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22,621 | Eight 1 Ft coins, one of which is fake. It's a very good counterfeit, differing from the others only in that it is lighter. Using a balance scale, how can you find the fake 1 Ft coin in 2 weighings? | 2 |
22,648 | 9. Find the number of natural numbers from 1 to 100 that have exactly four natural divisors, at least three of which do not exceed 10. | 8 |
22,658 | In a particular group of people, some always tell the truth, the rest always lie. There are 2016 in the group. One day, the group is sitting in a circle. Each person in the group says, "Both the person on my left and the person on my right are liars."
What is the difference between the largest and smallest number of people who could be telling the truth? | 336 |
22,677 | 3.174. $\sin 2 \alpha+\sin 4 \alpha-\sin 6 \alpha$. | 4 \sin \alpha \sin 2\alpha \sin 3\alpha |
22,714 | Rumcajs teaches Cipísek to write numbers. They started from one and wrote consecutive natural numbers. Cipísek pleaded to stop, and Rumcajs promised that they would stop writing when Cipísek had written a total of 35 zeros. What is the last number Cipísek writes? | 204 |
22,748 | 20. Let $f: \mathbb{Q} \backslash\{0,1\} \rightarrow \mathbb{Q}$ be a function such that
$$
x^{2} f(x)+f\left(\frac{x-1}{x}\right)=2 x^{2}
$$
for all rational numbers $x \neq 0,1$. Here $\mathbb{Q}$ denotes the set of rational numbers. Find the value of $f\left(\frac{1}{2}\right)$.
20. Answer: 1
Substituting $x=\frac{1}{2},-1,2$, we get
$$
\begin{aligned}
\frac{1}{4} f\left(\frac{1}{2}\right)+f(-1) & =\frac{1}{2}, \\
f\left(\frac{1}{2}\right)+f(-1)+f(2) & =2, \\
4 f(2) & =8 .
\end{aligned}
$$
Solving these equations, we get $f\left(\frac{1}{2}\right)=1$. In fact the same method can be used to determine $f$. Letting $x=z, \frac{z-1}{z}, \frac{1}{1-z}$, we get
$$
\begin{array}{l}
z^{2} f(z)+f\left(\frac{z-1}{z}\right)=2 z^{2}, \\
\left(\frac{z-1}{z}\right)^{2} f\left(\frac{z-1}{z}\right)+f\left(\frac{1}{11-z}\right)=2\left(\frac{z-1}{z}\right)^{2}, \\
f(z)+\frac{1}{(1-z)^{2}} f\left(\frac{1}{1-z}\right)=\frac{z}{2(1-z)^{2}} . \\
\end{array}
$$
Using Cramer's rule, we can solve this system of linear equations in the unknowns $f(z), f\left(\frac{z-1}{z}\right), f\left(\frac{1}{1-z}\right)$. We obtain
$$
f(z)=1+\frac{1}{(1-z)^{2}}-\frac{1}{z^{2}} .
$$
Indeed one can easily check that it satisfies the given functional equation. | 1 |
22,771 | From point \( P \) inside triangle \( \triangle ABC \), perpendiculars are drawn to sides \( BC \), \( CA \), and \( AB \) with the feet of the perpendiculars being \( D \), \( E \), and \( F \) respectively. Using \( AF \), \( BF \), \( BD \), \( CD \), \( CE \), \( AE \) as diameters, six semicircles are drawn outward. The areas of these six semicircles are denoted as \( S_{1}, S_{2}, S_{3}, S_{4}, S_{5}, S_{6} \). Given that \( S_{5} - S_{6} = 2 \) and \( S_{1} - S_{2} = 1 \), find \( S_{4} - S_{3} = \). | 3 |
22,805 | Suppose there are 128 ones written on a blackboard. At each step, you can erase any two numbers \(a\) and \(b\) from the blackboard and write \(ab + 1\). After performing this operation 127 times, only one number is left. Let \(A\) be the maximum possible value of this remaining number. Find the last digit of \(A\). | 2 |
22,807 | Let \( w = -\frac{1}{2} + \frac{\sqrt{3}}{2} i \) (where \( i^2 = -1 \)). Determine the number of distinct values that the algebraic expression \( w^m + w^n + w^l \) can take, given that \( m, n, l \) are pairwise distinct integers. | 10 |
22,821 | 6. In order to prevent Cinderella from attending the ball held by the prince, her stepmother and two sisters gave her a problem to solve. Calculate: $(2020-1)+(2019-2)+(2018-3)+\cdots \cdots+(1011-1010)$. Please help Cinderella solve this problem, the answer is $\qquad$ | 1020100 |
22,844 | 5. In isosceles triangle $A B C$ with base $A B$, the angle bisectors $C L$ and $A K$ are drawn. Find $\angle A C B$ of triangle $A B C$, given that $A K = 2 C L$. | 108^\circ |
22,845 | ## Task 3 - 340713
Franziska is looking for a four-digit natural number $z$ that satisfies the following statements (1), (2), and (3):
(1) The units digit of $z$ is 1 greater than the tens digit of $z$.
