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10,256 | **p1.** Steven has just learned about polynomials and he is struggling with the following problem: expand $(1-2x)^7$ as $a_0 +a_1x+...+a_7x^7$ . Help Steven solve this problem by telling him what $a_1 +a_2 +...+a_7$ is.**p2.** Each element of the set ${2, 3, 4, ..., 100}$ is colored. A number has the same color as any divisor of it. What is the maximum number of colors?**p3.** Fuchsia is selecting $24$ balls out of $3$ boxes. One box contains blue balls, one red balls and one yellow balls. They each have a hundred balls. It is required that she takes at least one ball from each box and that the numbers of balls selected from each box are distinct. In how many ways can she select the $24$ balls?**p4.** Find the perfect square that can be written in the form $\overline{abcd} - \overline{dcba}$ where $a, b, c, d$ are non zero digits and $b < c$ . $\overline{abcd}$ is the number in base $10$ with digits $a, b, c, d$ written in this order.**p5.** Steven has $100$ boxes labeled from $ 1$ to $100$ . Every box contains at most $10$ balls. The number of balls in boxes labeled with consecutive numbers differ by $ 1$ . The boxes labeled $1,4,7,10,...,100$ have a total of $301$ balls. What is the maximum number of balls Steven can have?**p6.** In acute $\vartriangle ABC$ , $AB=4$ . Let $D$ be the point on $BC$ such that $\angle BAD = \angle CAD$ . Let $AD$ intersect the circumcircle of $\vartriangle ABC$ at $X$ . Let $\Gamma$ be the circle through $D$ and $X$ that is tangent to $AB$ at $P$ . If $AP = 6$ , compute $AC$ .**p7.** Consider a $15\times 15$ square decomposed into unit squares. Consider a coloring of the vertices of the unit squares into two colors, red and blue such that there are $133$ red vertices. Out of these $133$ , two vertices are vertices of the big square and $32$ of them are located on the sides of the big square. The sides of the unit squares are colored into three colors. If both endpoints of a side are colored red then the side is colored red. If both endpoints of a side are colored blue then the side is colored blue. Otherwise the side is colored green. If we have $196$ green sides, how many blue sides do we have?**p8.** Carl has $10$ piles of rocks, each pile with a different number of rocks. He notices that he can redistribute the rocks in any pile to the other $9$ piles to make the other $9$ piles have the same number of rocks. What is the minimum number of rocks in the biggest pile?**p9.** Suppose that Tony picks a random integer between $1$ and $6$ inclusive such that the probability that he picks a number is directly proportional to the the number itself. Danny picks a number between $1$ and $7$ inclusive using the same rule as Tony. What is the probability that Tony’s number is greater than Danny’s number?**p10.** Mike wrote on the board the numbers $1, 2, ..., n$ . At every step, he chooses two of these numbers, deletes them and replaces them with the least prime factor of their sum. He does this until he is left with the number $101$ on the board. What is the minimum value of $n$ for which this is possible?
PS. You should use hide for answers. Collected [here](https://artofproblemsolving.com/community/c5h2760506p24143309). | -2 |
10,260 | A one-meter gas pipe has rusted through in two places. Determine the probability that all three resulting parts can be used as connectors to gas stoves, given that according to regulations, the stove must not be closer than 25 cm to the main gas pipe. | \dfrac{1}{16} |
10,270 | A magician has a set of $16^{2}$ unique cards. Each card has one red side and one blue side; on each card, there is a natural number between 1 and 16 written on both sides. We will call a card a "duplicate" if the numbers on both sides of the card are the same. The magician wants to draw two cards such that at least one of them is a duplicate and no number appears on both drawn cards at the same time. In how many ways can he do this? | 3480 |
10,311 | 10-8-1. There is a magical grid sheet of size $2000 \times 70$, initially all cells are gray. The painter stands on a certain cell and paints it red. Every second, the painter takes two steps: one cell to the left and one cell down, and paints the cell red where he ends up after the two steps. If the painter is in the leftmost column and needs to step left, he teleports to the rightmost cell of the same row; if the painter is in the bottom row and needs to step down, he teleports to the top cell of the same column. After several moves, the painter returns to the cell where he started. How many red cells are on the sheet at this moment?
The image below shows an example of the painter's moves: first the painter is in cell 1, then in cell 2, and so on.
