problem stringlengths 37 4.98k | answer stringlengths 1 141 | subject stringclasses 8
values | level stringclasses 7
values |
|---|---|---|---|
$\mathrm{Al}$ and Bert must arrive at a town $22.5 \mathrm{~km}$ away. They have one bicycle between them and must arrive at the same time. Bert sets out riding at $8 \mathrm{~km} / \mathrm{h}$, leaves the bicycle and then walks at $5 \mathrm{~km} / \mathrm{h}$. Al walks at $4 \mathrm{~km} / \mathrm{h}$, reaches the bi... | 75 | Algebra | olympiads |
Three of the roots of the equation $x^4 -px^3 +qx^2 -rx+s = 0$ are $\tan A, \tan B$, and $\tan C$, where $A, B$, and $C$ are angles of a triangle. Determine the fourth root as a function only of $p, q, r$, and $s.$ | \frac{r - p}{q - s - 1} | Algebra | aops_forum |
11.2. In the piggy bank, there are 1000 coins of 1 ruble, 2 rubles, and 5 rubles, with a total value of 2000 rubles. How many coins of each denomination are in the piggy bank, given that the number of 1-ruble coins is a prime number. | 1-ruble-3,2-ruble-996,5-ruble-1 | Number Theory | olympiads |
4. 188 Find all three-digit numbers that satisfy the following condition: the quotient when divided by 11 equals the sum of the squares of its digits. | 550 \text{ and } 803 | Number Theory | inequalities |
8. (15 points) Xiao Zhang set off from location A to location B at exactly 8:00 AM, with a speed of 60 kilometers per hour. At 9:00 AM, Xiao Wang set off from location B to location A. After reaching location B, Xiao Zhang immediately returned along the same route and both of them arrived at location A exactly at 12:00... | 96 | Other | olympiads |
3. (8 points) Four pirates, Jack, Jimmy, Tom, and Sang, share 280 gold coins. Jack says: "I get 11 fewer coins than Jimmy, 15 more coins than Tom, and 20 fewer coins than Sang." So, Sang gets $\qquad$ coins. | 86 | Algebra | olympiads |
5. As shown in Figure 1, in the right triangle $\triangle ABC$, the two legs are $AC=2, BC=3$, and $P$ is a point on the hypotenuse $AB$. The triangle is folded along $CP$ to form a right dihedral angle $A-CP-B$. When $AB=\sqrt{7}$, the value of the dihedral angle $P-AC-B$ is $\qquad$. | \arctan\sqrt{2} | Geometry | olympiads |
[ Arithmetic. Mental calculation, etc.]
For the book, 100 rubles were paid, and it remains to pay as much as it would remain to pay if it had been paid as much as it remains to pay. How much does the book cost? | 200 | Logic and Puzzles | olympiads |
Example 1 Try to find the interval of monotonic increase for the function $y=\log _{0.5}\left(x^{2}+4 x+4\right)$. | (-\infty,-2) | Algebra | olympiads |
3. The triangle $A B C$ is isosceles with $A B=B C$. The point $D$ is a point on $B C$, between $B$ and $C$, so that $A C=A D=B D$.
What is the size of angle $A B C$ ? | 36 | Geometry | olympiads |
16. In $\triangle A B C$
$$
A B=8, B C=7 \text {, }
$$
$C A=6$. Also, as shown in the figure, extend side $B C$ to point $P$, such that $\triangle P A B$ is similar to $\triangle P C A$. The length of $P C$ is
$$
\text { (A) } 7 \text {; (B) } 8 \text {; (C) } 9 \text {; (D) } 10 \text {; (E) } 11 .
