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stringlengths 33
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|---|---|---|---|
A group of cows and horses are randomly divided into two equal rows. Each animal in one row is directly opposite an animal in the other row. If 75 of the animals are horses and the number of cows opposite cows is 10 more than the number of horses opposite horses, determine the total number of animals in the group.
|
170
|
olympiads
| 0.25
|
In the set \( A = \{1, 2, 3, \cdots, 2011\} \), the number of elements whose last digit is 1 is ______.
|
202
|
olympiads
| 0.421875
|
Aline writes the number 1000000 on the board. Together with Hannah, they alternately replace the number written on the board with either \( n-1 \) or \( \left\lfloor \frac{n+1}{2} \right\rfloor \), starting with Hannah. The player who writes the number 1 wins. Who wins?
|
Hannah
|
olympiads
| 0.359375
|
Two workers, working together, can complete a certain job in 8 hours. The first worker, working alone, can complete the entire job 12 hours faster than the second worker, if he were to work alone. How many hours does it take each of them to complete the job when working separately?
|
24 \text{ and } 12 \text{ hours}
|
olympiads
| 0.359375
|
Find the number which, together with its square, forms the smallest sum.
|
-\frac{1}{2}
|
olympiads
| 0.328125
|
Calculate the limit of the function:
$$\lim _{x \rightarrow-1} \frac{\left(x^{2}+3 x+2\right)^{2}}{x^{3}+2 x^{2}-x-2}$$
|
0
|
olympiads
| 0.5
|
Solve the equation \(5p = q^3 - r^3\), where \(p, q, r\) are prime numbers.
|
q = 7, r = 2, p = 67
|
olympiads
| 0.171875
|
A three-digit number, when a decimal point is added in an appropriate place, becomes a decimal. This decimal is 201.6 less than the original three-digit number. What is the original three-digit number?
|
224
|
olympiads
| 0.15625
|
Martin and Olivier are playing a game. On a row containing $N$ squares, they each take turns placing a token in one of the squares, brown for Martin and orange for Olivier, such that two adjacent squares cannot contain tokens of the same color and each square can contain at most one token. The first player who cannot place a token loses. Martin starts. Who has a winning strategy?
|
ext{Olivier}
|
olympiads
| 0.125
|
200 people stand in a circle. Each of them is either a liar or a conformist. Liars always lie. A conformist standing next to two conformists always tells the truth. A conformist standing next to at least one liar can either tell the truth or lie. 100 of the standing people said: "I am a liar," the other 100 said: "I am a conformist." Find the maximum possible number of conformists among these 200 people.
|
150
|
olympiads
| 0.21875
|
Find the ratio \( m_{1} / m_{2} \) of the two hanging balls, given that the tension forces in the upper and lower threads differ by a factor of two.
|
1
|
olympiads
| 0.078125
|
A prime number $p$ is a **moderate** number if for every $2$ positive integers $k > 1$ and $m$ , there exists k positive integers $n_1, n_2, ..., n_k $ such that \[ n_1^2+n_2^2+ ... +n_k^2=p^{k+m} \]
If $q$ is the smallest **moderate** number, then determine the smallest prime $r$ which is not moderate and $q < r$ .
|
7
|
aops_forum
| 0.0625
|
Solve the equation
$$
x^{4} - 14x^{3} + 71x^{2} - 154x + 120 = 0
$$
given that the sum of two of its roots is 5, and the product of the other two roots is 20.
|
2, 3, 4, 5
|
olympiads
| 0.0625
|
In Moscow, a tennis tournament is being held. Each team consists of 3 players. Every team plays against every other team, and each participant from one team plays exactly one game against each participant from the other team. Due to time constraints, a maximum of 150 games can be played in the tournament. How many teams can participate in the tournament so that all games can be completed?
|
6
|
olympiads
| 0.125
|
In the regular quadrilateral pyramid $P-ABCD$, $G$ is the centroid of $\triangle PBC$. Find the value of $\frac{V_{G-PAD}}{V_{G-PAB}}$.