(2) The hundreds digit of $z$ is twice the tens digit of $z$.
(3) The number $z$ is twice a prime number.
Prove that there is exactly one such number; determine this number! | 9634 |
22,849 | 323. Find the area of the triangle formed by the intersection of a sphere of radius $R$ with a trihedral angle, the dihedral angles of which are equal to $\alpha, \beta$ and $\gamma$, and the vertex coincides with the center of the sphere. | (\alpha + \beta + \gamma - \pi) R^2 |
22,853 | A four digit number is called [i]stutterer[/i] if its first two digits are the same and its last two digits are also the same, e.g. $3311$ and $2222$ are stutterer numbers. Find all stutterer numbers that are square numbers. | 7744 |
22,892 | 3. A set of 55 dominoes contains all possible combinations of two numbers from 0 to 9, including dominoes with the same number on both ends. (In the image, three dominoes are shown: a domino with the numbers 3 and 4, a domino with the numbers 0 and 9, and a domino with the number 3 on both ends.) How many dots are there in total in the entire set of dominoes?
 | 495 |
22,904 | 4. In a convex quadrilateral $A B C D$, the angles at vertices $B, C$, and $D$ are $30^{\circ}, 90^{\circ}$, and $120^{\circ}$ respectively. Find the length of segment $A B$, if $A D=C D=2$. | 6 |
22,956 | [ Decimal number system ] [ Divisibility of numbers. General properties ]
In a 100-digit number 12345678901234...7890, all digits standing at odd positions were erased; in the resulting 50-digit number, all digits standing at odd positions were erased again, and so on. The erasing continued until there was nothing left to erase. What was the last digit to be erased? | 4 |
22,958 | 3. The tennis player and the badminton player got tired of sports tournaments and decided to have a mathematical duel. They wrote down the numbers from 25 to 57 inclusive in a row and agreed to place either “+” or “-” (in any order, but only these signs) between them. If the result of the expression turns out to be even when all the signs between the numbers are placed, the tennis player wins, and if it is odd - the badminton player wins. Who turned out to be the winner and why?
Olympiad for schoolchildren "Hope of Energy". Final stage.
When all the signs between the numbers are placed, the tennis player wins, and if it is odd - the badminton player wins. Who turned out to be the winner and why? | \text{Badminton player} |
22,961 | 11.2. Is the number $4^{2019}+6^{2020}+3^{4040}$ prime? | No |
22,966 | 4. Find the smallest integer $n \geqslant 1$ such that the equation
$$
a^{2}+b^{2}+c^{2}-n d^{2}=0
$$
admits the only integer solution $a=b=c=d=0$.