$$ | C | Geometry | cn_contest |
20. Given $a, b, c \in \mathbf{Z}_{+}, [a, b]=12, [b, c]$ $=15$. Then the minimum possible value of $[a, c]$ is ( ).
( A) 20
(B) 30
(C) 60
(D) 120
(E) 180 | A | Number Theory | cn_contest |
16. Let $F_{1}$ and $F_{2}$ be the left and right foci, respectively, of the hyperbola
$$
\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1 \quad(a>0, b>0)
$$
$O$ is the origin, point $P$ is on the left branch of the hyperbola, point $M$ is on the right directrix, and it satisfies
$$
F_{1} \boldsymbol{O}=\boldsymbol{P M}, O P... | y= \pm \sqrt{5} x-3 | Geometry | cn_contest |
7.194. $\log _{x} 2-\log _{4} x+\frac{7}{6}=0$. | \frac{1}{\sqrt[3]{4}};8 | Algebra | olympiads |
2. A natural number ends with zero, and the greatest of its divisors, not equal to itself, is a power of a prime number. Find the second-to-last digit of this number. | 1or5 | Number Theory | olympiads |
5. Given a regular tetrahedron $S-ABC$ with the base being an equilateral triangle of side length 1, and the side edges being 2. A section through $AB$ divides the volume of the tetrahedron into two equal parts. Then the cosine of the plane angle of the dihedral angle formed by the section and the base is $\qquad$ | \frac{2\sqrt{15}}{15} | Geometry | olympiads |
1. If $a, b$ are real numbers, satisfying $\frac{1}{a}-\frac{1}{b}=\frac{1}{a+b}$, then the value of $\frac{b}{a}-\frac{a}{b}$ is ( ).
(A) -1
(B) 0
(C) $\frac{1}{2}$
(D) 1 | D | Algebra | cn_contest |
4. As shown in Figure 5, positive real numbers $a_{1}, a_{2}, \cdots, a_{8}$ are marked at the corresponding vertices of a cube, such that the number at each vertex is the average of the three numbers marked at the adjacent vertices. Let
$$
\begin{array}{l}
M=\left(a_{1}+2 a_{2}+3 a_{3}+4 a_{4}\right) . \\
\quad\left(5... | \frac{65}{2} | Algebra | cn_contest |
167. To transport 60 tons of cargo from one place to another, a certain number of trucks were required. Due to the poor condition of the road, each truck had to carry 0.5 tons less than originally planned, which is why 4 additional trucks were required. How many trucks were initially required? | 20 | Algebra | olympiads |
5. Let $a^{b}=\frac{1}{8}$. What is the value of $a^{-3 b}$ ? | 512 | Algebra | olympiads |
Example 5 Determine all polynomials $p(x)$ that satisfy the following conditions:
$$
p\left(x^{2}+1\right)=[p(x)]^{2}+1, p(0)=0 .
$$ | p(x)=x | Algebra | olympiads |
12. (10 points) In a sphere with a radius of 10 cm, there is a cube with integer (cm) edge lengths. What is the maximum edge length of the cube?
untranslated part:
设半径为 10 厘米的球中有一个棱长为整数 (厘米) 的正方体, 则该正方体的棱长最大等于多少？
translated part:
In a sphere with a radius of 10 cm, there is a cube with integer (cm) edge lengths. Wha... | 11 | Geometry | olympiads |
6・180 Determine all functions $f: R \rightarrow R$, where $R$ is the set of real numbers, such that for all $x, y \in R$, we have
