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56,747 | 31. A black ant and a red ant start from two places 78 meters apart at the same time, moving towards each other. The black ant travels 6 meters per minute, and the red ant travels 7 meters per minute. After 1 minute, they both turn around and move away from each other, then after 3 minutes, they both turn around and move towards each other again, then after 5 minutes, they both turn around and move away from each other again, then after 7 minutes, they both turn around and move towards each other again, …‥ According to this pattern, they will first meet after $\qquad$ minutes. | 48 |
56,752 | 7. Let $P$ be a moving point on the ellipse $\frac{y^{2}}{4}+\frac{x^{2}}{3}=1$, and let points $A(1,1), B(0,-1)$. Then the maximum value of $|P A|+|P B|$ is $\qquad$ . | 5 |
56,764 | 1. (5 points) Find the degree measure of the angle
$$
\delta=\arccos \left(\left(\sin 2195^{\circ}+\sin 2196^{\circ}+\cdots+\sin 5795^{\circ}\right)^{\cos } 2160^{\circ}+\cos 2161^{\circ}+\cdots+\cos 5760^{\circ}\right)
$$ | 55 |
56,768 | Find the largest integer $x$ such that $4^{27}+4^{1010}+4^{x}$ is a perfect square. | 1992 |
56,803 | SG. 2 In figure $1, A B$ is parallel to $D C, \angle A C B$ is a right angle, $A C=C B$ and $A B=B D$. If $\angle C B D=b^{\circ}$, find the value of $b$. | 15 |
56,806 | Problem 11.6. The quadratic trinomial $P(x)$ is such that $P(P(x))=x^{4}-2 x^{3}+4 x^{2}-3 x+4$. What can $P(8)$ be? List all possible options. | 58 |
56,820 | 3. Determine the primitive $F: \mathbb{R} \longrightarrow \mathbb{R}$ of the function $f: \mathbb{R} \longrightarrow \mathbb{R}$
$$
f(x)=\frac{\sin x \cdot \sin \left(x-\frac{\pi}{4}\right)}{e^{2 x}+\sin ^{2} x}
$$
for which $F(0)=0$. | F(x)=\frac{\sqrt{2}}{2}x-\frac{\sqrt{2}}{4}\ln(e^{2x}+\sin^{2}x) |
56,829 | Example 2-31 Form an n-digit number using the 5 digits $1,3,5,7,9$, with the requirement that the digits 3 and 7 appear an even number of times, while the other 3 digits have no restrictions. How many such numbers are there? | a_{n}=(5^{n}+2\cdot3^{n}+1)/4 |
56,835 | [Lengths of sides, heights, medians, and bisectors]
In rhombus $A B C D$, point $Q$ divides side $B C$ in the ratio $1: 3$, counting from vertex $B$, and point $E$ is the midpoint of side $A B$. It is known that the median $C F$ of triangle $C E Q$ is $2 \sqrt{2}$, and $E Q=\sqrt{2}$. Find the radius of the circle inscribed in rhombus $A B C D$. | \frac{\sqrt{7}}{2} |
56,906 | Five boys and six girls are to be seated in a row of eleven chairs so that they sit one at a time from one end to the other. The probability that there are no more boys than girls seated at any point during the process is $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Evaluate $m + n$. | 9 |
56,914 | 15. Quanti sono i numeri di cinque cifre (cioè fra 10000 e 99999) che non contengono zeri e sono multipli di 12 ? | 4374 |
56,921 | A: There are 7 boxes arranged in a row and numbered 1 through 7 . You have a stack of 2015 cards, which you place one by one in the boxes. The first card is placed in box \#1, the second in box \#2, and so forth up to the seventh card which is placed in box \#7. You then start working back in the other direction, placing the eighth card in box \#6, the ninth in box \#5, up to the thirteenth card being placed in box \#1. The fourteenth card is then placed in box \#2, and this continues until every card is distributed. What box will the last card be placed in? | 3 |
56,952 | ## Task B-4.5.
