problem stringlengths 37 4.98k | answer stringlengths 1 141 | subject stringclasses 8
values | level stringclasses 7
values |
|---|---|---|---|
7. If the equation about $x$
$$
x^{3}+a x^{2}+b x-4=0\left(a 、 b \in \mathbf{N}_{+}\right)
$$
has a positive integer solution, then $|a-b|=$ | 1 | Algebra | cn_contest |
## Problem Statement
Calculate the limit of the numerical sequence:
$$
\lim _{n \rightarrow \infty} \sqrt{n+2}(\sqrt{n+3}-\sqrt{n-4})
$$ | \frac{7}{2} | Calculus | olympiads |
3. Solve the equation $\sqrt{6 \cos 4 x+15 \sin 2 x}=2 \cos 2 x$. | -\frac{1}{2}\arcsin\frac{1}{8}+\pin | Algebra | olympiads |
12. On the Cartesian plane, the number of lattice points (i.e., points with both integer coordinates) on the circumference of a circle centered at $(199,0)$ with a radius of 199 is $\qquad$ . | 4 | Geometry | olympiads |
## Task $8 / 85$
Determine the smallest natural number $n$ that can be represented both as the sum of 10 and as the sum of 794 consecutive (not necessarily natural) integers! | 1985 | Number Theory | olympiads |
[i]B stands for Begginners , A stands for Advanced[/i]
[b]B1.[/b] What is the last digit of $2022^2 + 2202^2$?
[b]B2 / A1.[/b] Find the area of the shaded region!
[img]https://cdn.artofproblemsolving.com/attachments/d/4/dd49bfe56ba77e8eec9f91220725ced15f61d8.png[/img]
[b]B3 / A2.[/b] If $\Psi (n^2 + k) = n - 2k$, ... | 36 | Other | aops_forum |
16. The sum of the lengths of the three sides of a right-angled triangle is $16 \mathrm{~cm}$. The sum of the squares of the lengths of the three sides of the triangle is $98 \mathrm{~cm}^{2}$.
What is the area, in $\mathrm{cm}^{2}$, of the triangle?
A 8
B 10
C 12
D 14
E 16 | 8 | Geometry | olympiads |
73. Seven monkeys share a box of chestnuts, each getting a different amount, with the monkey that gets the most receiving 50 nuts. Therefore, the box of chestnuts can have at most $\qquad$ nuts. | 329 | Number Theory | olympiads |
## Task 3 - 290523
A natural number $z$ is sought that satisfies the following conditions:
(1) The digit in the tens place of $z$ is 0.
(2) If one forms a new number $z^{\prime}$ by removing the digit 0 in the tens place from $z$ and then calculates the sum $z+z^{\prime}$, the result is 5174.
Show that there can on... | 4702 | Number Theory | olympiads |
[Pythagorean Theorem (direct and inverse) $]$ Law of Cosines $\quad]$
A circle inscribed in a right triangle with legs of 6 and 8 touches the hypotenuse at point $M$. Find the distance from point $M$ to the vertex of the right angle. | 2\sqrt{\frac{29}{5}} | Geometry | olympiads |
Let $f \in \mathbb Z[X]$. For an $n \in \mathbb N$, $n \geq 2$, we define $f_n : \mathbb Z / n \mathbb Z \to \mathbb Z / n \mathbb Z$ through $f_n \left( \widehat x \right) = \widehat{f \left( x \right)}$, for all $x \in \mathbb Z$.
(a) Prove that $f_n$ is well defined.
(b) Find all polynomials $f \in \mathbb Z[X... | f(X) = \pm X + b | Number Theory | aops_forum |
For which natural numbers $n$ is the value of the following expression a perfect square?