$$
f(x-f(y))=f(f(y))+x f(y)+f(x)-1
$$
holds. | f(x)=1-\frac{x^{2}}{2} | Algebra | olympiads |
5. Given $x, y \in \mathbf{R}_{+}$. Then the maximum value of $\frac{x}{2 x+y}+\frac{y}{x+2 y}$ is ( ).
| \frac{2}{3} | Algebra | cn_contest |
A bag contains 100 red and 100 blue balls. We draw balls randomly, one by one, without replacement, until we have drawn all 100 red balls. Determine the expected value of the number of balls left in the bag. | \frac{100}{201} | Combinatorics | olympiads |
(2) Let $n \geqslant 2, a_{1}, a_{2}, \cdots, a_{n}$ be positive real numbers, determine the largest real number $C_{n}$ such that the inequality: $\frac{a_{1}^{2}+a_{2}^{2}+\cdots+a_{n}^{2}}{n} \geqslant\left(\frac{a_{1}+a_{2}+\cdots+a_{n}}{n}\right)^{2}+C_{n}\left(a_{1}-a_{n}\right)^{2}$. (2010 Central European Mathe... | \frac{1}{2 n} | Inequalities | inequalities |
Part of an "n-pointed regular star" is shown. It is a simple closed polygon in which all $2n$ edges are congruent, angles $A_1,A_2,\cdots,A_n$ are congruent, and angles $B_1,B_2,\cdots,B_n$ are congruent. If the acute angle at $A_1$ is $10^\circ$ less than the acute angle at $B_1$, then $n=$
$\text{(A) } 12\quad \text{... | 36 | Geometry | amc_aime |
6. Person A draws five lines on a plane, with no three lines intersecting at the same point. If two lines determine an intersection point in the figure, A can get one piece of candy; if there is a set of parallel lines, A can also get one piece of candy. For example, in Figure 11, there are seven intersection points an... | 1,5,8,10 | Combinatorics | olympiads |
8. In the figure shown in Figure 3, on both sides of square $P$, there are $a$ and $b$ squares to the left and right, and $c$ and $d$ squares above and below, where $a$, $b$, $c$, and $d$ are positive integers, satisfying
$$
(a-b)(c-d)=0 \text {. }
$$
The shape formed by these squares is called a "cross star".
There i... | 13483236 | Combinatorics | cn_contest |
A bag contains exactly 15 marbles of which 3 are red, 5 are blue, and 7 are green. The marbles are chosen at random and removed one at a time from the bag until all of the marbles are removed. One colour of marble is the first to have 0 remaining in the bag. What is the probability that this colour is red?
## PART B
... | \frac{21}{40} | Combinatorics | olympiads |
# 9. Problem 9
Vasya throws three dice (cubes with numbers from 1 to 6 on the faces) and adds the numbers that come up. Additionally, if all three numbers are different, he can throw all three dice again and add the numbers that come up to the already accumulated sum, and continue doing so until at least two of the th... | 23.625 | Combinatorics | olympiads |
## Problem 1
Find the natural numbers that satisfy the relation:
$9 \leq \sqrt{\sqrt{2 n+2013}+\sqrt{2 n+2009}}+\sqrt{\sqrt{2 n+2013}-\sqrt{2 n+2009}}<10$. | n\in{0,1,\ldots,145} | Inequalities | olympiads |
Find all the primes between 1 and 15. | 2,3,5,7,11,13 | Number Theory | olympiads |
(1) (2007 - Guangdong) A bus travels from location A to location B at a constant speed of $60 \mathrm{~km} / \mathrm{h}$ for 1 hour,
stays in location B for half an hour, then travels to location C at a constant speed of $80 \mathrm{~km} / \mathrm{h}$ for 1 hour. Among the following graphs describing the relationship... | {\begin{pmatrix}60,&0\leqslant\leqslant1,\\60,&1<\leqslant1.5,\\80(-1.5} | Algebra | olympiads |
1. Let $O$ be a point inside $\triangle A B C$, and $\overrightarrow{O A}+2 \overrightarrow{O B}$ $+3 \overrightarrow{O C}=\mathbf{0}$, then the ratio of the area of $\triangle A O C$ to the area of $\triangle B O C$ is $\qquad$ | 2:1 | Geometry | olympiads |
8.3. Workers were laying a floor of size $n \times n$ using two types of tiles: $2 \times 2$ and $3 \times 1$. It turned out that they managed to completely cover the floor such that the same number of tiles of each type was used. For which $n$ could this have been possible? (Cutting tiles or overlapping them is not al... | ndivisible7 | Combinatorics | olympiads |
1. A palindrome is a number where the digits read the same forwards or backwards, such as 4774 or 505 . What is the smallest palindrome that is larger than 2015 ? | 2112 | Number Theory | olympiads |
3. For what value of the parameter $a$ does the domain of the function $f(x)=a^{2}+4 a+2-\ln ^{2}(x-a)$ and the set of its values not intersect and together cover the entire number line?
