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stringlengths 33
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float64 0.06
0.59
|
|---|---|---|---|
In a certain month, there were three Sundays that fell on even dates. What day of the week was the 20th of that month?
|
Thursday
|
olympiads
| 0.140625
|
Find all solutions to the equation \( x^2 - 8[x] + 7 = 0 \).
|
1, \sqrt{33}, \sqrt{41}, 7
|
olympiads
| 0.0625
|
Find all solutions to the equation
\[ m^2 - 2mn - 3n^2 = 5 \]
where \( m \) and \( n \) are integers.
|
(4, 1), (2, -1), (-4, -1), (-2, 1)
|
olympiads
| 0.078125
|
Suppose \(a, b\), and \(c\) are real numbers such that
\[
\begin{aligned}
a^{2} - bc &= 14, \\
b^{2} - ca &= 14, \\
c^{2} - ab &= -3.
\end{aligned}
\]
Compute \(|a + b + c|\).
|
5
|
olympiads
| 0.109375
|
Divide a square into 25 smaller squares, where 24 of these smaller squares are unit squares, and the remaining piece can also be divided into squares with a side length of 1. Find the area of the original square.
|
25
|
olympiads
| 0.359375
|
Ankit, Box, and Clark are taking the tiebreakers for the geometry round, consisting of three problems. Problem $k$ takes each $k$ minutes to solve. If for any given problem there is a $\frac13$ chance for each contestant to solve that problem first, what is the probability that Ankit solves a problem first?
|
\frac{19}{27}
|
aops_forum
| 0.078125
|
If a circle with the origin as its center and a radius of 1 is tangent to the line \(\frac{x}{a}+\frac{y}{b}=1\) (where \(a > 0, b > 0\)), then the minimum value of the product \(a b\) is _____.
|
2
|
olympiads
| 0.078125
|
As shown in the figure, $P$ is a point on the side $AB$ of a regular hexagon $ABCDEF$. Line $PM$ is parallel to $CD$ and intersects $EF$ at $M$. Line $PN$ is parallel to $BC$ and intersects $CD$ at $N$. A red and a blue sprite start simultaneously from point $N$ and walk at a constant speed along the perimeters of the pentagon $NPMED$ and the hexagon $CBFAED$ respectively, completing one full lap and returning to point $N$ at the same time. What is the ratio of the blue sprite's speed to the red sprite's speed?
|
1.2 \text{ times}
|
olympiads
| 0.125
|
What is the smallest natural number that leaves a remainder of 2 when divided by 3, a remainder of 4 when divided by 5, and a remainder of 4 when divided by 7?
|
74
|
olympiads
| 0.203125
|
An isosceles trapezoid is given, in which a circle is inscribed and around which a circle is circumscribed.
The ratio of the height of the trapezoid to the radius of the circumscribed circle is $\sqrt{2 / 3}$. Find the angles of the trapezoid.
|
45^{\circ}, 135^{\circ}
|
olympiads
| 0.0625
|
If \(a, b, c\) satisfy the equations \(a+b+c=1\), \(a^{2}+b^{2}+c^{2}=2\), and \(a^{3}+b^{3}+c^{3}=3\), what is \(a b c\)?
|
\frac{1}{6}
|
olympiads
| 0.109375
|
Let $n$ be a natural number greater than 2. $l$ is a line on a plane. There are $n$ distinct points $P_1$ , $P_2$ , …, $P_n$ on $l$ . Let the product of distances between $P_i$ and the other $n-1$ points be $d_i$ ( $i = 1, 2,$ …, $n$ ). There exists a point $Q$ , which does not lie on $l$ , on the plane. Let the distance from $Q$ to $P_i$ be $C_i$ ( $i = 1, 2,$ …, $n$ ). Find $S_n = \sum_{i = 1}^{n} (-1)^{n-i} \frac{c_i^2}{d_i}$ .
|
S_n = 0
|
aops_forum
| 0.265625
|
Convert the parametric equations \( x = 2 \cos t, y = 3 \sin t \) where \( 0 \leq t \leq 2 \pi \) to an equation with two variables.
|
\frac{x^2}{4} + \frac{y^2}{9} = 1
|
olympiads
| 0.5625
|
In triangle \(ABC\), a point \(D\) is marked on side \(AC\) such that \(BC = CD\). Find \(AD\) if it is known that \(BD = 13\) and angle \(CAB\) is three times smaller than angle \(CBA\).