Comment. Let's provide another, equivalent, formulation of this problem. A grid sheet of size $2000 \times 70$ is glued into a torus, as shown in the picture.
The painter walks on the torus "diagonally". After several moves, the painter returns to the cell where he started. How many red cells are on the sheet at this moment? | 14000 |
10,331 | 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 |
10,335 | 3. The number of real roots of the equation $x^{2}|x|-5 x|x|+2 x=0$ is $\qquad$.
The equation $x^{2}|x|-5 x|x|+2 x=0$ has $\qquad$ real roots. | 4 |
10,345 | On a table, there are 10 number cards labeled $0$ to $9$. Three people, A, B, and C, each take three of these cards. They then compute the sum of all different three-digit numbers that can be formed with the three cards they took. The results for A, B, and C are $1554$, $1688$, and $4662$ respectively. What is the remaining card? (Note: $6$ and $9$ cannot be considered as each other.) | 9 |
10,360 | When \( n \) is a positive integer, the function \( f \) satisfies:
$$
\begin{array}{l}
f(n+3)=\frac{f(n)-1}{f(n)+1}, \\
f(1) \neq 0 \text{ and } f(1) \neq \pm 1.
\end{array}
$$
Determine the value of \( f(11) f(2021) \). | -1 |
10,368 | 3. Find the number of natural numbers $k$, not exceeding 242400, such that $k^{2}+2 k$ is divisible by 303. | 3200 |
10,381 | Given distinct natural numbers \( k, l, m, n \), it is known that there exist three natural numbers \( a, b, c \) such that each of the numbers \( k, l, m, n \) is a root of either the equation \( a x^{2} - b x + c = 0 \) or the equation \( c x^{2} - 16 b x + 256 a = 0 \). Find \( k^{2} + l^{2} + m^{2} + n^{2} \). | 325 |
10,431 | Inside the triangle \(ABC\), a point \(M\) is taken such that \(\angle MBA = 30^\circ\) and \(\angle MAB = 10^\circ\). Find \(\angle AMC\) if \(\angle ACB = 80^\circ\) and \(AC = BC\). | 70^\circ |
10,447 | Esquecinaldo has a terrible memory for remembering numbers, but excellent for remembering sequences of operations. Therefore, to remember his five-digit bank code, he can remember that the code does not have repeated digits, none of the digits is zero, the first two digits form a power of 5, the last two digits form a power of 2, the middle digit is a multiple of 3, and the sum of all the digits is an odd number. Now he no longer needs to memorize the number, because he knows that his code is the largest number that satisfies these conditions. What is this code? | 25916 |
10,460 | Points \( P \) and \( Q \) are located on side \( BC \) of triangle \( ABC \), with \( BP: PQ: QC = 1: 2: 3 \). Point \( R \) divides side \( AC \) of this triangle such that \( AR: RC = 1: 2 \). What is the ratio of the area of quadrilateral \( PQST \) to the area of triangle \( ABC \), if \( S \) and \( T \) are the intersection points of line \( BR \) with lines \( AQ \) and \( AP \), respectively? | \dfrac{5}{24} |
10,465 | 5.1. Construct a rectangle, where each side is greater than 1, using six rectangles $7 \times 1, 6 \times 1, 5 \times 1, 4 \times 1, 3 \times 1, 2 \times 1$ and a square $1 \times 1$. | 7 \times 4 |
10,470 | Example 5. Find the sum of the series $f(x)=\sum_{n=0}^{\infty}\left(n^{2}+3 n-5\right) \cdot x^{n}$. | \dfrac{-7x^2 + 14x - 5}{(1 - x)^3} |
10,527 | 23. Find all such prime numbers, so that each of them can be represented as the sum and difference of two prime numbers. | 5 |
10,528 | In rectangle \(ABCD\), \(AB = 20 \, \text{cm}\) and \(BC = 10 \, \text{cm}\). Points \(M\) and \(N\) are taken on \(AC\) and \(AB\), respectively, such that the value of \(BM + MN\) is minimized. Find this minimum value. | 16 |
10,561 | Let \( A \) be a set of any 100 distinct positive integers. Define
\[ B = \left\{ \left.\frac{a}{b} \right|\, a, b \in A \text{ and } a \neq b \right\}, \]
and let \( f(A) \) denote the number of elements in the set \( B \). What is the sum of the maximum and minimum values of \( f(A) \)? | 10098 |
10,593 | 5. The Ivanovs' income as of the beginning of June:
$$
105000+52200+33345+9350+70000=269895 \text { rubles }
$$ | 269895 |