Mare has chosen 6 different digits from the set $\{1,2,3,4,5,6,7,8\}$. Using these digits, she wrote down on paper all possible six-digit numbers where the digits do not repeat. If $S$ is the sum of all the written numbers, determine the largest prime divisor of the number $S$. | 37 |
56,970 | 14. As shown in Figure 3, in $\triangle A B C$, $O$ is the midpoint of side $B C$, and a line through point $O$ intersects lines $A B$ and $A C$ at two distinct points $M$ and $N$ respectively. If
$$
\begin{array}{l}
\overrightarrow{A B}=m \overrightarrow{A M}, \\
\overrightarrow{A C}=n \overrightarrow{A N},
\end{array}
$$
then the value of $m+n$ is | 2 |
57,006 | 13.3 .8 * Given that $\odot C$ satisfies the following three conditions:
(1) $\odot C$ is tangent to the $x$-axis; (2) the center $C$ lies on the line $3x - y = 0$; (3) $\odot C$ intersects the line $x - y = 0$ at points $A$ and $B$, and the area of $\triangle ABC$ is $\sqrt{14}$. Find the equation of $\odot C$ that meets these conditions. | (x-1)^2+(y-3)^2=9or(x+1)^2+(y+3)^2=9 |
57,012 | 9.4. From Zlatoust to Miass, a "GAZ", a "MAZ", and a "KAMAZ" set off simultaneously. The "KAMAZ", having reached Miass, immediately turned back and met the "MAZ" 18 km from Miass, and the "GAZ" - 25 km from Miass. The "MAZ", having reached Miass, also immediately turned back and met the "GAZ" 8 km from Miass. What is the distance from Zlatoust to Miass? | 60 |
57,018 | 36. Given the lengths of the three sides of $\triangle A B C$ are $a$, $b$, and $c$ satisfying $\frac{4}{a}=\frac{1}{c}+\frac{3}{b}$, then $\angle A$ is $\qquad$ (fill in "acute angle", "right angle", "supplementary angle"). | acuteangle |
57,022 | 3. For the numbers $x$ and $y, 0<x<y$, the equation $x^{2}+4 y^{2}=5 x y$ holds. Calculate the value of the expression $\frac{x+2 y}{x-2 y}$. | -3 |
57,046 | 9. Is this possible? Can three people cover a distance of 60 km in 3 hours if they have a two-person motorcycle at their disposal? The motorcycle's speed is 50 km/h, and the pedestrian's speed is 5 km/h. | Yes |
57,050 | 4. In the rectangular prism $A^{\prime} C$, $A B=5, B C=4, B^{\prime} B=6$, and $E$ is the midpoint of $A A^{\prime}$. Find the distance between the skew lines $B E$ and $A^{\prime} C^{\prime}$. | \frac{60}{\sqrt{769}} |
57,072 | 1. Let $\triangle A B C$ be a given acute triangle, then the solution set of the equation $x^{2} \sin ^{2} B+x\left(\cos ^{2} C-\cos ^{2} A+\sin ^{2} B\right)$ $+\sin ^{2} A \cos ^{2} C=0$ with respect to $x$ is $\qquad$ | {-\frac{\sinA\cosC}{\sinB}} |
57,084 | 7. In the tetrahedron $P-ABC$, $PA=PB=a$, $PC=AB=BC=CA=b$, and $a<b$, then the range of $\frac{a}{b}$ is $\qquad$ | (\sqrt{2-\sqrt{3}},1) |
57,088 | 4. If $x_{1}, x_{2}, x_{3}, x_{4}, x_{5}, x_{6}$ are six different positive integers, taking values from $1, 2, 3, 4, 5, 6$. Let
$$
\begin{aligned}
S= & \left|x_{1}-x_{2}\right|+\left|x_{2}-x_{3}\right|+\left|x_{3}-x_{4}\right|+ \\
& \left|x_{4}-x_{5}\right|+\left|x_{5}-x_{6}\right|+\left|x_{6}-x_{1}\right| .