|
13
|
olympiads
| 0.078125
|
A point \( E \) is taken on the base \( AB \) of an isosceles triangle \( ABC \), and the incircles of triangles \( ACE \) and \( ECB \) touch the segment \( CE \) at points \( M \) and \( N \) respectively. Find the length of the segment \( MN \), given the lengths of segments \( AE \) and \( BE \).
|
MN = \frac{|AE - BE|}{2}
|
olympiads
| 0.078125
|
Find a polynomial of degree 2001 such that \( P(x) + P(1-x) = 1 \) for all \( x \).
|
P(x) = \frac{1}{2} + \left( x - \frac{1}{2} \right)^{2001}
|
olympiads
| 0.1875
|
Given \( a + b = 8 \) and \( a^2 b^2 = 4 \), find the value of \( \frac{a^2 + b^2}{2} - ab \).
|
28 \text{ or } 36
|
olympiads
| 0.515625
|
Some cards each have a pair of numbers written on them. There is just one card for each pair (a, b) with \(1 \leq a < b \leq 2003\). Two players play the following game. Each removes a card in turn and writes the product \(ab\) of its numbers on the blackboard. The first player who causes the greatest common divisor of the numbers on the blackboard to fall to 1 loses. Which player has a winning strategy?
|
The first player has a winning strategy
|
olympiads
| 0.078125
|
A deck of cards contains 52 cards, where each of the suits "diamonds", "clubs", "hearts", and "spades" has 13 cards, marked $2,3, \cdots, 10, J, Q, K, A$. Any two cards of the same suit and adjacent ranks are called "straight flush cards," and $A$ with 2 is also considered a consecutive pair (i.e., $A$ can be used as 1). Determine the number of ways to draw 13 cards from this deck so that each rank appears exactly once and no "straight flush cards" are included.
|
3^{13} - 3
|
olympiads
| 0.125
|
Xiaohu uses 6 equilateral triangles with side lengths of 1 to form shapes on a table without overlapping. Each triangle must share at least one side fully with another triangle. As shown in the image, there are two possible shapes formed. Among all the possible shapes, what is the smallest perimeter?
|
6
|
olympiads
| 0.59375
|
Boys and girls formed a circular arrangement such that the number of children whose right-hand neighbor is of the same gender equals the number of children whose right-hand neighbor is of a different gender. What could be the total number of children in the circular arrangement?
|
4k
|
olympiads
| 0.0625
|
Given positive integers \( x, y, z \) and real numbers \( a, b, c, d \) such that \( x \leqslant y \leqslant z, x^a = y^b = z^c = 70^d \), and \( \frac{1}{a} + \frac{1}{b} + \frac{1}{c} = \frac{1}{d} \), determine the relationship between \( x + y \) and \( z \). Fill in the blank with “ \( > \)”, “ \( < \)” or “ \( = \)”.
|
=
|
olympiads
| 0.09375
|
A football invitational tournament involves 16 cities, with each city sending two teams, Team A and Team B. According to the competition rules, after several days of matches, it is observed that except for Team A from City $A$, the number of matches played by all other teams are different from each other. Find the number of matches played by Team B from City $A$.
|
15
|
olympiads
| 0.09375
|
The sequence of functions \( F_{1}(x), F_{2}(x), \ldots \) satisfies the following conditions:
\[ F_{1}(x)=x, \quad F_{n+1}(x)=\frac{1}{1-F_{n}(x)} . \]
The integer \( C \) is a three-digit cube such that \( F_{C}(C)=C \).
What is the largest possible value of \( C \)?
|
343
|
olympiads
| 0.0625
|
If the student bought 11 pens, he would have 8 rubles left, but for 15 pens he is short of 12 rubles and 24 kopecks. How much money did the student have?
|
63.66
|
olympiads
| 0.265625
|
A $3n$ -table is a table with three rows and $n$ columns containing all the numbers $1, 2, …, 3n$ . Such a table is called *tidy* if the $n$ numbers in the first row appear in ascending order from left to right, and the three numbers in each column appear in ascending order from top to bottom. How many tidy $3n$ -tables exist?
|
\frac{(3n)!}{6^n \cdot n!}
|
aops_forum
| 0.46875
|
Increasing sequence of positive integers \( a_1, a_2, a_3, \ldots \) satisfies \( a_{n+2} = a_n + a_{n+1} \) (for \( n \geq 1 \)). If \( a_7 = 120 \), what is \( a_8 \) equal to?