10,597 | 1. Kolya came up with an entertainment for himself: he rearranges the digits in the number 2015, then places a multiplication sign between any two digits. In this process, none of the resulting two factors should start with zero. Then he calculates the value of this expression. For example: $150 \cdot 2=300$, or $10 \cdot 25=250$. What is the largest number he can get as a result of such a calculation? | 1050 |
10,607 | ## Problem Statement
Calculate the limit of the numerical sequence:
$\lim _{n \rightarrow \infty} \frac{\sqrt{\left(n^{4}+1\right)\left(n^{2}-1\right)}-\sqrt{n^{6}-1}}{n}$ | -\dfrac{1}{2} |
10,620 | 11. From $1,2, \cdots, 100$ choose $k$ numbers, among which there must be two numbers that are coprime, then the minimum value of $k$ is | 51 |
10,623 | ## Problem Statement
Calculate the indefinite integral:
$$
\int \frac{2 x^{3}+5}{x^{2}-x-2} d x
$$ | x^{2} + 2x + 7 \ln|x - 2| - \ln|x + 1| + C |
10,633 | ## Task 2 - 070832
The cross sum of a natural number is understood to be the sum of its digits: for example, 1967 has the cross sum $1+9+6+7=23$.
Determine the sum of all cross sums of the natural numbers from 1 to 1000 inclusive! | 13501 |
10,658 | The probability that a purchased light bulb will work is $0.95$.
How many light bulbs need to be bought so that with a probability of $0.99$ there are at least five working ones among them?
# | 7 |
10,667 | Example 7 (1982 Kyiv Mathematical Olympiad) Find the natural number $N$, such that it is divisible by 5 and 49, and including 1 and $N$, it has a total of 10 divisors. | 12005 |
10,679 | In a regular hexagon $A B C D E F$, the diagonals $A C$ and $C E$ are divided by points $M$ and $N$ respectively in the following ratios: $\frac{A M}{A C} = \frac{C N}{C E} = r$. If points $B$, $M$, and $N$ are collinear, determine the ratio $r$. | \dfrac{\sqrt{3}}{3} |
10,704 | A circle passes through the vertices \(A\) and \(C\) of triangle \(ABC\) and intersects its sides \(AB\) and \(BC\) at points \(K\) and \(T\) respectively, such that \(AK:KB = 3:2\) and \(BT:TC = 1:2\). Find \(AC\) if \(KT = \sqrt{6}\). | 3\sqrt{5} |
10,706 | ## Problem 3
Find all functions $f: \Re \rightarrow \Re$ such that
$$
x^{2} \cdot f(x)+f(1-x)=2 x-x^{4}
$$ | 1 - x^2 |
10,718 | 13. Find $f: \mathbf{R} \rightarrow \mathbf{R}$, for any $x, y \in \mathbf{R}$, such that $f\left(x^{2}-y^{2}\right)=(x-y)(f(x)+f(y))$. | f(x) = kx |
10,757 | 1. Simplify the fraction $\frac{\sqrt{-x}-\sqrt{-3 y}}{x+3 y+2 \sqrt{3 x y}}$.
Answer. $\frac{1}{\sqrt{-3 y}-\sqrt{-x}}$. | \dfrac{1}{\sqrt{-3 y}-\sqrt{-x}} |
10,770 | Example 3. Given points A, B and line l in the same plane, try to find a point P on l such that
$$
P A+P B \text { is minimal. }
$$ | P |
10,782 | 13. The wealthy Croesus buys 88 identical vases. The price of each, expressed in drachmas, is an integer (the same for all 88 vases). We know that Croesus pays a total of $a 1211 b$ drachmas, where $a, b$ are digits to be determined (which may be distinct or the same). How many drachmas does a single vase cost? | 1274 |
10,797 | Task 1. 15 points
Find the minimum value of the expression
$$
\sqrt{x^{2}-\sqrt{3} \cdot|x|+1}+\sqrt{x^{2}+\sqrt{3} \cdot|x|+3}
$$
as well as the values of $x$ at which it is achieved. | \sqrt{7} |
10,809 | 61. Arrange the numbers 1 to 9 in a row from left to right, such that every three consecutive numbers form a three-digit number that is a multiple of 3. There are $\qquad$ ways to do this. | 1296 |
10,820 | Two people, A and B, start climbing a mountain from the bottom at the same time along the same path. After reaching the top, they immediately start descending along the same path. It is known that their descending speeds are three times their ascending speeds. A and B meet 150 meters from the top. When A returns to the bottom, B just reaches the halfway point of the descent. Find the distance from the bottom to the top of the mountain. | 1550 |
10,834 | 3. In $\triangle A B C$, $\angle B=60^{\circ}, \angle C=75^{\circ}, S_{\triangle A B C}=\frac{1}{2}(3+\sqrt{3}), B C=$ | 2 |
10,837 | In the diagram, a go-kart track layout is shown. The start and finish are at point $A$, and the go-kart driver can return to point $A$ and re-enter the track as many times as desired.