|
2
|
olympiads
| 0.171875
|
There are 11 empty boxes. In one move, you can place one coin in any 10 of them. Two players take turns. The winner is the one who, after their move, first places the 21st coin in any one of the boxes. Who wins with optimal play?
|
Second player wins
|
olympiads
| 0.125
|
The set of pairs of positive reals (x, y) such that \( x^y = y^x \) form the straight line \( y = x \) and a curve. Find the point at which the curve cuts the line.
|
(e, e)
|
olympiads
| 0.109375
|
One day, Papa Smurf conducted an assessment for 45 Smurfs in the Smurf Village. After the assessment, he found that the average score of the top 25 Smurfs was 93 points, and the average score of the bottom 25 Smurfs was 89 points. By how many points does the total score of the top 20 Smurfs exceed that of the bottom 20 Smurfs?
|
100
|
olympiads
| 0.109375
|
Let $A$ and $B$ be two distinct points on the circle $\Gamma$ , not diametrically opposite. The point $P$ , distinct from $A$ and $B$ , varies on $\Gamma$ . Find the locus of the orthocentre of triangle $ABP$ .
|
\Gamma
|
aops_forum
| 0.078125
|
There are two small piles of candies. Two players take turns; the player on their turn will eat one of the small piles and divide the other pile into two (the two new piles can be equal or unequal in size). If the remaining pile consists of only one piece of candy and cannot be divided further, the player eats this piece and wins. Initially, the two piles contain 33 and 35 candies, respectively. Who will win? Will it be the starting player or the opponent? How should they play to ensure victory?
|
First player wins
|
olympiads
| 0.15625
|
There are $2n$ cards, each numbered from 1 to $n$ (each number appearing exactly on two cards). The cards are placed face down on the table. A set of $n$ cards is called good if each number appears exactly once in the set. Baron Munchausen claims that he can point to 80 sets of $n$ cards, at least one of which is definitely good. For what maximum value of $n$ can the baron's words be true?
|
7
|
olympiads
| 0.1875
|
Two convex polygons have a total of 33 sides and 243 diagonals. Find the number of diagonals in the polygon with the greater number of sides.
|
189
|
aops_forum
| 0.28125
|
On a line, there are blue and red points, with at least 5 red points. It is known that on any interval with endpoints at red points that contains a red point inside, there are at least 4 blue points. Additionally, on any interval with endpoints at blue points that contains 3 blue points inside, there are at least 2 red points. What is the maximum number of blue points that can be on an interval with endpoints at red points, not containing other red points inside?
|
4
|
olympiads
| 0.265625
|
In a circle with a radius of $10 \text{ cm}$, segment $AB$ is a diameter and segment $AC$ is a chord of $12 \text{ cm}$. Determine the distance between points $B$ and $C$.
|
16 \, \text{cm}
|
olympiads
| 0.109375
|
In a school there are $1200$ students. Each student is part of exactly $k$ clubs. For any $23$ students, they are part of a common club. Finally, there is no club to which all students belong. Find the smallest possible value of $k$ .
|
23
|
aops_forum
| 0.203125
|
Find all f:N $\longrightarrow$ N that:
[list]**a)** $f(m)=1 \Longleftrightarrow m=1 $ **b)** $d=gcd(m,n) f(m\cdot n)= \frac{f(m)\cdot f(n)}{f(d)} $ **c)** $ f^{2000}(m)=f(m) $ [/list]
|
f(n) = n
|
aops_forum
| 0.15625
|
Three workers can complete a certain task. The second and third worker together can complete it twice as fast as the first worker; the first and third worker together can complete it three times faster than the second worker. How many times faster can the first and second worker together complete the task compared to the third worker?
|
\frac{7}{5}
|
olympiads
| 0.21875
|
A and B bought the same number of sheets of stationery. A put 1 sheet of stationery into each envelope and had 40 sheets of stationery left after using all the envelopes. B put 3 sheets of stationery into each envelope and had 40 envelopes left after using all the sheets of stationery. How many sheets of stationery did they each buy?