|
194
|
olympiads
| 0.078125
|
A circle is inscribed in a right angle. The chord connecting the points of tangency is equal to 2. Find the distance from the center of the circle to this chord.
|
1
|
olympiads
| 0.171875
|
Let $\left\{x_{n}\right\}$ be a sequence of positive real numbers. If $x_{1}=\frac{3}{2}$ and $x_{n+1}^{2}-x_{n}^{2}=\frac{1}{(n+2)^{2}}-\frac{1}{n^{2}}$ for all positive integers $n$, find $x_{1}+x_{2}+\cdots+x_{2009}$.
|
\frac{4040099}{2010}
|
olympiads
| 0.125
|
Let $ n$ be a given positive integer. Find the smallest positive integer $ u_n$ such that for any positive integer $ d$ , in any $ u_n$ consecutive odd positive integers, the number of them that can be divided by $ d$ is not smaller than the number of odd integers among $ 1, 3, 5, \ldots, 2n \minus{} 1$ that can be divided by $ d$ .
|
u_n = 2n-1
|
aops_forum
| 0.0625
|
Find the number of positive integers $n$ such that the highest power of $7$ dividing $n!$ is $8$ .
|
7
|
aops_forum
| 0.171875
|
Given a line \( l \) and a point \( M \) on this line. Using a two-sided ruler (a ruler with parallel edges) and without a compass, construct a perpendicular from point \( M \) to line \( l \).
|
PM \perp l
|
olympiads
| 0.0625
|
Shuai Shuai memorized words for 7 days. Starting from the second day, he memorized 1 more word each day than the previous day. The sum of the number of words memorized in the first 4 days is equal to the sum of the number of words memorized in the last 3 days. How many words did Shuai Shuai memorize in total over these 7 days?
|
84
|
olympiads
| 0.53125
|
A natural number \( n \) is called unlucky if it cannot be represented as \( n = x^2 - 1 \) or \( n = y^2 - 1 \) with natural \( x, y > 1 \). Is the number of unlucky numbers finite or infinite?
|
Infinite
|
olympiads
| 0.46875
|
In a village, there are seven people. Some of them are liars (always lie), and the rest are knights (always tell the truth). Each person made a statement about each of the others, saying whether they are a knight or a liar. Out of 42 statements, 24 were "He is a liar." What is the minimum number of knights that can live in the village?
|
3
|
olympiads
| 0.375
|
In a regular triangular pyramid, the side of the base is equal to \(a\), and the angle between the slant height (apothem) and a lateral face is \(\frac{\pi}{4}\). Find the height of the pyramid.
|
\frac{a \sqrt{6}}{6}
|
olympiads
| 0.09375
|
As shown in the figure, in quadrilateral $ABCD$, the diagonals $AC$ and $BD$ are perpendicular. The lengths of the four sides are $AB = 6$, $BC = m$, $CD = 8$, and $DA = n$. Find the value of $m^{2} + n^{2}$.
|
100
|
olympiads
| 0.203125
|
Person A and Person B start from points A and B, respectively, at the same time and walk towards each other. They meet after 4 hours. If both increase their speeds by 3 kilometers per hour, they would meet after 3 hours and 30 minutes. What is the distance between points A and B in kilometers?
|
168
|
olympiads
| 0.40625
|
Let \(\triangle ABC\) be inscribed in the unit circle \(\odot O\), with the center \(O\) located within \(\triangle ABC\). If the projections of point \(O\) onto the sides \(BC\), \(CA\), and \(AB\) are points \(D\), \(E\), and \(F\) respectively, find the maximum value of \(OD + OE + OF\).
|
\frac{3}{2}
|
olympiads
| 0.09375
|
Cement was delivered to the warehouse in bags of $25 \mathrm{~kg}$ and $40 \mathrm{~kg}$. There were twice as many smaller bags as larger ones. The warehouse worker reported the total number of bags delivered but did not specify how many of each type there were. The manager assumed all bags weighed $25 \mathrm{~kg}$. Therefore, he multiplied the reported number of bags by 25 and recorded this as the total weight of the cement delivery. Overnight, thieves stole 60 of the larger bags, and the remaining amount of cement in the warehouse was exactly the same as what the manager had recorded. How many kilograms of cement were left?
|
12000 \text{ kg}
|
olympiads
| 0.296875
|
\( p \) and \( q \) are primes such that the numbers \( p+q \) and \( p+7q \) are both squares. Find the value of \( p \).
|
2
|
olympiads
| 0.171875
|
Find the distance from the point $M_{0}$ to the plane passing through the three points $M_{1}, M_{2}, M_{3}$.