\end{aligned}
$$
Then the minimum value of $S$ is $\qquad$ | 10 |
57,105 | Example 4 Given two points $M(-1,0), N(1,0)$, and point $P$ such that $\overrightarrow{M P} \cdot \overrightarrow{M N}, \overrightarrow{P M} \cdot \overrightarrow{P N}, \overrightarrow{N M} \cdot \overrightarrow{N P}$ form an arithmetic sequence with a common difference less than zero, find the curve that is the locus of point $P$. (2002 National College Entrance Examination Question) | x^2+y^2=3 |
57,111 | In a class with $23$ students, each pair of students have watched a movie together. Let the set of movies watched by a student be his [i]movie collection[/i]. If every student has watched every movie at most once, at least how many different movie collections can these students have? | 23 |
57,146 | 20. How many subsets of the set $\{1,2,3, \ldots, 9\}$ do not contain consecutive odd integers? | 208 |
57,150 | Problem 4. The operating time of a radio device element follows the law $f(x)=\lambda e^{-\lambda x}$. In the table
| $x_{2}$ | 2.5 | 7.5 | 12.5 | 17.5 | 22.5 | 27.5 | $\Sigma$ |
| :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: |
| $n_{2}$ | 133 | 45 | 15 | 4 | 2 | 1 | 200 |
the empirical distribution of the average operating time of $n=200$ elements is provided: $x_{i}$ - the average operating time of the element in hours, $n_{i}$ - the number of elements that operated for $x_{i}$ hours. Find $\lambda$. | \lambda=0.2 |
57,160 | Shapovalov A.V.
On the table, 28 coins of the same size are arranged in a triangular shape (see figure). It is known that the total mass of any three coins that touch each other pairwise is 10 g. Find the total mass of all 18 coins on the boundary of the triangle.
 | 60 |
57,178 | 4.1. In an $11 \times 11$ square, the central cell is painted black. Maxim found a rectangular grid of the largest area that is entirely within the square and does not contain the black cell. How many cells does it have? | 55 |
57,196 | # 2. Solution:
According to the condition, the center of the second blot is at a distance of no less than 2 cm from the edge of the sheet, i.e., inside a rectangle of 11 cm by 16 cm. Consider the event A "The blots intersect." For this to happen, the center of the second blot must fall within a circle of radius 4 cm with the same center as the first blot.
The probability of this event is $P(A)=\frac{S_{\mathrm{kp}}}{S_{\text {mp }}}=\frac{16 \pi}{11 \cdot 16}=\frac{\pi}{11}$. | \frac{\pi}{11} |
57,203 | 3. A square and an equaliteral triangle together have the property that the area of each is the perimeter of the other. Find the square's area. | 12\sqrt[3]{4} |
57,208 | 28. A three-digit number with all distinct digits, when 2022 is written in front of it, becomes a seven-digit number, which is an integer multiple of the original three-digit number. The smallest original three-digit number is $\qquad$ . | 120 |
57,210 | 【Question 12】Given three natural numbers $1,2,3$, perform an operation on these three numbers, replacing one of the numbers with the sum of the other two, and perform this operation 9 times. After these operations, the maximum possible value of the largest number among the three natural numbers obtained is $\qquad$ _. | 233 |
57,221 | What can be the maximum area of a triangle if the length of any of its sides is not greater than 2? | \sqrt{3} |
57,226 | 12. (2004 High School Competition - Liaoning Preliminary) The sum of the maximum and minimum values of the function $f(x)=\frac{\sqrt{2} \sin \left(x+\frac{\pi}{4}\right)+2 x^{2}+x}{2 x^{2}+\cos x}$ is . $\qquad$ | 2 |
