problem stringlengths 37 4.98k | answer stringlengths 1 141 | subject stringclasses 8
values | level stringclasses 7
values |
|---|---|---|---|
## Problem Statement
Find the distance from point $M_{0}$ to the plane passing through three points $M_{1}, M_{2}, M_{3}$.
$M_{1}(0 ;-3 ; 1)$
$M_{2}(-4 ; 1 ; 2)$
$M_{3}(2 ;-1 ; 5)$
$M_{0}(-3 ; 4 ;-5)$ | \frac{90}{\sqrt{194}} | Geometry | olympiads |
9.215. Find the domain of the function
$$
y=\sqrt[4]{\frac{x^{2}-6 x-16}{x^{2}-12 x+11}}+\frac{2}{x^{2}-49}
$$ | x\in(-\infty;-7)\cup(-7;-2]\cup(1;7)\cup(7;8]\cup(11;\infty) | Algebra | olympiads |
Emily Thorne is throwing a Memorial Day Party to kick off the Summer in the Hamptons, and she is trying to figure out the seating arrangment for all of her guests. Emily saw that if she seated $4$ guests to a table, there would be $1$ guest left over (how sad); if she seated $5$ to a table, there would be $3$ guests le... | 193 | Number Theory | aops_forum |
$\underline{\text { Zhenedarov R.G. }}$
Two pirates, Bill and John, each having 74 gold coins, decided to play such a game: they will take turns laying coins on the table, one, two, or three at a time, and the one who places the hundredth coin in sequence will win. Bill starts. Who can win this game, regardless of how... | John | Logic and Puzzles | olympiads |
(Example 5 Write a sufficient but not necessary condition and a necessary but not sufficient condition for the quadratic equation with real coefficients $x^{2}-2 a x+a=0$ to have two distinct positive real roots.
) | >2\geqslant1 | Algebra | olympiads |
There are $10000$ trees in a park, arranged in a square grid with $100$ rows and $100$ columns. Find the largest number of trees that can be cut down, so that sitting on any of the tree stumps one cannot see any other tree stump.
| 2500 | Combinatorics | aops_forum |
1. Given $a+\frac{1}{b}=\frac{2}{a}+2 b \neq 0$. Then $\frac{a}{b}$ is ( ).
(A) -1
(B) 1
(C) 2
(D) cannot be determined | C | Algebra | cn_contest |
5. Let the function
$$
f(x)=x \log _{2} x+(a-x) \log _{2}(a-x)
$$
be symmetric about the line $x=\frac{1}{2}$. Then for any real numbers $x_{i} \in(0,1)(1 \leqslant i \leqslant 4)$ satisfying $\sum_{i=1}^{4} x_{i}=1$, the minimum value of $s=\sum_{i=1}^{4} x_{i} \log _{2} x_{i}$ is . $\qquad$ | -2 | Algebra | cn_contest |
Problem 2. The quadratic function $f(x)=-x^{2}+4 p x-p+1$ is given. Let $S$ be the area of the triangle with vertices at the intersection points of the parabola $y=f(x)$ with the $x$-axis and the vertex of the same parabola. Find all rational $p$, for which $S$ is an integer.
| 0,1,\frac{1}{4},-\frac{3}{4} | Algebra | olympiads |
Example 1. In how many different ways can three people be selected for three different positions from ten candidates? | 720 | Combinatorics | olympiads |
Which of the following is equal to $9^{4}$ ?
(A) $3^{2}$
(B) $3^{4}$
(C) $3^{6}$
(D) $3^{8}$
(E) $3^{10}$ | D | Algebra | olympiads |
The set $ S$ consists of $ n > 2$ points in the plane. The set $ P$ consists of $ m$ lines in the plane such that every line in $ P$ is an axis of symmetry for $ S$. Prove that $ m\leq n$, and determine when equality holds. | m \leq n | Geometry | aops_forum |
$18 \cdot 30$ Among the following propositions, which one is true?
(A) A quadrilateral is a parallelogram if one pair of opposite sides are equal and one diagonal bisects the other diagonal.
(B) A quadrilateral is a parallelogram if one pair of opposite sides are equal and both diagonals are equal.
