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Consider four function graphs, labelled (2) through (5). The domain of function (3) is now $$\{-5, -4, -3, -2, -1, 0, 1, 2, 3\}.$$ Determine the product of the labels of the functions which are invertible. The function expressions and domains for the other functions are:
- Function (2): $f(x) = x^2 - 4x + 3$ over the domain $[-1, 4]$
- Function (4): $g(x) = -\tan(x)$ over the domain $(-\frac{\pi}{2}, \frac{\pi}{2})$
- Function (5): $h(x) = 5/x$ over the domain $(-\infty, -0.2) \cup (0.2, \infty)$
|
60
|
4/8
|
In triangle $ABC$, $AB$ is congruent to $AC$, and the measure of angle $ABC$ is $60^{\circ}$. Segment $BD$ bisects angle $ABC$ with point $D$ on side $AC$. Point $E$ is on side $BC$ such that segment $DE$ is parallel to side $AB$, and point $F$ is on side $AC$ such that segment $EF$ is parallel to segment $BD$ and $EF$ bisects $\angle DEC$. How many isosceles triangles are present?
|
7
|
0/8
|
A botanical garden map shows that one inch represents 1200 feet on the ground. A line segment on the map is 6.5 inches long. An error in the map indicates a part of the path that doesn’t exist, which is shown as a 0.75-inch line segment. How many feet on the ground does the correct path cover?
|
6900 \text{ feet}
|
5/8
|
Suppose we have an equilateral triangle with vertices at \( (0, 0) \), \( (6, 0) \), and \( (3, 3\sqrt{3}) \), and we place 36 points equally spaced along the perimeter (including the vertices and dividing each side into 12 congruent sections). If \( P \), \( Q \), and \( R \) are any three non-collinear points among these, how many different possible positions are there for the centroid of \( \triangle PQR \)?
|
361
|
0/8
|
It takes 18 men working steadily 3 days to build a wall. If 3 of the men work twice as efficiently as the others, how many days would it take 30 men to build the same wall, assuming 5 of them are twice as efficient as the others?
|
1.8
|
0/8
|
A square and a regular hexagon are coplanar and share a common side $\overline{AD}$. Determine the degree measure of exterior angle $BAC$. Use a diagram for reference if needed.
|
150^\circ
|
3/8
|
A region is bounded by four quarter-circle arcs constructed at each corner of a square whose sides measure $4/\pi$. Calculate the perimeter of this region.
|
4
|
2/8
|
A frustum of a right circular cone was generated by slicing a smaller cone from the top of a larger cone. The larger cone originally had a height of 30 cm before the top was cut off to create the frustum. Now, the frustum that remains has a height of 18 cm and the radius of its upper base is 6 cm, while the radius of its lower base is 10 cm. Determine the altitude of the smaller cone that was cut off.
|
12 \text{ cm}
|
2/8
|
1. Determine all pairs $(p, q)$ of real numbers such that the equation $x^{2}+p x+q=0$ has a solution $v$ in the set of real numbers, and if $t$ is a root of this equation, then $|2 t-15|$ is also a root.
|
(-30, 225)
|
0/8
|
9. Given $z \in \mathbf{C}$. If the equation in $x$
$$
4 x^{2}-8 z x+4 i+3=0
$$
has real roots. Then the minimum value of $|z|$ is $\qquad$
|
1
|
2/8
|
A circle passes through vertices $A$ and $B$ of triangle $A B C$ and is tangent to line $A C$ at point $A$. Find the radius of the circle if $\angle B A C=\alpha, \angle A B C=\beta$ and the area of triangle $A B C$ is $S$.
|
\sqrt{ \dfrac{S \sin(\alpha + \beta)}{2 \sin^3 \alpha \sin \beta} }
|
1/8
|
8. Given that $A B C D$ and $A^{\prime} B^{\prime} C^{\prime} D^{\prime}$ are two rhombuses with side lengths of $\sqrt{3}+1$. If $A C \perp A^{\prime} C^{\prime}$, $\angle A B C=\angle A^{\prime} B^{\prime} C^{\prime}=120^{\circ}$, then the perimeter of the shaded part is $\qquad$
|
8
|
0/8
|
1. Arrange the consecutive natural numbers from 1 to 99 in sequence to form a large number:
1234567891011…979899,
By extracting four consecutive digits, you can get a four-digit number, such as 5678, 1011, etc. Among the four-digit numbers obtained by this method, the largest is . $\qquad$
|
9909
|
0/8
|
## Task 2 - 010932
Kurt is riding a tram along a long straight street. Suddenly, he sees his friend walking in the opposite direction on this street at the same level. After one minute, the tram stops. Kurt gets off and runs after his friend at twice the speed of his friend, but only at a quarter of the average speed of the tram.