57,232 | 10. 2019 circles split a plane into a number of parts whose boundaries are arcs of those circles. How many colors are needed to color this geographic map if any two neighboring parts must be coloured with different colours? | 2 |
57,315 | Example 3 Arrange the positive integers that are coprime with 105 in ascending order, and find the 1000th term of this sequence. | 2186 |
57,344 | 2. Find all positive integers $n$ for which the number $n^{2}+6 n$ is a perfect square of an integer. | 2 |
57,369 | 10. (20 points) Given the function
$$
f(x)=x^{4}+a x^{3}+b x^{2}+a x+1(a, b \in \mathbf{R})
$$
has at least one root. Find the minimum value of $a^{2}-b$. | 1 |
57,378 | 12. For a residential building, the construction investment is 250 yuan per square meter, considering a lifespan of 50 years, and an annual interest rate of $5 \%$, then the monthly rent per square meter should be $\qquad$ yuan to recover the entire investment. | 1.14 |
57,390 | 3.48 A young man was returning home from vacation on a bicycle. At first, after traveling several kilometers, he spent one day more than half the number of days remaining after this until the end of his vacation. Now the young man has two options to travel the remaining distance to arrive home on time: to travel $h$ km more daily than originally planned, or to maintain the original daily travel distance, exceeding it only once - on the last day of the journey by $2 h$ km. How many days before the end of his vacation did the young man set out for home? | 4 |
57,413 | 3. If $x \in(-1,1)$, then
$$
f(x)=x^{2}-a x+\frac{a}{2}
$$
is always positive, the range of the real number $a$ is $\qquad$ $ـ$. | (0,2] |
57,431 | Let's determine the vertices of a square whose diagonals lie on the coordinate axes and whose sides are tangent to the ellipse
$$
4 x^{2}+9 y^{2}=36
$$
What is the area of the rectangle defined by the points of tangency? | \frac{144}{13} |
57,437 | 3. Calculate the value of the product $\frac{2^{3}-1}{2^{3}+1} \cdot \frac{3^{3}-1}{3^{3}+1} \cdot \frac{4^{3}-1}{4^{3}+1} \cdot \ldots \cdot \frac{300^{3}-1}{300^{3}+1}$. | \frac{90301}{135450} |
57,441 | ## Problem Statement
Calculate the volumes of solids formed by rotating the figures bounded by the graphs of the functions. The axis of rotation is $O y$.
$$
y=\arccos \frac{x}{3}, y=\arccos x, y=0
$$ | 2\pi^{2} |
57,443 | 4. Let $a, b, c$ be any real numbers, and satisfy: $a>b>c,(a-b)(b-c)(c-a)=-16$. Then the minimum value of $\frac{1}{a-b}+\frac{1}{b-c}-\frac{1}{c-a}$ is $\qquad$ | \frac{5}{4} |
57,474 | Which is the highest fourth-degree $p(x)$ polynomial, which has $x_{1}=-3$ and $x_{2}=5$ as its zeros, and they are also points of local minimum? We also know that the polynomial $q(x)=p(x+1)$ is even, and the value of its local maximum is 256. | x^{4}-4x^{3}-26x^{2}+60x+225 |
57,482 | ## Task Condition
Find the derivative.
$$
y=\left(3 x^{2}-4 x+2\right) \sqrt{9 x^{2}-12 x+3}+(3 x-2)^{4} \cdot \arcsin \frac{1}{3 x-2}, 3 x-2>0
$$ | 12(3x-2)^{3}\cdot\arcsin\frac{1}{3x-2} |
57,498 | 2. In triangle $A B C$, the median $B K$ is twice as small as side $A B$ and forms an angle of $32^{\circ}$ with it. Find the angle $A B C$. | 106 |
57,522 | 41. A pair of two-digit numbers, like 18 and 81, whose tens and units digits are reversed, are considered a "family." Their sum is 99. There are $\qquad$ pairs of such "families" whose sum is 99. | 4 |
57,536 | 4. The turtle crawled out of its house and moved in a straight line at a constant speed of 5 m/hour. After an hour, it turned $90^{\circ}$ (right or left) and continued moving, then crawled for another hour, then turned $90^{\circ}$ (right or left) again... and so on. It crawled for 11 hours, turning $90^{\circ}$ at the end of each hour. What is the shortest distance from the house it could have been? | 5 |