If there are multiple possible answers, list them in any order separated by a semicolon. | -1,-2 | Algebra | olympiads |
55. A person worked for 24 consecutive days and earned 1900 yuan. He works full days from Monday to Friday, with a daily wage of 100 yuan; works half days on Saturday, with a wage of 50 yuan; and does not work on Sunday, with no wage. It is known that he started working from a certain day in late March. Given that Marc... | 18 | Logic and Puzzles | olympiads |
9. Let $[x]$ be the greatest integer not exceeding the real number $x$. Given the sequence $\left\{a_{n}\right\}$ satisfies: $a_{1}=\frac{1}{2}, a_{n+1}=a_{n}^{2}+3 a_{n}+1, \quad n \in N^{*}$, find $\left[\sum_{k=1}^{2017} \frac{a_{k}}{a_{k}+2}\right]$.
| 2015 | Algebra | olympiads |
1. Given the quadratic function
$$
f(x)=x^{2}+b x+8(b \neq 0)
$$
has two distinct real roots $x_{1}, x_{2}$, and $x_{1}+\frac{1}{x_{2}}, x_{2}+\frac{1}{x_{1}}$ are the two zeros of a quadratic function $g(x)$ with a leading coefficient of 1. If $g(1)=f(1)$, find all possible values of $g(1)$. | -8 | Algebra | olympiads |
5. Calculate: $1+3 \frac{1}{6}+5 \frac{1}{12}+7 \frac{1}{20}+9 \frac{1}{30}+11 \frac{1}{42}+13 \frac{1}{56}+15 \frac{1}{72}+17 \frac{1}{90}$ | 81\frac{2}{5} | Algebra | olympiads |
2. Solve in $\mathbb{R}$ the equation:
$$
4^{x} \cdot 9^{\frac{1}{x}}+9^{x} \cdot 4^{\frac{1}{x}}+6^{x+\frac{1}{x}}=108
$$ | 1 | Algebra | olympiads |
1. If $\frac{20122012 \cdots 201215}{n \uparrow}$ is divisible by 15, then the minimum value of $n$ is ( ).
(A) 3
(B) 4
(C) 5
(D) 6 | A | Number Theory | cn_contest |
Folkpor
Five identical balls are moving in one direction in a straight line at some distance from each other, and five other identical balls are moving towards them. The speeds of all the balls are the same. When any two balls collide, they fly apart in opposite directions with the same speed they had before the colli... | 25 | Combinatorics | olympiads |
The number of distinct pairs $(x,y)$ of real numbers satisfying both of the following equations:
\[x=x^2+y^2 \ \ y=2xy\]
is
$\textbf{(A) }0\qquad \textbf{(B) }1\qquad \textbf{(C) }2\qquad \textbf{(D) }3\qquad \textbf{(E) }4$ | 4 | Algebra | amc_aime |
Example 1. Construct a table of differences of various orders for the following values of $x$ and $f(x)$:
\[
\begin{gathered}
x_{0}=-1, x_{1}=-2, x_{2}=1, x_{3}=2, x_{4}=3 \\
y_{0}=0, y_{1}=7, y_{2}=30, y_{3}=-16, y_{4}=-45
\end{gathered}
\] | 171 | Algebra | olympiads |
5. On the day the Ice Palace opened, the children made plans to go there together to escape the heat. In front of the Ice Palace, there was a staircase. Alice took 20 steps in 120 seconds. At the same speed, Alice walked for a total of 180 seconds, completing all the steps. The total number of steps to the Ice Palace i... | 30 | Algebra | olympiads |
Find all functions $f(x)$ defined on the set of rational numbers whose function values are rational numbers, satisfying $f(x-f(y))=$ $f(x) \cdot f(y), x, y \in \mathbf{Q}$. | f(x)=0f(x)=1 | Algebra | olympiads |
