problem stringlengths 37 4.98k | answer stringlengths 1 141 | subject stringclasses 8
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B1. The floor function of any real number $a$ is the integer number denoted by $\lfloor a\rfloor$ such that $\lfloor a\rfloor \leq a$ and $\lfloor a\rfloor>a-1$. For example, $\lfloor 5\rfloor=5,\lfloor\pi\rfloor=3$ and $\lfloor-1.5\rfloor=-2$. Find the difference between the largest integer solution of the equation $\... | 614 | Number Theory | olympiads |
4. Given that the three sides of $\triangle A B C$ are exactly three consecutive positive integers, and its perimeter and area are $p_{1}$ and $S_{1}$, respectively. If the three sides of $\triangle A B C$ are each increased by 10, the new $\triangle A^{\prime} B^{\prime} C^{\prime}$ has a perimeter and area of $p_{2}$... | \frac{3}{5} | Geometry | cn_contest |
8. In $\triangle A B C$, the medians $A E, B F$, and $C D$ intersect at $M$. Given that $E, C, F$, and $M$ are concyclic, and $C D=n$. Find the length of segment $A B$.
(18th All-Russian Competition Problem) | AB=\frac{2}{3}\sqrt{3}n | Geometry | olympiads |
2. (15 points) A satellite is launched vertically from the pole of the Earth at the first cosmic speed. To what maximum distance from the Earth's surface will the satellite travel? (The acceleration due to gravity at the Earth's surface $g=10 \mathrm{m} / \mathrm{c}^{2}$, radius of the Earth $R=6400$ km). | 6400 | Algebra | olympiads |
3. One lap of a standard running track is $400 \mathrm{~m}$.
How many laps does each athlete run in a $5000 \mathrm{~m}$ race?
A 4
B 5
C 8
D 10
E $12 \frac{1}{2}$ | 12\frac{1}{2} | Algebra | olympiads |
2. In the cube $A B C D-A_{1} B_{1} C_{1} D_{1}$, the angle formed by $B C_{1}$ and the section $B B_{1} D_{1} D$ is ( ).
(A) $\frac{\pi}{6}$
(B) $\frac{\pi}{4}$
(C) $\frac{\pi}{3}$
(D) $\frac{\pi}{2}$ | A | Geometry | cn_contest |
4. Philatelist Andrey decided to distribute all his stamps equally into 3 envelopes, but it turned out that one stamp was extra. When he distributed them equally into 5 envelopes, 3 stamps were extra; finally, when he distributed them equally into 7 envelopes, 5 stamps remained. How many stamps does Andrey have in tota... | 208 | Number Theory | olympiads |
Ostap Bender and Kisa Vorobyaninov divided the revenue from selling elephants to the population between themselves. Ostap thought: if I had taken 40% more money, Kisa's share would have decreased by 60%. And how would Vorobyaninov's share change if Ostap had taken 50% more money? | 75 | Algebra | olympiads |
4. The height of a ball thrown vertically upwards from the ground is a quadratic function of its motion time. Xiao Hong throws two balls vertically upwards, 1 second apart. Assuming the two balls are thrown from the same height above the ground, and they reach the same maximum height above the ground 1.1 seconds after ... | 1.6 | Algebra | cn_contest |
8. Niu Niu walks along a straight road. She walks from the 1st tree to the 5th tree in exactly 5 minutes. If the distance between adjacent trees is equal, at this speed, it will take $\qquad$ more minutes for Niu Niu to reach the 17th tree. | 15 | Other | olympiads |