(C) If the quadrilat... | D | Geometry | olympiads |
Example, Let the function $f(x)=\sqrt{x^{2}+1}-a x$, where $a>0$. Find the range of values for $a$ such that the function $f(x)$ is monotonic on $[0,+\infty)$. | \geqslant1 | Calculus | olympiads |
## Task Condition
Calculate the volumes of solids formed by rotating figures bounded by the graphs of functions. The axis of rotation is $O y$.
$$
y=x^{2}-2 x+1, x=2, y=0
$$ | \frac{7\pi}{6} | Calculus | olympiads |
15 A finite set of points $M$ on the plane satisfies: for any two points $A, B$ in $M$, there must exist a third point $C$ such that $\triangle A B C$ is an equilateral triangle. Find the maximum number of elements in $M$. | 3 | Geometry | olympiads |
Example 6. Find the integral $\int \frac{\ln x}{x} d x$. | \frac{1}{2}\ln^{2}x+C | Calculus | olympiads |
6. Let $|\overrightarrow{A B}|=10$, if a point $P$ on the plane satisfies, for any $t \in \mathbf{R}$, $|\overrightarrow{A P}-t \overrightarrow{A B}| \geq 3$, then the minimum value of $\overrightarrow{P A} \cdot \overrightarrow{P B}$ is
$\qquad$ , and at this time $|\overrightarrow{P A}+\overrightarrow{P B}|=$ $\qquad... | -16,6 | Geometry | olympiads |
Given that $\log_{10} \sin x + \log_{10} \cos x = -1$ and that $\log_{10} (\sin x + \cos x) = \textstyle \frac{1}{2} (\log_{10} n - 1)$, find $n$. | 12 | Algebra | aops_forum |
Solve the equation
$$
e^{x}+2 e^{-x}=3
$$
where $e$ is the base of the natural logarithm (see the function table in your school textbook). | x_{1}=0,\,x_{2}\approx0.6931 | Algebra | olympiads |
The [quadratic equation](https://artofproblemsolving.com/wiki/index.php/Quadratic_equation) $x^2+mx+n$ has roots twice those of $x^2+px+m$, and none of $m,n,$ and $p$ is zero. What is the value of $n/p$?
$\textbf{(A) }\ {{{1}}} \qquad \textbf{(B) }\ {{{2}}} \qquad \textbf{(C) }\ {{{4}}} \qquad \textbf{(D) }\ {{{8}}} \q... | 8 | Algebra | amc_aime |
Problem 3. A natural number is written on the board. Nikolai noticed that he can append a digit to the right of it in two ways so that the resulting number is divisible by 9. In how many ways can he append a digit to the right of the given number so that the resulting number is divisible by 3? | 4 | Number Theory | olympiads |
Example 3 Add three digits after 764 to make the resulting six-digit number divisible by 8, 9, and 11. | 764280 | Number Theory | olympiads |
Determine the greatest number, who is the product of some positive integers, and the sum of these numbers is $1976.$ | 2 \cdot 3^{658} | Number Theory | aops_forum |
## Task B-4.1.
For which natural numbers $n$ will the expression $\left(\frac{3+i}{2-i}\right)^{n}$ be a real number? How many such numbers $n$ are less than 2016? | 503 | Algebra | olympiads |
22) In a golf tournament, 256 participants are competing. The tournament is structured such that in each round, 4 participants compete: the winner advances to the next round while the other 3 participants are eliminated. How many rounds are necessary to determine the absolute winner of the tournament?
(A) 16
(B) 64
$\b... | 85 | Combinatorics | olympiads |
14) How many positive integers $n$ are there for which $8 n+50$ is a multiple of $2 n+1$?
(A) None
(B) 1 (C) 2
(D) 5
(E) 10. | 1 | Number Theory | olympiads |
A lake contains $250$ trout, along with a variety of other fish. When a marine biologist catches and releases a sample of $180$ fish from the lake, $30$ are identified as trout. Assume that the ratio of trout to the total number of fish is the same in both the sample and the lake. How many fish are there in the lake?
