$ of rectangle $ABCD$ is 20 cm, and the width $BC$ is 16 cm. Inside the rectangle, there are two overlapping squares $DEFG$ and $BHIJ$. It is known that the perimeters of the three shaded rectangles are equal. Therefore, the area of rectangle INFM is $\qquad$ square ... | 32 | Geometry | olympiads |
The number
$\frac 2{\log_4{2000^6}} + \frac 3{\log_5{2000^6}}$
can be written as $\frac mn$ where $m$ and $n$ are relatively prime positive integers. Find $m + n$. | 7 | Algebra | amc_aime |
2.48 For the equations $x^{2}+c x+2=0$ and $x^{2}+2 x+c=2$, which of the following conclusions is correct?
(A) At least one equation has real roots.
(B) At most one equation has real roots.
(C) Both have real roots.
(D) Neither has real roots.
(3rd "Five Sheep Cup" Junior High School Mathematics Competition, 1991) | A | Algebra | olympiads |
A movie theatre has eleven rows of seats. The rows are numbered from 1 to 11 . Oddnumbered rows have 15 seats and even-numbered rows have 16 seats. How many seats are there in the theatre?
(A) 176
(B) 186
(C) 165
(D) 170
(E) 171 | 170 | Number Theory | olympiads |
4. (5 points) If a two-digit number divisible by 5 is neither divisible by 3 nor by 4, its 97 times is an even number, and the tens digit is not less than 6, then this two-digit number is $\qquad$ | 70 | Number Theory | olympiads |
10. Place 11 identical balls into six distinct boxes so that at most three boxes are empty. The number of ways to do this is $\qquad$. | 4212 | Combinatorics | cn_contest |
## 206. Math Puzzle $7 / 82$
We know that the continents move on the Earth's crust and that, for example, South America drifts about $3 \mathrm{~cm}$ per year.
By how much has the landmass moved in the 6000 years of recorded history? | 180\mathrm{~} | Logic and Puzzles | olympiads |
10. Let the sequence $\left\{a_{n}\right\}$ have the sum of the first $n$ terms $S_{n}$ satisfying
$$
S_{n}+a_{n}=\frac{n-1}{n(n+1)}(n=1,2, \cdots) \text {. }
$$
Then the general term $a_{n}=$ | a_{n}=\frac{1}{2^{n}}-\frac{1}{n(n+1)} | Algebra | cn_contest |
15. The sequence $\left\{a_{n}\right\}$ satisfies: $a_{0}=1, a_{n}=\left[\sqrt{S_{n-1}}\right]$ $(n=1,2, \cdots,[x]$ represents the greatest integer not greater than $x$, $\left.S_{k}=\sum_{i=0}^{k} a_{i}(k=0,1, \cdots)\right)$. Find the value of $a_{2006}$. | 998 | Number Theory | cn_contest |
8.5. In the city, there are 9 bus stops and several buses. Any two buses have no more than one common stop. Each bus has exactly three stops. What is the maximum number of buses that can be in the city? | 12 | Combinatorics | olympiads |
5. In the coordinate plane, a point whose both coordinates are integers is called an integer point. We use $I$ to denote the set of all lines, $M$ to denote the set of lines passing through exactly one integer point, $N$ to denote the set of lines not passing through any integer point, and $P$ to denote the set of line... | D | Number Theory | olympiads |
26・19 Let the moving point $M(x, y)$ be such that the ratio of its distance to the point $F(4,0)$ to its distance to the line $x=3$ is 2, then the equation of the locus of $M(x, y)$ is
(A) $\frac{x^{2}}{12}-\frac{y^{2}}{4}=1$.
(B) $\frac{x^{2}}{4}-\frac{y^{2}}{12}=1$.
(C) $3 x^{2}-y^{2}-16 x+20=0$.
(D) $3 y^{2}-x^{2}-1... | 3x^{2}-y^{2}-16x+20=0 | Geometry | olympiads |
9. Given is a regular tetrahedron of volume 1 . We obtain a second regular tetrahedron by reflecting the given one through its center. What is the volume of their intersection? | \frac{1}{2} | Geometry | olympiads |