1. Let $n$ be a positive integer, then the minimum value of $|n-1|+|n-2|+\cdots \cdots+|n-100|$ is ).
A. 2500
B. 4950
C. 5050
D. 5150 | 2500 | Algebra | olympiads |
9. (6 points) The difference between two numbers, A and B, is 144, and A is 14 less than 3 times B. Therefore, A is $\qquad$ | 223 | Algebra | olympiads |
A cooperative has 5 sites with the following known distances (the distances are understood to be between the entrances of the sites): From Almás, Bárkány is 2 km away, from there Cseresznye is $1650 \mathrm{~m}$ away. From Cseresznye to Dinnye, the distance is 8 and a half km, and from there to Epres, it is 3 and $3 / ... | 3100 | Geometry | olympiads |
In each cell of a table $8\times 8$ lives a knight or a liar. By the tradition, the knights always say the truth and the liars always lie. All the inhabitants of the table say the following statement "The number of liars in my column is (strictly) greater than the number of liars in my row". Determine how many possible... | 255 | Logic and Puzzles | aops_forum |
578. Check the Pythagorean theorem for a right triangle with legs 1 and 2. | 5 | Geometry | olympiads |
Task 3. Write down all four-digit numbers that are written with two fives and two sevens. Calculate the difference between the largest and smallest number of the obtained numbers. | 2178 | Combinatorics | olympiads |
If $ |x| + x + y = 10$ and $x + |y| - y = 12$, find $x + y$.
$ \textbf{(A)}\ -2\qquad\textbf{(B)}\ 2\qquad\textbf{(C)}\ \frac{18}{5}\qquad\textbf{(D)}\ \frac{22}{3}\qquad\textbf{(E)}\ 22 $ | \frac{18}{5} | Logic and Puzzles | aops_forum |
What is the minimum value of $f(x)=\left|x-1\right| + \left|2x-1\right| + \left|3x-1\right| + \cdots + \left|119x - 1 \right|$?
$\textbf{(A)}\ 49 \qquad \textbf{(B)}\ 50 \qquad \textbf{(C)}\ 51 \qquad \textbf{(D)}\ 52 \qquad \textbf{(E)}\ 53$ | 49 | Algebra | amc_aime |
Two integers have a sum of $26$. when two more integers are added to the first two, the sum is $41$. Finally, when two more integers are added to the sum of the previous $4$ integers, the sum is $57$. What is the minimum number of even integers among the $6$ integers?
$\textbf{(A)}\ 1\qquad\textbf{(B)}\ 2\qquad\textbf{... | 1 | Algebra | amc_aime |
21st BMO 1985 Problem 6 Find all non-negative integer solutions to 5 a 7 b + 4 = 3 c . | 5^17^0+4=3^2 | Number Theory | olympiads |
Four, (30 points) Let natural numbers $a, b, c, d$ satisfy $\frac{a}{b}+\frac{c}{d}<1$ and $a+c=20$. Find the maximum value of $\frac{a}{b}+\frac{c}{d}$.
| \frac{1385}{1386} | Inequalities | cn_contest |
19.2.15 ** Let $s$ be the set of all rational numbers $r$ that satisfy the following conditions: $0<r<1$, and $r$ can be expressed as a repeating decimal of the form $0 . a b c a b c a b c \cdots=0 . \dot{a} b \dot{c}$, where the digits $a, b, c$ do not necessarily have to be distinct. Write the elements of $s$ as redu... | 660 | Number Theory | olympiads |
3.191. $\frac{\sin \left(\frac{5}{2} \pi+\frac{\alpha}{2}\right)\left(1+\operatorname{tg}^{2}\left(\frac{3}{4} \alpha-\frac{\pi}{2}\right)\right)}{\cos ^{-2} \frac{\alpha}{4}\left(\operatorname{tg}^{2}\left(\frac{3}{2} \pi-\frac{\alpha}{4}\right)-\operatorname{tg}^{2}\left(\frac{3}{4} \alpha-\frac{7}{2} \pi\right)\righ... | \frac{1}{8} | Algebra | olympiads |
4. A positive integer, if added to 100 and 168 respectively, can result in two perfect squares. This positive integer is $\qquad$ | 156 | Number Theory | cn_contest |
3. The value of the radical $\sqrt{16 \sqrt{8 \sqrt{4}}}$ is $(\quad)$.
(A) 4
(B) $4 \sqrt{2}$
(C) 8
(D) $8 \sqrt{2}$
(E) 16 | C | Algebra | cn_contest |
Adi the Baller is shooting hoops, and makes a shot with probability $p$. He keeps shooting hoops until he misses. The value of $p$ that maximizes the chance that he makes between 35 and 69 (inclusive) buckets can be expressed as $\frac{1}{\sqrt[b]{a}}$ for a prime $a$ and positive integer $b$. Find $a+b$.