How many minutes will it take for him to catch up? How did you arrive at your result?
|
9
|
5/8
|
Example 10. In the rectangular prism $A B C D-A_{1} B_{1} C_{1} D_{1}$, $B D=10 \mathrm{~cm}, \angle D_{1} B D=30^{\circ}, \angle A O D=60^{\circ}$. Find the distance between $A C$ and $B D_{1}$ (Figure 18).
|
\dfrac{5\sqrt{39}}{13}
|
3/8
|
The medians $A A^{\prime}$ and $B B^{\prime}$ of triangle $A B C$ intersect at point $M$, and $\angle A M B = 120^{\circ}$. Prove that the angles $A B^{\prime} M$ and $B A^{\prime} M$ cannot both be acute or both be obtuse.
|
\text{The angles } AB'M \text{ and } BA'M \text{ cannot both be acute or both be obtuse.}
|
2/8
|
Let $n$ be a positive integer. Ana and Banana are playing the following game:
First, Ana arranges $2n$ cups in a row on a table, each facing upside-down. She then places a ball under a cup
and makes a hole in the table under some other cup. Banana then gives a finite sequence of commands to Ana,
where each command consists of swapping two adjacent cups in the row.
Her goal is to achieve that the ball has fallen into the hole during the game. Assuming Banana has no information
about the position of the hole and the position of the ball at any point, what is the smallest number of commands
she has to give in order to achieve her goal?
|
n(2n - 1)
|
3/8
|
XXXV OM - I - Problem 9
Three events satisfy the conditions:
a) their probabilities are equal,
b) any two of them are independent,
c) they do not occur simultaneously.
Determine the maximum value of the probability of each of these events.
|
\dfrac{1}{2}
|
4/8
|
1. In some cells of a $1 \times 2021$ strip, one chip is placed in each. For each empty cell, the number equal to the absolute difference between the number of chips to the left and to the right of this cell is written. It is known that all the written numbers are distinct and non-zero. What is the minimum number of chips that can be placed in the cells?
|
1347
|
2/8
|
Nine cells of a $10 \times 10$ diagram are infected. At each step, a cell becomes infected if it was already infected or if it had at least two infected neighbors (among the 4 adjacent cells).
(a) Can the infection spread everywhere?
(b) How many initially infected cells are needed to spread the infection everywhere?
|
10
|
1/8
|
5. As shown in Figure 1, the plane containing
square $A B C D$ and the plane containing
square $A B E F$ form a $45^{\circ}$ dihedral angle.
Then the angle formed by the skew lines $A C$ and
$B F$ is $\qquad$
|
\arccos \frac{2 - \sqrt{2}}{4}
|
0/8
|
4. Among 25 coins, two are counterfeit. There is a device into which two coins can be placed, and it will show how many of them are counterfeit. Can both counterfeit coins be identified in 13 tests.
|
Yes
|
0/8
|
Problem 3. Inside the cube $A B C D A^{\prime} B^{\prime} C^{\prime} D^{\prime}$, consider the regular quadrilateral pyramid $S A B C D$ with base $A B C D$, such that $\angle\left(\left(S A^{\prime} B^{\prime}\right),\left(S C^{\prime} D^{\prime}\right)\right)=30^{\circ}$. Let point $M$ be on the edge $A^{\prime} D^{\prime}$ such that $\angle A^{\prime} B^{\prime} M=30^{\circ}$.
a) Find the angle between the apothem of the pyramid $S A B C D$ and the plane $(A B C)$.
b) Determine the tangent of the angle between the planes $(MAB')$ and $(S A B)$.
|
60^\circ
|
0/8
|
209. "Fibonacci Tetrahedron". Find the volume of the tetrahedron whose vertices are located at the points with coordinates $\left(F_{n}, F_{n+1}, F_{n+2}\right), \quad\left(F_{n+3}, F_{n+4}, F_{n+5}\right), \quad\left(F_{n+6}, F_{n+7}, F_{n+8}\right)$ and $\left(F_{n+9}, F_{n+10}, F_{n+11}\right)$, where $F_{i}$ is the $i$-th term of the Fibonacci sequence: $1,1,2,3,5,8 \ldots$.
|
0
|
2/8
|
II. Positive integers $x_{1}, x_{2}, \cdots, x_{n}\left(n \in \mathbf{N}_{+}\right)$ satisfy $x_{1}^{2}+x_{2}^{2}+\cdots+x_{n}^{2}=111$.