On the way from $A$ to $B$ or back, the young racer Yura spends one minute. Yura also spends one minute on the loop. The loop can only be driven counterclockwise (the arrows indicate the possible directions of travel). Yura does not turn back halfway and does not stop. The total duration of the race is 10 minutes. Find the number of possible different routes (sequences of sections traveled). | 34 |
10,857 | \( z_{1}, z_{2}, z_{3} \) are the three roots of the polynomial
\[ P(z) = z^{3} + a z + b \]
and satisfy the condition
\[ \left|z_{1}\right|^{2} + \left|z_{2}\right|^{2} + \left|z_{3}\right|^{2} = 250 \]
Moreover, the three points \( z_{1}, z_{2}, z_{3} \) in the complex plane form a right triangle. Find the length of the hypotenuse of this right triangle. | 5\sqrt{15} |
10,872 | 3. Given an integer $k$ satisfying $1000<k<2020$, and such that the system of linear equations in two variables $\left\{\begin{array}{l}7 x-5 y=11 \\ 5 x+7 y=k\end{array}\right.$ has integer solutions, then the number of possible values for $k$ is $\qquad$. | 13 |
10,875 | 7. The diagonals of a trapezoid are perpendicular to each other, and one of them is 17. Find the area of the trapezoid if its height is 15.
Answer. $\frac{4335}{16}$. | \dfrac{4335}{16} |
10,891 | A box contains 111 balls: red, blue, green, and white. It is known that if you draw 100 balls without looking into the box, you will definitely get at least four balls of different colors. What is the minimum number of balls that need to be drawn, without looking into the box, to ensure that you get at least three balls of different colors? | 88 |
10,914 | A given rectangle $ R$ is divided into $mn$ small rectangles by straight lines parallel to its sides. (The distances between the parallel lines may not be equal.) What is the minimum number of appropriately selected rectangles’ areas that should be known in order to determine the area of $ R$? | m + n - 1 |
10,917 | 8. The value of $4 \sin 40^{\circ}-\tan 40^{\circ}$ is | \sqrt{3} |
10,923 | 16. (6 points) In the rabbit figure formed by the tangram, the area of the rabbit's ears (shaded part) is 10 square centimeters. Then the area of the rabbit figure is $\qquad$ square centimeters. | 80 |
10,940 | How many positive integers \( x \) less than 10000 are there, for which the difference \( 2^x - x^2 \) is not divisible by 7? | 7142 |
10,949 | 119(988). In a flask, there is a solution of table salt. From the flask, $\frac{1}{5}$ of the solution is poured into a test tube and evaporated until the percentage of salt in the test tube doubles. After this, the evaporated solution is poured back into the flask. As a result, the salt content in the flask increases by $3 \%$. Determine the initial percentage of salt. | 27 |
10,963 | We start with 5000 forints in our pocket to buy gifts, visiting three stores. In each store, we find a gift that we like and purchase it if we have enough money. The prices in each store are independently 1000, 1500, or 2000 forints, each with a probability of $\frac{1}{3}$. What is the probability that we are able to purchase gifts from all three stores and still have money left? | \dfrac{17}{27} |
11,018 | Two spheres of one radius and two spheres of another radius are arranged so that each sphere touches the three other spheres and a plane. Find the ratio of the radius of the larger sphere to the radius of the smaller sphere. | 2 + \sqrt{3} |
11,054 | 3. In a convex quadrilateral $A B C D$, diagonals $A C$ and $D B$ are perpendicular to sides $D C$ and $A B$ respectively. A perpendicular is drawn from point $B$ to side $A D$, intersecting $A C$ at point $O$. Find $A O$, if $A B=4, O C=6$. | 2 |
11,057 | If the function
$$
f(x) = 3 \cos \left(\omega x + \frac{\pi}{6}\right) - \sin \left(\omega x - \frac{\pi}{3}\right) \quad (\omega > 0)
$$
has a minimum positive period of \(\pi\), then the maximum value of \(f(x)\) on the interval \(\left[0, \frac{\pi}{2}\right]\) is \(\qquad\) | 2\sqrt{3} |