Proposed by... | 37 | Calculus | aops_forum |
$32 \cdot 64$ A triangle has integer side lengths and area, one side length is 21, and the perimeter is 48, then the length of the shortest side is
(A) 8.
(B) 10.
(C) 12.
(D) 14.
(E) 16.
(14th American High School Mathematics Examination, 1963) | 10 | Geometry | olympiads |
2. Given a triangle with one angle greater than $120^{\circ}$ and side lengths $m$, $m+1$, and $m+2$. Then the range of the real number $m$ is ( ).
(A) $m>1$
(B) $m>3$
(C) $\frac{3}{2}<m<3$
(D) $1<m<\frac{3}{2}$ | D | Geometry | cn_contest |
6 As shown in the figure, in the unit cube $A B C D-A_{1} B_{1} C_{1} D_{1}$, $E, F, G$ are the midpoints of edges $A A_{1}, C_{1} D_{1}, D_{1} A_{1}$, respectively. Then the distance from point $B_{1}$ to the plane containing $E F G$ is $\qquad$ | \frac{\sqrt{3}}{2} | Geometry | olympiads |
How many $3$-digit positive integers have digits whose product equals $24$?
$\textbf{(A)}\ 12 \qquad \textbf{(B)}\ 15 \qquad \textbf{(C)}\ 18 \qquad \textbf{(D)}\ 21 \qquad \textbf{(E)}\ 24$ | 21 | Combinatorics | amc_aime |
Points $R$, $S$ and $T$ are vertices of an equilateral triangle, and points $X$, $Y$ and $Z$ are midpoints of its sides. How many noncongruent triangles can be
drawn using any three of these six points as vertices?
$\text{(A)}\ 1 \qquad \text{(B)}\ 2 \qquad \text{(C)}\ 3 \qquad \text{(D)}\ 4 \qquad \text{(E)}\ 20$ | 4 | Geometry | amc_aime |
Twenty kilograms of cheese are on sale in a grocery store. Several customers are lined up to buy this cheese. After a while, having sold the demanded portion of cheese to the next customer, the salesgirl calculates the average weight of the portions of cheese already sold and declares the number of customers for whom t... | 10 | Logic and Puzzles | aops_forum |
G9.2 If $x=1.9 \dot{8} \dot{9}$ and $x-1=\frac{K}{99}$, find $K$.
G9.3 The average of $p, q$ and $r$ is 18. The average of $p+1, q-2, r+3$ and $t$ is 19. Find $t$. | 20 | Algebra | olympiads |
Example 6 Find all functions $f$, whose domain is all positive real numbers, and whose values are positive real numbers, satisfying the following conditions: (1) $f[x f(y)]=y f(x)$, for all $x, y \in \mathbf{R}^{+}$; (2) as $x \rightarrow \infty$, $f(x) \rightarrow 0$. | f(x)=\frac{1}{x} | Algebra | olympiads |
Task B-4.3. How many rational terms are there in the expansion of the binomial $(\sqrt{2013}+\sqrt[3]{2013})^{2012}$? | 336 | Algebra | olympiads |
2. (10 points) The little rabbit and the little turtle start from location $A$ to the forest amusement park at the same time. The little rabbit jumps forward 36 meters per minute, and after every 3 minutes of jumping, it plays on the spot. The first time it plays for 0.5 minutes, the second time for 1 minute, the third... | 12 | Algebra | olympiads |
Example 1 Let $a, b$ be positive constants, $x_{1}, x_{2}, \cdots, x_{n}$ be positive real numbers, and $n \geqslant 2$ be a positive integer. Find:
$$
y=\frac{x_{1} x_{2} \cdots x_{n}}{\left(a+x_{1}\right)\left(x_{1}+x_{2}\right) \cdots\left(x_{n-1}+x_{n}\right)\left(x_{n}+b\right)}
$$
the maximum value. | \frac{1}{(\sqrt[n+1]{a}+\sqrt[n+1]{b})^{n+1}} | Inequalities | cn_contest |
Kymbrea's comic book collection currently has $30$ comic books in it, and she is adding to her collection at the rate of $2$ comic books per month. LaShawn's collection currently has $10$ comic books in it, and he is adding to his collection at the rate of $6$ comic books per month. After how many months will LaShawn's... | 25 | Algebra | amc_aime |
$p(n) $ is a product of all digits of n.Calculate:
$ p(1001) + p(1002) + ... + p(2011) $ | 91125 | Number Theory | aops_forum |
5. Determine the number of ways to represent a natural number $n$ as the sum of several (two or more) natural numbers, where the order matters. (For example, for $n=4$, we have the following possibilities: $3+1, 2+2, 1+3, 2+1+1$, $1+2+1, 1+1+2, 1+1+1+1$, i.e., a total of 7 desired ways.)