Find the maximum possible value of $S=\frac{x_{1}+x_{2}+\cdots+x_{n}}{n}$.
|
\dfrac{21}{4}
|
3/8
|
Given the side \( a \) and angle \( \alpha \) of the triangle \( ABC \). Additionally, it is known that the length of the angle bisector from vertex \( C \) is also \( a \). Can the triangle be constructed using these data with Euclidean construction methods?
|
\text{No}
|
1/8
|
1. Given $\min _{x \in R} \frac{a x^{2}+b}{\sqrt{x^{2}+1}}=3$.
(1) Find the range of $b$;
(2) For a given $b$, find $a$.
|
[3, \infty)
|
0/8
|
A regular quadrilateral pyramid has a circumscribed sphere whose center coincides with the center of the inscribed sphere. What is the angle between two adjacent side edges?
|
90^\circ
|
0/8
|
Problem 8.1. (15 points) In a $5 \times 5$ grid, each cell is painted one of three colors: red, blue, or green. To the right of each row, the total number of blue and red cells in that row is written, and below each column, the total number of blue and green cells in that column is written.
To the right of the table, the numbers $1,2,3,4,5$ appear in some order. Could the numbers $1,2,3,4,5$ also appear below the table in some order?
|
Yes
|
2/8
|
4. In a water-filled and tightly sealed aquarium in the shape of a rectangular parallelepiped measuring 3 m $\times 4$ m $\times 2$ m, there are two small balls: an aluminum one and a wooden one. At the initial moment, the aquarium is at rest, and the distance between the balls is 2 m. What is the greatest distance between the balls that can be observed if the aquarium starts to move with constant acceleration? Provide an example of the motion in which the maximum distance is achieved.
|
\sqrt{29}
|
3/8
|
Three, (50 points) In an $8 \times 8$ grid, each cell contains a knight or a knave. Knights always tell the truth, and knaves always lie. It is known that each resident in the grid says: “The number of knaves in my column is (strictly) greater than the number of knaves in my row.” Determine the number of arrangements of residents that satisfy this condition.
|
255
|
1/8
|
14.9. (SFRY, 74). For arbitrary $n \geqslant 3$ points $A_{1}$, $A_{2}, \ldots, A_{n}$ on a plane, no three of which lie on the same line, let $\alpha$ denote the smallest of the angles $\angle A_{i} A_{j} A_{k}$ formed by triples $A_{i}, A_{j}, A_{k}$ of distinct points. For each value of $n$, find the greatest value of $\alpha$. Determine for which arrangements of points this value is achieved.
|
\dfrac{180^\circ}{n}
|
4/8
|
49. In $\triangle A B C$, the sides opposite to $\angle A, \angle B, \angle C$ are $a, b, c$ respectively. There is an inequality $\left|\frac{\sin A}{\sin A+\sin B}+\frac{\sin B}{\sin B+\sin C}+\frac{\sin C}{\sin C+\sin A}-\frac{3}{2}\right|<m$, then $m=\frac{8 \sqrt{2}-5 \sqrt{5}}{6}$ is the smallest upper bound.
|
\dfrac{8\sqrt{2} - 5\sqrt{5}}{6}
|
2/8
|
11.1. Anya left the house, and after some time Vanya left from the same place and soon caught up with Anya. If Vanya had walked twice as fast, he would have caught up with Anya three times faster. How many times faster would Vanya have caught up with Anya (compared to the actual time) if, in addition, Anya walked twice as slow
|
7
|
1/8
|
Right triangle $A B C$ is divided by the height $C D$, drawn to the hypotenuse, into two triangles: $B C D$ and $A C D$. The radii of the circles inscribed in these triangles are 4 and 3, respectively. Find the radius of the circle inscribed in triangle $A B C$.
|
5
|
5/8
|
Let $ A \equal{} \{(a_1,\dots,a_8)|a_i\in\mathbb{N}$ , $ 1\leq a_i\leq i \plus{} 1$ for each $ i \equal{} 1,2\dots,8\}$.A subset $ X\subset A$ is called sparse if for each two distinct elements $ (a_1,\dots,a_8)$,$ (b_1,\dots,b_8)\in X$,there exist at least three indices $ i$,such that $ a_i\neq b_i$.