|
120 Letters
|
olympiads
| 0.09375
|
Place eight spheres, each with a radius of 1, in two layers inside a cylinder. Ensure that each sphere is tangent to its four neighboring spheres, as well as tangent to one base and the lateral surface of the cylinder. Determine the height of the cylinder.
|
\sqrt[4]{8} + 2
|
olympiads
| 0.1875
|
If two stagecoaches travel daily from Bratislava to Brașov, and likewise, two stagecoaches travel daily from Brașov to Bratislava, and considering that the journey takes ten days, how many stagecoaches will you encounter on your way when traveling by stagecoach from Bratislava to Brașov?
|
20
|
olympiads
| 0.265625
|
A convex polyhedron \(\mathcal{P}\) has 26 vertices, 60 edges, and 36 faces. 24 faces are triangular and 12 are quadrilateral. A spatial diagonal is a line segment joining two vertices that do not belong to the same face. How many spatial diagonals does \(\mathcal{P}\) have?
|
241
|
olympiads
| 0.390625
|
If the three points (1, a, b), (a, 2, b), and (a, b, 3) are collinear in 3-space, what is the value of \(a + b\)?
|
4
|
olympiads
| 0.359375
|
Let \( X \backslash Y = \{a \mid a \in X, a \notin Y\} \) denote the difference of sets \( X \) and \( Y \). Define the symmetric difference of sets \( A \) and \( B \) as \( A \Delta B = (A \backslash B) \cup (B \backslash A) \). Given two non-empty finite sets \( S \) and \( T \) such that \( |S \Delta T| = 1 \), find the minimum value of \( k = |S| + |T| \).
|
3
|
olympiads
| 0.4375
|
If \( H \) is the orthocenter of triangle \( ABC \), then
\[ H A \cdot h_{a} + H B \cdot h_{b} + H C \cdot h_{c} = \frac{a^{2} + b^{2} + c^{2}}{2} \]
|
\frac{a^2 + b^2 + c^2}{2}
|
olympiads
| 0.484375
|
Moor has $2016$ white rabbit candies. He and his $n$ friends split the candies equally amongst themselves, and they find that they each have an integer number of candies. Given that $n$ is a positive integer (Moor has at least $1$ friend), how many possible values of $n$ exist?
|
35
|
aops_forum
| 0.46875
|
Find the integral \(\int \operatorname{ch}^{3} x \operatorname{sh} x \, dx\).
|
\frac{\cosh^4(x)}{4} + C
|
olympiads
| 0.140625
|
Aerith picks five numbers and for every three of them, takes their product, producing ten products. She tells Bob that the nine smallest positive divisors of sixty are among her products. Can Bob figure out the last product?
|
30
|
olympiads
| 0.0625
|
Let there be positive integers $a, c$ . Positive integer $b$ is a divisor of $ac-1$ . For a positive rational number $r$ which is less than $1$ , define the set $A(r)$ as follows. $$ A(r) = \{m(r-ac)+nab| m, n \in \mathbb{Z} \} $$ Find all rational numbers $r$ which makes the minimum positive rational number in $A(r)$ greater than or equal to $\frac{ab}{a+b}$ .
|
\frac{a}{a + b}
|
aops_forum
| 0.109375
|
Simplify the expression:
$$
\cos a \cdot \cos 2a \cdot \cos 4a \cdot \ldots \cdot \cos 2^{n-1}a
$$
|
rac{ ext{sin}(2^n a)}{2^n ext{sin} a}
|
olympiads
| 0.078125
|
The king called two wise men. He gave the first one 100 blank cards and ordered him to write a positive number on each (the numbers do not have to be different) without showing them to the second wise man. Then, the first wise man can communicate several different numbers to the second wise man, each of which is either written on one of the cards or is the sum of the numbers on some of the cards (without specifying how each number is obtained). The second wise man must determine which 100 numbers are written on the cards. If he cannot do this, both will be beheaded; otherwise, each will lose as many beard hairs as the numbers the first wise man communicated to the second. How can the wise men, without colluding, stay alive and lose the minimum number of hairs?