## Fourth grade - B category | 2^{n-1}-1 | Combinatorics | olympiads |
In the following drawing, $ABCD$ is a square and points $E$ and $F$ are on sides $BC$ and $CD$ such that $AEF$ is a right triangle, $AE=4$ and $EF=3$. What is the area of the square?
 | \frac{256}{17} | Geometry | olympiads |
14. On the $x y$-plane, let $S$ denote the region consisting of all points $(x, y)$ for which
$$
\left|x+\frac{1}{2} y\right| \leq 10, \quad|x| \leq 10 \text { and } \quad|y| \leq 10 .
$$
The largest circle centred at $(0,0)$ that can be fitted in the region $S$ has area $k \pi$. Find the value of $k$. | 80 | Geometry | olympiads |
9. (16 points) Given a positive number $p$ and the parabola $C: y^{2}=2 p x(p>0), A\left(\frac{p}{6}, 0\right)$ is a point on the axis of symmetry of the parabola $C$, $O$ is the vertex of the parabola $C$, and $M$ is any point on the parabola $C$. Find the maximum value of $\frac{|O M|}{|A M|}$. | \frac{3\sqrt{2}}{4} | Geometry | olympiads |
6. How many ordered 5-tuples $(a, b, c, d, e)$ of integers satisfy $10<a<b<c<d<e<20$ ? | 126 | Combinatorics | olympiads |
6. An isosceles obtuse triangle $P Q T$ with base $P T$ is inscribed in a circle $\Omega$. Chords $A B$ and $C D$, parallel to the line $P T$, intersect the side $Q T$ at points $K$ and $L$ respectively, and $Q K=K L=L T$. Find the radius of the circle $\Omega$ and the area of triangle $P Q T$, if $A B=2 \sqrt{66}, C D... | R=12.5,S=108 | Geometry | olympiads |
If $x+3=10$, what is the value of $5 x+15$ ?
(A) 110
(B) 35
(C) 80
(D) 27
(E) 50 | 50 | Algebra | olympiads |
4. Determine all functions $f: \mathbb{N} \rightarrow \mathbb{N}$ such that for every $n \in \mathbb{N}$,
$$
2 n+2001 \leqslant f(f(n))+f(n) \leqslant 2 n+2002
$$
(Romania) | f(n)=n+667 | Number Theory | olympiads |
Let's denote by $s(n)$ the sum of the digits of the number $n$. For example, $s(2345) = 2 + 3 + 4 + 5 = 14$. Observe that:
$40 - s(40) = 36 = 9 \times 4; 500 - s(500) = 495 = 9 \times 55; 2345 - s(2345) = 2331 = 9 \times 259$.
(a) What can we say about the number $n - s(n)$?