Find the maximal possible number of elements in a sparse subset of set $ A$.
|
5040
|
1/8
|
43. There are 100 consecutive natural numbers. Please arrange them in a certain order, then calculate the sum of every three adjacent numbers, and find the maximum number of sums that are even. $\qquad$
|
97
|
0/8
|
11.6. A stack consists of 300 cards: 100 white, 100 black, and 100 red. For each white card, the number of black cards lying below it is counted; for each black card, the number of red cards lying below it is counted; and for each red card, the number of white cards lying below it is counted. Find the maximum possible value of the sum of the three hundred resulting numbers.
|
20000
|
2/8
|
Triangle $ABC$ satisfies $AB=104$, $BC=112$, and $CA=120$. Let $\omega$ and $\omega_A$ denote the incircle and $A$-excircle of $\triangle ABC$, respectively. There exists a unique circle $\Omega$ passing through $A$ which is internally tangent to $\omega$ and externally tangent to $\omega_A$. Compute the radius of $\Omega$.
|
49
|
4/8
|
22. Try to write two sets of integer solutions for the indeterminate equation $x^{2}-2 y^{2}=1$ as $\qquad$ .
|
(3, 2)
|
0/8
|
7、The volunteer service team delivers newspapers to elderly people with mobility issues in the community. Xiao Ma is responsible for an elderly person living on the 7th floor. It takes 14 seconds to go up or down one floor, so it takes Xiao Ma a total of $\qquad$ seconds for a round trip up and down.
|
168
|
4/8
|
Example 6. Find the one-sided limits of the function $f(x)=\frac{6}{x-3}$ as $x \rightarrow 3$ from the left and from the right.
|
-\infty
|
3/8
|
Shaovalov A.v.
For which $n>2$ can the integers from 1 to $n$ be arranged in a circle so that the sum of any two adjacent numbers is divisible by the next number in the clockwise direction?
|
3
|
2/8
|
In an isosceles triangle, the legs are equal to \(a\) and the base is equal to \(b\). An inscribed circle in this triangle touches its sides at points \(M\), \(N\), and \(K\). Find the area of triangle \(MNK\).
|
\dfrac{b^2 (2a - b) \sqrt{4a^2 - b^2}}{16 a^2}
|
0/8
|
Yamenniko i.v.
The numbers 2, 3, 4, ..., 29, 30 are written on the board. For one ruble, you can mark any number. If a number is already marked, you can freely mark its divisors and numbers that are multiples of it. What is the minimum number of rubles needed to mark all the numbers on the board?
|
5
|
0/8
|
8. It is known that 99 wise men are seated around a large round table, each wearing a hat of one of two different colors. Among them, 50 people's hats are of the same color, and the remaining 49 people's hats are of the other color. However, they do not know in advance which 50 people have the same color and which 49 people have the other color. They can only see the colors of the hats on others' heads, but not their own. Now they are required to simultaneously write down the color of their own hat on the paper in front of them. Question: Can they pre-arrange a strategy to ensure that at least 74 of them write the correct color?
|
74
|
3/8
|
22. There are four cards, each with a number written on both sides. The first card has 0 and 1, the other three cards have 2 and 3, 4 and 5, 7 and 8 respectively. Now, any three of these cards are taken out and placed in a row, forming a total of $\qquad$ different three-digit numbers.
|
168
|
3/8
|
2. On a very long road, a race was organized. 22 runners started at different times, each running at a constant speed. The race continued until each runner had overtaken all the slower ones. The speeds of the runners who started first and last were the same, and the speeds of the others were different from theirs and distinct from each other.
What could be the number of overtakes, if each involved exactly two people? In your answer, indicate the largest and smallest possible numbers in any order, separated by a semicolon.
Example of answer format:
$10 ; 20$
|
20 ; 210
|
0/8
|
## Task 4 - 260924
For a right triangle $A B C$ with the right angle at $C$, it is required that this right angle be divided into four equal angles by the median of side $A B$, the angle bisector of $\angle A C B$, and the altitude perpendicular to side $A B$.
Investigate whether there exists a triangle $A B C$ that meets these requirements, and whether all such triangles are similar to each other!
Determine, if this is the case, the measures of the angles $\angle B A C$ and $\angle A B C$!
|
22.5^\circ
|
2/8
|
5. For different real numbers $m$, the equation $y^{2}-6 m y-4 x+$ $9 m^{2}+4 m=0$ represents different parabolas. A line intersects all these parabolas, and the length of the chord intercepted by each parabola is $\frac{8 \sqrt{5}}{9}$. Then the equation of this line is $\qquad$ .
|
y = 3x - \dfrac{1}{3}
|
4/8
|
7. Dijana used sticks of length $5 \mathrm{~cm}$ to form a row of equilateral triangles (as shown in the figure). If Dijana used 99 sticks, what is the distance between the two farthest points in the row of triangles formed?
The use of a pocket calculator or any reference materials is not allowed.
