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50,350 | Let $a_1,a_2,a_3,a_4,a_5$ be distinct real numbers. Consider all sums of the form $a_i + a_j$ where $i,j \in \{1,2,3,4,5\}$ and $i \neq j$. Let $m$ be the number of distinct numbers among these sums. What is the smallest possible value of $m$? | 7 |
18,525 | Mommy preserves plums in jars so that the plums from one jar are enough for either 16 cups, or 4 pies, or half a sheet of fruit slices.
In the pantry, she has 4 such jars and wants to bake one sheet of fruit slices and 6 pies. $\mathrm{On}$ how many cups will the remaining plums be enough?
(M. Petrová) | 8 |
25,021 | Find the smallest \( n > 4 \) for which we can find a graph on \( n \) points with no triangles and such that for every two unjoined points we can find just two points joined to both of them. | 16 |
63,491 | 9. To plant willow trees and peach trees along the edges of a rectangular school playground, it is known that the playground is 150 meters long and 60 meters wide. It is required that the distance between every two willow trees is 10 meters, and the distance between every two peach trees is also 10 meters. Additionally, each willow tree must have peach trees on both sides, and each peach tree must have willow trees on both sides. How many willow trees and how many peach trees are needed? | 42 |
53,128 | 9. (16 points) Given $a, b>0, a+b=1$. Find
$$
y=\left(a+\frac{1}{2015 a}\right)\left(b+\frac{1}{2015 b}\right)
$$
the minimum value. | \frac{2\sqrt{2016}-2}{2015} |
17,301 | Let $ABC$ be equilateral , and $D, E,$ and $F$ be the midpoints of $\overline{BC}, \overline{CA},$ and $\overline{AB},$ respectively. There exist points $P, Q,$ and $R$ on $\overline{DE}, \overline{EF},$ and $\overline{FD},$ respectively, with the property that $P$ is on $\overline{CQ}, Q$ is on $\overline{AR},$ and $R$ is on $\overline{BP}.$ The ratio of the area of triangle $ABC$ to the area of triangle $PQR$ is $a + b\sqrt {c},$ where $a, b$ and $c$ are integers, and $c$ is not divisible by the square of any prime . What is $a^{2} + b^{2} + c^{2}$ | 83 |
54,356 | [Chain (continuous) fractions]
Solve the equation in positive integers
$$
1-\frac{1}{2+\frac{1}{3+\ldots+\frac{1}{n-1+\frac{1}{n}}}}=\frac{1}{x_{1}+\frac{1}{x_{2}+\ldots \ddots+\frac{1}{x_{n-1}+\frac{1}{x_{n}}}}}
$$ | x_{1}=1,x_{2}=1,x_{3}=3,x_{4}=4,\ldots,x_{n}=n |
11,756 | The domain of the function $f$ is the set of real numbers. For every number $a$, if $a<x<a+100$, then $a \leq f(x) \leq a+100$. Determine the function $f$! | f(x) = x |
28,595 | 8. A line $l$ is drawn through the right focus of the hyperbola $x^{2}-\frac{y^{2}}{2}=1$ intersecting the hyperbola at points $A$ and $B$. If a real number $\lambda$ makes $|A B|=\lambda$ such that there are exactly 3 such lines $l$, then $\lambda=$ $\qquad$. | 4 |
8,270 | 14. (15 points) The main staircase of a mall hall is shown in the figure. There are 15 steps from the 1st floor to the 2nd floor, each step is 16 cm high and 26 cm deep. It is known that the staircase is 3 meters wide. To lay a carpet on the stairs from the 1st floor to the 2nd floor, which costs 80 yuan per square meter, how much money is needed at least to buy the carpet? | 1512 |
11,446 | The sequence $\left\{a_{n}\right\}$ satisfies $a_{1}=15$ and $a_{n+1}=\frac{\sqrt{a_{n}^{2}+1}-1}{a_{n}}$ for $n=1,2,\cdots$. Prove that
$$
a_{n}>\frac{3}{2^{n}}, \quad n=1,2,\cdots.