|
101
|
olympiads
| 0.0625
|
Given $P=\{x \mid |x-a| < 4, a \in \mathbf{R}\}$ and $Q=\{x \mid x^{2} - 4x + 3 < 0\}$, and $x \in P$ being a necessary condition for $x \in Q$, find the range of values for $a$.
|
a \in [-1, 5]
|
olympiads
| 0.09375
|
A lady made 3 round doilies with radii of 2, 3, and 10 inches, respectively. She placed them on a round table so that each doily touches the two others and the edge of the table. What is the radius of the table?
|
R = 15
|
olympiads
| 0.109375
|
Let \( x_{i} (i=1, 2, 3, 4) \) be real numbers such that \( \sum_{i=1}^{4} x_{i} = \sum_{i=1}^{4} x_{i}^{7} = 0 \). Find the value of the following expression:
\[ u = x_{4} \left( x_{4} + x_{1} \right) \left( x_{4} + x_{2} \right) \left( x_{4} + x_{3} \right). \]
|
0
|
olympiads
| 0.25
|
What is the minimum number of weights required to weigh any number of grams from 1 to 100 on a balance scale, if the weights can only be placed on one pan?
|
7
|
olympiads
| 0.515625
|
Given a sequence of positive numbers \(\left\{a_{n}\right\}(n \geqslant 0)\) that satisfies \(a_{n}=\frac{a_{n-1}}{m a_{n-2}}\) for \(n = 2, 3, \dots\), where \(m\) is a real parameter. If \(a_{2009}=\frac{a_{0}}{a_{1}}\), find the value of \(m\).
|
m = 1
|
olympiads
| 0.5
|
Let $k$ and $a$ are positive constants. Denote by $V_1$ the volume of the solid generated by a rotation of the figure enclosed
by the curve $C: y=\frac{x}{x+k}\ (x\geq 0)$ , the line $x=a$ and the $x$ -axis around the $x$ -axis, and denote by $V_2$ that of
the solid by a rotation of the figure enclosed by the curve $C$ , the line $y=\frac{a}{a+k}$ and the $y$ -axis around the $y$ -axis.
Find the ratio $\frac{V_2}{V_1}.$
|
k
|
aops_forum
| 0.078125
|
Find the number of non-negative integer solutions of the following inequality:
\[ x + y + z + u \leq 20. \]
|
10626
|
olympiads
| 0.0625
|
The side of an isosceles triangle is equal to 2, and the angle at the vertex is $120^{\circ}$. Find the diameter of the circumscribed circle.
|
4
|
olympiads
| 0.078125
|
For a positive integer $n$ and any real number $c$ , define $x_k$ recursively by :
\[ x_0=0,x_1=1 \text{ and for }k\ge 0, \;x_{k+2}=\frac{cx_{k+1}-(n-k)x_k}{k+1} \]
Fix $n$ and then take $c$ to be the largest value for which $x_{n+1}=0$ . Find $x_k$ in terms of $n$ and $k,\; 1\le k\le n$ .
|
x_k = \binom{n-1}{k-1}
|
aops_forum
| 0.078125
|
Find all integers \( m \) such that \( m^{2} + 6m + 28 \) is a perfect square.
|
6 \text{ and } -12
|
olympiads
| 0.234375
|
The sides of a triangle are 11, 13, and 12. Find the median drawn to the longest side.
|
\frac{19}{2}
|
olympiads
| 0.53125
|
Let \( \mathbb{R} \) be the set of real numbers and \( k \) a non-negative real number. Find all continuous functions \( f : \mathbb{R} \to \mathbb{R} \) such that \( f(x) = f(x^2 + k) \) for all \( x \).
|
The solution is f(x) = C, where C is a constant real number.
|
olympiads
| 0.421875
|
For which values of \( x \) does the expression \( \cos^2(\pi \cos x) + \sin^2(2 \pi \sqrt{3} \sin x) \) attain its minimum value?