57,545 | 3. In an equilateral triangle $\mathrm{ABC}$, the height $\mathrm{BH}$ is drawn. On the line $\mathrm{BH}$, a point $\mathrm{D}$ is marked such that $\mathrm{BD}=\mathrm{AB}$. Find $\angle \mathrm{CAD}$. | 15 |
57,548 | 11. Let the sequence $\left\{a_{n}\right\}$ have the sum of the first $n$ terms $S_{n}$ satisfying: $S_{n}+a_{n}=\frac{n-1}{n(n+1)}, n=1,2, \cdots$, then the general term $a_{n}=$ $\qquad$ | a_{n}=\frac{1}{2^{n}}-\frac{1}{n(n+1)} |
57,558 | 1. Arrange the natural numbers $1,2,3, \cdots, 999$. 1.5 into the number $N=1234 \cdots 998999$. Then, the sum of the digits of $N$ is $\qquad$ | 13500 |
57,568 | Problem 5. Petya and Vasya are playing a game. On the far left field of a strip consisting of 13 cells, there is a pile of 2023 stones. The players take turns, with Petya starting. Each player, on their turn, can move any stone one or two fields to the right. The player who first places any stone on the far right cell wins. Who among the boys can win regardless of the opponent's play? (20 points)
 | Vasya |
57,588 | Example 1 Given point $A(0,2)$ and two points $B, C$ on the parabola $y^{2}=x+4$, such that $A B \perp B C$. Find the range of the y-coordinate of point $C$.
(2002 National High School League Question) | (-\infty,0]\cup[4,+\infty) |
57,603 | $3+$
Avor: Bakayeva.v.
Forty children were playing in a circle. Of them, 22 were holding hands with a boy and 30 were holding hands with a girl. How many girls were in the circle? | 24 |
57,636 | A store sells shirts, water bottles, and chocolate bars. Shirts cost $\$ 10$ each, water bottles cost $\$ 5$ each, and chocolate bars cost $\$ 1$ each. On one particular day, the store sold $x$ shirts, $y$ water bottles, and $z$ chocolate bars. The total revenue from these sales was $\$ 120$. If $x, y$ and $z$ are integers with $x>0, y>0$ and $z>0$, how many possibilities are there for the ordered triple $(x, y, z)$ ? | 121 |
57,639 | 2.051. $\frac{\left(a^{2}-b^{2}\right)\left(a^{2}+\sqrt[3]{b^{2}}+a \sqrt[3]{b}\right)}{a \sqrt[3]{b}+a \sqrt{a}-b \sqrt[3]{b}-\sqrt{a b^{2}}}: \frac{a^{3}-b}{a \sqrt[3]{b}-\sqrt[6]{a^{3} b^{2}}-\sqrt[3]{b^{2}}+a \sqrt{a}} ;$
$$
a=4.91 ; b=0.09
$$ | 5 |
57,642 | Problem 2. In a $3 \times 3$ table, natural numbers (not necessarily distinct) are placed such that the sums in all rows and columns are different. What is the minimum value that the sum of the numbers in the table can take? | 17 |
57,674 | 7. As shown in Figure 1, a cross-section of the cube $A B C D-E F G H$ passes through vertices $A, C$, and a point $K$ on edge $E F$, dividing the cube into two parts with a volume ratio of 3:1. Then the value of $\frac{E K}{K F}$ is $\qquad$ | \sqrt{3} |
57,705 | Let $a < b < c < d < e$ be real numbers. We calculate all possible sums in pairs of these 5 numbers. Of these 10 sums, the three smaller ones are 32, 36, 37, while the two larger ones are 48 and 51. Determine all possible values that $e$ can take. | 27.5 |
57,709 | 28. A problem solved by Poisson (1781-1840) in his youth. Someone has 12 pints (a unit of volume) of honey and wants to pour out half of this amount, but he does not have a container with a capacity of 6 pints. He has 2 containers: one with a capacity of 8 pints, and the other with a capacity of 5 pints. How can he pour 6 pints of honey into the 8-pint container? What is the minimum number of transfers required to do this? (This problem determined Poisson's life path: he dedicated his entire life to mathematics.) | 6 |
57,713 | Sonkin $M$.