2. Find all solutions to the equation
$$
\sqrt[3]{x}+\sqrt[3]{2 x-3}=\sqrt[3]{12(x-1)}
$$ | x_{1}=1,x_{2,3}=3 | Algebra | olympiads |
3. The number of positive integer solutions to the equation $x^{4004}+y^{400 x}=z^{2002}$ is $(\quad)$.
A. 0
B. 1
C. finite
D. infinite | A | Number Theory | olympiads |
We know that two distinct points determine a unique line. How many lines are determined by any two of the nine points marked on the given grid? | 20 | Combinatorics | olympiads |
## Problem Statement
Calculate the limit of the function:
$\lim _{x \rightarrow \frac{\pi}{6}} \frac{2 \sin ^{2} x+\sin x-1}{2 \sin ^{2} x-3 \sin x+1}$ | -3 | Calculus | olympiads |
# 2. Task 2
In a dumpling shop, you can order dumplings in portions of 6, 9, and 20 pieces. Thus, not every number of dumplings can be ordered with these sets, for example, 1, 2, 3, 4, 5, 7, and 8 cannot be bought. What is the largest number of dumplings that cannot be ordered in the dumpling shop? | 43 | Number Theory | olympiads |
2. Let
$$
f(x)=\frac{\sqrt{2+\sqrt{2}} x+\sqrt{2-\sqrt{2}}}{-\sqrt{2-\sqrt{2}} x+\sqrt{2+\sqrt{2}}} .
$$
Determine
$$
\underbrace{f(f \ldots(f}_{1987}(x)) \ldots)
$$ | f_{1987}(x)=\frac{\sqrt{2-\sqrt{2}}x+\sqrt{2+\sqrt{2}}}{-\sqrt{2+\sqrt{2}}x+\sqrt{2-\sqrt{2}}} | Algebra | olympiads |
The side of the base of a regular triangular prism $A B C A_1 B_1 C_1$ is 4, and the lateral edge is 3. On the edge $B B_1$, a point $F$ is taken, and on the edge $C C_1$, a point $G$ is taken such that $B_1 F=1, C G=\frac{2}{3}$. Points $E$ and $D$ are the midpoints of edges $A C$ and $B_1 C_1$ respectively. Find the ... | \sqrt{\frac{51}{2}} | Geometry | olympiads |
Example 8 Let $x \in \mathbf{R}$, find the minimum value of the function $f(x)=\sqrt{x^{2}-4 x+13}+$ $\sqrt{x^{2}-10 x+26}$. | 5 | Algebra | olympiads |
26.41 Let $O A B C$ be a unit square in the $x y$ plane, where $O(0,0)$, $A(1,0)$, $B(1,1)$, and $C(0,1)$. Let $u=x^{2}-y^{2}$, $v=2 x y$ be a transformation from the $x y$ plane to the $u v$ plane. The image of the square under this transformation is
(20th American High School Mathematics Examination, 1969) | D | Algebra | olympiads |
What is the value of $2^{0}+20^{0}+201^{0}+2016^{0}$ ? | 4 | Number Theory | olympiads |
## Task 3 - 040513
The school garden of a city school has an area of 0.15 ha. The garden is divided into 9 plots, each with an area of $150 \mathrm{~m}^{2}$ or $200 \mathrm{~m}^{2}$.
How many plots of each size are there in the garden? | 6\cdot150+3\cdot200=1500 | Algebra | olympiads |
A basin is filled by three pipes. The first and second pipes fill the basin in 70 minutes, the first and third in 84 minutes, and the second and third in 140 minutes. How many minutes does it take to fill the basin through each pipe individually, and how many minutes does it take if all three pipes are open at the same... | 60 | Algebra | olympiads |
3.89. Two cones have concentric bases and the same angle, equal to $\alpha$, between the height and the slant height. The radius of the base of the outer cone is $R$. The lateral surface area of the inner cone is half the total surface area of the outer cone. Find the volume of the inner cone. | \frac{1}{3}\piR^{3}\cos^{3}(\frac{\pi}{4}-\frac{\alpha}{2})\operatorname{ctg}\alpha | Geometry | olympiads |
$17 \cdot 140$ An isosceles triangle with base $\sqrt{2}$, the medians to the two equal sides are perpendicular to each other, then the area of this triangle is
(A) 1.5 .
(B) 2 .
(C) 2.5 .
(D) 3.5 .
(E) 4 .