## Solutions | 7 |
22,975 | When
$$
x_{1}^{x_{1}} x_{2} x_{2} \ldots x_{n}^{x_{n}} \quad \text{and} \quad x_{1} = x_{2} = \cdots = x_{n} = \frac{1}{n},
$$
find the minimum value. | \dfrac{1}{n} |
22,990 | Three people played. $A$ had $10 \mathrm{~K}$, $B$ had $57 \mathrm{~K}$, and $C$ had $29 \mathrm{~K}$. At the end of the game, $B$ had three times as much money as $A$, and $C$ had three times as much money as what $A$ won. How much did $C$ win or lose? | -5 |
23,017 | How many convex 33-sided polygons can be formed by selecting their vertices from the vertices of a given 100-sided polygon, such that the two polygons do not share any common sides? | \dbinom{67}{33} + \dbinom{66}{32} |
23,030 | Prove that if \( a \) and \( b \) are positive numbers, then
$$
\frac{(a-b)^{2}}{2(a+b)} \leq \sqrt{\frac{a^{2}+b^{2}}{2}}-\sqrt{a b} \leq \frac{(a-b)^{2}}{a+b}
$$ | \frac{(a-b)^{2}}{2(a+b)} \leq \sqrt{\frac{a^{2}+b^{2}}{2}}-\sqrt{a b} \leq \frac{(a-b)^{2}}{a+b} |
23,083 | Task 13. Find the minimum value of the function
$$
\psi(x)=\sqrt{15-12 \cos x}+\sqrt{7-4 \sqrt{3} \sin x}
$$
on the interval $[0 ; 0.5 \pi]$. | 4 |
23,086 | For which values of \(a\) and \(b\) do the equations
\[
19 x^{2} + 19 y^{2} + a x + b y + 98 = 0
\]
and
\[
98 x^{2} + 98 y^{2} + a x + b y + 19 = 0
\]
have a common solution? | a^2 + b^2 \geq 13689 |
23,087 | # 4.1. Condition:
In front of the elevator stand people weighing 50, 51, 55, 57, 58, 59, 60, 63, 75, and 140 kg. The elevator's load capacity is 180 kg. What is the minimum number of trips needed to get everyone up? | 4 |
23,105 | Real numbers \(a\), \(b\), and \(c\) and positive number \(\lambda\) make \(f(x) = x^3 + ax^2 + b x + c\) have three real roots \(x_1\), \(x_2\), \(x_3\), such that:
(1) \(x_2 - x_1 = \lambda\);
(2) \(x_3 > \frac{1}{2}(x_1 + x_2)\).
Find the maximum value of \(\frac{2 a^3 + 27 c - 9 a b}{\lambda^3}\). | \dfrac{3\sqrt{3}}{2} |
23,110 | Determine all positive real numbers $x$ and $y$ satisfying the equation
$$
x+y+\frac{1}{x}+\frac{1}{y}+4=2 \cdot(\sqrt{2 x+1}+\sqrt{2 y+1}) .
$$ | 1 + \sqrt{2} |
23,182 | Triangle $ABC$ has $BC=1$ and $AC=2$ . What is the maximum possible value of $\hat{A}$ . | 30^\circ |
23,185 |
Let positive numbers \( x, y, z \) satisfy the following system of equations:
\[
\left\{
\begin{array}{l}
x^{2} + x y + y^{2} = 75 \\
y^{2} + y z + z^{2} = 16 \\
z^{2} + x z + x^{2} = 91
\end{array}
\right.
\]
Find the value of the expression \( xy + yz + xz \). | 40 |
23,201 | Putnam 1998 Problem A1 A cone has circular base radius 1, and vertex a height 3 directly above the center of the circle. A cube has four vertices in the base and four on the sloping sides. What length is a side of the cube? Solution | \dfrac{9\sqrt{2} - 6}{7} |
23,202 | 10. (20 points) Find the function $f: \mathbf{R}_{+} \rightarrow \mathbf{R}_{+}$, such that $f(f(x))=6 x-f(x)$. | 2x |
23,224 | 1. Write the 4 numbers $1,9,8,8$ in a row and perform the following operation: For each pair of adjacent numbers, subtract the left number from the right number, and write the result between the two numbers. This completes one operation. Then, perform the same operation on the 7 numbers that are now in a row, and continue this process for a total of 100 operations. Find the sum of the final row of numbers.
（Problem from the 51st Moscow Mathematical Olympiad） | 726 |
23,228 | E csudaszép emlék födi szent porait Diophantosnak $S$ éveinek számát hirdeti a fölirat.
This beautiful monument covers the sacred ashes of Diophantus, the inscription on the monument proclaims the number of his years $S$.
Éltének hatodát boldog gyermekkorda kapta, tizenkettede szőtt álmokat ifjúkorán.
He spent one-sixth of his life in a joyful childhood, and one-twelfth of his life dreaming in his youth.
Majd hetedét tölté el, mennyekzője mikor lőn; $\mathrm{S}$ az ötödik tavaszon kis fia is született.
Then he spent one-seventh of his life, when he became a groom; $S$ on the fifth spring, his little son was born.
Hajh, de szegény éppen csak még félannyi időt élt, Mint a bús apa, kit ... bánata sírba vive ...
Alas, but the poor child lived only half as long as his grieving father, who ... was carried to the grave by his sorrow ...
Négy évig hordván gyötrelmét a szerető szív. Élete hosszát ím: - látod e bölcs sorokon.