$$ | a_n > \frac{3}{2^n} |
54,288 | Ex. 103. A diameter divides a circle into two parts, one of which contains a smaller inscribed circle touching the larger circle at point $M$, and the diameter at point $K$. The ray $MK$ intersects the larger circle a second time at point
$N$. Find the length of $MN$, if the sum of the distances from point $M$ to the ends of the diameter is 6. | 3\sqrt{2} |
30,939 | 12. Let $\mathbb{N}$ be the set of all positive integers. A function $f: \mathbb{N} \rightarrow \mathbb{N}$ satisfies $f(m+$ $n)=f(f(m)+n)$ for all $m, n \in \mathbb{N}$, and $f(6)=2$. Also, no two of the values $f(6), f(9), f(12)$ and $f(15)$ coincide. How many three-digit positive integers $n$ satisfy $f(n)=f(2005)$ ? | 225 |
51,134 | 8. (10 points) In the inscribed quadrilateral $A B C D$, the degree measures of the angles are in the ratio $\angle A: \angle B: \angle C=2: 3: 4$. Find the length of $A C$, if $C D=8, B C=7.5 \sqrt{3}-4$. | 17 |
56,898 | Given four points not lying in the same plane. How many planes are there from which all four points are equidistant? | 7 |
59,281 | 16. Team A and Team B each send out 7 players to compete in a Go broadcast tournament, following a predetermined order. The competition starts with the No. 1 players from both teams. The loser is eliminated, and the winner then faces the No. 2 player from the losing team, $\cdots$, until all players from one team are eliminated, and the other team wins, forming a competition process. It is known that Team A only used their first 5 players to consecutively defeat all 7 players from Team B. How many such competition processes are possible? | 210 |
53,661 | [b]p1.[/b] How many real solutions does the following system of equations have? Justify your answer.
$$x + y = 3$$
$$3xy -z^2 = 9$$
[b]p2.[/b] After the first year the bank account of Mr. Money decreased by $25\%$, during the second year it increased by $20\%$, during the third year it decreased by $10\%$, and during the fourth year it increased by $20\%$. Does the account of Mr. Money increase or decrease during these four years and how much?
[b]p3.[/b] Two circles are internally tangent. A line passing through the center of the larger circle intersects it at the points $A$ and $D$. The same line intersects the smaller circle at the points $B$ and $C$. Given that $|AB| : |BC| : |CD| = 3 : 7 : 2$, find the ratio of the radiuses of the circles.
[b]p4.[/b] Find all integer solutions of the equation $\frac{1}{x}+\frac{1}{y}=\frac{1}{19}$
[b]p5.[/b] Is it possible to arrange the numbers $1, 2, . . . , 12$ along the circle so that the absolute value of the difference between any two numbers standing next to each other would be either $3$, or $4$, or $5$? Prove your answer.
[b]p6.[/b] Nine rectangles of the area $1$ sq. mile are located inside the large rectangle of the area $5$ sq. miles. Prove that at least two of the rectangles (internal rectangles of area $1$ sq. mile) overlap with an overlapping area greater than or equal to $\frac19$ sq. mile
PS. You should use hide for answers. | 0 |
63,102 | 8. Given $a, b, c \in R, a^{2}+b^{2}+c^{2}=1$, then the minimum value of $3 a b-3 b c+2 c^{2}$ is $\qquad$
Translate the above text into English, please retain the original text's line breaks and format, and output the translation result directly. | -\frac{1+\sqrt{7}}{2} |
25,608 | 6. Given that $f(x)$ is an odd function defined on $\mathbf{R}$, $f(1)=1$, and for any $x<0$, we have
$$
\begin{array}{l}
f\left(\frac{1}{x}\right)=x f\left(\frac{1}{1-x}\right) . \\
\text { Then } \sum_{k=1}^{1009} f\left(\frac{1}{k}\right) f\left(\frac{1}{2019-k}\right)=
\end{array}
$$
$\qquad$ | \frac{2^{2016}}{2017!} |
27,483 | 30. Several families lived in one house. In total, there are more children in these families than adults: there are more adults than boys; more boys than girls; and more girls than families. There are no childless families, and no families have the same number of children. Each girl has at least one brother and at most one sister. In one of the families, there are more children than in all the others combined. How many families lived in the house? How many boys and how many girls are there in each of these families? | 3 |
57,589 | 1.38
$$
\frac{\sqrt{1+\left(\frac{x^{2}-1}{2 x}\right)^{2}}}{\left(x^{2}+1\right) \frac{1}{x}}
$$ | \frac{x}{2|x|} |
2,328 | In the figure, \( L_{1} \) and \( L_{2} \) are tangents to the three circles. If the radius of the largest circle is 18 and the radius of the smallest circle is \( 4b \), find \( c \), where \( c \) is the radius of circle \( W \). | 12 |
26,039 | 6.64*. In a regular $n$-gon ( $n \geqslant 3$ ), the midpoints of all sides and diagonals are marked. What is the maximum number of marked points that can lie on one circle? | n |