(b) Using the previous item, calculate $s... | 5 | Number Theory | olympiads |
The area of parallelogram $ABCD$ is $51\sqrt{55}$ and $\angle{DAC}$ is a right angle. If the side lengths of the parallelogram are integers, what is the perimeter of the parallelogram? | 90 | Geometry | aops_forum |
19. (12 points) In response to the national strategy of "targeted poverty alleviation, industrial poverty alleviation", Harbin City is recruiting volunteers for the voluntary promotion of the "Poverty Alleviation Policy" across the city. From 500 volunteers aged between $[20,45]$, 100 are randomly selected, and their a... | 1.8 | Combinatorics | olympiads |
4. Xiao Pang, Xiao Dingding, Xiao Ya, and Xiao Qiao's four families, a total of 8 parents and 4 children, went to the amusement park together. The amusement park's ticket pricing is: 100 yuan per adult; 50 yuan per child; for 10 people or more, a group ticket is available at 70 yuan per person. They need to spend at le... | 800 | Logic and Puzzles | olympiads |
19th Australian 1998 Problem A1 Find all the real roots of (x + 1998) (x + 1999) (x + 2000) (x + 2001) + 1 = 0. | -\frac{3999}{2}\\frac{1}{2}\sqrt{5} | Algebra | olympiads |
Find all pairs $(x, y)$ of integers which satisfy the equation
$$
(x+y)^{2}\left(x^{2}+y^{2}\right)=2009^{2}
$$ | (40,9),(9,40),(-40,-9),(-9,-40) | Number Theory | olympiads_ref |
Four, (50 points) Find all prime numbers $p$, such that there exist integers $m, n$, satisfying: $p=m^{2}+n^{2}, p \mid m^{3}+n^{3}-4$.
| 2,5,13 | Number Theory | olympiads |
## Task A-3.4.
Inside the triangle $ABC$ there is a point $T$ such that $|AT|=56,|BT|=40,|CT|=$ 35. The feet of the perpendiculars from point $T$ to the sides of triangle $ABC$ are the vertices of an equilateral triangle. Determine the angle $\varangle ABC$. | 60 | Geometry | olympiads |
How many times do we need to lift the pencil if we want to draw the grid of a chessboard on a piece of paper (its perimeter and internal lines), if we are only allowed to traverse each line once?
If we are allowed to traverse some segments more than once, what is the shortest path that allows us to cover the grid of t... | 13 | Logic and Puzzles | olympiads |
The diagonals $AC$ and $BD$ of a convex quadrilateral $ABCD$ with $S_{ABC} = S_{ADC}$ intersect at $E$. The lines through $E$ parallel to $AD$, $DC$, $CB$, $BA$
meet $AB$, $BC$, $CD$, $DA$ at $K$, $L$, $M$, $N$, respectively. Compute the ratio $\frac{S_{KLMN}}{S_{ABC}}$ | 1 | Geometry | aops_forum |
4. Given an isosceles trapezoid \(ABCD (AD \parallel BC, AD > BC)\). A circle \(\Omega\) is inscribed in angle \(BAD\), touches segment \(BC\) at point \(C\), and intersects \(CD\) again at point \(E\) such that \(CE = 7\), \(ED = 9\). Find the radius of the circle \(\Omega\) and the area of trapezoid \(ABCD\). | R=2\sqrt{7},S_{ABCD}=56\sqrt{7} | Geometry | olympiads |
The numbers $1,2,\dots,10$ are written on a board. Every minute, one can select three numbers $a$, $b$, $c$ on the board, erase them, and write $\sqrt{a^2+b^2+c^2}$ in their place. This process continues until no more numbers can be erased. What is the largest possible number that can remain on the board at this point?... | 8\sqrt{6} | Other | aops_forum |
8. Let $I$ be the incenter of $\triangle A B C$, and $3 \overrightarrow{I A}+4 \overrightarrow{I B}+5 \overrightarrow{I C}=\mathbf{0}$. Then the size of angle $C$ is $\qquad$ | 90 | Geometry | olympiads |
6. Variant 1. Grisha thought of such a set of 10 different natural numbers that their arithmetic mean is 16. What is the maximum possible value of the largest of the numbers he thought of? | 115 | Number Theory | olympiads |
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