In trapezoid $A B C D$ with area 1, the bases $B C$ and $A D$ are in the ratio $1: 2$. Let $K$ be the midpoint of diagonal $A C$. Line $D K$ intersects side $A B$ at point $L$. Find the area of quadrilateral $B C K L$. | \frac{2}{9} |
57,722 | 3. A 101-gon is inscribed in a circle with diameter $\mathrm{XY}=6$ and has an axis of symmetry perpendicular to this diameter. Find the sum of the squares of the distances from the vertices of the 101-gon to the point $\mathrm{X}$. | 1818 |
57,723 | 3. Josh takes a walk on a rectangular grid of $n$ rows and 3 columns, starting from the bottom left corner. At each step, he can either move one square to the right or simultaneously move one square to the left and one square up. In how many ways can he reach the center square of the topmost row? | 2^{n-1} |
57,732 | 10. Let $S_{n}=1+2+\cdots+n$. Then among $S_{1}, S_{2}$, $\cdots, S_{2015}$, there are. $\qquad$ that are multiples of 2015. | 8 |
57,734 | 8 If the three lines from the line system $C: x \cos t+(y+1) \sin t=2$ form an equilateral triangle region $D$, then the area of region $D$ is $\qquad$ . | 12\sqrt{3} |
57,746 | 2. Given $x, y, z > 1$, satisfying
$$
\log _{x} 2+\log _{y} 4+\log _{z} 8=1 \text {. }
$$
Then the minimum value of $x y^{2} z^{3}$ is $\qquad$ | 2^{36} |
57,748 | 6. Calculate $\cos \frac{2 \pi}{7} \cdot \cos \frac{4 \pi}{7} \cdot \cos \frac{6 \pi}{7}$
The value is | \frac{1}{8} |
57,763 | 7. Let $x_{i} \in \mathbf{R}, x_{i} \geqslant 0(i=1,2,3,4,5), \sum_{i=1}^{5} x_{i}=1$, then $\max \left\{x_{1}+x_{2}, x_{2}+x_{3}, x_{3}+x_{4}, x_{4}\right.$ $\left.+x_{5}\right\}$ the minimum value equals $\qquad$ . | \frac{1}{3} |
57,775 | 4. At a point $R$ on a line, there is a robot that moves along this line to the left or right as it wishes. It is programmed to take 2 steps on the first move, 4 steps on the second move, 6 steps on the third move, and in general, $2n$ steps on the $n$-th move.
a) Describe a variant of the robot's movement such that it starts from point $R$ and ends its movement at point $R$ after exactly 4 moves.
b) What is the minimum number of moves the robot must make so that it starts from point $R$ and ends up back at $R$ at the end of its movement?
c) Prove that there is a variant of the robot's movement such that it starts from $R$ and ends up at $R$ after 179 moves.
## NOTE
- All questions are mandatory;
- Each question is worth 7 points;
- No points are awarded by default;
- The actual working time is 2 hours from the moment the question is received.
## Mathematics Olympiad Regional Phase - February 16, 2013
## Grade 6 - grading rubric | 3 |
57,801 | A group of cows and horses are randomly divided into two equal rows. (The animals are welltrained and stand very still.) 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 |
57,811 | 4. "24 Points" is a familiar math game to many people, the game process is as follows: arbitrarily draw 4 cards from 52 cards (excluding the jokers). Use the numbers on these 4 cards (from 1 to 13, where $A=1, J=11, Q=12, K=13$) to derive 24 using the four basic arithmetic operations. The person who finds the algorithm first wins. The game rules state that all 4 cards must be used, and each card can only be used once. For example, with $2,3,4, Q$, the algorithm $2 \times Q \times(4-3)$ can be used to get 24.