## SCHOOL/CITY COMPETITION IN MATHEMATICS
|
125
|
1/8
|
The sequence \(\{a_n\}\) is defined as follows: \(a_1\) is any positive integer. For integers \(n \geq 1\), \(a_{n+1}\) is the smallest positive integer that is coprime with \(\sum_{i=1}^{n} a_i\) and is different from \(a_1, \cdots, a_n\). Prove that every positive integer appears in the sequence \(\{a_n\}\).
|
\text{Every positive integer appears in the sequence }\{a_n\}
|
0/8
|
Problem 4. A field in the shape of a rectangle, with a length $20 \mathrm{~m}$ longer than its width, is fenced with 3 strands of wire, using a total of $840 \mathrm{~m}$ of wire. What is the perimeter and what are the lengths of the sides of the field?
|
280
|
1/8
|
A kindergarten group consisting of 5 girls and 7 boys is playing "house." They select from among themselves a bride and groom, a mother, two flower girls, a best man for the bride and one for the groom, a bridesmaid for the bride and one for the groom. We know that three of the girls each have a brother in the group and there are no other sibling pairs. How many ways can the selection be made if siblings cannot be the bride and groom and the mother cannot be a sibling of either the bride or the groom? (Flower girls can only be girls, best men can only be boys, and there are no gender restrictions for the witnesses.)
|
626400
|
0/8
|
10.228. One end of the diameter of a semicircle coincides with the vertex of the angle at the base of an isosceles triangle, and the other lies on this base. Find the radius of the semicircle if it touches one lateral side and divides the other into segments of lengths 5 and 4 cm, measured from the base.
|
\dfrac{15\sqrt{11}}{11}
|
1/8
|
9. Given that $D$ is a point on the side $BC$ of the equilateral $\triangle ABC$ with side length 1, the inradii of $\triangle ABD$ and $\triangle ACD$ are $r_{1}$ and $r_{2}$, respectively. If $r_{1}+r_{2}=\frac{\sqrt{3}}{5}$, then there are two points $D$ that satisfy this condition, denoted as $D_{1}$ and $D_{2}$. The distance between $D_{1}$ and $D_{2}$ is $\qquad$.
|
\dfrac{\sqrt{6}}{5}
|
4/8
|
One, (20 points) As shown in Figure 3, goods are to be transported from a riverside city $A$ to a location $B$ 30 kilometers away from the riverbank. According to the distance along the river, the distance from $C$ to $A$, $AC$, is 40 kilometers.
If the waterway transportation cost is half of the highway transportation cost, how should the point $D$ on the riverbank be determined, and a road be built from $B$ to $D$, so that the total transportation cost from $A$ to $B$ is minimized?
|
40 - 10\sqrt{3}
|
4/8
|
4. A rectangle $11 \times 12$ is cut into several strips $1 \times 6$ and $1 \times 7$. What is the minimum total number of strips?
|
20
|
0/8
|
2. Clever Dusya arranges six cheat sheets in four secret pockets so that the 1st and 2nd cheat sheets end up in the same pocket, the 4th and 5th cheat sheets also end up in the same pocket, but not in the same pocket as the 1st. The others can be placed anywhere, but only one pocket can remain empty (or all can be filled). In how many different ways can this be done?
#
|
144
|
1/8
|
II. (16 points) As shown in the figure, extend the sides $AB, BC, CD, DA$ of quadrilateral $ABCD$ to $E, F, G, H$ respectively, such that $\frac{BE}{AB} = \frac{CF}{BC} = \frac{DG}{CD} = \frac{AH}{DA} = m$. If $S_{EFGH} = 2 S_{ABCD}$ ($S_{EFGH}$ represents the area of quadrilateral $EFGH$), find the value of $m$.
---
Please note that the figure mentioned in the problem is not provided here.
|
\dfrac{\sqrt{3} - 1}{2}
|
4/8
|
An integer is called parity-monotonic if its decimal representation $a_{1}a_{2}a_{3}\cdots a_{k}$ satisfies $a_{i}<a_{i+1}$ if $a_{i}$ is odd , and $a_{i}>a_{i+1}$ if $a_{i}$ is even . How many four-digit parity-monotonic integers are there?
Please give the answer directly without any intermediate steps.
|
640
|
2/8
|
8. (10 points) When withdrawing money from an $A T M$ machine, one needs to input the bank card password to proceed to the next step. The password is a 6-digit number ranging from 000000 to 999999. A person forgot the password but remembers that it contains the digits 1, 3, 5, 7, 9 and no other digits. If there is no limit to the number of incorrect password attempts, the person can input $\qquad$ different passwords at most to proceed to the next step.
|
1800
|
4/8
|
Example 1. Find the solution $(x, y)$ that satisfies the equation $\frac{1}{x}-\frac{1}{y}=\frac{1}{12}$, and makes $y$ the largest positive integer.
|
(11, 132)
|
5/8
|
Let $F = \max_{1 \leq x \leq 3} |x^3 - ax^2 - bx - c|$. When $a$, $b$, $c$ run over all the real numbers, find the smallest possible value of $F$.
|
\dfrac{1}{4}
|
2/8
|
1. Find all real numbers $x$ such that $\left[x^{3}\right]=4 x+3$. Here $[y]$ denotes the greatest integer not exceeding the real number $y$.