For four years he bore his grief in a loving heart. The length of his life is thus: - see it in these wise lines. | 84 |
23,275 | 5. The solution set of the inequality $x^{2}+\sqrt{2-x} \leqslant 2$ is | \left[ \frac{1 - \sqrt{5}}{2}, 1 \right] |
23,282 | A geometric sequence has its first element as 6, the sum of the first $n$ elements is $\frac{45}{4}$, and the sum of the reciprocals of these elements is $\frac{5}{2}$. Which is this geometric sequence? | 6, 3, \frac{3}{2}, \frac{3}{4} |
23,312 | 206. Распределение орехов. Тетушка Марта купила орехов. Томми она дала один орех и четверть оставшихся, и Бесси получила один орех и четверть оставшихся, Боб тоже получил один орех и четверть оставшихся, и, наконец, Джесси получила один орех и четверть оставшихся. Оказалось, что мальчики получили на 100 орехов больше, чем девочки.
Сколько орехов тетушка Марта оставила себе | 321 |
23,321 | How many positive two-digit integers are factors of $2^{24}-1$ | 12 |
23,332 | 4. In the Cartesian coordinate system $x O y$, points $A, B$ are two moving points on the right branch of the hyperbola $x^{2}-y^{2}=1$. Then the minimum value of $\overrightarrow{O A} \cdot \overrightarrow{O B}$ is
(Wang Huixing provided the problem) | 1 |
23,358 | 3-ча 2. See problem 2 for 9th grade.
3-ча 3. Given 100 numbers \(a_{1}, a_{2}, a_{3}, \ldots, a_{100}\), satisfying the conditions:
\[
\begin{array}{r}
a_{1}-4 a_{2}+3 a_{3} \geqslant 0, \\
a_{2}-4 a_{3}+3 a_{4} \geqslant 0, \\
a_{3}-4 a_{4}+3 a_{5} \geqslant 0, \\
\ldots \\
\ldots \\
\ldots \\
a_{99}-4 a_{100}+3 a_{1} \geqslant 0, \\
a_{100}-4 a_{1}+3 a_{2} \geqslant 0
\end{array}
\]
It is known that \(a_{1}=1\), determine \(a_{2}, a_{3}, \ldots, a_{100}\). | 1 |
23,367 | Given \(\frac{1}{1-x-x^{2}-x^{3}}=\sum_{n=0}^{\infty} a_{n} x^{n}\), if \(a_{n-1}=n^{2}\), find the set of possible values for \(n\). | \{1, 9\} |
23,379 | Let \( l \) and \( m \) be two skew lines. On line \( l \), there are three points \( A \), \( B \), and \( C \) such that \( AB = BC \). From points \( A \), \( B \), and \( C \), perpendiculars \( AD \), \( BE \), and \( CF \) are drawn to line \( m \) with feet \( D \), \( E \), and \( F \) respectively. Given that \( AD = \sqrt{15} \), \( BE = \frac{7}{2} \), and \( CF = \sqrt{10} \), find the distance between lines \( l \) and \( m \). | \sqrt{6} |
23,400 | Find the number of eight-digit numbers for which the product of the digits equals 7000. The answer must be given as an integer. | 5600 |
23,429 | 64. The sum of the digits of the result of the expression $999999999-88888888+7777777-666666+55555-4444+333-22+1$ is $\qquad$ | 45 |
23,432 | Given that point \( \mathrm{G} \) is the centroid of \( \triangle ABC \) and point \( \mathrm{P} \) is an interior point of \( \triangle GBC \) (excluding the boundary), if \( AP = \lambda AB + \mu AC \), then what is the range of values for \(\lambda + \mu\)? | \left( \dfrac{2}{3}, 1 \right) |
23,436 | Given two skew lines \( l \) and \( m \). On \( l \), there are three points \( A, B, \) and \( C \) such that \( AB = BC \). From points \( A, B,\) and \( C \), perpendicular lines \( AD, BE, \) and \( CF \) are dropped to \( m \) with feet \( D, E, \) and \( F \), respectively. It is known that \( AD = \sqrt{15}, BE = \frac{7}{2}, \) and \( CF = \sqrt{10} \). Find the distance between lines \( l \) and \( m \). | \sqrt{6} |
23,438 | 10.256. The lengths of two sides of an acute triangle are $\sqrt{13}$ and $\sqrt{10} \mathrm{~cm}$. Find the length of the third side, knowing that this side is equal to the height drawn to it. | 3 |
23,446 | 7. Let $\theta_{1} 、 \theta_{2}$ be acute angles, and
$$
\frac{\sin ^{2020} \theta_{1}}{\cos ^{2018} \theta_{2}}+\frac{\cos ^{2020} \theta_{1}}{\sin ^{2018} \theta_{2}}=1 \text {. }
$$
Then $\theta_{1}+\theta_{2}=$ $\qquad$ . | \dfrac{\pi}{2} |
23,484 | 1. Determine the smallest natural number that is divisible by 225 and whose decimal representation consists only of the digits 0 and 1. | 11111111100 |