$$
n^{5}-n^{4}-2 n^{3}+2 n^{2}+n-1
$$ | k^{2}+1 | Algebra | olympiads |
4. Among the following 3 geometric propositions,
(1) Two similar triangles, if their perimeters are equal, then these two triangles are congruent;
(2) Two similar triangles, if two sets of sides are equal, then these two triangles are congruent;
(3) Two similar triangles, the angles (not obtuse) formed by their corresp... | B | Geometry | cn_contest |
## Task 32/79
It is given that $9 \cdot 45=405$. For which products of a one-digit and a two-digit number does it hold that the product can be obtained by inserting a zero between the first and the second digit of the two-digit factor? | 6\cdot18=108,7\cdot15=105,9\cdot45=405 | Number Theory | olympiads |
## Problem Statement
Calculate the definite integral:
$$
\int_{-3}^{0}\left(x^{2}+6 x+9\right) \sin 2 x \, d x
$$ | -\frac{17+\cos6}{4} | Calculus | olympiads |
38. The combat power of Beidou Xing Si and Nan Xizi are both three-digit numbers $\overline{a b c}$. When the Ultra Ring lights up, the two merge to transform into an Ultra Man, and the transformed Ultra Man's combat power rises to a six-digit number $\overline{d e f a b c}$, and $\overline{d e f a b c}=\overline{a b c... | 390625 | Number Theory | olympiads |
Find the number of ways to choose 2005 red, green, and yellow balls such that the number of red balls is even or the number of green balls is odd. | \binom{2007}{2}-\binom{1004}{2} | Combinatorics | olympiads |
4. Calculate: $2017 \times 2016 + 2016 \times 2014 - 2015 \times 2016 - 2015 \times 2017$. | 1 | Algebra | olympiads |
On the plane, there is an angle of $60^{\circ}$. A circle touches one side of this angle, intersects the other side at points $A$ and $B$, and intersects the angle bisector at points $C$ and $D$. $A B = C D = \sqrt{6}$. Find the area of the circle bounded by this circle. | \pi\sqrt{3} | Geometry | olympiads |
5. For $n$ consecutive natural numbers, if each number is written in its standard prime factorization form, and each prime factor appears an odd number of times, such $n$ consecutive natural numbers are called a “consecutive $n$ strange group” (for example, when $n=3$, $22=2^{1} \times 11^{1}$, $23=23^{1}$, $24=2^{3} \... | 7 | Number Theory | cn_contest |
3. In the cells of a $3 \times 3$ square, the numbers $1,2,3, \ldots, 9$ are arranged. It is known that any two consecutive numbers are located in adjacent (by side) cells. Which number can be in the central cell if the sum of the numbers in the corner cells is $18?$ | 7 | Logic and Puzzles | olympiads |
Example 5 What is the minimum degree of the highest term of a polynomial with rational coefficients that has $\sqrt{2}$ and $1-\sqrt[3]{2}$ as roots?
(2013, Joint Autonomous Admission Examination of Peking University and Other Universities) | 5 | Algebra | cn_contest |
4. If non-zero vectors $\boldsymbol{\alpha}, \boldsymbol{\beta}$ satisfy $|\boldsymbol{\alpha}+\boldsymbol{\beta}|=|\boldsymbol{\alpha}-\boldsymbol{\beta}|$, then the angle between $\boldsymbol{\alpha}$ and $\boldsymbol{\beta}$ is | 90 | Algebra | olympiads |
13.111. The train was delayed by $t$ hours. By increasing the speed by $m$ km/h, the driver eliminated the delay on a section of $s$ km. Determine what speed the train should have had on this section if there had been no delay. | \frac{\sqrt{(4+)}-}{2} | Algebra | olympiads |
11. In the sequence $\left\{a_{n}\right\}$, $a_{1}, a_{2}$ are given non-zero integers, $a_{n+2}=\left|a_{n+1}-a_{n}\right|$.
(1) If $a_{16}=4, a_{17}=1$, find $a_{2018}$;
(2) Prove: From $\left\{a_{n}\right\}$, it is always possible to select infinitely many terms to form two different constant subsequences. | 1 | Number Theory | olympiads |
6. [6] Let $\pi$ be a permutation of the numbers from 1 through 2012 . What is the maximum possible number of integers $n$ with $1 \leq n \leq 2011$ such that $\pi(n)$ divides $\pi(n+1)$ ? | 1006 | Number Theory | olympiads |
5. Two players, A and B, are playing a table tennis match, with the agreement: the winner of each game gets 1 point, and the loser gets 0 points; the match stops when one player is 2 points ahead or after 6 games have been played. Suppose the probability of A winning each game is $\frac{3}{4}$, and the probability of B... | \frac{97}{32} | Algebra | cn_contest |
6. There are 100 equally divided points on a circle. The number of obtuse triangles formed by these points as vertices is $\qquad$ . | 117600 | Combinatorics | cn_contest |
7. In a square $A B C D$ with side length 2, a segment $M N$ of length 1 is constrained to have its endpoint $M$ on side $A B$ and its endpoint $N$ on side $B C$. This segment divides the square into a triangle $T$ and a pentagon $P$. What is the maximum value that the ratio of the area of $T$ to that of $P$ can assume... | \frac{1}{15} | Geometry | olympiads |
Solve the equation $3^{x}=2^{x} y+1, x, y \in \mathbf{Z}^{+}$.