$... | 1500 | Algebra | amc_aime |
Teacher Kadrnozkova was buying tickets at the cash register of the zoo for her students and for herself. The ticket for an adult was more expensive than for a schoolchild, but not more than twice as much. Teacher Kadrnozkova paid a total of 994 CZK. Teacher Hnizdo had three more students with him than his colleague, so... | 25 | Algebra | olympiads |
Example 16 Divide a circle into $n(\geqslant 2)$ sectors $S_{1}, S_{2}, \cdots, S_{n}$. Now, color these sectors using $m(\geqslant 2)$ colors, with each sector being colored one color and requiring that adjacent sectors have different colors. How many different coloring methods are there? | a_{n}=(-1)^{n}+(-1)^{n}(-1) | Combinatorics | olympiads |
Let $m \ge n$ be positive integers. Frieder is given $mn$ posters of Linus with different integer dimensions of $k \times l$ with $1 \ge k \ge m$ and $1 \ge l \ge n$. He must put them all up one by one on his bedroom wall without rotating them. Every time he puts up a poster, he can either put it on an empty spot on th... | m \cdot \frac{n(n+1)}{2} | Combinatorics | aops_forum |
7. Given $z \in \mathrm{C}$, if the equation $x^{2}-2 z x+\frac{3}{4}+\mathrm{i}=0$ (i being the imaginary unit) has real roots, then the minimum value of the modulus of the complex number $z$ $|z|$ is $\qquad$ . | 1 | Algebra | olympiads |
72. Diagonals of two faces are drawn from the same vertex of a cube. Find the magnitude of the angle between these diagonals. | 60 | Geometry | olympiads |
5. Given the function
$$
f(x)=4 \sin 2 x+3 \cos 2 x+2 a \sin x+4 a \cos x
$$
has a minimum value of -6. Then the value of the real number $a$ is $\qquad$. | \\sqrt{2} | Algebra | olympiads |
Alice and Bob live on the same road. At time $t$, they both decide to walk to each other's houses at constant speed. However, they were busy thinking about math so that they didn't realize passing each other. Alice arrived at Bob's house at $3:19\text{pm}$, and Bob arrived at Alice's house at $3:29\text{pm}$. Charlie, ... | 179 | Algebra | aops_forum |
Task 1. Determine the value of the numerical expression
$$
8004^{2}-8003^{2}+8002^{2}-8001^{2}+\ldots+4^{2}-3^{2}+2^{2}-1^{2}
$$ | 32036010 | Algebra | olympiads |
3. The area of the closed figure enclosed by the curve $x^{2}+2 y^{2} \leqslant 4|y|$ is . $\qquad$ | 2\sqrt{2}\pi | Geometry | olympiads |
11.42. If five points are given on a plane, then by considering all possible triples of these points, one can form 30 angles. Let the smallest of these angles be $\alpha$. Find the maximum value of $\alpha$.
If five points are given on a plane, then by considering all possible triples of these points, one can form 30 ... | 36 | Geometry | olympiads |
For a certain task, $B$ needs 6 more days than $A$, and $C$ needs 3 more days than $B$. If $A$ works for 3 days and $B$ works for 4 days together, they accomplish as much as $C$ does in 9 days. How many days would each take to complete the task alone? | 18,24,27 | Algebra | olympiads |
3. Denote, that there exist such points $D$ on side $A B$ and $E$ on side $A C$ of triangle $A B C$, such that $|A E|=|E D|=|D B|$ and $|A D|=|D C|=|C B|$. Determine the sizes of the angles of triangle $A B C$. | \angleBAC=36,\angleCBA=72,\angleACB=72 | Geometry | olympiads |
374. Form the equation of the plane passing through the points $M(1 ; 2 ; 0), N(1 ; -1 ; 2), P(0 ; 1 ; -1)$ and find the angles of its normal with the coordinate axes. | 5x-2y-3z-1=0,\alpha\approx3548^{\},\beta\approx10856^{\},\gamma\approx1197^{\} | Algebra | olympiads |
Let $\mathcal{P}$ be a set of monic polynomials with integer coefficients of the least degree, with root $k \cdot \cos\left(\frac{4\pi}{7}\right)$, as $k$ spans over the positive integers. Let $P(x) \in \mathcal{P}$ be the polynomial so that $|P(1)|$ is minimized. Find the remainder when $P(2017)$ is divided by $1000$.... | 167 | Algebra | aops_forum |
Example 14. Find $\int(1+2 \sin x)^{2} d x$. | 3x-4\cosx-\sin2x+C | Calculus | olympiads |
Find the lowest possible values from the function
\[ f(x) \equal{} x^{2008} \minus{} 2x^{2007} \plus{} 3x^{2006} \minus{} 4x^{2005} \plus{} 5x^{2004} \minus{} \cdots \minus{} 2006x^3 \plus{} 2007x^2 \minus{} 2008x \plus{} 2009\]
for any real numbers $ x$. | 1005 | Calculus | aops_forum |
4. For any given irrational numbers $a, b$ and real number $r$ $(r>0)$, the number of rational points on the circle $(x-a)^{2}+(y-b)^{2}=r^{2}$ is ( ).