（Yang Wenpeng）
|
-1
|
2/8
|
The 6th problem of the 3rd China Northern Hope Star Mathematics Summer Camp:
Given non-negative real numbers $a, b, c, x, y, z$ satisfying $a+b+c=x+y+z=1$.
Find the maximum value of $\left(a-x^{2}\right)\left(b-y^{2}\right)\left(c-z^{2}\right)$.
|
\dfrac{1}{16}
|
2/8
|
5. We need to fill a $3 \times 3$ table with nine given numbers such that in each row and column, the largest number is the sum of the other two. Determine whether it is possible to complete this task with the numbers
a) 1, 2, 3, 4, 5, 6, 7, 8, 9;
b) 2, 3, 4, 5, 6, 7, 8, 9, 10.
If yes, find out how many ways the task can be completed such that the largest number is in the center of the table.
(Jaromír Šimša)
|
0
|
0/8
|
A regular triangle and a square are inscribed into a circle of radius \(R\) such that they share a common vertex. Calculate the area of their common part.
|
\dfrac{(8\sqrt{3} - 9)}{4} R^2
|
0/8
|
26th CanMO 1994 Problem 4 AB is the diameter of a circle. C is a point not on the line AB. The line AC cuts the circle again at X and the line BC cuts the circle again at Y. Find cos ACB in terms of CX/CA and CY/CB. Solution
|
\sqrt{ \dfrac{CX}{CA} \cdot \dfrac{CY}{CB} }
|
0/8
|
6 A regular tetrahedron $D-ABC$ has a base edge length of 4 and a side edge length of 8. A section $\triangle AEF$ is made through point $A$ intersecting side edges $DB$ and $DC$. What is the minimum perimeter of $\triangle AEF$? $\qquad$ .
|
11
|
4/8
|
Highimiy and.
In Anchuria, a day can be either clear, when the sun shines all day, or rainy, when it rains all day. And if today is not like yesterday, the Anchurians say that the weather has changed today. Once, Anchurian scientists established that January 1 is always clear, and each subsequent day in January will be clear only if the weather changed exactly a year ago on that day. In 2015, January in Anchuria was quite varied: sunny one day, rainy the next. In which year will the weather in January first change in exactly the same way as it did in January 2015?
|
2047
|
0/8
|
3. Find the area of the region defined by the inequality: $|y-| x-2|+| x \mid \leq 4$.
Translate the above text into English, please retain the original text's line breaks and format, and output the translation result directly.
|
32
|
2/8
|
1. Isaac writes down a three-digit number. None of the digits is a zero. Isaac gives his sheet with the number to Dilara, and she writes under Isaac's number all the three-digit numbers that can be obtained by rearranging the digits of Isaac's number. Then she adds up all the numbers on the sheet. The result is 1221. What is the largest number that Isaac could have written down?
|
911
|
4/8
|
Example 12. Find $\int\left(2^{3 x}-1\right)^{2} \cdot 4^{x} d x$.
|
\frac{2^{8x}}{8 \ln 2} - \frac{2 \cdot 2^{5x}}{5 \ln 2} + \frac{2^{2x}}{2 \ln 2} + C
|
1/8
|
Given a positive integer $N$. There are three squirrels that each have an integer. It is known that the largest integer and the least one differ by exactly $N$. Each time, the squirrel with the second largest integer looks at the squirrel with the largest integer. If the integers they have are different, then the squirrel with the second largest integer would be unhappy and attack the squirrel with the largest one, making its integer decrease by two times the difference between the two integers. If the second largest integer is the same as the least integer, only of the squirrels would attack the squirrel with the largest integer. The attack continues until the largest integer becomes the same as the second largest integer. What is the maximum total number of attacks these squirrels make?
Proposed by USJL, ST.
|
N
|
1/8
|
A fair 100-sided die is rolled twice, giving the numbers $a$ and $b$ in that order. If the probability that $a^2 - 4b$ is a perfect square is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers, compute $100m + n$.
|
6900
|
0/8
|
On a natural number $n$, you are allowed two operations:
1. Multiply $n$ by $2$, or
2. Subtract $3$ from $n$.
For example, starting with $8$, you can reach $13$ as follows: $8 \longrightarrow 16 \longrightarrow 13$. You need two steps, and you cannot do it in less than two steps.