(Romania 2005 Selection Exam) | (x,y)=(1,1),(2,2),(4,5) | Number Theory | olympiads |
Austin and Joshua play a game. Austin chooses a random number equal to 1, 2, 3, 4, or 5 . Joshua then chooses randomly from the remaining four numbers. Joshua's first round score is equal to the product of his number and Austin's number. Austin then chooses randomly from the remaining three numbers, and his first round... | \frac{2}{3} | Combinatorics | olympiads |
23. Calculate the sum $1 \cdot x+2 x^{2}+3 x^{3}+\ldots+n x^{n}$. | \frac{nx^{n+1}}{x-1}-\frac{x(x^{n}-1)}{(x-1)^{2}} | Algebra | olympiads |
## Task A-4.6.
Given is the point $A(0,2)$ on the parabola $y^{2}=x+4$. Find all points $B$ different from $A$ on the given parabola for which there exists a point $C$, also on the parabola, such that the angle $\varangle A C B$ is a right angle. | (b^{2}-4,b),whereb\in(-\infty,0]\cup[4,+\infty) | Geometry | olympiads |
2. There are three numbers arranged in sequence: $3, 9, 8$. For any two adjacent numbers, the difference between the right number and the left number is written between these two numbers, resulting in a new sequence $3, 6, 9, -1, 8$, which is called the first operation; after the second similar operation, a new sequenc... | 520 | Number Theory | cn_contest |
10.215. In a right-angled triangle, the medians of the legs are $\sqrt{52}$ and $\sqrt{73}$. Find the hypotenuse of the triangle. | 10 | Geometry | olympiads |
Let $a_{10} = 10$, and for each integer $n >10$ let $a_n = 100a_{n - 1} + n$. Find the least $n > 10$ such that $a_n$ is a multiple of $99$. | 45 | Number Theory | aops_forum |
Yvon has 4 different notebooks and 5 different pens. He must bring exactly one notebook and exactly one pen to his class. How many different possible combinations of notebooks and pens could he bring?
(A) 9
(B) 16
(C) 20
(D) 10
(E) 5 | 20 | Combinatorics | olympiads |
## Problem Statement
Calculate the volumes of bodies bounded by the surfaces.
$$
\frac{x^{2}}{4}+\frac{y^{2}}{9}-\frac{z^{2}}{36}=-1, z=12
$$ | 48\pi | Calculus | olympiads |
If we write every day of 2014 in the form of an eight-digit number, for example, 20140125 represents January 25, 2014, how many eight-digit numbers have the digits '1', '2', '0' appearing the same number of times? | 43 | Combinatorics | olympiads |
Compute the number of positive integers $n \leq 50$ such that there exist distinct positive integers $a,b$ satisfying
\[
\frac{a}{b} +\frac{b}{a} = n \left(\frac{1}{a} + \frac{1}{b}\right).
\] | 18 | Number Theory | aops_forum |
13. (ROM) Let $P$ be a polynomial of degree $n$ satisfying
$$ P(k)=\binom{n+1}{k}^{-1} \quad \text { for } k=0,1, \ldots, n $$
Determine $P(n+1)$. | P(n+1)= \begin{cases}1, & 2 \mid n ; \\ 0, & 2 \nmid n .\end{cases} | Combinatorics | olympiads_ref |
13. Given $\frac{1}{3} \leqslant a \leqslant 1$. If $f(x)=a x^{2}-2 x+1$ has a maximum value $M(a)$ and a minimum value $N(a)$ on $[1,3]$, let $g(a)=M(a)-N(a)$.