(A) at most one
(B) at most two
(C) at least two, finite in number
(D) infinitely many | B | Geometry | cn_contest |
10.1 Move a particle, and when it travels the second mile and subsequent miles, its speed is inversely proportional to the number of miles traveled. And in each subsequent mile, the speed is constant. If it takes 2 hours to travel the second mile, then the time required to travel the $n$-th mile (in hours) is
(A) $\fra... | 2(n-1) | Algebra | olympiads |
2. Using the digits 0 and 5, write all ten-digit numbers that are divisible by 9. How many such numbers are there? | 9 | Number Theory | olympiads |
Determine all prime numbers $p, q$ and $r$ such that $p+q^{2}=r^{4}$.
(Karl Czakler)
Answer. The only solution is $p=7, q=3, r=2$. | p=7,q=3,r=2 | Number Theory | olympiads |
[ Inscribed and Circumscribed Circles ]
The hypotenuse of a right triangle is 4 m. Find the radius of the circumscribed circle.
# | 2 | Geometry | olympiads |
1. Given the function $f(x)$ satisfies $f(1)=2$, and
$$
f(x+1)=\frac{1+f(x)}{1-f(x)}
$$
for any $x$ in its domain. Then $f(2016)=$
$\qquad$ | \frac{1}{3} | Algebra | cn_contest |
## Task 5 - 080835
Fritz is supposed to multiply a three-digit natural number $z$ by itself. By mistake, he writes the first factor as a number 5 less. When this is pointed out to him, he says: "I'll just take the second factor as a number 5 greater, then the result will be correct."
a) Is this statement true?
b) As... | -25 | Algebra | olympiads |
Find all prime number $p$, such that there exist an infinite number of positive integer $n$ satisfying the following condition:
$p|n^{ n+1}+(n+1)^n.$
(September 29, 2012, Hohhot) | 2 | Number Theory | aops_forum |
II. (16 points) Find all three-digit numbers such that the last three digits of their square are the original two-digit number.
The above text has been translated into English, retaining the original text's line breaks and format. | 625 \text{ or } 376 | Number Theory | cn_contest |
In the right triangle shown the sum of the distances $BM$ and $MA$ is equal to the sum of the distances $BC$ and $CA$.
If $MB = x, CB = h$, and $CA = d$, then $x$ equals:
$\textbf{(A)}\ \frac{hd}{2h+d}\qquad\textbf{(B)}\ d-h\qquad\textbf{(C)}\ \frac{1}{2}d\qquad\textbf{(D)}\ h+d-\sqrt{2d}\qquad\textbf{(E)}\ \sqrt{h^... | \frac{hd}{2h+} | Geometry | amc_aime |
## Task 4 - 100734
According to the legend, the Bohemian Queen Libussa made the granting of her hand dependent on the solution of a riddle that she gave to her three suitors:
"If I give the first suitor half of the contents of this basket of plums and one more plum, the second suitor half of the remaining contents an... | 30 | Logic and Puzzles | olympiads |
4. On the base plane of a cone with height $h$ and base radius $R$, a point (outside the cone) is given, which is at a distance of $aR$ from the circumference of the base. Find the angle between the tangent planes to the cone passing through the given point.
First method:
Let $L$ be the given point, $B$ and $C$ be th... | 2\operatorname{arctg}(\sqrt{5}/2) | Geometry | olympiads |
An equilateral triangle of side $n$ is divided into equilateral triangles of side $1$. Find the greatest possible number of unit segments with endpoints at vertices of the small triangles that can be chosen so that no three of them are sides of a single triangle. | n(n+1) | Combinatorics | aops_forum |
[Sum of angles in a triangle. Theorem of the exterior angle.] Counting in two ways
In a square, 20 points were marked and connected by non-intersecting segments to each other and to the vertices of the square, such that the square was divided into triangles. How many triangles were formed? | 42 | Geometry | olympiads |
Let $ A$, $ B$ be the number of digits of $ 2^{1998}$ and $ 5^{1998}$ in decimal system. $ A \plus{} B \equal{} ?$
$\textbf{(A)}\ 1998 \qquad\textbf{(B)}\ 1999 \qquad\textbf{(C)}\ 2000 \qquad\textbf{(D)}\ 3996 \qquad\textbf{(E)}\ 3998$ | 1999 | Number Theory | aops_forum |
3. In the expansion of $(x+y+z)^{8}$, the sum of the coefficients of all terms of the form $x^{2} y^{a} z^{b}(a, b \in \mathbf{N})$ is