|
x = \pm \frac{\pi}{3} + \pi t, \quad t \in \mathbb{Z}
|
olympiads
| 0.0625
|
A palace has $32$ rooms and $40$ corridors. As shown in the figure below, each room is represented by a dot, and each corridor is represented by a segment connecting the rooms. Put $n$ robots in these rooms, such that there is at most one robot in each room. Each robot is assigned to a corridor connected to its room. And the following condition is satisfied: $\bullet$ Let all robots move along the assigned corridors at the same time. They will arrive at the rooms at another ends of the corridors at the same time. During this process, any two robots will not meet each other. And each robot will arrive at a different room in the end.
Assume that the maximum value of the positive integer $n$ we can take as above is $N$ . How many ways to put $N$ robots in the palace and assign the corridors to them satisfying the above condition?
Remark: Any two robot are not distinguished from each other.
|
\binom{32}{16}
|
aops_forum
| 0.171875
|
Calculate $\sin 18^{\circ}$, using only the first two terms of series (3), and estimate the resulting error.
|
0.3090
|
olympiads
| 0.203125
|
The numbers \( a \) and \( b \) are such that the polynomial \( x^{4} + 3x^{3} + x^{2} + ax + b \) is the square of some other polynomial. Find \( b \).
|
\frac{25}{64}
|
olympiads
| 0.515625
|
Gwen, Eli, and Kat take turns flipping a coin in their respective order. The first one to flip heads wins. What is the probability that Kat will win?
|
\frac{1}{7}
|
aops_forum
| 0.203125
|
Given the circumference \( C \) of a circle, it is straightforward to find its center and determine the radius. Typically, the center is obtained by drawing perpendicular bisectors to two chords, and then the radius is determined. While constructing the perpendicular bisector to a chord requires drawing a pair of arcs, you can limit yourself to three arcs when constructing the perpendicular bisectors of two adjacent chords by using the same arc for both bisectors. Using a compass and a straightedge, this method can be optimized by drawing only two arcs and using the straightedge once. This method is known as "Shale's method".
Take a point \( O \) on the circumference \( C \) as the center of a circle \( D \). Find the points of intersection \( P \) and \( Q \) of circle \( D \) with the circumference \( C \). Draw another circle with center \( Q \) having the same radius, and note the point \( R \) where it intersects circle \( D \) inside \( C \). Let the line \( PR \) intersect circumference \( C \) at point \( L \). Then \( QL \) is the radius of the circle \( C \), as is \( LR \).
|
r = QL
|
olympiads
| 0.234375
|
How many positive integers a less than $100$ such that $4a^2 + 3a + 5$ is divisible by $6$ .
|
32
|
aops_forum
| 0.0625
|
Find the locus of points that are equidistant from the point \( A(4, 1) \) and from the y-axis.
|
(y - 1)^2 = 16(x - 2)
|
olympiads
| 0.28125
|
Simplify the expression
$$
\frac{\sin 11^{\circ} \cos 15^{\circ}+\sin 15^{\circ} \cos 11^{\circ}}{\sin 18^{\circ} \cos 12^{\circ}+\sin 12^{\circ} \cos 18^{\circ}}
$$
|
2 \sin 26^{\circ}
|
olympiads
| 0.578125
|
Given a regular $4n$-gon $A_{1} A_{2} \ldots A_{4n}$ with area $S$, where $n > 1$. Find the area of the quadrilateral $A_{1} A_{n} A_{n+1} A_{n+2}$.
|
\frac{S}{2}
|
olympiads
| 0.09375
|
Three lines intersect at a point \( O \). A point \( M \) is taken outside these lines, and perpendiculars are dropped from it to the lines. Points \(\mathrm{H}_{1}, \mathrm{H}_{2}, \mathrm{H}_{3}\) are the feet of these perpendiculars. Find the ratio of the length \( OM \) to the radius of the circumscribed circle around triangle \(\mathrm{H}_{1} \mathrm{H}_{2} \mathrm{H}_{3}\).