(10th American High School Mathematics Examination, 1959) | \frac{3}{2} | Geometry | olympiads |
2. In the Cartesian coordinate system $x O y$, the graph of the function $f(x)=a \sin a x+\cos a x(a>0)$ over an interval of the smallest positive period length and the graph of the function $g(x)=\sqrt{a^{2}+1}$ enclose a closed figure whose area is $\qquad$ . | \frac{2\pi}{}\sqrt{^{2}+1} | Algebra | olympiads |
11.195. The radius of the base of the cone is $R$, and the angle of the sector of its lateral surface is $90^{\circ}$. Determine the volume of the cone. | \frac{\piR^3\sqrt{15}}{3} | Geometry | olympiads |
Problem 5.1. Dasha calls a natural number special if four different digits are used to write it. For example, the number 3429 is special, while the number 3430 is not special.
What is the smallest special number greater than 3429? | 3450 | Number Theory | olympiads |
The natural numbers $m$ and $k$ satisfy the equality $$1001 \cdot 1002 \cdot ... \cdot 2010 \cdot 2011 = 2^m (2k + 1)$$. Find the number $m$. | 1008 | Number Theory | aops_forum |
There are $ 7$ boxes arranged in a row and numbered $1$ through $7$. You have a stack of $2015$ cards, which you place one by one in the boxes. The first card is placed in box #$1$, the second in box #$2$, and so forth up to the seventh card which is placed in box #$7$. You then start working back in the other directi... | \text{box 3} | Combinatorics | aops_forum |
Problem 9.1. Find all values of $a$ such that the equation
$$
\sqrt{\left(4 a^{2}-4 a-1\right) x^{2}-2 a x+1}=1-a x-x^{2}
$$
has exactly two solutions.
Sava Grozdev, Svetlozar Doychev | =\frac{1}{3},=\frac{1}{2},\in(\frac{5}{6},\frac{3}{2})\backslash{1} | Algebra | olympiads |
Given two similar polygons, construct a third one that is similar to the previous ones and whose area is equal to the sum of the areas of the previous ones. | ^{2}=^{2}+b^{2} | Geometry | olympiads |
3. Among the 1000 natural numbers from 1 to 1000, the number of natural numbers that cannot be divided by 4 or 6 is ( )
A. 416
B. 584
C. 625
D. 667 | 667 | Number Theory | olympiads |
Example 7 A cube of a positive integer is a four-digit number, and the fourth power of this positive integer is a six-digit number. The 10 digits used here are exactly $0,1,2,3,4,5,6,7,8,9$, each appearing once, without repetition or omission. Try to find this positive integer.
Translate the above text into English, p... | 18 | Number Theory | olympiads |
1. Find all complex numbers $z$ such that
$$
P(x)=(x-z)\left(x-z^{2}\right)\left(x-z^{3}\right)
$$
has real coefficients. | z\in{\frac{-1\\sqrt{3}\mathrm{i}}{2},\\mathrm{i}}\cup\mathbb{R} | Algebra | olympiads |
In a triangle with sides of lengths $a$, $b$, and $c$, $(a+b+c)(a+b-c) = 3ab$. The measure of the angle opposite the side length $c$ is
$\textbf{(A)}\ 15^\circ\qquad\textbf{(B)}\ 30^\circ\qquad\textbf{(C)}\ 45^\circ\qquad\textbf{(D)}\ 60^\circ\qquad\textbf{(E)}\ 150^\circ$ | 60 | Geometry | amc_aime |