A. 112
B. 448
C. 1792
D. 14336 | 1792 | Combinatorics | olympiads |
## Task 1.
Determine all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ such that for all real numbers $x$ and $y$,
$$
f\left(x^{2}\right)+x f(y)=f(x) f(x+f(y)) .
$$ | f(x)=0f(x)=x | Algebra | olympiads |
For what integers $n>2$ is the following statement true? "Any convex $n$-gon has a side such that neither of the angles at the endpoints of this side is acute." | n\geq7 | Geometry | olympiads |
Cyclic quadrilateral $ABCD$ has $AC\perp BD$, $AB+CD=12$, and $BC+AD=13$. FInd the greatest possible area of $ABCD$. | 36 | Geometry | aops_forum |
3. Let the sequence of positive numbers $a_{0}, a_{1}, \cdots, a_{n}, \cdots$ satisfy
$$
\sqrt{a_{n} a_{n-2}}-\sqrt{a_{n-1} a_{n-2}}=2 a_{n-1}(n \geqslant 2) \text {, }
$$
and $a_{0}=a_{1}=1$. Find the general term formula for the sequence $\left\{a_{n}\right\}$. | a_{n}={\begin{pmatrix}1,\quadn=0,\\0\prod_{k=1}^{n}(2^{k}-1)^{2},n\in{N}_{+}0)\end{pmatrix}.} | Algebra | olympiads |
Let $n_1,n_2,\ldots,n_s$ be distinct integers such that
$$(n_1+k)(n_2+k)\cdots(n_s+k)$$is an integral multiple of $n_1n_2\cdots n_s$ for every integer $k$. For each of the following assertions give a proof or a counterexample:
$(\text a)$ $|n_i|=1$ for some $i$
$(\text b)$ If further all $n_i$ are positive, then
$$\{n... | \{n_1, n_2, \ldots, n_s\} = \{1, 2, \ldots, s\} | Number Theory | aops_forum |
14. Determine the smallest integer $n$ with $n \geq 2$ such that
$$
\sqrt{\frac{(n+1)(2 n+1)}{6}}
$$
is an integer. | 337 | Number Theory | olympiads |
4. (10 points) The sum of all simplest proper fractions with 100 as the denominator equals $\qquad$
Translate the text above into English, please retain the original text's line breaks and format, and output the translation result directly. | 20 | Logic and Puzzles | olympiads |
A box contains 105 black balls, 89 gray balls, and 5 white balls. Outside the box, there are white balls in sufficient quantity to repeatedly perform the following procedure until only two balls remain in the box:
- two balls are removed from the box without looking;
- if the balls removed are of different colors, the... | D | Logic and Puzzles | olympiads |
[ Symmetry properties and center of symmetry ] [ Varignon parallelogram $]$
Point $O$, located inside a convex quadrilateral of area $S$, is reflected symmetrically with respect to the midpoints of its sides.
Find the area of the quadrilateral with vertices at the obtained points. | 2S | Geometry | olympiads |
18.50 As shown in the figure, in rectangle $A B C D$, it is known that $A B=8, \quad B C=22$, $E, F$ are points on $B C$, and $A E=E F=F D$. Then the length of $E F$ is
(A) 9.
(B) 10.
(C) 11.
(D) 12.
(China Sichuan Province Junior High School Mathematics Competition, 1991) | 10 | Geometry | olympiads |
13.94 For which natural numbers $n$ and $k$, are the binomial coefficients
$$
C_{n}^{k-1}, C_{n}^{k}, C_{n}^{k+1}
$$
in arithmetic progression? | n=u^2-2,k=C_{u}^{2}-1ork=C_{u+1}^{2}-1,u\geqslant3 | Combinatorics | olympiads |
7 . 14 Condition A: $\sqrt{1+\sin \theta}=a$. Condition B: $\sin \frac{\theta}{2}+\cos \frac{\theta}{2}=a$. Then
(A) A is a necessary and sufficient condition for B.