23,502 | Determine all positive real number arrays \(\left(x_{1}, x_{2}, x_{3}, x_{4}, x_{5}\right)\) that satisfy the following inequalities:
$$
\begin{array}{l}
\left(x_{1}^{2}-x_{3} x_{5}\right)\left(x_{2}^{2}-x_{3} x_{5}\right) \leqslant 0, \\
\left(x_{2}^{2}-x_{4} x_{1}\right)\left(x_{3}^{2}-x_{4} x_{1}\right) \leqslant 0, \\
\left(x_{3}^{2}-x_{5} x_{2}\right)\left(x_{4}^{2}-x_{5} x_{2}\right) \leqslant 0, \\
\left(x_{4}^{2}-x_{1} x_{3}\right)\left(x_{5}^{2}-x_{1} x_{3}\right) \leqslant 0, \\
\left(x_{5}^{2}-x_{2} x_{4}\right)\left(x_{1}^{2}-x_{2} x_{4}\right) \leqslant 0
\end{array}
$$ | (x, x, x, x, x) |
23,510 | $C$ is the midpoint of segment $A B$, $P$ is a point on the circle with diameter $A B$, and $Q$ is a point on the circle with diameter $A C$. What is the relationship between the lengths of segments $A P$ and $A Q$ if $P Q$ is perpendicular to $A B$? | AP = \sqrt{2} \, AQ |
23,514 | 3. Given $A(-1,2)$ is a point on the parabola $y=2 x^{2}$, and line $l$ passes through point $A$, and is tangent to the parabola, then the equation of line $l$ is $\qquad$ . | y = -4x - 2 |
23,524 | For each value \( n \in \mathbb{N} \), determine how many solutions the equation \( x^{2} - \left\lfloor x^{2} \right\rfloor = \{x\}^{2} \) has on the interval \([1, n]\). | n^2 - n + 1 |
23,526 | The faces of a 12-sided die are numbered 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, and 12 such that the sum of the numbers on opposite faces is 13. The die is meticulously carved so that it is biased: the probability of obtaining a particular face \( F \) is greater than \( \frac{1}{12} \), the probability of obtaining the face opposite \( F \) is less than \( \frac{1}{12} \) while the probability of obtaining any one of the other ten faces is \( \frac{1}{12} \).
When two such dice are rolled, the probability of obtaining a sum of 13 is \( \frac{29}{384} \).
What is the probability of obtaining face \( F \)? | \dfrac{7}{48} |
23,528 | 19. An ant moves along the edges of a unit cube at a speed of 1 unit per second, starting from any vertex. Assuming that each time the ant reaches a vertex, it turns to continue moving in any direction with equal probability (allowing for backtracking), the expected time for the ant to return to its starting point for the first time is $\qquad$ | 8 |
23,541 | Twelve pencils are sharpened so that all of them have different lengths. Masha wants to arrange the pencils in a box in two rows of 6 pencils each, such that the lengths of the pencils in each row decrease from left to right, and each pencil in the second row lies on a longer pencil. How many ways can she do this? | 132 |
23,544 | 6. Given positive real numbers $x, y$ satisfy
$\left(2 x+\sqrt{4 x^{2}+1}\right)\left(\sqrt{y^{2}+4}-2\right) \geqslant y$, then the minimum value of $x+y$ is $\qquad$. | 2 |
23,548 | 5. Given a sequence $x_{n}$ such that $x_{1}=1, x_{2}=2, x_{n+2}=\left|x_{n+1}\right|-x_{n}$. Find $x_{2015}$. | 0 |
23,565 | Let $a,b,c,d$ be real numbers such that $b-d \ge 5$ and all zeros $x_1, x_2, x_3,$ and $x_4$ of the polynomial $P(x)=x^4+ax^3+bx^2+cx+d$ are real. Find the smallest value the product $(x_1^2+1)(x_2^2+1)(x_3^2+1)(x_4^2+1)$ can take. | 16 |
23,615 | There are 294 distinct cards with numbers \(7, 11, 7^{2}, 11^{2}, \ldots, 7^{147}, 11^{147}\) (each card has exactly one number, and each number appears exactly once). How many ways can two cards be selected so that the product of the numbers on the selected cards is a perfect square? | 15987 |
23,622 | A rectangular table of size \( x \) cm \( \times 80 \) cm is covered with identical sheets of paper of size 5 cm \( \times 8 \) cm. The first sheet is placed in the bottom-left corner, and each subsequent sheet is placed 1 cm higher and 1 cm to the right of the previous one. The last sheet is adjacent to the top-right corner. What is the length \( x \) in centimeters? | 77 |
23,626 | Given the sequence $\left\{a_{n}\right\}$ that satisfies