8. At the opening ceremony of a sports meet, a large and a small square formation merged into a 15 by 15 square formation. Therefore, the original large square formation had
people, and the small square formation had
people. | 144 | Geometry | olympiads |
25. For a real number $x$, let $\lfloor x\rfloor$ denote the greatest integer not exceeding $x$. Consider the function
$$
f(x, y)=\sqrt{M(M+1)}(|x-m|+|y-m|),
$$
where $M=\max (\lfloor x\rfloor,\lfloor y\rfloor)$ and $m=\min (\lfloor x\rfloor,\lfloor y\rfloor)$. The set of all real numbers $(x, y)$ such that $2 \leq x, ... | 2021 | Algebra | olympiads |
9. If the graph of the function $g(x)$ is symmetric to the graph of the function $f(x)=\frac{1-2^{x}}{1+2^{x}}$ with respect to the line $y=x$, then the value of $g\left(\frac{2}{7}\right)$ is $\qquad$ . | \log_{2}\frac{5}{9} | Algebra | olympiads |
2.129. $\frac{25 \cdot \sqrt[4]{2}+2 \sqrt{5}}{\sqrt{250}+5 \sqrt[4]{8}}-\sqrt{\frac{\sqrt{2}}{5}+\frac{5}{\sqrt{2}}+2}=-1$. | -1 | Algebra | olympiads |
Problem 5. On the side $A D$ of the square $A B C D$, point $K$ is marked, and on the extension of ray $A B$ beyond point $B$ - point $L$. It is known that $\angle L K C=45^{\circ}, A K=1, K D=2$. Find $L B$. | 2 | Geometry | olympiads |
14. N1 (JAP) Determine all positive integers $n \geq 2$ that satisfy the following condition: For all integers $a, b$ relatively prime to $n$, $a \equiv b(\bmod n) \quad$ if and only if $\quad a b \equiv 1(\bmod n)$. | n\mid24 | Number Theory | olympiads |
II. (25 points) As shown in Figure 4, the incircle $\odot I$ of $\triangle ABC$ touches $AB$ and $AC$ at points $D$ and $E$, respectively. Extend $DI$ to $M$ and $EI$ to $N$ such that $IM = IN = 9IE$. Points $B$, $I$, $C$, $M$, and $N$ are concyclic. The ratio of the perimeter of $\triangle ABC$ to side $BC$ is the sim... | 2009 | Geometry | cn_contest |
To be factored into the product of three factors:
$$
\left(x^{2}+x y+y^{2}\right)^{2}-\left(x^{2} y^{2}+y^{2} z^{2}+z^{2} x^{2}\right)
$$ | (x^{2}+y^{2})(x+y+z)(x+y-z) | Algebra | olympiads |
1. Find the smallest positive integer $k$, such that there exist positive integers $m, n$, satisfying $k=19^{n}-5^{m}$. | 14 | Number Theory | olympiads |
[ Rhombi. Properties and Characteristics ]
In rhombus $A B C D$, the angle $\angle A B C=60^{\circ}$. A circle is tangent to the line $A D$ at point $A$, and the center of the circle lies inside the rhombus. The tangents to the circle, drawn from point $C$, are perpendicular. Find the ratio of the perimeter of the rho... | \frac{\sqrt{3}+\sqrt{7}}{\pi} | Geometry | olympiads |
4. On each field of the chessboard, a number is written. The sum of the numbers written on any four fields that form a knight's path (in the shape of the letter Г) is constant. How many different numbers are written on the board? Explain your answer. | 2 | Logic and Puzzles | olympiads |
## Condition of the problem
To derive the equation of the normal to the given curve at the point with abscissa $x_{0}$.