$M_{1}(-3 ; 4 ;-7)$
$M_{2}(1 ; 5 ;-4)$
$M_{3}(-5 ;-2 ; 0)$
$M_{0}(-12 ; 7 ;-1)$
|
\frac{459}{\sqrt{2265}}
|
olympiads
| 0.34375
|
Let the germination rate of rye seeds be $90 \%$. What is the probability that out of 7 sown seeds, 5 will germinate?
|
0.124
|
olympiads
| 0.171875
|
A certain team's current winning percentage is 45%. If the team wins 6 out of the next 8 games, their winning percentage will increase to 50%. How many games has the team won so far?
|
18
|
olympiads
| 0.546875
|
Rosa received a bottle of perfume in the shape of a cylinder with a base radius of $7 \mathrm{~cm}$ and a height of $10 \mathrm{~cm}$. After using the perfume for two weeks, 0.45 liters remain in the bottle. What fraction represents the volume of perfume Rosa has already used?
|
\frac{49 \pi - 45}{49 \pi}
|
olympiads
| 0.265625
|
The graph of the function \( y = x^2 + ax + b \) is shown in the figure. It is known that line \( AB \) is perpendicular to the line \( y = x \).
Find the length of the line segment \( OC \).
|
1
|
olympiads
| 0.109375
|
Calculate \(\left\lfloor \sqrt{2}+\sqrt[3]{\frac{3}{2}}+\sqrt[4]{\frac{4}{3}}+\ldots+\sqrt[2009]{\frac{2009}{2008}} \right\rfloor\), where \(\left\lfloor x \right\rfloor\) denotes the greatest integer less than or equal to \(x\).
|
2008
|
olympiads
| 0.0625
|
Osvaldo bought a cheese in the shape of an equilateral triangle. He wants to divide the cheese equally between himself and his four cousins. Create a diagram showing how he should make this division.
|
Assim, cada pessoa recebe exatamente 5 triângulos menores.
|
olympiads
| 0.203125
|
The shorter side of a rectangle is equal to 1, and the acute angle between the diagonals is $60^\circ$. Find the radius of the circle circumscribed around the rectangle.
|
1
|
olympiads
| 0.09375
|
Find the smallest positive integer \( b \) such that \( 1111_b \) (1111 in base \( b \)) is a perfect square. If no such \( b \) exists, write "No solution".
|
7
|
olympiads
| 0.1875
|
There are \( n \) different positive integers, each one not greater than 2013, with the property that the sum of any three of them is divisible by 39. Find the greatest value of \( n \).
|
52
|
olympiads
| 0.109375
|
Comparison of fractions. Let \( x \) and \( y \) be positive numbers. Which of the fractions is larger:
$$
\frac{x^{2}+y^{2}}{x+y} \quad \text{or} \quad \frac{x^{2}-y^{2}}{x-y} ?
$$
|
The fraction \frac{x^2 - y^2}{x-y} \text{ is greater than } \frac{x^2 + y^2}{x+y}
|
olympiads
| 0.125
|
Either in 24 minutes or in 72 minutes.
|
24 \text{ minutes } \text{or } 72 \text{ minutes}
|
olympiads
| 0.4375
|
The circle touches the sides of an angle at the points \(A\) and \(B\). A point \(M\) is chosen on the circle. The distances from \(M\) to the sides of the angle are 24 and 6. Find the distance from \(M\) to the line \(AB\).
|
12
|
olympiads
| 0.0625
|
Perpendiculars $AP$ and $AK$ are drawn from vertex $A$ to the angle bisectors of the external angles $\angle ABC$ and $\angle ACB$ of triangle $ABC$. Find the length of segment $PK$ if the perimeter of triangle $ABC$ is $p$.
|
\frac{p}{2}
|
olympiads
| 0.078125
|
In a triangle, the lengths of two sides are 6 cm and 3 cm. Find the length of the third side if the half-sum of the heights drawn to these sides equals the third height.