11,065 | We randomly place points $A, B, C$, and $D$ on the circumference of a circle, independently of each other. What is the probability that the chords $AB$ and $CD$ intersect? | \dfrac{1}{3} |
11,078 | In an acute-angled triangle \( \triangle ABC \), given \( AC = 1 \), \( AB = c \), and the circumradius of \( \triangle ABC \) is \( R \leq 1 \), prove that \( \cos A < c \leq \cos A + \sqrt{3} \sin A \). | \cos A < c \leq \cos A + \sqrt{3} \sin A |
11,125 | 1. Let's call two positive integers almost adjacent if each of them is divisible (without a remainder) by their difference. During a math lesson, Vova was asked to write down in his notebook all numbers that are almost adjacent to $2^{10}$. How many numbers will he have to write down? | 21 |
11,141 | ## Task A-4.4.
Players $A, B$ and $C$ take turns rolling a die. What is the probability that $C$ gets a higher number than the other two players? | \dfrac{55}{216} |
11,147 | Problem 6.8. A natural number $n$ is called good if 2020 gives a remainder of 22 when divided by $n$. How many good numbers exist? | 10 |
11,162 | 3. The number of zeros of the function $f(x)=x^{2} \ln x+x^{2}-2$ is $\qquad$ . | 1 |
11,166 | Given that points \( M \) and \( N \) are the midpoints of sides \( AB \) and \( BC \) respectively in an equilateral triangle \( \triangle ABC \). Point \( P \) is outside \( \triangle ABC \) such that \( \triangle APC \) is an isosceles right triangle with \( \angle APC = 90^\circ \). Connect points \( PM \) and \( AN \), and let their intersection be point \( I \). Prove that line \( CI \) bisects \( \angle ACM \). | CI \text{ bisects } \angle ACM |
11,202 | 38. Using 80 squares with a side length of 2 cm, you can form $\qquad$ types of rectangles with an area of 320 square cm. | 5 |
11,225 | Sedrakyan $H$.
Two parabolas with different vertices are the graphs of quadratic trinomials with leading coefficients $p$ and $q$. It is known that the vertex of each parabola lies on the other parabola. What can $p+q$ be? | 0 |
11,234 | 4. An item is priced at 80 yuan. If you buy one such item and pay with 10 yuan, 20 yuan, and 50 yuan denominations, there are $\qquad$ different ways to pay. | 7 |
11,257 | 314. Find the derivative of the function $y=\sin \left(x^{3}-3 x^{2}\right)$. | (3x^2 - 6x)\cos\left(x^3 - 3x^2\right) |
11,260 | In preparation for the family's upcoming vacation, Tony puts together five bags of jelly beans, one bag for each day of the trip, with an equal number of jelly beans in each bag. Tony then pours all the jelly beans out of the five bags and begins making patterns with them. One of the patterns that he makes has one jelly bean in a top row, three jelly beans in the next row, five jelly beans in the row after that, and so on:
\[\begin{array}{ccccccccc}&&&&*&&&&\\&&&*&*&*&&&\\&&*&*&*&*&*&&\\&*&*&*&*&*&*&*&\\ *&*&*&*&*&*&*&*&*\\&&&&\vdots&&&&\end{array}\]
Continuing in this way, Tony finishes a row with none left over. For instance, if Tony had exactly $25$ jelly beans, he could finish the fifth row above with no jelly beans left over. However, when Tony finishes, there are between $10$ and $20$ rows. Tony then scoops all the jelly beans and puts them all back into the five bags so that each bag once again contains the same number. How many jelly beans are in each bag? (Assume that no marble gets put inside more than one bag.) | 45 |
11,310 | Find all \( a_{0} \in \mathbb{R} \) such that the sequence \( a_{0}, a_{1}, \cdots \) determined by \( a_{n+1}=2^{n}-3a_{n}, n \in \mathbb{Z}^{+} \) is increasing. | \dfrac{1}{5} |
11,320 | 12. (3 points) The school organized a spring outing, renting boats for students to row. If each boat seats 3 people, 16 people are left without a boat; if each boat seats 5 people, one boat is short of 4 people. The school has a total of $\qquad$ students. | 46 |
11,328 | 3. Find the number of natural numbers $k$, not exceeding 353500, such that $k^{2}+k$ is divisible by 505. | 2800 |
11,353 | 397. Given the distribution function $F(x)$ of a random variable $X$. Find the distribution function $G(y)$ of the random variable $Y=3X+2$.