(1) Find the function expression for $g(a)$;
(2) Prove that $g(a) \geqslant \frac{1}{2}$ always holds. | \frac{1}{2} | Algebra | cn_contest |
2. Given $x, y \in\left[-\frac{\pi}{4}, \frac{\pi}{4}\right], a \in R$, and $\left\{\begin{array}{l}x^{3}+\sin x-2 a=0 \\ 4 y^{3}+\sin y \cos y+a=0\end{array}\right.$, then $\cos (x+2 y)=$ | 1 | Algebra | olympiads |
Problem 2. A group of adventurers is showing off their loot. It is known that exactly 13 adventurers have rubies; exactly 9 have emeralds; exactly 15 have sapphires; exactly 6 have diamonds. In addition, it is known that
- if an adventurer has sapphires, then they have either emeralds or diamonds (but not both at the ... | 22 | Combinatorics | olympiads |
There are 12 people such that for every person A and person B there exists a person C that is a friend to both of them. Determine the minimum number of pairs of friends and construct a graph where the edges represent friendships. | 20 | Combinatorics | aops_forum |
3.174. Find $\operatorname{tg} 2 \alpha$, if it is known that $\cos \left(\alpha-90^{\circ}\right)=0.2$ and $90^{\circ}<\alpha<180^{\circ}$. | -\frac{4\sqrt{6}}{23} | Algebra | olympiads |
Example 3.7. Is the function $z=$ $=f(x, y)=\sqrt{4-x^{2}-y^{2}}$ bounded above (below)? | 0\leqslantf(x,y)\leqslant2 | Calculus | olympiads |
14. (17th Japan Mathematical Olympiad) $n$ is a four-digit number with a non-zero tens digit. If the first two digits and the last two digits of $n$ are considered as two two-digit numbers, find all $n$ that satisfy the condition that the product of these two two-digit numbers is a divisor of $n$.
| 1734or1352 | Number Theory | olympiads |
【Example 5】 6 boys and 4 girls are to serve as attendants on 5 buses, with two people per bus. Assuming boys and girls are separated, and the buses are distinguishable, how many ways are there to assign them? | 5400 | Combinatorics | olympiads |
Let $a_1=2021$ and for $n \ge 1$ let $a_{n+1}=\sqrt{4+a_n}$. Then $a_5$ can be written as $$\sqrt{\frac{m+\sqrt{n}}{2}}+\sqrt{\frac{m-\sqrt{n}}{2}},$$ where $m$ and $n$ are positive integers. Find $10m+n$. | 45 | Other | aops_forum |
Consider equation $I: x+y+z=46$ where $x, y$, and $z$ are positive integers, and equation $II: x+y+z+w=46$,
where $x, y, z$, and $w$ are positive integers. Then
$\textbf{(A)}\ \text{I can be solved in consecutive integers} \qquad \\ \textbf{(B)}\ \text{I can be solved in consecutive even integers} \qquad \\ \textbf{(C... | \textbf{(C)} | Number Theory | amc_aime |
165. On a straight line, there are 6 points, and on a parallel line, there are 8 points. How many triangles exist with vertices at these points? | 288 | Combinatorics | olympiads |
Let $\mathbb{Z}^{+}$be the set of all positive integers. Find all functions $f: \mathbb{Z}^{+} \rightarrow \mathbb{Z}^{+}$ satisfying the following conditions for all $x, y \in \mathbb{Z}^{+}$:
$$
\begin{aligned}
f(x, x) & =x, \\
f(x, y) & =f(y, x), \\
(x+y) f(x, y) & =y f(x, x+y) .
\end{aligned}
$$ | f(x,y)=\operatorname{lcm}(x,y) | Algebra | olympiads |
A game with three piles of stones. There are three piles of stones: the first has 10, the second has 15, and the third has 20. On a turn, it is allowed to split any pile into two smaller parts; the player who cannot make a move loses.
# | 42 | Combinatorics | olympiads |
8.1. For all triples $(x, y, z)$ satisfying the system
$$
\left\{\begin{array}{l}
2 \sin x=\operatorname{tg} y \\
2 \cos y=\operatorname{ctg} z \\
\sin z=\operatorname{tg} x
\end{array}\right.