|
2
|
olympiads
| 0.1875
|
How many real numbers are roots of the polynomial \[x^9 - 37x^8 - 2x^7 + 74x^6 + x^4 - 37x^3 - 2x^2 + 74x?\]
|
5
|
aops_forum
| 0.0625
|
Let \( a, b, c \) be the three roots of \( X^3 - 3X + 1 \). Calculate \( a^4 + b^4 + c^4 \).
|
18
|
olympiads
| 0.578125
|
Let \( ABCDEF \) be a regular hexagon. Let \( P \) be the circle inscribed in \( \triangle BDF \). Find the ratio of the area of circle \( P \) to the area of rectangle \( ABDE \).
|
\frac{\pi \sqrt{3}}{12}
|
olympiads
| 0.109375
|
Consider the polynomial with blanks:
\[ T = \_ X^2 + \_ X + \_ \]
Tic and Tac play the following game. In one turn, Tic chooses a real number and Tac places it in one of the 3 blanks. After 3 turns, the game ends. Tic wins if the resulting polynomial has 2 distinct rational roots, and Tac wins otherwise.
Who has a winning strategy?
|
Tic
|
olympiads
| 0.25
|
Seyed has 998 white coins, a red coin, and an unusual coin with one red side and one white side. He can not see the color of the coins instead he has a scanner which checks if all of the coin sides touching the scanner glass are white. Is there any algorithm to find the red coin by using the scanner at most 17 times?
|
f(1000) \leq 17
|
aops_forum
| 0.078125
|
If we define a polygon as a closed broken line (allowing for self-intersections of the line), then is the following definition of a parallelogram correct: "A quadrilateral in which two opposite sides are equal and parallel is a parallelogram"?
|
Нет
|
olympiads
| 0.125
|
Alvita is planning a garden patio to be made from identical square paving stones laid out in a rectangle measuring $x$ stones by $y$ stones. She finds that when she adds a border of width one stone around the patio, the area of the border is equal to the original area of the patio. How many possible values for $x$ are there?
|
4
|
olympiads
| 0.1875
|
There are 2011 white and 2012 black balls in a box. Two balls are randomly drawn. If they are of the same color, they are discarded, and a black ball is placed in the box. If they are of different colors, the black ball is discarded, and the white ball is placed back in the box. The process continues until only one ball remains in the box. What color is the last ball?
|
The last remaining ball is white.
|
olympiads
| 0.25
|
A tree has 10 pounds of apples at dawn. Every afternoon, a bird comes and eats x pounds of apples. Overnight, the amount of food on the tree increases by 10%. What is the maximum value of x such that the bird can sustain itself indefinitely on the tree without the tree running out of food?
|
\frac{10}{11}
|
aops_forum
| 0.140625
|
Convex quadrilateral \(E F G H\) has its vertices \(E, F, G, H\) located on the sides \(A B, B C, C D, D A\) respectively of convex quadrilateral \(A B C D\), such that
\[
\frac{A E}{E B} \cdot \frac{B F}{F C} \cdot \frac{C G}{G D} \cdot \frac{D H}{H A} = 1
\]
Points \(A, B, C, D\) are located on the sides \(H_{1} E_{1}, E_{1} F_{1}, F_{1} G_{1}, G_{1} H_{1}\) respectively of convex quadrilateral \(E_{1} F_{1} G_{1} H_{1}\), with the conditions \(E_{1} F_{1} \parallel E F, F_{1} G_{1} \parallel F G, G_{1} H_{1} \parallel G H, H_{1} E_{1} \parallel H E\). Given \(\frac{E_{1} A}{A H_{1}} = \lambda\), find the value of \(\frac{F_{1} C}{C G_{1}}\).
|
\lambda
|
olympiads
| 0.078125
|
Given a regular pentagon, determine the point for which the sum of distances to the vertices of the pentagon is minimal.
|
The point is the center of the pentagon.