If in one game, you happen to draw $2,5, \mathrm{~J}, \mathrm{Q}$, then your algorithm is: $\qquad$ | 2\times(11-5)+12 |
57,828 | 18. In $\triangle A B C$, the three sides $a$, $b$, $c$ satisfy $2 b=a+c$. Find the value of $5 \cos A-4 \cos A \cos C+5 \cos C$. | 4 |
57,843 | 3. Hua Hua writes letters to Yuan Yuan with a ballpoint pen. When the 3rd pen refill is used up, she is writing the 4th letter; when she finishes the 5th letter, the 4th pen refill is not yet used up; if Hua Hua uses the same amount of pen refill for each letter, then to finish writing 16 letters, Hua Hua needs to prepare at least $\qquad$ pen refills. | 13 |
57,844 | 8. Six people are playing a coin-tossing game in a circle (the coin is fair), with each person tossing the coin once. The rule is: those who get the coin landing tails up have to perform a show, while those who get heads up do not have to perform. What is the probability that no two performers are adjacent? $\qquad$ . | \frac{9}{32} |
57,874 | 5. In the country, there are 100 cities, and several non-stop air routes are in operation between them, such that one can travel from any city to any other, possibly with layovers. For each pair of cities, the minimum number of flights required to travel from one to the other was calculated. The transportation difficulty of the country is defined as the sum of the squares of these 4950 numbers. What is the maximum value that the transportation difficulty can take? The answer should be given as a number (in decimal notation). | 8332500 |
57,875 | Problem 4. A student has 144 balls, which they distribute into four boxes, following these rules:
a) the number of balls in the first box differs by 4 from the number of balls in the second box;
b) the number of balls in the second box differs by 3 from the number of balls in the third box;
c) the number of balls in the third box differs by 2 from the number of balls in the fourth box;
d) the first box contains the most balls.
What is the maximum number of balls the student can put in the second box?
Working time 2 hours.
Each problem is worth 7 points.
## National Mathematics Olympiad Local Stage, Iaşi | 37 |
57,901 | In a circle with a diameter of $10 \, \text{cm}$, there are two perpendicular chords $AB$ and $CD$, whose lengths are $9 \, \text{cm}$ and $8 \, \text{cm}$.
Calculate the distance from the intersection of lines $AB$ and $CD$ to the center of the circle.
(L. Hozová) | \frac{\sqrt{55}}{2} |
57,907 | # Variant №1
№1. The number of male students in the university is $35 \%$ more than the number of female students. All students are distributed between two buildings, with $\frac{2}{5}$ of all male students in the first building and $\frac{4}{7}$ of all female students in the second building. How many students are there in total in the university, if it is known that fewer than 2500 students are in the first building, and more than 2500 students are in the second building? | 4935 |
57,911 | ## Task 2 - V10712
One of the largest man-made lakes is the Zimljansker Reservoir in the Soviet Union. It has a surface area of about $2600 \mathrm{~km}^{2}$. In contrast, the area of the Müggelsee is about 750 ha.
How many times larger is the area of the Zimljansker Reservoir? | 347 |
57,915 | 57. In a magical country, there live two types of people, type A who only tell the truth, and type B who only tell lies. One day, 2014 citizens of this country lined up, and each one said: “There are more B type people behind me than A type people in front of me.” Therefore, among these 2014 citizens, there are $\qquad$ type A people. | 1007 |
57,919 | 5.85 It is known that a safe is managed by an 11-member committee, and several locks are added to the safe. The keys to these locks are distributed among the committee members. To ensure that any 6 members present can open the safe, but any 5 cannot, what is the minimum number of locks that should be added to the safe?
---
The translation maintains the original text's line breaks and format. | 462 |
57,935 | Given are a rectangular board of size $13 \times 2$ and arbitrarily many dominoes of sizes $2 \times 1$ and $3 \times 1$. The board is to be completely covered by such dominoes without gaps or overlaps, and no domino is allowed to protrude beyond the board. Furthermore, all dominoes must be aligned in the same direction, i.e., their long sides must be parallel to each other.
How many such coverings are possible?