Starting from $11$, what is the least number of steps required to reach $121$?
|
10
|
0/8
|
Michael writes down all the integers between $1$ and $N$ inclusive on a piece of paper and discovers that exactly $40\%$ of them have leftmost digit $1$. Given that $N > 2017$, find the smallest possible value of $N$.
|
1481480
|
2/8
|
A football tournament is played between 5 teams, each pair playing exactly one match. Points are awarded as follows:
- 5 points for a victory
- 0 points for a loss
- 1 point each for a draw with no goals scored
- 2 points each for a draw with goals scored
In the final ranking, the five teams have points that are 5 consecutive numbers. Determine the least number of goals that could be scored in the tournament.
|
6
|
0/8
|
Given triangle $ABC$ and a point $P$ inside it, $\angle BAP=18^\circ$, $\angle CAP=30^\circ$, $\angle ACP=48^\circ$, and $AP=BC$. If $\angle BCP=x^\circ$, find $x$.
|
6
|
5/8
|
A quadruple $(a, b, c, d)$ of distinct integers is said to be $\text{balanced}$ if $a + c = b + d$. Let $\mathcal{S}$ be any set of quadruples $(a, b, c, d)$ where $1 \leq a < b < d < c \leq 20$ and where the cardinality of $\mathcal{S}$ is $4411$. Find the least number of balanced quadruples in $\mathcal{S}$.
|
91
|
3/8
|
Let $S(n)$ be the sum of the squares of the positive integers less than and coprime to $n$. For example, $S(5) = 1^2 + 2^2 + 3^2 + 4^2$, but $S(4) = 1^2 + 3^2$. Let $p = 2^7 - 1 = 127$ and $q = 2^5 - 1 = 31$ be primes. The quantity $S(pq)$ can be written in the form \( \frac{p^2q^2}{6}\left(a - \frac{b}{c} \right) \) where $a$, $b$, and $c$ are positive integers, with $b$ and $c$ coprime and $b < c$. Find $a$.
|
7561
|
3/8
|
Let $ABCD$ be a trapezoid with $AB \parallel CD$ and $AD = BD$. Let $M$ be the midpoint of $AB$, and let $P \neq C$ be the second intersection of the circumcircle of $\triangle BCD$ and the diagonal $AC$. Suppose that $BC = 27$, $CD = 25$, and $AP = 10$. If $MP = \frac{a}{b}$ for relatively prime positive integers $a$ and $b$, compute $100a + b$.
|
2705
|
1/8
|
21. Let $S_{n}$ be the sum of the first $n$ terms of an arithmetic sequence. Assume that $S_{3}=9, S_{20}$ $>0$, and $S_{21}<0$. Then the range of the common difference $d$ is $\qquad$ , the maximum term of the sequence $S_{1}, S_{2}, S_{3}, \cdots$ is $\qquad$
Note: term - one item, arithmetic sequence - arithmetic sequence, common difference - common difference, maximum term - maximum (value) term
|
S_{10}
|
2/8
|
6. 2 teachers are taking 30 students across a river by boat, but there is only one small boat that can carry 8 people (no boatman). Each time they cross the river, 2 teachers are needed to act as boatmen. To get everyone across, the boat must make at least $\qquad$ trips. (A round trip counts as 2 trips)
|
9
|
5/8
|
## Task A-4.5. (4 points)
A rectangular path with a width of $1.5 \mathrm{~m}$ and a length of $20 \mathrm{~m}$ needs to be tiled with identical tiles in the shape of an isosceles right triangle with legs of length $50 \mathrm{~cm}$, such that the legs are parallel to the sides of the rectangle. Determine the number of ways this can be done.
|
2^{120}
|
0/8
|
# Problem 4. (10 points)
On December 31 at 16:35, Misha realized he had no New Year's gifts for his entire family. He wants to give different gifts to his mother, father, brother, and sister. Each of the gifts is available in 4 stores: Romashka, Odynachik, Nezabudka, and Lysichka, which close at 20:00. The journey from home to each store and between any two stores takes 30 minutes.
The table below shows the cost of the gifts in all four stores and the time Misha will need to spend shopping in each store. What is the minimum amount of money Misha can spend if he must definitely manage to buy all 4 gifts?
| | mother | father | brother | sister | Time spent in the store (min.) |
| :--- | :--- | :--- | :--- | :--- | :--- |
| Romashka | 1000 | 750 | 930 | 850 | 35 |
| Odynachik | 1050 | 790 | 910 | 800 | 30 |
| Nezabudka | 980 | 810 | 925 | 815 | 40 |
| :--- | :--- | :--- | :--- | :--- | :--- |
| Lysichka | 1100 | 755 | 900 | 820 | 25 |
|
3435
|
0/8
|
Shapovalov A.V.