$$
find the smallest value of the expression $\cos x-\sin z$. | -\frac{5\sqrt{3}}{6} | Algebra | olympiads |
## 1. Jaja
Baka Mara has four hens. The first hen lays one egg every day. The second hen lays one egg every other day. The third hen lays one egg every third day. The fourth hen lays one egg every fourth day. If on January 1, 2023, all four hens laid one egg, how many eggs in total will Baka Mara's hens lay throughout... | 762 | Number Theory | olympiads |
G7.1 Let $M=\frac{78^{3}+22^{3}}{78^{2}-78 \times 22+22^{2}}$. Find $M$. | 100 | Algebra | olympiads |
Exercise 3. Let $x, y, z$ be non-zero real numbers such that $x+y+z=0$. Suppose that
$$
\frac{x}{y}+\frac{y}{z}+\frac{z}{x}=\frac{x}{z}+\frac{z}{y}+\frac{y}{x}+1
$$
Determine the value of $\frac{x}{y}+\frac{y}{z}+\frac{z}{x}$. | -1 | Algebra | olympiads |
3. From the eight numbers $-3,-2,-1,0,1,2,3,4$, any three different numbers are taken as the coefficients of the quadratic function $f(x)=a x^{2}+b x+c(a \neq 0)$. If the graph of the quadratic function passes through the origin, and its vertex is in the first quadrant or the third quadrant, how many such quadratic fun... | 24 | Algebra | olympiads |
7 In $\triangle A B C$, $A B=10, A B$ side's height is 3, when $A C \cdot B C$ is the smallest,
$A C+B C=$ $\qquad$ | 4\sqrt{10} | Geometry | olympiads |
If facilities for division are not available, it is sometimes convenient in determining the decimal expansion of $1/a$, $a>0$, to use the iteration $$x_{k+1}=x_k(2-ax_k), \quad \quad k=0,1,2,\dots ,$$ where $x_0$ is a selected “starting” value. Find the limitations, if any, on the starting values $x_0$, in order that t... | 0 < x_0 < \frac{2}{a} | Calculus | aops_forum |
## 73. Man and Dog.
- Walking the dog, - a mathematician friend once told me, - gives me plenty of food for thought. Once, for example, my dog, after waiting for me to go out, looked to see which way I was going to head, and when I started down the path, he raced to the end of it. Then he returned to me, ran to the en... | 16 | Algebra | olympiads |
$13 \cdot 10$ On the coordinate plane, a point with both coordinates as integers is called an integer point. For any natural number $n$, connect the origin $O$ with the point $A_{n}(n, n+3)$. Let $f(n)$ denote the number of integer points on the line segment $O A_{n}$, excluding the endpoints. Find the value of $f(1)+f... | 1326 | Number Theory | olympiads |
288. $2 \sqrt{8}-7 \sqrt{18}+5 \sqrt{72}-\sqrt{50}$. | 8\sqrt{2} | Algebra | olympiads |
5. The solution set of the inequality $f(x)=a x^{2}-x-c>0$ is $\{x \mid-2<x<1\}$, then the graph of the function $y=f(-x)$ (Figure 6-2) is $(\quad)$. | B | Algebra | inequalities |
5. Solve the inequality $\frac{|x-2|-\left|x^{2}-4 x+2\right|}{2 \sqrt{2 x^{2}+7 x+3}-3 x-4} \geq 0$. | x\in[-0.5;0]\cup[1;2)\cup(2;3]\cup[4;+\infty) | Inequalities | olympiads |
3. (2000 National High School Competition Question) Given that $A$ is the left vertex of the hyperbola $x^{2}-y^{2}=1$, and points $B$ and $C$ are on the right branch of the hyperbola, $\triangle A B C$ is an equilateral triangle, then the area of $\triangle A B C$ is | 3\sqrt{3} | Geometry | olympiads |
In the diagram, points $B, C$ and $D$ lie on a line. Also, $\angle A B C=90^{\circ}$ and $\angle A C D=150^{\circ}$. The value of $x$ is
(A) 30
(B) 45
(C) 90
(D) 150
(E) 60
 | 60 | Geometry | olympiads |
10.2. The number $a$ is a root of the quadratic equation $x^{2}-x-50=0$. Find the value of $a^{4}-101 a$. | 2550 | Algebra | olympiads |
20. (12 points) The number of staff and the distribution of male and female staff in two departments, A and B, of a certain unit are shown in Table 1. Now, a stratified sampling method (using simple random sampling without replacement within strata) is adopted to select three staff members from the two departments for ... | \frac{9}{5} | Combinatorics | olympiads |
## Problem Statement
Calculate the limit of the numerical sequence:
$$
\lim _{n \rightarrow \infty} \frac{n \sqrt[6]{n}+\sqrt[3]{32 n^{10}+1}}{(n+\sqrt[4]{n}) \sqrt[3]{n^{3}-1}}
$$ | \infty | Calculus | olympiads |