$$
a_{n-1} = a_{n} + a_{n-2} \quad (n \geqslant 3),
$$
let $S_{n}$ be the sum of the first $n$ terms. If $S_{2018} = 2017$ and $S_{2019} = 2018$, then find $S_{20200}$. | 1010 |
23,627 | Quadrilateral $ABCD$ satisfies $\angle ABC = \angle ACD = 90^{\circ}, AC=20,$ and $CD=30.$ Diagonals $\overline{AC}$ and $\overline{BD}$ intersect at point $E,$ and $AE=5.$ What is the area of quadrilateral $ABCD?$ | 360 |
23,629 | A rectangular piece of cardboard was cut along its diagonal. On one of the obtained pieces, two cuts were made parallel to the two shorter sides, at the midpoints of those sides. In the end, a rectangle with a perimeter of $129 \mathrm{~cm}$ remained. The given drawing indicates the sequence of cuts.
What was the perimeter of the original sheet before the cut? | 258 |
23,644 | A packet of seeds was passed around the table. The first person took 1 seed, the second took 2 seeds, the third took 3 seeds, and so on: each subsequent person took one more seed than the previous one. It is known that on the second round, a total of 100 more seeds were taken than on the first round. How many people were sitting at the table? | 10 |
23,662 | If \(\cos ^{4} \theta + \sin ^{4} \theta + (\cos \theta \cdot \sin \theta)^{4} + \frac{1}{\cos ^{4} \theta + \sin ^{4} \theta} = \frac{41}{16}\), find the value of \(\sin ^{2} \theta\). | \dfrac{1}{2} |
23,668 | Let $A$ and $B$ be two opposite vertices of a cube with side length 1. What is the radius of the sphere centered inside the cube, tangent to the three faces that meet at $A$ and to the three edges that meet at $B$? | 2 - \sqrt{2} |
23,675 | Determine the minimum of the following function defined in the interval $45^{\circ}<x<90^{\circ}$:
$$
y=\tan x+\frac{\tan x}{\sin \left(2 x-90^{\circ}\right)}
$$ | 3\sqrt{3} |
23,694 | In the isosceles triangle \(ABC\) with the sides \(AB = BC\), the angle \(\angle ABC\) is \(80^\circ\). Inside the triangle, a point \(O\) is taken such that \(\angle OAC = 10^\circ\) and \(\angle OCA = 30^\circ\). Find the angle \(\angle AOB\). | 70^\circ |
23,706 | Calculate the surface area of the part of the paraboloid of revolution \( 3y = x^2 + z^2 \) that is located in the first octant and bounded by the plane \( y = 6 \). | \dfrac{39}{4}\pi |
23,716 | Calculate the lengths of the arcs of the curves given by the equations in the rectangular coordinate system.
\[ y = \ln \frac{5}{2 x}, \quad \sqrt{3} \leq x \leq \sqrt{8} \] | 1 + \dfrac{1}{2} \ln \dfrac{3}{2} |
23,732 | Circle $C_0$ has radius $1$ , and the point $A_0$ is a point on the circle. Circle $C_1$ has radius $r<1$ and is internally tangent to $C_0$ at point $A_0$ . Point $A_1$ lies on circle $C_1$ so that $A_1$ is located $90^{\circ}$ counterclockwise from $A_0$ on $C_1$ . Circle $C_2$ has radius $r^2$ and is internally tangent to $C_1$ at point $A_1$ . In this way a sequence of circles $C_1,C_2,C_3,\ldots$ and a sequence of points on the circles $A_1,A_2,A_3,\ldots$ are constructed, where circle $C_n$ has radius $r^n$ and is internally tangent to circle $C_{n-1}$ at point $A_{n-1}$ , and point $A_n$ lies on $C_n$ $90^{\circ}$ counterclockwise from point $A_{n-1}$ , as shown in the figure below. There is one point $B$ inside all of these circles. When $r = \frac{11}{60}$ , the distance from the center $C_0$ to $B$ is $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Find $m+n$
[asy] draw(Circle((0,0),125)); draw(Circle((25,0),100)); draw(Circle((25,20),80)); draw(Circle((9,20),64)); dot((125,0)); label("$A_0$",(125,0),E); dot((25,100)); label("$A_1$",(25,100),SE); dot((-55,20)); label("$A_2$",(-55,20),E); [/asy] | 110 |
23,735 | In triangle $MPQ$, a line parallel to side $MQ$ intersects side $MP$, the median $MM_1$, and side $PQ$ at points $D$, $E$, and $F$ respectively. It is known that $DE = 5$ and $EF = 7$. What is the length of $MQ$? | 17 |
23,739 | Find the sum of all distinct possible values of $x^2-4x+100$ , where $x$ is an integer between 1 and 100, inclusive.