(Walther Janous)
Answer. There are 257 possible coverings. | 257 |
57,950 | ## Problem Statement
Calculate the limit of the numerical sequence:
$\lim _{n \rightarrow \infty} \frac{\sqrt{n+2}-\sqrt[3]{n^{3}+2}}{\sqrt[7]{n+2}-\sqrt[5]{n^{5}+2}}$ | 1 |
57,966 | Problem 3. In the vertices of a regular 2019-gon, numbers are placed such that the sum of the numbers in any nine consecutive vertices is 300. It is known that the 19th vertex has the number 19, and the 20th vertex has the number 20. What number is in the 2019th vertex? | 61 |
57,971 | 30th IMO 1989 shortlist Problem 18 Five points are placed on a sphere of radius 1. That is the largest possible value for the shortest distance between two of the points? Find all configurations for which the maximum is attained. Solution | \sqrt{2} |
57,980 | In a production of Hamlet, several roles are paired, for example, Gertrude and the Queen in the Play can be played by two actors. Before the performance, they decide by drawing lots which of the two will play the Queen in the Play and which will play Gertrude on that day. Similar lotteries decide for the other pairs as well. Sári has already seen one performance, but she wants to see the Gertrude/Queen in the Play, Claudius/Player King, and Ophelia/Fortinbras roles in the other version as well, although not necessarily in the same performance. How many more performances should she buy tickets for so that there is at least a $90 \%$ probability that she will see all three roles in the other casting as well? | 5 |
58,019 | Transform the fraction
$$
\frac{5 \sqrt[3]{6}-2 \sqrt[3]{12}}{4 \sqrt[3]{12}+2 \sqrt[3]{6}}
$$
so that the denominator is a rational number. | \frac{24\sqrt[3]{4}-12\sqrt[3]{2}-11}{34} |
58,023 | 3. $\mathrm{ABCD}$ is a trapezoid with bases $\mathrm{AD}=6$ and $\mathrm{BC}=10$. It turns out that the midpoints of all four sides of the trapezoid lie on the same circle. Find its radius.
If there are multiple correct answers, list them in any order separated by a semicolon. | 4 |
58,047 | 11. Use the digits 1-9 each once to form a two-digit perfect square, a three-digit perfect square, and a four-digit perfect square. What is the smallest four-digit perfect square among them? $\qquad$ | 1369 |
58,050 | 1. If real numbers $x, y$ satisfy $x^{2}-2 x y+5 y^{2}=4$, then the range of $x^{2}+y^{2}$ is $\qquad$ | [3-\sqrt{5},3+\sqrt{5}] |
58,076 | 57. Point $K$ is the midpoint of edge $A A_{1}$ of the cube $A B C D A_{1} B_{1} C_{1} D_{1}$, point $L$ lies on edge $B C$. Segment $K L$
is tangent to the sphere inscribed in the cube. In what ratio does the point of tangency divide segment $K L$? | \frac{4}{5} |
58,090 | 4. (7 points) The numbers $a, b, c, d$ belong to the interval $[-5.5 ; 5.5]$. Find the maximum value of the expression $a+2 b+c+2 d-a b-b c-c d-d a$. | 132 |
58,102 | 8. There are three segments of length $2^{n}(n=0,1, \cdots, 1009)$. Then the number of non-congruent triangles that can be formed by these 3030 segments is (answer in digits). | 510555 |
58,142 | 7. The range of the function $f(x)$ $=\frac{x-x^{3}}{\left(1+x^{2}\right)^{2}}$ is $\qquad$ _. | [-\frac{1}{4},\frac{1}{4}] |
58,176 | ## Task A-2.2.
Two circles with radii 1 and 3 touch each other externally at point $A$, and their external common tangent touches them at points $B$ and $C$. Determine the sum of the squares of the lengths of the sides of triangle $A B C$. | 24 |
58,179 | 6.146. $\frac{(x-1)(x-2)(x-3)(x-4)}{(x+1)(x+2)(x+3)(x+4)}=1$. | 0 |
58,181 | Eliane wants to choose her schedule for swimming. She wants to attend two classes per week, one in the morning and one in the afternoon, not on the same day, nor on consecutive days. In the morning, there are swimming classes from Monday to Saturday, at $9 \mathrm{~h}$, $10 \mathrm{~h}$, and $11 \mathrm{~h}$, and in the afternoon, from Monday to Friday, at $17 \mathrm{~h}$ and $18 \mathrm{~h}$. In how many distinct ways can Eliane choose her schedule? | 96 |
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