In [trapezoid](https://artofproblemsolving.com/wiki/index.php/Trapezoid) $ABCD$ with $\overline{BC}\parallel\overline{AD}$, let $BC = 1000$ and $AD = 2008$. Let $\angle A = 37^\circ$, $\angle D = 53^\circ$, and $M$ and $N$ be the [midpoints](https://artofproblemsolving.com/wiki/index.php/Midpoint) of $\overline{BC}$ an... | 504 | Geometry | amc_aime |
Find all functions from $\mathbb{R}_{>0}$ to $\mathbb{R}_{>0}$ such that:
$$
f(x) f(y)=2 f(x+y f(x))
$$ | f(x)=2 | Algebra | olympiads |
If the perimeter of a rectangle is $p$ and its diagonal is $d$, the difference between the length and width of the rectangle is:
$\textbf{(A)}\ \frac {\sqrt {8d^2 - p^2}}{2} \qquad \textbf{(B)}\ \frac {\sqrt {8d^2 + p^2}}{2} \qquad \textbf{(C)}\ \frac{\sqrt{6d^2-p^2}}{2}\qquad\\ \textbf{(D)}\ \frac {\sqrt {6d^2 + p^2}... | \frac{\sqrt{8d^2-p^2}}{2} | Geometry | amc_aime |
505. Solve the system of equations:
$$
\begin{gathered}
x+y+z=9 \\
\frac{1}{x}+\frac{1}{y}+\frac{1}{z}=1 \\
x y+x z+y z=27
\end{gathered}
$$
Problem from "Mathesis". | x=y=z=3 | Algebra | olympiads |
The squares of two positive integers differ by 2016. Find the maximum possible sum of the two integers.
[i]Proposed by Clive Chan | 1008 | Number Theory | aops_forum |
Given $a 、 b 、 c \geqslant 0, a+b+c=5$, let $S=2 a+2 a b+a b c$. Find the maximum value of $S$.
Translate the above text into English, please retain the original text's line breaks and format, and output the translation result directly. | 18 | Algebra | olympiads |
## Problem Statement
Based on the definition of the derivative, find $f^{\prime}(0)$:
$$
f(x)=\left\{\begin{array}{c}
x^{2} e^{|x|} \sin \frac{1}{x^{2}}, x \neq 0 \\
0, x=0
\end{array}\right.
$$ | 0 | Calculus | olympiads |
Solve the following equation:
$$
\frac{3}{(x+2)(x-1)}=\frac{1}{x(x-1)^{2}}+\frac{3}{x(x-3)}
$$ | \frac{19}{13} | Algebra | olympiads |
10.319. The diagonals of an isosceles trapezoid are perpendicular to each other, and its area is $a^{2}$. Determine the height of the trapezoid. | a | Geometry | olympiads |
8. Xiao Yu's family consists of five people: Dad, Mom, Brother, Sister, and Xiao Yu. From the 1st to the 5th of this month, there are two chores every day: one person cooks, and another person washes the dishes, and everyone has to cook once and wash the dishes once during these 5 days. One day, they had the following ... | 54132 | Logic and Puzzles | olympiads |
9. Let $A B C$ be a triangle, and let $B C D E, C A F G, A B H I$ be squares that do not overlap the triangle with centers $X, Y, Z$ respectively. Given that $A X=6, B Y=7$, and $C Z=8$, find the area of triangle $X Y Z$. | \frac{21\sqrt{15}}{4} | Geometry | olympiads |
For some positive integer $n$, there exists $n$ different positive integers $a_1, a_2, ..., a_n$ such that
$(1)$ $a_1=1, a_n=2000$
$(2)$ $\forall i\in \mathbb{Z}$ $s.t.$ $2\le i\le n, a_i -a_{i-1}\in \{-3,5\}$
Determine the maximum value of n. | 1996 | Combinatorics | aops_forum |
4. (5 points) The last three digits of the expression $1 \times 1+11 \times 11+111 \times 111+\cdots+111 \cdots 111$ (2010 1’s) $\times 111 \cdots 111$ (2010 1’s) are $\qquad$ . | 690 | Number Theory | olympiads |
Shenelle has some square tiles. Some of the tiles have side length $5\text{ cm}$ while the others have side length $3\text{ cm}$. The total area that can be covered by the tiles is exactly $2014\text{ cm}^2$. Find the least number of tiles that Shenelle can have. | 94 | Number Theory | aops_forum |
13.398 Two friends decided to go hunting. One of them lives 46 km from the hunting base, the other, who has a car, lives 30 km from the base between the base and his friend's house. They set off at the same time, with the car owner driving towards his friend who was walking. Upon meeting, they drove together to the bas... | 60 | Algebra | olympiads |
Let $n$ be a positive integer. Determine the smallest positive integer $k$ such that for any colouring of the cells of a $2n\times k$ table with $n$ colours there are two rows and two columns which intersect in four squares of the same colour. | 2n^2 - n + 1 | Combinatorics | aops_forum |