*Proposed by Robin Park* | 328053 |
23,740 | In triangle \( \triangle ABC \), the sides opposite to angles \( A \), \( B \), and \( C \) are \( a \), \( b \), and \( c \) respectively. If the angles \( A \), \( B \), and \( C \) form a geometric progression, and \( b^{2} - a^{2} = ac \), then the radian measure of angle \( B \) is equal to ________. | \dfrac{2\pi}{7} |
23,749 | There are four distinct codes $A, B, C, D$ used by an intelligence station, with one code being used each week. Each week, a code is chosen randomly with equal probability from the three codes that were not used the previous week. Given that code $A$ is used in the first week, what is the probability that code $A$ is also used in the seventh week? (Express your answer as a simplified fraction.) | \dfrac{61}{243} |
23,750 | Square $ABCD$ has sides of length 1. Points $E$ and $F$ are on $\overline{BC}$ and $\overline{CD},$ respectively, so that $\triangle AEF$ is equilateral. A square with vertex $B$ has sides that are parallel to those of $ABCD$ and a vertex on $\overline{AE}.$ The length of a side of this smaller square is $\frac{a-\sqrt{b}}{c},$ where $a, b,$ and $c$ are positive integers and $b$ is not divisible by the square of any prime. Find $a+b+c.$ | 12 |
23,766 | Seven students count from 1 to 1000 as follows:
Alice says all the numbers, except she skips the middle number in each consecutive group of three numbers. That is, Alice says 1, 3, 4, 6, 7, 9, . . ., 997, 999, 1000.
Barbara says all of the numbers that Alice doesn't say, except she also skips the middle number in each consecutive group of three numbers.
Candice says all of the numbers that neither Alice nor Barbara says, except she also skips the middle number in each consecutive group of three numbers.
Debbie, Eliza, and Fatima say all of the numbers that none of the students with the first names beginning before theirs in the alphabet say, except each also skips the middle number in each of her consecutive groups of three numbers.
Finally, George says the only number that no one else says.
What number does George say? | 365 |
23,772 | Let us call a ticket with a number from 000000 to 999999 excellent if the difference between some two neighboring digits of its number is 5. Find the number of excellent tickets. | 409510 |
23,791 | Given that point \( P \) lies on the hyperbola \( \Gamma: \frac{x^{2}}{463^{2}} - \frac{y^{2}}{389^{2}} = 1 \). A line \( l \) passes through point \( P \) and intersects the asymptotes of hyperbola \( \Gamma \) at points \( A \) and \( B \), respectively. If \( P \) is the midpoint of segment \( A B \) and \( O \) is the origin, find the area \( S_{\triangle O A B} = \quad \). | 180107 |
23,806 | Let $A_0=(0,0)$ . Distinct points $A_1,A_2,\dots$ lie on the $x$ -axis, and distinct points $B_1,B_2,\dots$ lie on the graph of $y=\sqrt{x}$ . For every positive integer $n,\ A_{n-1}B_nA_n$ is an equilateral triangle. What is the least $n$ for which the length $A_0A_n\geq100$ | 17 |
23,807 | $a$ , $b$ , $c$ are real. What is the highest value of $a+b+c$ if $a^2+4b^2+9c^2-2a-12b+6c+2=0$ | \dfrac{17}{3} |
23,819 | Given a convex quadrilateral \(ABCD\), \(X\) is the midpoint of the diagonal \(AC\). It is known that \(CD \parallel BX\). Find \(AD\), given that \(BX = 3\), \(BC = 7\), and \(CD = 6\). | 14 |
23,822 | Calculate the lengths of arcs of curves given by equations in polar coordinates.