In a small town, there is only one tram line. It is a circular line, and trams run in both directions. There are stops called Circus, Park, and Zoo on the loop. The journey from Park to Zoo via Circus is three times longer than the journey not via Circus. The journey from Circus to Zoo via Park is half as short as the journey not via Park. Which route from Park to Circus is shorter - via Zoo or not via Zoo - and by how many times?
|
11
|
3/8
|
## Task 6 - 080736
The great German mathematician Carl Friedrich Gauss was born on April 30, 1777, in Braunschweig.
On which day of the week did his birthday fall?
(April 30, 1967, was a Sunday; the years 1800 and 1900 were not leap years).
|
Wednesday
|
4/8
|
4・ 203 There are two coal mines, A and B. Coal from mine A releases 4 calories when burned per gram, and coal from mine B releases 6 calories when burned per gram. The price of coal at the origin is: 20 yuan per ton for mine A, and 24 yuan per ton for mine B. It is known that: the transportation cost of coal from mine A to city N is 8 yuan per ton. If the coal from mine B is to be transported to city N, how much should the transportation cost per ton be to make it more economical than transporting coal from mine A?
|
18
|
3/8
|
4. A warehouse stores 400 tons of cargo, with the weight of each being a multiple of a centner and not exceeding 10 tons. It is known that any two cargos have different weights. What is the minimum number of trips that need to be made with a 10-ton truck to guarantee the transportation of these cargos from the warehouse?
|
51
|
0/8
|
Among all pairs of real numbers $(x, y)$ such that $\sin \sin x = \sin \sin y$ with $-10 \pi \le x, y \le 10 \pi$, Oleg randomly selected a pair $(X, Y)$. Compute the probability that $X = Y$.The answer is in the form rac{m}{n}, where gcd(m, n) = 1. Please provide the value of m + n.
|
21
|
2/8
|
An arithmetic sequence consists of $ 200$ numbers that are each at least $ 10$ and at most $ 100$. The sum of the numbers is $ 10{,}000$. Let $ L$ be the least possible value of the $ 50$th term and let $ G$ be the greatest possible value of the $ 50$th term. What is the value of $ G - L$?The answer is in the form rac{m}{n}, where gcd(m, n) = 1. Please provide the value of m + n.
|
8279
|
5/8
|
Find $XY$ in the triangle below.
[asy]
unitsize(1inch);
pair P,Q,R;
P = (0,0);
Q= (1,0);
R = (0,1);
draw (P--Q--R--P,linewidth(0.9));
draw(rightanglemark(Q,P,R,3));
label("$X$",P,S);
label("$Y$",Q,S);
label("$Z$",R,N);
label("$12\sqrt{2}$",R/2,W);
label("$45^\circ$",(0.7,0),N);
[/asy]The answer is in the form k\sqrt{m}+n,. Please provide the value of k + m + n.
|
14
|
2/8
|
2. Given $x \geqslant 0, x^{2}+(y-4)^{2} \leqslant 4$, let $u=$ $\frac{x^{2}+\sqrt{3} x y+2 y^{2}}{x^{2}+y^{2}}$. Then the range of $u$ is $\qquad$ .
|
[2, \dfrac{5}{2}]
|
3/8
|
Given that \( O \) is the origin, \( N(1,0) \), \( M \) is a moving point on the line \( x=-1 \), and the angle bisector of \(\angle M O N \) intersects the line segment \( MN \) at point \( P \):
(1) Find the equation of the locus of point \( P \);
(2) Draw a line \( l \) passing through point \( Q \left(-\frac{1}{2}, -\frac{1}{2}\right) \) with slope \( k \). If line \( l \) intersects the curve \( E \) at exactly one point, determine the range of values for \( k \).
|
y^2 = x
|
0/8
|
24. Find the sum of all even factors of 1152 .
|
3302
|
4/8
|
3. Calculate $\frac{1}{1 \times 3 \times 5}+\frac{1}{3 \times 5 \times 7}+\frac{1}{5 \times 7 \times 9}+\cdots+\frac{1}{2001 \times 2003 \times 2005}$.
|
\dfrac{1}{12} - \dfrac{1}{4 \times 2003 \times 2005}
|
0/8
|
Given two points on the sphere and the circle $k$, which passes through exactly one of the two points. How many circles are there on the sphere that pass through both points and are tangent to $k$?
|
1
|
0/8
|
Twenty-seven balls labelled from $1$ to $27$ are distributed in three bowls: red, blue, and yellow. What are the possible values of the number of balls in the red bowl if the average labels in the red, blue and yellow bowl are $15$, $3$, and $18$, respectively?
|
11
|
0/8
|
Subsets and Splits
Filtered Answers A-D
Retrieves 100 rows where the answer is a single letter from A to D, providing basic filtering of the dataset.