|
4 \text{ cm}
|
olympiads
| 0.125
|
The coordinates of the vertices of a triangle are $A(p, q), B(q, r), C(r, p)$, where $p<q<r$. What are the bounds of the angles at each vertex?
|
45^
\circ < \alpha, \beta < 90^
\circ
|
olympiads
| 0.0625
|
Given \( n \geq 2 \), find the remainder of the Euclidean division of \( X^{n} - X^{n-1} + 1 \) by \( X^{2} - 3X + 2 \).
|
2^{n-1} X + (1 - 2^{n-1})
|
olympiads
| 0.0625
|
Let \( a, b, c \) be the sides opposite to the interior angles \( A, B, C \) of triangle \( \triangle ABC \), and let the area of the triangle be \( S = \frac{1}{2} c^2 \). If \( ab = \sqrt{2} \), then the maximum value of \( a^2 + b^2 + c^2 \) is .
|
4
|
olympiads
| 0.140625
|
Michelle is drawing segments in the plane. She begins from the origin facing up the $y$ -axis and draws a segment of length $1$ . Now, she rotates her direction by $120^\circ$ , with equal probability clockwise or counterclockwise, and draws another segment of length $1$ beginning from the end of the previous segment. She then continues this until she hits an already drawn segment. What is the expected number of segments she has drawn when this happens?
|
5
|
aops_forum
| 0.109375
|
Given prime numbers \( p \) and \( q \) such that \( p^{2} + 3pq + q^{2} \) is a perfect square, what is the maximum possible value of \( p+q \)?
|
10
|
olympiads
| 0.078125
|
Mitya is going to fold a square sheet of paper $ABCD$. He calls the fold beautiful if side $AB$ intersects side $CD$ and the four resulting right triangles are equal. Before this, Vanya randomly chooses a point $F$ on the sheet. Find the probability that Mitya can make a beautiful fold passing through point $F$.
|
\frac{1}{2}
|
olympiads
| 0.09375
|
The third chick received as much porridge as the first two chicks combined. The fourth chick received as much porridge as the second and third chicks combined. The fifth chick received as much porridge as the third and fourth chicks combined. The sixth chick received as much porridge as the fourth and fifth chicks combined. The seventh chick did not receive any porridge because it ran out. It is known that the fifth chick received 10 grams of porridge. How much porridge did the magpie cook?
|
40 grams
|
olympiads
| 0.15625
|
There are two girls: Eva and Margaret. Someone asks me: "Is it true that if you love Eva, then you also love Margaret?" I answer: "If this is true, then I love Eva, and if I love Eva, then it is true."
Which girl can be said with certainty that I love?
|
I love Eva and I love Margaret
|
olympiads
| 0.109375
|
There is a natural 1001-digit number \( A \). The 1001-digit number \( Z \) is the same number \( A \), written in reverse order (for example, for four-digit numbers, these might be 7432 and 2347). It is known that \( A > Z \). For which \( A \) will the quotient \( A / Z \) be the smallest (but strictly greater than 1)?
|
\underbrace{999 \ldots 999}_{499}800...00
|
olympiads
| 0.0625
|
Decompose the polynomial \( y = x^{3} - 2 x^{2} + 3 x + 5 \) into positive integer powers of the binomial \( x-2 \).
|
11 + 7(x - 2) + 4(x-2)^2 + (x-2)^3
|
olympiads
| 0.59375
|
Arrange the fractions $\frac{1}{2}$, $\frac{1}{3}$, $\frac{1}{4}$, $\frac{1}{5}$, $\frac{1}{6}$, $\frac{1}{7}$, and the average of these 6 fractions in ascending order. Determine the position of this average value in the order.
|
5
|
olympiads
| 0.078125
|
A train was traveling from point A to point B at a constant speed. Halfway through the journey, it experienced a breakdown and stopped for 15 minutes. After this, the driver had to increase the train’s speed by 4 times in order to arrive at point B on schedule. How many minutes does the train travel from point A to point B according to the schedule?
|
40
|
olympiads
| 0.0625
|
Find \(\lim_ {x \rightarrow \pi} \frac{\sin^{2} x}{1+\cos^{3} x}\).
|
\frac{2}{3}
|
olympiads
| 0.09375
|
Given that $\left\{a_{n}\right\}$ is a geometric series and $a_{1} a_{2017}=1$. If $f(x)=\frac{2}{1+x^{2}}$, then find the value of $f\left(a_{1}\right)+f\left(a_{2}\right)+f\left(a_{3}\right)+\cdots+f\left(a_{2017}\right)$.