$$
\rho = 3(1 + \sin \varphi), -\frac{\pi}{6} \leq \varphi \leq 0
$$ | 6(\sqrt{3} - \sqrt{2}) |
23,827 | At a bus stop near Absent-Minded Scientist's house, two bus routes stop: #152 and #251. Both go to the subway station. The interval between bus #152 is exactly 5 minutes, and the interval between bus #251 is exactly 7 minutes. The intervals are strictly observed, but these two routes are not coordinated with each other and their schedules do not depend on each other. At a completely random moment, the Absent-Minded Scientist arrives at the stop and gets on the first bus that arrives, in order to go to the subway. What is the probability that the Scientist will get on bus #251? | \dfrac{5}{14} |
23,832 | Arthur, Bob, and Carla each choose a three-digit number. They each multiply the digits of their own numbers. Arthur gets 64, Bob gets 35, and Carla gets 81. Then, they add corresponding digits of their numbers together. The total of the hundreds place is 24, that of the tens place is 12, and that of the ones place is 6. What is the difference between the largest and smallest of the three original numbers?
*Proposed by Jacob Weiner* | 182 |
23,845 | If the sum of the digits of a natural number \( n \) is subtracted from \( n \), the result is 2016. Find the sum of all such natural numbers \( n \). | 20245 |
23,847 | Find the smallest natural number ending in the digit 2 such that it doubles when this digit is moved to the beginning. | 105263157894736842 |
23,848 | It is known that the variance of each of the given independent random variables does not exceed 4. Determine the number of such variables for which the probability that the deviation of the arithmetic mean of the random variable from the arithmetic mean of their mathematical expectations by no more than 0.25 exceeds 0.99. | 6400 |
23,857 | In a round-robin tournament with 6 teams, each team plays one game against each other team, and each game results in one team winning and one team losing. At the end of the tournament, the teams are ranked by the number of games won. What is the maximum number of teams that could be tied for the most wins at the end of the tournament? | 5 |
23,863 | Find the only value of \( x \) in the open interval \((- \pi / 2, 0)\) that satisfies the equation
$$
\frac{\sqrt{3}}{\sin x} + \frac{1}{\cos x} = 4.
$$ | -\dfrac{4\pi}{9} |
23,885 | Triangle $ABC$ has a right angle at $C$ , and $D$ is the foot of the altitude from $C$ to $AB$ . Points $L, M,$ and $N$ are the midpoints of segments $AD, DC,$ and $CA,$ respectively. If $CL = 7$ and $BM = 12,$ compute $BN^2$ . | 193 |
23,890 | Find the maximum value of the product \(x^{2} y^{2} z^{2} u\) given the condition that \(x, y, z, u \geq 0\) and:
\[ 2x + xy + z + yz u = 1 \] | \dfrac{1}{512} |
23,891 | Gabor wanted to design a maze. He took a piece of grid paper and marked out a large square on it. From then on, and in the following steps, he always followed the lines of the grid, moving from grid point to grid point. Then he drew some lines within the square, totaling 400 units in length. These lines became the walls of the maze. After completing the maze, he noticed that it was possible to reach any unit square from any other unit square in exactly one way, excluding paths that pass through any unit square more than once. What is the side length of the initially drawn large square? | 21 |
23,895 | In \(\triangle ABC\), \(DC = 2BD\), \(\angle ABC = 45^\circ\), and \(\angle ADC = 60^\circ\). Find \(\angle ACB\) in degrees. | 75 |
23,901 | Given the random variables \( X \sim N(1,2) \) and \( Y \sim N(3,4) \), if \( P(X < 0) = P(Y > a) \), find the value of \( a \). | 3 + \sqrt{2} |
23,905 | Find all irreducible positive fractions which increase threefold if both the numerator and the denominator are increased by 12. | \dfrac{2}{9} |
23,907 | A thin diverging lens with an optical power of $D_{p} = -6$ diopters is illuminated by a beam of light with a diameter $d_{1} = 10$ cm. On a screen positioned parallel to the lens, a light spot with a diameter $d_{2} = 20$ cm is observed. After replacing the thin diverging lens with a thin converging lens, the size of the spot on the screen remains unchanged. Determine the optical power $D_{c}$ of the converging lens. | 18 |
23,912 | A positive integer \( A \) divided by \( 3! \) gives a result where the number of factors is \(\frac{1}{3}\) of the original number of factors. What is the smallest such \( A \)? | 12 |
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