$y=\sqrt{x}-3 \sqrt[3]{x}, x_{0}=64$ | 64 | Calculus | olympiads |
6. Let $P_{1}, P_{2}, \ldots, P_{6}$ be points in the complex plane, which are also roots of the equation $x^{6}+6 x^{3}-216=0$. Given that $P_{1} P_{2} P_{3} P_{4} P_{5} P_{6}$ is a convex hexagon, determine the area of this hexagon. | 9\sqrt{3} | Algebra | olympiads |
4. Given $z \in C$ and the ratio of the real part to the imaginary part of $(z+5)^{2}$ is $-\frac{3}{4}$, then the range of values for $z$ that satisfy the condition is $\qquad$ . | {x+yi\lvert\,(x+2y+5)(x-\frac{y}{2}+5)=0\cap((x+5)y\neq0).} | Algebra | olympiads |
10.244. A circle with a radius of 3 cm is inscribed in a triangle. Calculate the lengths of the sides of the triangle if one of them is divided by the point of tangency into segments of 4 and $3 \mathrm{~cm}$. | 24 | Geometry | olympiads |
5. Let $x_{1}, x_{2}, \cdots, x_{n}$ be a sequence of integers satisfying the following conditions: (i) $-1 \leqslant x_{i} \leqslant 2$, for $i=1,2,3, \cdots, n$. (ii) $x_{1}+x_{2}+\cdots+x_{n}=19$ and (iii) $x_{1}^{2}+x_{2}^{2}+\cdots+x_{m}^{2}=99$. Let $m$ and $M$ be the minimum and maximum values of $x_{1}^{3}+x_{2... | 7 | Algebra | olympiads |
3. A thermally insulated vessel is divided by a heat-conducting partition into two parts of different volumes. In the first part, there is helium at a temperature of $127{ }^{\circ} \mathrm{C}$ in an amount of $v_{1}=0.2$ moles. In the second part, there is helium at a temperature of $7{ }^{\circ} \mathrm{C}$ in an amo... | 31\mathrm{C},0.76 | Algebra | olympiads |
Example 6. $N$ is a positive integer, in the square $R$ with vertices at $(N, 0)$, $(0, N)$, $(-N, 0)$, $(0, -N)$ (including the boundary), how many integer points are there? | 2 N^{2}+2 N+1 | Combinatorics | cn_contest |
4. If $2016+3^{n}$ is a perfect square, then the positive integer $n=$ . $\qquad$ | 2 | Number Theory | cn_contest |
## Problem Statement
Find the angle between the planes:
$$
\begin{aligned}
& 3 x-y+2 z+15=0 \\
& 5 x+9 y-3 z-1=0
\end{aligned}
$$ | \frac{\pi}{2} | Geometry | olympiads |
Exercise 6. Find the largest integer $n \geqslant 3$, satisfying:
"for all integers $k \in\{2,3, \cdots, \mathrm{n}\}$ if $k$ and $\mathrm{n}$ are coprime then $\mathrm{k}$ is a prime number." | 30 | Number Theory | olympiads |
\section*{Problem 1}
(a) Each of \(\mathrm{x}_{1}, \ldots, \mathrm{x}_{\mathrm{n}}\) is \(-1,0\) or 1 . What is the minimal possible value of the sum of all \(\mathrm{x}_{\mathrm{i}} \mathrm{x}_{\mathrm{j}}\) with \(1<=\mathrm{i}<\mathrm{j}<=\mathrm{n}\) ? (b) Is the answer the same if the \(\mathrm{x}_{\mathrm{i}}\) ... | -[n/2] | Combinatorics | olympiads |
## Problem Statement
Calculate the limit of the function:
$\lim _{x \rightarrow 1} \frac{3-\sqrt{10-x}}{\sin 3 \pi x}$ | -\frac{1}{18\pi} | Calculus | olympiads |
6. Spheres with radii $1,2,3$ are externally tangent to each other, and planes $\alpha$ and $\beta$ are tangent to all three spheres. Then the dihedral angle formed by plane $\alpha$ and plane $\beta$ is $\qquad$ . | 2 \arccos \frac{\sqrt{23}}{6} | Geometry | cn_contest |
Example 8. A merchant has a 40-pound weight, which broke into 4 pieces after falling to the ground. Later, it was found that the weight of each piece is an integer number of pounds, and these 4 pieces can be used to weigh any integer weight from 1 to 40 pounds. What are the weights of these 4 pieces of the weight? | 1, 3, 9, 27 | Logic and Puzzles | cn_contest |
## Task B-1.5.
In a right-angled triangle $A B C$, with a right angle at vertex $C$, the length of the hypotenuse is 12. Squares $A B D E$ and $A C G F$ are constructed outward on sides $\overline{A B}$ and $\overline{A C}$. If points $D, E$, $F$ and $G$ lie on the same circle, calculate the perimeter of triangle $A B... | 12\sqrt{2}+12 | Geometry | olympiads |
When $1999^{2000}$ is divided by $5$, the remainder is
$\text{(A)}\ 0 \qquad \text{(B)}\ 1 \qquad \text{(C)}\ 2 \qquad \text{(D)}\ 3 \qquad \text{(E)}\ 4$ | 1 | Number Theory | amc_aime |
Problem 9-6. In a cubic chest with a side of $2^{n}$ dm, there are $8^{n}$ different spices: it contains eight closed cubic boxes with a side of $2^{n-1}$ dm, each of which contains eight closed cubic boxes with a side of $2^{n-2}$ dm, and so on down to boxes with a side of 1 dm, each containing its own spice.