翻译结果如下：
397. Given the distribution function $F(x)$ of a random variable $X$. Find the distribution function $G(y)$ of the random variable $Y=3X+2$. | F\left(\frac{y - 2}{3}\right) |
11,395 | In a right triangle \( ABC \) with \( AC = 16 \) and \( BC = 12 \), a circle with center at \( B \) and radius \( BC \) is drawn. A tangent to this circle is constructed parallel to the hypotenuse \( AB \) (the tangent and the triangle lie on opposite sides of the hypotenuse). The leg \( BC \) is extended to intersect this tangent. Determine by how much the leg is extended. | 15 |
11,431 | Suppose \( f(x) = \lg \frac{1 + 2^x + 4^x a}{3} \), where \( a \in \mathbf{R} \). Determine the range of values for \( a \) if \( f(x) \) is defined for \( x \in (-\infty, 1] \). | (-\frac{3}{4}, +\infty) |
11,440 | 3. At the robot running competition, a certain number of mechanisms were presented. The robots were released on the same distance in pairs. The protocol recorded the differences in the finish times of the winner and the loser in each of the races. All of them turned out to be different: 1 sec., 2 sec., 3 sec., 4 sec., 5 sec., 7 sec. It is known that during the races, each robot competed with each other exactly once, and that each robot always ran at the same speed. In your answer, indicate the time in seconds of the slowest mechanism, if the best time to complete the distance was 30 seconds. | 37 |
11,441 | 求函数 \( f(x) = \frac{(x^2 + 1)\sin(x)}{e^{x}} \) 在一般点的导数。 | \frac{(2x - x^2 - 1)\sin(x) + (x^2 + 1)\cos(x)}{e^x} |
11,448 | Gavrila found that the front tires of the car last for 21,000 km, and the rear tires last for 28,000 km. Therefore, he decided to swap them at some point so that the car would travel the maximum possible distance. Find this maximum distance (in km). | 24000 |
11,492 | 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 |
11,502 | In $\triangle ABC$, $AD \perp BC$ at point $D$, $DE \perp AC$ at point $E$, $M$ is the midpoint of $DE$, and $AM \perp BE$ at point $F$. Prove that $\triangle ABC$ is an isosceles triangle. | \triangle ABC \text{ is an isosceles triangle} |
11,523 | Given a function \( f: \mathbf{R} \rightarrow \mathbf{R} \) such that for any real numbers \( x \) and \( y \), \( f(2x) + f(2y) = f(x+y) f(x-y) \). Additionally, \( f(\pi) = 0 \) and \( f(x) \) is not identically zero. What is the period of \( f(x) \)? | 4\pi |
11,529 | 15 If $m, n \in\left\{x \mid x=a_{2} \times 10^{2}+a_{1} \times 10+a_{0}\right\}$, where $a_{i} \in\{1,2,3, 4,5,6,7\}, i=0,1,2$, and $m+n=636$, then the number of distinct points $(m, n)$ representing points on the plane is $\qquad$ | 90 |
11,563 | For some angle $x$, the equation
$$
a \cos ^{2} x+b \cos x+c=0
$$
holds. Write down the quadratic equation that $\cos 2 x$ satisfies. Apply your result to the case $a=4, b=2$, $c=-1$. | 4 \cos^2 2x + 2 \cos 2x - 1 = 0 |
11,573 | 5-3. Solve the inequality
$$
\sqrt{5 x-11}-\sqrt{5 x^{2}-21 x+21} \geqslant 5 x^{2}-26 x+32
$$
In your answer, specify the sum of all integer values of $x$ that satisfy the inequality. | 3 |
11,583 | 13. Let $A B C D$ be a parallelogram. It is known that side $A B$ measures 6, angle $B \widehat{A} D$ measures $60^{\circ}$, and angle $A \widehat{D} B$ is a right angle. Let $P$ be the centroid of triangle $A C D$. Calculate the value of the product of the areas of triangle $A B P$ and quadrilateral $A C P D$. | 27 |