|
2017
|
olympiads
| 0.171875
|
On the side \(AC\) of triangle \(ABC\), points \(E\) and \(K\) are taken such that \(E\) lies between \(A\) and \(K\), and \(AE:EK:KC = 3:5:4\). Median \(AD\) intersects segments \(BE\) and \(BK\) at points \(L\) and \(M\) respectively. Find the ratio of the areas of triangles \(BLM\) and \(ABC\). The answer is \(\frac{1}{5}\).
|
\frac{1}{5}
|
olympiads
| 0.328125
|
In a certain kingdom, every two people differ by their set of teeth. What could be the maximum population of the kingdom if the maximum number of teeth is 32?
|
4294967296
|
olympiads
| 0.484375
|
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
|
olympiads
| 0.15625
|
Two arithmetic progressions are given. The first and fifth terms of the first progression are 7 and -5, respectively. The first term of the second progression is 0, and the last term is $7 / 2$. Find the sum of the terms of the second progression, given that the third terms of both progressions are equal.
|
14
|
olympiads
| 0.59375
|
In a meadow, ladybugs have gathered. If a ladybug has six spots on its back, it always tells the truth. If it has four spots, it always lies. There are no other types of ladybugs in the meadow. The first ladybug said, "Each of us has the same number of spots on our back." The second said, "Together we have 30 spots in total." The third disagreed, saying, "Altogether, we have 26 spots on our backs." "Of these three, exactly one told the truth," declared each of the remaining ladybugs. How many ladybugs were there in total in the meadow?
|
5
|
olympiads
| 0.140625
|
Solve the following system of equations:
$$
\begin{aligned}
a^{3} x+a^{2} y+a z+u & =0 \\
b^{3} x+b^{2} y+b z+u & =0 \\
c^{3} x+c^{2} y+c z+u & =0 \\
d^{3} x+d^{2} y+d z+u & =1
\end{aligned}
$$
|
\begin{aligned}
x &= \frac{1}{(d-a)(d-b)(d-c)} \\
y &= -\frac{a+b+c}{(d-a)(d-b)(d-c)} \\
z &= \frac{a b + b c + c a}{(d-a)(d-b)(d-c)} \\
u &= -\frac{a b c}{(d-a)(d-b)(d-c)}
\end{aligned}
|
olympiads
| 0.0625
|
Solve the equation \(x^{3} - [x] = 3\), where \([x]\) denotes the integer part of the number \(x\).
|
x = \sqrt[3]{4}
|
olympiads
| 0.15625
|
Find all integer values that the expression
$$
\frac{p q + p^{p} + q^{q}}{p + q}
$$
can take, where \( p \) and \( q \) are prime numbers.
|
3
|
olympiads
| 0.140625
|
A "crocodile" is a figure whose move consists of jumping to a cell, which can be reached by shifting one cell vertically or horizontally, and then shifting $N$ cells in the perpendicular direction (with $N=2$, the "crocodile" is a chess knight).
For which values of $N$ can a "crocodile" move from any cell on an infinite chessboard to any other cell?
|
Even N
|
olympiads
| 0.140625
|
Petya wrote all natural numbers from 1 to 16 into the cells of a $4 \times 4$ table such that any two consecutive numbers were in cells adjacent by side. Then he erased some numbers. Select all the pictures that could have resulted.
|
4 \text{ and } 5
|
olympiads
| 0.203125
|
Let $a_1,a_2,a_3,\dots,a_6$ be an arithmetic sequence with common difference $3$ . Suppose that $a_1$ , $a_3$ , and $a_6$ also form a geometric sequence. Compute $a_1$ .
|
12
|
aops_forum
| 0.40625
|
Is there a convex 1000-gon in which all angles are expressed in whole numbers of degrees?
|
Does not exist
|
olympiads
| 0.359375
|
\[
\log _{a^{m}} N^{n}=\frac{n}{m} \log _{a} N \quad (N>0, \, m \neq 0, \, n \text{ are any real numbers}).