In one ... | 2\cdot(8^{n+1}-1)/7 | Combinatorics | olympiads |
Find the number of positive integers $m$ for which there exist nonnegative integers $x_0$, $x_1$ , $\dots$ , $x_{2011}$ such that
\[m^{x_0} = \sum_{k = 1}^{2011} m^{x_k}.\] | 16 | Number Theory | amc_aime |
SG. 3 Let $a \oplus b=a b+10$. If $C=(1 \oplus 2) \oplus 3$, find the value of $C$. | 46 | Algebra | olympiads |
2. The following equation is to be solved in the set of natural numbers
$$
2^{x}+2^{y}+2^{z}=2336
$$ | 5,8,11 | Number Theory | olympiads |
Solve the following system of equations:
$$
2 \sqrt{\sqrt{x}+\sqrt{y}}+\sqrt{x}+\sqrt{y}=8, \quad \sqrt{x^{3}}+\sqrt{y^{3}}=40
$$ | x_1=6+4\sqrt{2},\quady_1=6-4\sqrt{2};\quadx_2=6-4\sqrt{2},\quady_2=6+4\sqrt{2} | Algebra | olympiads |
1020. Investigate the function for extremum
$$
f(x, y)=x^{2}+y^{2}-4 y+4
$$ | 0 | Algebra | olympiads |
5. Egor wrote a number on the board and encrypted it according to the rules of letter puzzles (different letters correspond to different digits, the same letters correspond to the same digits). The result was the word "GUATEMALA". How many different numbers could Egor have initially written if his number was divisible ... | 18480 | Logic and Puzzles | olympiads |
Example 6. Compute the integral
$$
I=\int_{|z|=2} \frac{d z}{1+z^{4}}
$$ | 0 | Calculus | olympiads |
Example 4 Try to find the unit digit of the integer part of $(\sqrt{2}+\sqrt{3})^{2012}$.
[2] | 7 | Number Theory | cn_contest |
7.263. $\left\{\begin{array}{l}4^{\frac{x}{y}+\frac{y}{x}}=32, \\ \log _{3}(x-y)=1-\log _{3}(x+y)\end{array}\right.$
The system of equations is:
\[
\left\{\begin{array}{l}
4^{\frac{x}{y}+\frac{y}{x}}=32, \\
\log _{3}(x-y)=1-\log _{3}(x+y)
\end{array}\right.
\] | (2;1) | Algebra | olympiads |
Example 1 Find the equation of the line $\iota$ passing through the intersection point of the two lines $x-2 y+4=0$ and $x+y-2=0$, and satisfying the following conditions.
(1) Passing through the point $(3,-2)$.
(2) Perpendicular to the line $3 x-4 y+7=0$. | 4x+3y-6=0 | Algebra | olympiads |
[ $\underline{\text { U }}$ equations in integers ]
Solve the equations in natural numbers:
a) $x^{2}-y^{2}=31$
b) $x^{2}-y^{2}=303$.
# | (16,15);(152,151),(52,49) | Number Theory | olympiads |
II. (16 points) Given the sequence $\left\{F_{n}\right\}$ satisfies
$$
\begin{array}{l}
F_{1}=F_{2}=1, \\
F_{n+2}=F_{n+1}+F_{n}\left(n \in \mathbf{Z}_{+}\right) .