11,600 | 某建筑公司希望优化其施工项目，目标是最大化施工进度 \( P \)，同时确保施工成本在预算之内。已知：
- 施工进度 \( P = k \cdot W \cdot L \cdot T \)，其中 \( k \) 是常数，\( W \) 是工人数量，\( L \) 是施工设备数量，\( T \) 是施工时间。
- 施工成本 \( C = p_W \cdot W + p_L \cdot L + p_T \cdot T \)，其中 \( p_W \)、\( p_L \) 和 \( p_T \) 分别为单位工人数量、施工设备数量和施工时间的成本。
- 总预算不超过8000万元。
- 工人数量在50至500人之间，施工设备数量在10至100台之间，施工时间在1至24个月之间。
如何确定工人数量 \( W \)、施工设备数量 \( L \) 和施工时间 \( T \)，以使施工进度 \( P \) 最大化？使用拉格朗日乘数法进行求解。 | W = \frac{8000}{3p_W}, \quad L = \frac{8000}{3p_L}, \quad T = \frac{8000}{3p_T} |
11,604 | 140. Find the product of two approximate numbers: $0.3862 \times$ $\times 0.85$. | 0.33 |
11,606 | If any two adjacent digits of a three-digit number have a difference of at most 1, it is called a "steady number". How many steady numbers are there? | 75 |
11,612 | A point \(A_{1}\) is taken on the side \(AC\) of triangle \(ABC\), and a point \(C_{1}\) is taken on the extension of side \(BC\) beyond point \(C\). The length of segment \(A_{1}C\) is 85% of the length of side \(AC\), and the length of segment \(BC_{1}\) is 120% of the length of side \(BC\). What percentage of the area of triangle \(ABC\) is the area of triangle \(A_{1}BC_{1}\)? | 102 |
11,625 | Let $n$ be an integer. Determine the largest constant $C$ possible so that for all $a_{1} \ldots, a_{n} \geqslant 0$ we have $\sum a_{i}^{2} \geqslant C \sum_{i<j} a_{i} a_{j}$. | \dfrac{2}{n-1} |
11,664 | 11. In the triangle $A B C, A B=A C=1, D$ and $E$ are the midpoints of $A B$ and $A C$ respectively. Let $P$ be a point on $D E$ and let the extensions of $B P$ and $C P$ meet the sides $A C$ and $A B$ at $G$ and $F$ respectively. Find the value of $\frac{1}{B F}+\frac{1}{C G}$. | 3 |
11,669 |
Friends Vasya, Petya, and Kolya live in the same house. One day, Vasya and Petya set out on foot to go fishing at the lake. Kolya stayed home but promised to meet his friends on the way back on his bicycle. Vasya was the first to head home, and at the same time, Kolya set out on his bicycle to meet him. Petya started his journey home from the lake with the same speed as Vasya at the moment when Kolya and Vasya met. Kolya, upon meeting Vasya, immediately turned around and gave him a ride home on his bicycle. Then, Kolya set off again on his bicycle towards the lake to meet Petya. After meeting Petya, Kolya turned around once more and took him home. As a result, the time Petya spent traveling from the lake to home was $4 / 3$ of the time Vasya spent on the same journey. How many times slower would Vasya's journey be to get home if he had walked the entire way? (8 points) | 3 |
11,682 | A jar contains $2$ yellow candies, $4$ red candies, and $6$ blue candies. Candies are randomly drawn out of the jar one-by-one and eaten. The probability that the $2$ yellow candies will be eaten before any of the red candies are eaten is given by the fraction $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Find $m + n$ . | 16 |
11,693 | 14. Harry Potter used a wand to change a fraction, and after the change, he found that the numerator increased by $20 \%$, and the denominator decreased by $19 \%$, then the new fraction is increased by $\qquad$ \%. (rounded to 1\%)
| 48 |
11,704 | Prove that \(\left(a^{2}+b^{2}+c^{2}-a b-b c-a c\right)\left(x^{2}+y^{2}+z^{2}-x y-y z-x z\right)=X^{2}+Y^{2}+Z^{2}-X Y-Y Z-X Z\),