1. Calculate: $22 \times 33+44 \times 55+66 \times 77+88 \times 99=$ | 16940 | Algebra | olympiads |
## Task A-2.3.
Determine all quadruples $(a, b, c, d)$ of natural numbers such that
$$
a^{3}=b^{2}, \quad c^{5}=d^{4} \quad \mathrm{and} \quad a-c=9
$$ | (25,125,16,32) | Number Theory | olympiads |
Problem 1. At a round table, 60 people are sitting. Each of them is either a knight, who always tells the truth, or a liar, who always lies. Each person at the table said: "Among the next 3 people sitting to my right, there is no more than one knight." How many knights could have been sitting at the table? List all pos... | 30 | Logic and Puzzles | olympiads |
7. Let $f(n)$ be the integer closest to $\sqrt[4]{n}$, then $\sum_{k=1}^{2017} \frac{1}{f(k)}=$ | \frac{2822}{7} | Number Theory | olympiads |
Find all natural $a,b$ such that $\left. {a(a + b) + 1} \right|(a + b)(b + 1) - 1$. | (a, b) = (1, b) | Number Theory | aops_forum |
(1) If $4^{a}-3 a^{b}=16, \log _{2} a=\frac{a+1}{b}$, then $a^{b}=$ | 16 | Algebra | olympiads |
4. If $\left\{\begin{array}{l}x^{2}+x y+y=14, \\ y^{2}+x y+x=28,\end{array}\right.$ find $x+y=$ $\qquad$
(2001, TI Cup National Junior High School Mathematics Competition) | 6 \text{ or } -7 | Algebra | cn_contest |
# Task 7. (14 points)
In the analysis of bank accounts, it was found that the remaining balance on each of them is more than 10 rubles. It also turned out that there is a group of clients, each of whom has the same amount of money on their account. This amount is a number consisting of only ones. If you add up all the... | 101 | Number Theory | olympiads |
20th APMC 1997 Problem 3 The 97 numbers 49/1, 49/2, 49/3, ... , 49/97 are written on a blackboard. We repeatedly pick two numbers a, b on the board and replace them by 2ab - a - b + 1 until only one number remains. What are the possible values of the final number? | 1 | Algebra | olympiads |
5. A square and a regular hexagon are drawn with the same side length. If the area of the square is $\sqrt{3}$, what is the area of the hexagon? | \frac{9}{2} | Geometry | olympiads |
22. We use the notation $\overline{a b}$ for the two-digit number with digits $a$ and $b$. Let $a, b, c$ be different digits. How many ways can you choose the digits $a, b, c$ such that $\overline{a b}<\overline{b c}<\overline{c a}$ ?
A 84
B 96
C 504
D 729
E 1000 | 84 | Combinatorics | olympiads |
There are four houses, located on the vertices of a square. You want to draw a road network, so that you can go from any house to any other. Prove that the network formed by the diagonals is not the shortest. Find a shorter network.
| 1 + \sqrt{3} | Geometry | aops_forum |
2. The range of the function $y=\frac{\sin 2 x-3}{\sin x+\cos x-2}$ is | [2-\sqrt{2}, 2+\sqrt{2}] | Algebra | cn_contest |
10.4. We will call a number greater than 25 semi-prime if it is the sum of some two different prime numbers. What is the maximum number of consecutive natural numbers that can be semi-prime | 5 | Number Theory | olympiads |
Example 7 $n$ cells are arranged in a row, to be painted with red, white, and black colors, with each cell painted one color, and it is required that the number of cells painted red must be even. How many different ways are there to paint the cells?
| b_{n}=\frac{1}{2}(3^{n}+1) | Combinatorics | olympiads |
4. The maximum value of the function $f(x)=\frac{\frac{1}{6} \cdot(-1)^{1+C_{2 x}^{x}} \cdot P_{x+2}^{5}}{1+C_{3}^{2}+C_{4}^{2}+\cdots+C_{x-1}^{2}}$ is ( ).
(A) 20
(B) 10
(C) -10
(D) -20 | D | Combinatorics | cn_contest |
Sir Alex is coaching a soccer team of $n$ players of distinct heights. He wants to line them up so that for each player $P$, the total number of players that are either to the left of $P$ and taller than $P$ or to the right of $P$ and shorter than $P$ is even. In terms of $n$, how many possible orders are there?
[i]Mi... | \lfloor \frac{n}{2} \rfloor! \cdot \lceil \frac{n}{2} \rceil! | Combinatorics | aops_forum |
Two positive integers $a_{1}, a_{2}, \cdots, a_{2006}$ (which can be the same) are such that $\frac{a_{1}}{a_{2}}, \frac{a_{2}}{a_{3}}, \cdots, \frac{a_{2005}}{a_{2006}}$ are all distinct. How many different numbers are there at least among $a_{1}$, $a_{2}, \cdots, a_{2006}$?