(B) A is a necessary condition for B.
(C) A is a sufficient condition for B.
(D) A is neither a necessary condition nor a sufficient condition for B.
(Chi... | D | Algebra | olympiads |
Determine all functions $ f$ mapping the set of positive integers to the set of non-negative integers satisfying the following conditions:
(1) $ f(mn) \equal{} f(m)\plus{}f(n)$,
(2) $ f(2008) \equal{} 0$, and
(3) $ f(n) \equal{} 0$ for all $ n \equiv 39\pmod {2008}$. | f(n) = 0 | Number Theory | aops_forum |
1. If $x^{2}-12 x+1=0$, then the unit digit of $x^{4}+x^{-4}$ is ( ).
(A) 1
(B) 2
(C) 3
(D) 4 | B | Algebra | cn_contest |
For which natural numbers $n$ is it true that the fraction
$$
\frac{n^{2}+2}{n(n+1)}
$$
is not simplifiable? | n=6k+1n=6k+3 | Number Theory | olympiads |
14. Taran thought of a whole number and then multiplied it by either 5 or 6 . Krishna added 5 or 6 to Taran's answer. Finally Eshan subtracted either 5 or 6 from Krishna's answer. The final result was 73 . What number did Taran choose?
A 10
B 11
C 12
D 13
E 14 | 12 | Algebra | olympiads |
Find all solutions $(a, b) \in \mathbb{Z}^{2}$ for each of the following equations
$$
3 a+5 b=1, \quad 3 a+5 b=4, \quad 183 \cdot a+117 \cdot b=3
$$ | (2-5k,-1+3k),(8-5k,-4+3k),(16-39k,-25+61k) | Number Theory | olympiads |
7.53. A light ray falls on a flat mirror at an angle $\alpha$. The mirror is rotated by an angle $\beta$ around the projection of the ray on the mirror. By what angle will the reflected ray deviate in this case? | \sin(\varphi/2)=\sin\alpha\sin\beta | Geometry | olympiads |
6. It is known that the sum of three fractions is $\frac{10}{11}$, and they have the same denominator, with the ratio of their numerators being $2: 3: 4$. Therefore, the largest of these three fractions is $\qquad$ . | \frac{40}{99} | Algebra | olympiads |
7. In Rt $\triangle A B C$, $\angle C=90^{\circ}$. Which of the following equations is not necessarily true? ( ).
(A) $\sin A=\sin B$
(B) $\cos A=\sin B$
(C) $\sin A=\cos B$
(D) $\sin (A+B)=\sin C$ | A | Geometry | cn_contest |
2. Four people, A, B, C, and D, planted a total of 60 trees. It is known that, the number of trees planted by A is half of the rest three, the number of trees planted by B is one-third of the rest three, and the number of trees planted by C is one-fourth of the rest three. How many trees did D plant $\qquad$? | 13 | Algebra | olympiads |
Example 3.3.2 (League 2004) In an acute triangle $ABC$, the altitude $CE$ from $C$ to $AB$ intersects the altitude $BD$ from $B$ to $AC$ at point $H$. The circle with diameter $DE$ intersects $AB$ and $AC$ at points $F$ and $G$, respectively. $FG$ intersects $AH$ at point $K$. Given $BC=25, BD=20, BE=7$, find the lengt... | \frac{216}{25} | Geometry | olympiads |
4. If $\triangle A B C$ satisfies $\cot A, \cot B, \cot C$ forming an arithmetic sequence, then the maximum value of $\angle B$ is $\qquad$ . | \frac{\pi}{3} | Geometry | olympiads |
3. Given a natural number $k>1$, the sum of $k$ consecutive natural numbers is 2000. Then the number of values of $k$ that satisfy the condition is ( ).