if \(X=a x+c y+b z, \quad Y=c x+b y+a z, \quad Z=b x+a y+c z\). | \left(a^{2}+b^{2}+c^{2}-ab-bc-ac\right)\left(x^{2}+y^{2}+z^{2}-xy-yz-xz\right)=X^{2}+Y^{2}+Z^{2}-XY-YZ-XZ |
11,722 | In a rectangular coordinate system, a circle centered at the point $(1,0)$ with radius $r$ intersects the parabola $y^2 = x$ at four points $A$, $B$, $C$, and $D$. If the intersection point $F$ of diagonals $AC$ and $BD$ is exactly the focus of the parabola, determine $r$. | \dfrac{\sqrt{15}}{4} |
11,727 | 8,9 |
The center of a circle with a radius of 5, circumscribed around an isosceles trapezoid, lies on the larger base, and the smaller base is equal to 6. Find the area of the trapezoid. | 32 |
11,729 | Find a factor of 1464101210001 that lies between 1210000 and 1220000. | 1211101 |
11,732 | The median \(AD\) of an acute-angled triangle \(ABC\) is 5. The orthogonal projections of this median onto the sides \(AB\) and \(AC\) are 4 and \(2\sqrt{5}\), respectively. Find the side \(BC\). | 2\sqrt{10} |
11,759 | There are six rock specimens with weights of 8.5 kg, 6 kg, 4 kg, 4 kg, 3 kg, and 2 kg. They need to be distributed into three backpacks such that the heaviest backpack is as light as possible. What is the weight of the rock specimens in the heaviest backpack? | 10 |
11,812 | Dima and Sergey were picking raspberries from a bush that had 900 berries. Dima alternated his actions while picking: he put one berry in the basket, and then he ate the next one. Sergey also alternated: he put two berries in the basket, and then he ate the next one. It is known that Dima picks berries twice as fast as Sergey. At some point, the boys collected all the raspberries from the bush.
Who ended up putting more berries in the basket? What will be the difference? | 100 |
11,816 | A stack of A4 sheets was folded in half and then folded again to create a booklet of A5 format. The pages of this booklet were then renumbered as $1, 2, 3, \ldots$ It turned out that the sum of the numbers on one of the sheets is 74. How many sheets were in the stack? | 9 |
11,823 | ## Problem Statement
Write the equation of the plane passing through point $A$ and perpendicular to vector $\overrightarrow{B C}$.
$A(-3 ; 6 ; 4)$
$B(8 ;-3 ; 5)$
$C(0 ;-3 ; 7)$ | 4x - z + 16 = 0 |
11,839 | Example 2. Given $a_{\mathrm{n}}=3 n-1$, find $S_{\mathrm{n}}$.
| \dfrac{3n^2 + n}{2} |
11,840 | Let $X_{1}, \ldots, X_{n}$ ($n \geqslant 1$) be independent and identically distributed non-negative random variables with density $f=f(x)$,
$$
S_{n}=X_{1}+\ldots+X_{n} \quad \text {and} \quad M_{n}=\max \left\{X_{1}, \ldots, X_{n}\right\}
$$
Prove that the Laplace transform $\phi_{n}=\phi_{n}(\lambda), \lambda \geqslant 0$, of the ratio $S_{n} / M_{n}$ is given by
$$
\phi_{n}(\lambda)=n e^{-\lambda} \int_{0}^{\infty}\left(\int_{0}^{x} e^{-\lambda y / x} f(y) d y\right)^{n-1} f(x) d x
$$ | \phi_{n}(\lambda)=n e^{-\lambda} \int_{0}^{\infty}\left(\int_{0}^{x} e^{-\lambda y / x} f(y) d y\right)^{n-1} f(x) d x |
11,849 | Given two sets
$$
\begin{array}{l}
A=\{(x, y) \mid |x|+|y|=a, a>0\}, \\
B=\{(x, y) \mid |xy|+1=|x|+|y|\}.
\end{array}
$$
If \( A \cap B \) is the set of vertices of a regular octagon in the plane, determine the value of \( a \). | 2 + \sqrt{2} |
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