|
olympiads
| 0.09375
|
In the future, each country in the world produces its Olympic athletes via cloning and strict training
programs. Therefore, in the fi nals of the 200 m free, there are two indistinguishable athletes from each
of the four countries. How many ways are there to arrange them into eight lanes?
|
2520
|
aops_forum
| 0.3125
|
Let point $O$ be the origin, and points $A$ and $B$ have coordinates $(a, 0)$ and $(0, a)$ respectively, where $a$ is a positive constant. Point $P$ lies on line segment $AB$, such that $\overrightarrow{AP}=t \cdot \overrightarrow{AB}$ for $0 \leqslant t \leqslant 1$. What is the maximum value of $\overrightarrow{OA} \cdot \overrightarrow{OP}$?
|
a^2
|
olympiads
| 0.53125
|
A square frame is placed on the ground, and a vertical pole is installed at the center of the square. When fabric is stretched over this structure from above, a small tent is formed. If two such frames are placed side by side, and a vertical pole of the same length is placed in the center of each, and fabric is stretched over the entire structure, a larger tent is formed. The small tent uses 4 square meters of fabric. How much fabric is needed for the large tent?
|
8 \text{ square meters}
|
olympiads
| 0.21875
|
How many $n$-digit numbers can be written using the digits 1 and 2, where no two adjacent digits are both 1? Let the count of such $n$-digit numbers be denoted by $f(n)$. Find $f(10)$.
|
144
|
olympiads
| 0.4375
|
The system
$$
\begin{aligned}
& (x - y)\left( x^2 - y^2 \right) = 160 \\
& (x + y)\left( x^2 + y^2 \right) = 580
\end{aligned}
$$
has a pair of solutions $x_{1} = 3, y_{1} = 7$. What other pair of solutions should this system have?
|
(7, 3)
|
olympiads
| 0.109375
|
Petya bought one cake, two muffins, and three bagels; Anya bought three cakes and one bagel; and Kolya bought six muffins. They all paid the same amounts of money. Lena bought two cakes and two bagels. How many muffins could she have bought for the same amount she spent?
|
5 \, \text{cakes}
|
olympiads
| 0.109375
|
The difference between the squares of two numbers is equal to the square of 10. What are these numbers?
|
26 \text{ and } 24
|
olympiads
| 0.109375
|
Let $S$ be a real number. It is known that however we choose several numbers from the interval $(0, 1]$ with sum equal to $S$ , these numbers can be separated into two subsets with the following property: The sum of the numbers in one of the subsets doesn’t exceed 1 and the sum of the numbers in the other subset doesn’t exceed 5.
Find the greatest possible value of $S$ .
|
5.5
|
aops_forum
| 0.09375
|
For what real values of \( p \) will the graph of the parabola \( y = x^2 - 2px + p + 1 \) be on or above that of the line \( y = -12x + 5 \)?
|
5 \leq p \leq 8
|
olympiads
| 0.234375
|
A road 28 kilometers long was divided into three unequal parts. The distance between the midpoints of the outer parts is 16 km. Find the length of the middle part.
|
4 \text{ km}
|
olympiads
| 0.0625
|
Find all rational numbers $k$ such that $0 \le k \le \frac{1}{2}$ and $\cos k \pi$ is rational.
|
k = 0, \frac{1}{2}, \frac{1}{3}
|
aops_forum
| 0.0625
|
Find the number of non-negative integer solutions to the equation \(x_{1}^{2} + x_{2}^{2} + \cdots + x_{2023}^{2} = 2 + x_{1} x_{2} + x_{2} x_{3} + \cdots + x_{2022} x_{2023}\). Express your answer using combinatorics if possible.
|
2 \binom{2024}{4}
|
olympiads
| 0.140625
|
This year, the Second Son of the Gods’ age is 4 times the age of Chen Xiang. Eight years later, the Second Son’s age is 8 years less than three times Chen Xiang's age. How old was Chen Xiang when he saved his mother?
|
16
|
olympiads
| 0.0625
|
Divide the given square into 2 squares, provided that the side length of one of them is given.
|
Solution is verified.
|
olympiads
| 0.0625
|
$A$ says to $B$: "I will tell you how old you are if you answer the following question. Multiply the digit in the tens place of your birth year by 5, add 2 to this product, and multiply the resulting sum by 2. Add the digit in the ones place of your birth year to this new product and tell me the resulting number."