(A) 1
(B) 2
(C) 3
(D) 6 | 3 | Number Theory | olympiads |
36. Find all positive integer triples $(a, m, n)$ such that
$$\left(a^{m}+1\right) \mid(a+1)^{n}$$ | (a, m, n)=(1, m, n), (a, 1, n), (2,3, r) | Number Theory | number_theory |
14. There is a ten-digit number. Counting from left to right, if its first digit is a certain number, then this ten-digit number contains that many 0s; if its second digit is a certain number, then this ten-digit number contains that many 1s; if its third digit is a certain number, then this ten-digit number contains t... | 6210001000 | Logic and Puzzles | olympiads |
# 3. Variant 1
Vasya cut out a triangle from cardboard and numbered its vertices with the digits $1,2,3$. It turned out that if Vasya's triangle is rotated 12 times clockwise around its vertex numbered 1 by an angle equal to the angle at this vertex, it will return to its original position. If Vasya's triangle is rota... | 4 | Geometry | olympiads |
11.3. Find all real numbers $a$ for which there exist three distinct real numbers $x, y, z$ such that $a=x+\frac{1}{y}=y+\frac{1}{z}=z+\frac{1}{x}$. | \1 | Algebra | olympiads |
Example 5 The base of the pyramid is an isosceles triangle, with the base and the equal sides being 12 and 10, respectively. Moreover, the projection of the pyramid's apex onto the base lies within the triangle, and the dihedral angles between each lateral face and the base are all $30^{\circ}$. Then the height of this... | \sqrt{3} | Geometry | cn_contest |
3. If $a, b, c$ are real numbers, $x=a^{2}-2 b+\frac{\pi}{3}, y=$ $b^{2}-2 c+\frac{\pi}{6}, z=c^{2}-2 a+\frac{\pi}{2}$, then among $x, y, z$
(A) at least one is greater than 0
(B) at least one is not less than 0
(C) at least one is not greater than 0
(D) at least one is less than 0 | A | Algebra | cn_contest |
$2+$ |
| [ | ]
For the angles of triangle $ABC$, it is known that $\sin A + \cos B = \sqrt{2}$ and $\cos A + \sin B = \sqrt{2}$. Find the measure of angle $C$. | 90 | Algebra | olympiads |
Anita plays the following single-player game: She is given a circle in the plane. The center of this circle and some point on the circle are designated “known points”. Now she makes a series of moves, each of which takes one of the following forms:
(i) She draws a line (infinite in both directions) between two “known p... | 5 | Geometry | aops_forum |
What is the smallest perfect square larger than $1$ with a perfect square number of positive integer factors?
[i]Ray Li[/i] | 36 | Number Theory | aops_forum |
19.43 If the sum of the measures of angles $A$, $B$, $C$, $D$, $E$, and $F$ in the figure on the right is $90 n^{\circ}$, then $n$ equals
(A) 2.
(B) 3.
(C) 4.
(D) 5.
(E) 6. | 4 | Geometry | olympiads |
Example 1 Calculate the sum $S=\sum_{n=1}^{509}\left[\frac{511 n}{509}\right]$. | 130051 | Number Theory | olympiads |
If $A=20^\circ$ and $B=25^\circ$, then the value of $(1+\tan A)(1+\tan B)$ is
$\mathrm{(A)\ } \sqrt{3} \qquad \mathrm{(B) \ }2 \qquad \mathrm{(C) \ } 1+\sqrt{2} \qquad \mathrm{(D) \ } 2(\tan A+\tan B) \qquad \mathrm{(E) \ }\text{none of these}$ | 2 | Algebra | amc_aime |
5.60 Solve the system of equations
$$
\left\{\begin{array}{l}
\sin x \sin y=0.75 \\
\operatorname{tg} x \operatorname{tg} y=3
\end{array}\right.
$$ | \\frac{\pi}{3}+\pi(k+n),\\frac{\pi}{3}+\pi(n-k),n,k\in\boldsymbol{Z} | Algebra | olympiads |
[Circles](https://artofproblemsolving.com/wiki/index.php/Circle) $C_1$ and $C_2$ are externally [tangent](https://artofproblemsolving.com/wiki/index.php/Tangent), and they are both internally tangent to circle $C_3.$ The radii of $C_1$ and $C_2$ are 4 and 10, respectively, and the [centers](https://artofproblemsolving... | 405 | Geometry | amc_aime |
9.1. Several consecutive natural numbers are written on the board. Exactly $52 \%$ of them are even. How many even numbers are written on the board | 13 | Number Theory | olympiads |
[Examples and Counterexamples. Constructions] [ Proof by Contradiction ]
## Auto: Zassavsiy A.A.
In a volleyball tournament, each team met every other team once. Each match consisted of several sets - until one of the teams won three times. If a match ended with a score of $3: 0$ or $3: 1$, the winning team received ... | 6 | Combinatorics | olympiads |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.