\end{array}
$$
If $F_{a} 、 F_{b} 、 F_{c} 、 F_{d}(a<b<c<d)$ are the side lengths of a convex quadrilateral, find the value of $d-b$. | 2 | Number Theory | cn_contest |
Simplify the following expression:
$$
\left[\frac{(a+b)^{2}+2 b^{2}}{a^{3}-b^{3}}-\frac{1}{a-b}+\frac{a+b}{a^{2}+a b+b^{2}}\right] \cdot\left(\frac{1}{b}-\frac{1}{a}\right) \cdot
$$ | \frac{1}{} | Algebra | olympiads |
## Subject II. (20 points)
Determine the first and last five digits of the number:
$$
\begin{array}{r}
2^{2014} \cdot 25^{1006} + 2^{2014} : \left[2^{1000} \cdot 2^{1012} + \left(2^{200} \cdot 2^{205}\right)^{5} : \left(2^{7}\right)^{2} + \left(5^{2014} : 5^{2013} - 1^{2014}\right)^{1005} \cdot 2\right] \cdot 1007 \\... | 40000\ldots000002014 | Number Theory | olympiads |
4. If the lengths of two sides of $\triangle A B C$ are $a$ and $b$, then the area of $\triangle A B C$ cannot be equal to ( ).
(A) $\frac{1}{4}\left(a^{2}+b^{2}\right)$
(B) $\frac{1}{2}\left(a^{2}+b^{2}\right)$
(C) $\frac{1}{8}(a+b)^{2}$
(D) $\frac{1}{4} a b$ | B | Geometry | cn_contest |
5. Let $a$ be a natural number with 2019 digits and divisible by 9. Let $b$ be the sum of the digits of $a$, let $c$ be the sum of the digits of $b$, and let $d$ be the sum of the digits of $c$. Determine the number $d$.
## Third grade - B category | 9 | Number Theory | olympiads |
Quadrilateral $ABCD$ is inscribed in a circle, $I$ is the center of the inscribed circle of triangle $ABD$. Find the minimum value of $BD$, if $AI=BC=CD=2$. | 2\sqrt{3} | Geometry | olympiads |
Subject (2). Consider the following natural numbers:
$$
a=1 \cdot 3 \cdot 5 \cdot 7 \cdots 27 \cdot 29 \cdot 31 \text { and } b=1 \cdot 3 \cdot 5 \cdot 7 \cdots 27 \cdot 29
$$
a) Prove that the number $a$ is divisible by 2015.
b) Find the largest natural number $n$ such that the number $a+b$ is divisible by $10^n$.
... | 4 | Number Theory | olympiads |
A recipe calls for $4 \frac{1}{2}$ cups of flour. If you only make half of the recipe, then how many cups of flour do you need?
(A) $2 \frac{1}{2}$
(B) $2 \frac{1}{4}$
(C) 9
(D) 2
(E) $2 \frac{3}{4}$ | 2\frac{1}{4} | Algebra | olympiads |
## [
The Law of Sines
A circle inscribed in an isosceles triangle \(ABC\) touches the base \(AC\) at point \(D\) and the lateral side \(AB\) at point \(E\). Point \(F\) is the midpoint of side \(AB\), and point \(G\) is the intersection of the circle and segment \(FD\), different from \(D\). The tangent to the circle... | \arccos\frac{3}{4} | Geometry | olympiads |
In how many ways can we distribute 14 identical candies among three children so that each child receives at least three candies? | 21 | Combinatorics | olympiads |
1. The imaginary part of the complex number $z=(1+\mathrm{i})^{2}(2+\mathrm{i})$ is ( ).
(A) -2 i
(B) -2
(C) 4 i
(D) 4 | D | Algebra | cn_contest |
34th Eötvös 1930 Problem 1 How many integers (1) have 5 decimal digits, (2) have last digit 6, and (3) are divisible by 3? | 3000 | Number Theory | olympiads |
a) For the given triangle $ABC$, all angles of which are less than $120^{\circ}$, find the point for which the sum of distances to the vertices is minimal.
b) Inside the triangle $ABC$, all angles of which are less than $120^{\circ}$, a point $O$ is taken from which its sides are seen at an angle of $120^{\circ}$. Pro... | \frac{^2+b^2+^2}{2}+2\sqrt{3}S | Geometry | olympiads |
7. (4 points) Find what $x+y$ can be equal to, given that $x^{3}+6 x^{2}+16 x=-15$ and $y^{3}+6 y^{2}+16 y=-17$. | -4 | Algebra | olympiads |
Joshua rolls two dice and records the product of the numbers face up. The probability that this product is composite can be expressed as $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Compute $m+n$.