$B$ responds: "43", to which $A$ immediately replies: "You are 16 years old!" How did $A$ know this?
|
16
|
olympiads
| 0.28125
|
Find all odd prime numbers $p$ for which there exists a natural number $g$ for which the sets \[A=\left\{ \left( {{k}^{2}}+1 \right)\,\bmod p|\,k=1,2,\ldots ,\frac{p-1}{2} \right\}\] and \[B=\left\{ {{g}^{k}}\bmod \,p|\,k=1,2,...,\frac{p-1}{2} \right\}\] are equal.
|
p = 3
|
aops_forum
| 0.140625
|
Find the maximum constant \( k \) such that for all real numbers \( a, b, c, d \) in the interval \([0,1]\), the inequality \( a^2 b + b^2 c + c^2 d + d^2 a + 4 \geq k(a^3 + b^3 + c^3 + d^3) \) holds.
|
2
|
olympiads
| 0.078125
|
Given the function
\[ f(x) = 256x^9 - 576x^7 + 432x^5 - 120x^3 + 9x \]
find the range of the function \( f(x) \) over the interval \( x \in [-1, 1] \).
|
[-1, 1]
|
olympiads
| 0.0625
|
A nine-sided polygon is circumscribed around a circle such that all its sides have integer lengths, and the first and third sides are equal to 1. Into what segments does the point of tangency divide the second side?
|
\frac{1}{2}
|
olympiads
| 0.125
|
The real numbers \(a, b, c\) satisfy \(5^{a}=2^{b}=\sqrt{10^{c}}\), and \(ab \neq 0\). Find the value of \(\frac{c}{a}+\frac{c}{b}\).
|
2
|
olympiads
| 0.296875
|
For which values of the parameter \(a\) is the sum of the squares of the roots of the equation \(x^{2}+2ax+2a^{2}+4a+3=0\) the largest? What is this sum? (The roots are considered with their multiplicity.)
|
18 \text{ (при } a = -3)
|
olympiads
| 0.125
|
For each non-empty subset \( A \) of \( S_{n} = \{1, 2, \cdots, n\} \), multiply each element \( k \) in \( A \) by \( (-1)^{k} \) and then sum the results. Find the total sum of all these sums.
|
(-1)^n \left( n + \frac{1-(-1)^n}{2} \right) \cdot 2^{n-2}
|
olympiads
| 0.125
|
Indicate all pairs $(x, y)$ for which the equality $\left(x^{4}+1\right)\left(y^{4}+1\right)=4x^2y^2$ holds.
|
(1, 1), (1, -1), (-1, 1), (-1, -1)
|
olympiads
| 0.109375
|
There are 30 crickets and 30 grasshoppers in a cage. Each time the red-haired magician performs a trick, he transforms 4 grasshoppers into 1 cricket. Each time the green-haired magician performs a trick, he transforms 5 crickets into 2 grasshoppers. After the two magicians have performed a total of 18 tricks, there are only grasshoppers and no crickets left in the cage. How many grasshoppers are there at this point?
|
6
|
olympiads
| 0.0625
|
How many sides in a convex polygon can be equal in length to the longest diagonal?
|
не более двух
|
olympiads
| 0.171875
|
Given a parabola \( C: y^2 = 4x \), with \( F \) as the focus of \( C \). A line \( l \) passes through \( F \) and intersects the parabola \( C \) at two points \( A \) and \( B \). The slope of the line is 1. Determine the angle between the vectors \( \overrightarrow{OA} \) and \( \overrightarrow{OB} \).
|
\pi - \arctan \left(\frac{4\sqrt{2}}{3}\right)
|
olympiads
| 0.078125
|
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