（Chen Yonggao, problem contributor） | 46 | Number Theory | cn_contest |
As the Kubiks head out of town for vacation, Jerry takes the first driving shift while Hannah and most of the kids settle down to read books they brought along. Tony does not feel like reading, so Alexis gives him one of her math notebooks and Tony gets to work solving some of the problems, and struggling over others.... | 132 | Geometry | aops_forum |
8. Given a quadratic equation with real coefficients
$$
x^{2}+(1+a) x+a+b+1=0
$$
with two real roots $x_{1}, x_{2}$, and $0<x_{1}<1<x_{2}$. Then the range of $\frac{b}{a}$ is $(\quad)$.
(A) $\left(-1,-\frac{1}{2}\right]$
(B) $\left(-1,-\frac{1}{2}\right)$
(C) $\left(-2,-\frac{1}{2}\right]$
(D) $\left(-2,-\frac{1}{2}\r... | D | Algebra | cn_contest |
2.14. The base of a right parallelepiped is a rhombus with an area of $Q$. The areas of the diagonal sections are $S_{1}$ and $S_{2}$. Determine the volume and lateral surface area of the parallelepiped. | \sqrt{\frac{S_{1}S_{2}Q}{2}};2\sqrt{S_{1}^{2}+S_{2}^{2}} | Geometry | olympiads |
Example 3.48. Find the extremum of the function
\[
f(x, y)=x^{2}+y^{2}-2 x-y
\]
subject to the condition that the variables \(x\) and \(y\) are related by the equation \(\varphi(x, y)=x+y-1=0\). | -1.125 | Calculus | olympiads |
Example 1 As shown, in Rt $\triangle ABC$, the hypotenuse $AB=5, CD \perp AB$. It is known that $BC, AC$ are the two roots of the quadratic equation $x^{2}-(2 m-1) x+4(m-1)=0$. Then the value of $m$ is $\qquad$. | 4 | Algebra | cn_contest |
[ Linear inequalities and systems of inequalities ]
$$
\text { [ The extremal principle (miscellaneous). ] }
$$
Nine digits: $1,2,3, \ldots, 9$ are written in some order (forming a nine-digit number). Consider all triples of consecutive digits, and find the sum of the corresponding seven three-digit numbers. What is ... | 4648 | Number Theory | olympiads |
How far are the corners of a $1 \mathrm{~m}$ edge length cube from the body diagonal? | \frac{\sqrt{6}}{3} | Geometry | olympiads |
6. As shown in Figure 2, circles $\odot O_{1}$ and $\odot O_{2}$ are externally separated, and their line of centers $O_{1} O_{2}$ intersects $\odot O_{1}$ at points $A$ and $B$, and intersects $\odot O_{2}$ at points $C$ and $D$. Circle $\odot O_{3}$ is internally tangent to $\odot O_{1}$ at point $B$, and circle $\od... | B | Geometry | cn_contest |
On the last day of school, Mrs. Wonderful gave jelly beans to her class. She gave each boy as many jelly beans as there were boys in the class. She gave each girl as many jelly beans as there were girls in the class. She brought $400$ jelly beans, and when she finished, she had six jelly beans left. There were two more... | 28 | Algebra | amc_aime |
64. Calculate the scores of ten students in a math test. It is known that the average score of the top four students is 95 points, and the average score of the last six students is 6 points less than the overall average score of the ten students. The average score of these ten students is $\qquad$ points. | 86 | Algebra | olympiads |
5. $a$ is a parameter, the function
$$
f(x)=(x+a) 3^{x-2+a^{2}}-(x-a) 3^{8-x-3 a}
$$
is an even function. Then the set of values that $a$ can take is ( ).
(A) $\{0,5\}$
(B) $\{-2,5\}$
(C) $\{-5,2\}$
(D) $\{1,2009\}$ | C | Algebra | cn_contest |
2. 100 We know that $12^{2}=144$ ends with two 4s, and $38^{2}=1444$ ends with three 4s. For a natural number not ending in zero, what is the maximum number of identical digits at the end of its square? | 3 | Number Theory | olympiads |
[Linear element relationships of similar triangles] [Rectangles and squares. Properties and characteristics]
In a triangle with a base of 30 and a height of 10, an isosceles right triangle is inscribed such that its hypotenuse is parallel to the base of the given triangle, and the vertex of the right angle lies on thi... | 12 | Geometry | olympiads |
8 For the real-coefficient quadratic equation $x^{2}+(a+1) x+a+b+1=0$ with two real roots $x_{1} 、 x_{2}$. If $0<x_{1}<1$ and $x_{2}>1$, then the range of $\frac{b}{a}$ is ( ).