6.40 Given a quadratic function $y=f(x)$ whose graph has a vertex at $(-1,1)$ and intersects the $y$-axis at $(0,2)$.
(1) Find the expression for this quadratic function;
(2) When $x=8, y=$ ?
(3) For any given $y$ value, can we always find an $x$ value? Why? Explain using a graph. | x^{2}+2x+2 | Algebra | olympiads |
Find all polynomials $P(x)$ with integer coefficients, such that for all positive integers $m, n$, $$m+n \mid P^{(m)}(n)-P^{(n)}(m).$$
[i]Proposed by Navid Safaei, Iran[/i] | P(x) \equiv c | Number Theory | aops_forum |
2. Solve the equation $\sin ^{4}(2025 x)+\cos ^{2019}(2016 x) \cdot \cos ^{2018}(2025 x)=1$.
(12 points) | \frac{\pi}{4050}+\frac{\pin}{2025},n\inZ,\frac{\pik}{9},k\inZ | Algebra | olympiads |
Let $m=999 \ldots 99$ be the number formed by 77 digits all equal to 9 and let $n=777 \ldots 77$ be the number formed by 99 digits all equal to 7. What is the number of digits of $m \cdot n$? | 176 | Number Theory | olympiads |
21.1.1 * Let the arbitrary 3 diagonals of a convex $n$-sided polygon not intersect at the same point inside the polygon. Find the number of intersection points of the diagonals inside the polygon. | \mathrm{C}_{n}^{4} | Combinatorics | olympiads |
Example 7. $y=\ln \left(x+\sqrt{x^{2}+1}\right)$. | \frac{1}{\sqrt{x^{2}+1}} | Calculus | olympiads |
3. How many of the integers from $2^{10}$ to $2^{18}$ inclusive are divisible by $2^{9}$ ? | 511 | Number Theory | olympiads |
11.12 Find the domain of the function $y=\sqrt{2^{x}-3^{x}}$.
Translate the text above into English, please retain the original text's line breaks and format, and output the translation result directly. | (-\infty,0] | Algebra | olympiads |
The smallest number in the set $\left\{\frac{1}{2}, \frac{2}{3}, \frac{1}{4}, \frac{5}{6}, \frac{7}{12}\right\}$ is
(A) $\frac{1}{2}$
(B) $\frac{2}{3}$
(C) $\frac{1}{4}$
(D) $\frac{5}{6}$
(E) $\frac{7}{12}$ | \frac{1}{4} | Number Theory | olympiads |
Problem 2. Solve the system $\left\{\begin{array}{l}\sqrt[4]{x \sqrt{x}}-\sqrt[4]{y \sqrt{y}}=7 \\ \sqrt[14]{x \sqrt{x \sqrt{x}}}+\sqrt[14]{y \sqrt{y \sqrt{y}}}=3\end{array}\right.$.
Mathematical Gazette $n$ r. 9/2013 | 256,1 | Algebra | olympiads |
Solve the following equation:
$$
\frac{\log \left(35-x^{3}\right)^{3}}{\log (5-x)}=9
$$ | x_{1}=2,x_{2}=3 | Algebra | olympiads |
8.5. Each digit of the natural number $N$ is strictly greater than the one to its left. What is the sum of the digits of the number $9 N$? | 9 | Number Theory | olympiads |
5. There were 5 times more strawberry bushes on the first bed than on the second. When 22 bushes were transplanted from the first bed to the second, the number of strawberry bushes on each bed became the same. How many bushes were there on each bed? | 11 | Algebra | olympiads |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.