\]
|
\log_{a^m} N^n = \frac{n}{m} \log_a N
|
olympiads
| 0.390625
|
Find the area of the triangle (see the diagram) on graph paper. (Each side of a square is 1 unit.)
|
1.5
|
olympiads
| 0.09375
|
In any permutation of the numbers \(1, 2, 3, \ldots, 18\), we can always find a set of 6 consecutive numbers whose sum is at least \(m\). Find the maximum value of the real number \(m\).
|
57
|
olympiads
| 0.0625
|
200 participants arrived at the international table tennis championship. The matches are elimination matches, meaning each match involves two players; the loser is eliminated from the tournament, and the winner remains. Find the maximum possible number of participants who have won at least three matches.
|
66
|
olympiads
| 0.078125
|
For which values of \( m \) does the inequality \(\frac{x^{2}-mx-2}{x^{2}-3x+4} > -1\) hold for all \( x \)?
|
m \in (-7, 1)
|
olympiads
| 0.3125
|
All edges of a polyhedron are equal and touch a sphere. Do its vertices necessarily belong to one sphere?
|
Нет
|
olympiads
| 0.296875
|
There are mittens in a bag: right and left. A total of 12 pairs: 10 red and 2 blue. How many mittens do you need to pull out to surely get a pair of mittens of the same color?
|
13
|
olympiads
| 0.0625
|
What is the greatest possible number of rays in space emanating from a single point and forming obtuse angles pairwise?
|
4
|
olympiads
| 0.234375
|
The graphs of the functions \( y = ax^2 \), \( y = bx \), and \( y = c \) intersect at a point located above the x-axis. Determine how many roots the equation \( ax^2 + bx + c = 0 \) can have.
|
0
|
olympiads
| 0.375
|
What is the greatest possible area of a triangle with sides $a, b, c$ bounded by the intervals: $0<a \leq 1, 1 \leq b \leq 2, 2 \leq c \leq 3$?
|
1
|
olympiads
| 0.140625
|
A mother is walking with a stroller around a lake and completes one lap in 12 minutes. Vanya rides a scooter on the same path in the same direction and overtakes the mother every 12 minutes. After what intervals of time will Vanya meet the mother if he rides at the same speed but in the opposite direction?
|
4
|
olympiads
| 0.078125
|
A secret facility is a rectangle measuring $200 \times 300$ meters. Outside the facility, at each of the four corners, there is a guard. An intruder approached the perimeter of the secret facility, and all the guards ran towards the intruder along the shortest paths on the external perimeter (the intruder stayed in place). Three guards ran a combined total of 850 meters to reach the intruder. How many meters did the fourth guard run to reach the intruder?
|
150
|
olympiads
| 0.1875
|
Find the distance from the point $M_0$ to the plane that passes through the three points $M_1$, $M_2$, and $M_3$.
$M_1(1, 0, 2)$
$M_2(1, 2, -1)$
$M_3(2, -2, 1)$
$M_0(-5, -9, 1)$
|
\sqrt{77}
|
olympiads
| 0.171875
|
Write the number 1997 using 10 threes and arithmetic operations.
|
1997 = 3 \times 333 + 3 \times 333 - 3 \div 3
|
olympiads
| 0.328125
|
Find all positive integer tuples \((x, y, z)\) such that \(z\) is a prime number and \(z^{x} = y^{3} + 1\).
|
\{(1,1,2), (2,2,3)\}
|
olympiads
| 0.078125
|
Given an even function \( y = f(x) \) defined on \(\mathbf{R}\) that satisfies \( f(x) = f(2 - x) \), and for \( x \in [0, 1] \), \( f(x) = x^2 \), determine the expression for \( f(x) \) when \( 2k - 1 \leqslant x \leqslant 2k + 1 \) (\(k \in \mathbf{Z}\)).
|
f(x) = (x - 2k)^2 \text{ for } 2k-1 \leq x \leq 2k+1 \text{ and } k \in \mathbf{Z}
|
olympiads
| 0.296875
|
Sérgio chooses two positive integers \(a\) and \(b\). He writes 4 numbers in his notebook: \(a\), \(a+2\), \(b\), and \(b+2\). Then, all 6 products of two of these numbers are written on the board. Let \(Q\) be the number of perfect squares written on the board. Determine the maximum value of \(Q\).
|
2
|
olympiads
| 0.078125
|
Find all positive integers \( x \) for which \( p(x) = x^2 - 10x - 22 \), where \( p(x) \) denotes the product of the digits of \( x \).
|
12
|
olympiads
| 0.078125
|
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