(A) $\left(-2,-\frac{1}{2}\right)$
(B) $\left(-2, \frac{1}{2}\right)$
(C) $\left(-1,-\frac{1}{2}\right)$
(D) $\left(-1, \frac{1}{2}\right)$ | (-2,-\frac{1}{2}) | Algebra | olympiads |
3. Given the set $M=\{(a, b) \mid a \leqslant-1, b \leqslant m\}$. If for any $(a, b) \in M$, it always holds that $a \cdot 2^{b}-b-3 a \geqslant 0$, then the maximum value of the real number $m$ is $\qquad$ | 1 | Inequalities | olympiads |
5. Given two points $A(0,1), B(6,9)$. If there is an integer point $C$ (Note: A point with both coordinates as integers is called an integer point), such that the area of $\triangle A B C$ is minimized. Then the minimum value of the area of $\triangle A B C$ is $\qquad$ | 1 | Geometry | cn_contest |
Task 1. Determine the unit digit of the sum
$$
1+1 \cdot 2+1 \cdot 2 \cdot 3+1 \cdot 2 \cdot 3 \cdot 4+1 \cdot 2 \cdot 3 \cdot 4 \cdot 5+\ldots+1 \cdot 2 \cdot 3 \cdot \ldots \cdot 2017
$$ | 3 | Number Theory | olympiads |
6. Let $[x]$ denote the greatest integer not exceeding the real number $x$. If $A=\left[\frac{7}{8}\right]+\left[\frac{7^{2}}{8}\right]+\left[\frac{7^{2019}}{8}\right]+\left[\frac{7^{2020}}{8}\right]$, then the remainder when $A$ is divided by 50 is $\qquad$ . | 40 | Number Theory | olympiads |
5. In the plane, there are 200 points, no three of which are collinear, and each point is labeled with one of the numbers $1, 2, 3$. All pairs of points labeled with different numbers are connected by line segments, and each line segment is labeled with a number 1, 2, or 3, which is different from the numbers at its en... | 199 | Combinatorics | olympiads |
Problem 3. In a surgical department, there are 4 operating rooms: A, B, V, and G. In the morning, they were all empty. At some point, an operation began in operating room A, after some time - in operating room B, then after some more time - in V, and then in $\Gamma$.
All four operations ended simultaneously, and the ... | 31 | Logic and Puzzles | olympiads |
23.39 A large cube is formed by stacking 27 small cubes. A plane is perpendicular to one of the large cube's diagonals and bisects this diagonal. The number of small cubes intersected by this plane is
(A) 16.
(B) 17.
(C) 18.
(D) 19.
(E) 20
(46th American High School Mathematics Examination, 1995) | 19 | Geometry | olympiads |
17. Find all positive solutions of the following system of indeterminate equations:
(i) $2 x_{1}+x_{2}+x_{3}=100, 3 x_{1}+5 x_{2}+15 x_{3}=270$;
(ii) $x_{1}+x_{2}+x_{3}=31, x_{1}+2 x_{2}+3 x_{3}=41$. | (i) x_{1}=40, x_{2}=15, x_{3}=5; (ii) \{22,8,1\},\{23,6,2\},\{24,4,3\},\{25,2,4\} | Algebra | number_theory |
Problem 9.8. On the side $CD$ of trapezoid $ABCD (AD \| BC)$, a point $M$ is marked. A perpendicular $AH$ is dropped from vertex $A$ to segment $BM$. It turns out that $AD = HD$. Find the length of segment $AD$, given that $BC = 16$, $CM = 8$, and $MD = 9$.
$ as follows: if $n=0$ or 1, then $\operatorname{rad}(n)=1$; if the prime factors of $n$ are $p_{1}, p_{2}, \cdots, p_{k}\left(p_{1}<p_{2}<\cdots<p_{k}\right)$, then $\operatorname{rad}(n)=p_{1} p_{2} \cdots p_{k}$. Find all non-negative integer coeffi... | f(x)=a x^{m} | Number Theory | cn_contest |
18. (12 points) As shown in Figure 4, in the equilateral $\triangle ABC$, points $D$ and $E$ are on sides $AC$ and $AB$ respectively, and $AD = \frac{1}{3} AC$, $AE = \frac{2}{3} AB$, $BD$ intersects $CE$ at point $F$.
(1) Prove that points $A$, $E$, $F$, and $D$ are concyclic;
(2) If the side length of the equilateral... | \frac{2}{3} | Geometry | cn_contest |
$14 \cdot 23$ A die is rolled six times. The probability of getting at least 5 on at least five of the rolls is
(A) $\frac{13}{729}$.
(B) $\frac{12}{729}$.
(C) $\frac{2}{729}$.
(D) $\frac{3}{729}$.
(E) None of these.
(25th American High School Mathematics Examination, 1974) | \frac{13}{729} | Combinatorics | olympiads |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.