[i]Proposed by Nathan Xiong[/i] | 65 | Combinatorics | aops_forum |
Problem 21. In a triangle with a perimeter of $2 \sqrt{3}$, the product of its three angle bisectors is 1, and the radius of the inscribed circle is $\frac{1}{3}$. Find the angles of the triangle. | 60;60;60 | Geometry | olympiads |
[Central Angle. Arc Length and Circumference]
On the coordinate plane ($x ; y$), a circle with radius 4 and center at the origin is drawn. The line given by the equation $y=\sqrt{3} x-4$ intersects it at points $A$ and $B$. Find the sum of the lengths of segment $A B$ and the larger arc $A B$. | \frac{16\pi}{3}+4\sqrt{3} | Geometry | olympiads |
2. At the Olympiad, for each solved problem, one could receive 3, 8, or 10 points. Vasya scored 45 points. What is the smallest number of problems he could have solved? (It is necessary to explain why he could not have solved fewer problems.) | 6 | Number Theory | olympiads |
How many integers $n$ are there for which the value of expression (1) is an integer? How many positive integers $n$ are there for which the value of $K$ is a positive integer?
$$
K=\frac{2610+201 n+110 n^{2}-n^{3}}{10+n}
$$ | 27 | Algebra | olympiads |
[Integer and fractional parts. Archimedes' principle]
How many solutions in natural numbers does the equation $\left[{ }^{x} / 10\right]=\left[{ }^{x} / 11\right]+1$ have? | 110 | Number Theory | olympiads |
1. In triangle $D E F$, the heights (emanating, respectively, from vertices $D, E$, and $F$) measure, in order, 84, 80, and 81 meters. Indicating with $d, e, f$ the lengths, respectively, of sides $E F, F D, D E$, which of these inequalities is correct?
(A) $e<f<d$
(D) $f<e<d$
(B) $d<f<e$
(E) $e<d<f$
(C) $d<e<f$ | <f<e | Geometry | olympiads |
For positive integer $n$, define $S_n$ to be the minimum value of the sum \[ \sum_{k=1}^n \sqrt{(2k-1)^2+a_k^2}, \] where $a_1,a_2,\ldots,a_n$ are positive real numbers whose sum is 17. There is a unique positive integer $n$ for which $S_n$ is also an integer. Find this $n$. | 12 | Calculus | aops_forum |
16. (13 points) A
schematic diagram of a computer device is shown in Figure 4, where $J_{1}$ and $J_{2}$ represent data inputs, and $C$ is the output for the computation result. The computation process involves inputting natural numbers $m$ and $n$ through $J_{1}$ and $J_{2}$ respectively, and after computation, a nat... | 2^{2001}+16 | Algebra | cn_contest |
8.293. $\frac{4 \sin \left(\frac{\pi}{6}+x\right) \sin \left(\frac{5 \pi}{6}+x\right)}{\cos ^{2} x}+2 \tan x=0$. | x_{1}=-\operatorname{arctg}\frac{1}{3}+\pik;x_{2}=\frac{\pi}{4}(4n+1) | Algebra | olympiads |
# Problem 1. (2 points)
Let $x, y, z$ be pairwise coprime three-digit natural numbers. What is the greatest value that the GCD $(x+y+z, x y z)$ can take? | 2994 | Number Theory | olympiads |
4.3. In a triangle $A B C$ the following relation holds:
$$
\sin ^{23} \frac{\alpha}{2} \cdot \cos ^{48} \frac{\beta}{2}=\sin ^{23} \frac{\beta}{2} \cdot \cos ^{48} \frac{\alpha}{2}
$$
where $\alpha$ and $\beta$ are the corresponding angles at $A$ and $B$. Find the ratio $A C / B C$.
(Cyprus) | 1 | Geometry | olympiads |
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