6,566 | The center of a balloon is observed by two ground observers at angles of elevation of $45^{\circ}$ and $22.5^{\circ}$, respectively. The first observer is to the south, and the second one is to the northwest of the point directly under the balloon. The distance between the two observers is 1600 meters. How high is the balloon floating above the horizontal ground? | 500 |
22,696 | Find the value of $\cos \theta+2 \cos 2 \theta+3 \cos 3 \theta+\cdots+n \cos n \theta$. | \dfrac{(n + 1)\cos n\theta - n\cos(n + 1)\theta - 1}{4\sin^2\left(\dfrac{\theta}{2}\right)} |
11,286 | Given a cube \( ABCDA_1B_1C_1D_1 \). What is the smaller part of the volume separated by the plane passing through the midpoint of edge \(AD\) and points \(X\) and \(Y\) on edges \(AA_1\) and \(CC_1\) such that \( \frac{AX}{A_1X} = \frac{CY}{C_1Y} = \frac{1}{7} \)? | \dfrac{25}{192} |
54,800 | 2. Gari took a 6 -item multiple choice test with 3 choices per item, labelled $A, B$, and $C$. After the test, he tried to recall his answers to the items. He only remembered that he never answered three consecutive A's, he never answered three consecutive $B$ 's, and he did not leave any item blank. How many possible sets of answers could Gari have had? | 569 |
28,249 | Given an isosceles triangle $A B C$ with base $A C$. A circle of radius $R$ with center at point $O$ passes through points $A$ and $B$ and intersects the line $B C$ at point $M$, distinct from $B$ and $C$. Find the distance from point $O$ to the center of the circumcircle of triangle $A C M$. | R |
56,506 | Problem 5.6. A three-digit number and two two-digit numbers are written on the board. The sum of the numbers that contain a seven in their notation is 208. The sum of the numbers that contain a three in their notation is 76. Find the sum of all three numbers. | 247 |
66,279 | We want to color the three-element parts of $\{1,2,3,4,5,6,7\}$, such that if two of these parts have no element in common, then they must be of different colors. What is the minimum number of colors needed to achieve this goal? | 3 |
65,968 | 16. Fill in each square in $\square \square+\square \square=\square \square \square$ with a digit from $0, 1, 2, \cdots \cdots, 9$ (digits in the squares can be the same, and the leading digit of any number cannot be 0), so that the equation holds.
There are | 4860 |
66,654 | I5.2 If $1-\frac{4}{x}+\frac{4}{x^{2}}=0, b=\frac{a}{x}$, find $b$. | 1 |
992 |
The rays \( AB \) and \( DC \) intersect at point \( P \), and the rays \( BC \) and \( AD \) intersect at point \( Q \). It is known that the triangles \( ADP \) and \( QAB \) are similar (vertices are not necessarily listed in corresponding order), and the quadrilateral \( ABCD \) can be inscribed in a circle with radius 4.
a) Find \( AC \).
b) Additionally, it is known that the circles inscribed in triangles \( ABC \) and \( ACD \) touch the segment \( AC \) at points \( K \) and \( T \) respectively, and \( CK: KT: TA = 3:1:4 \) (point \( T \) lies between \( K \) and \( A \)). Find \(\angle DAC \) and the area of the quadrilateral \( ABCD \). | 8 |
67,172 | (6) In an isosceles right triangle $\triangle ABC$, $CA = CB = 1$, and point $P$ is any point on the boundary of $\triangle ABC$. Find the maximum value of $PA \cdot PB \cdot PC$. (Li Weiguo) | \frac{\sqrt{2}}{4} |
50,274 | 16. If a positive integer cannot be written as the difference of two square numbers, then the integer is called a "cute" integer. For example, 1,2 and 4 are the first three "cute" integers. Find the $2010^{\text {th }}$ "cute" integer.
(Note: A square number is the square of a positive integer. As an illustration, 1,4,9 and 16 are the first four square numbers.) | 8030 |
58,333 | 4. Ana chose the digits $1,2,3,4,5,6,7$ and 9. She decided to form groups of 4 prime numbers and use all the chosen digits for each group of prime numbers. What is the sum of the prime numbers in each group?
Naloge rešuj samostojno. Za reševanje imaš na voljo 210 minut.
Uporaba zapiskov, literature ali žepnega računala ni dovoljena.
48th Mathematical Competition
for high school students of Slovenia
Ljutomer, April 17, 2004
## Tasks for 2nd year students | 190 |
23,045 | 【Question 15】Junjun starts from $\mathrm{A}$ and travels at a constant speed to $\mathrm{B}$. When Junjun starts, Aping starts from $\mathrm{B}$ and travels at a constant speed to $\mathrm{A}$. They meet at point $\mathrm{C}$ along the way. After meeting, Junjun walks another 100 meters and then turns around to chase Aping, catching up to Aping 360 meters from $\mathrm{C}$. When Junjun catches up to Aping, he immediately turns around and heads to $\mathrm{B}$. As a result, when Junjun arrives at $\mathrm{B}$, Aping also arrives at $\mathrm{A}$. What is the distance between $\mathrm{A}$ and $\mathrm{B}$ in meters? | 1656 |
62,578 | An infinite sheet of paper is divided into equal squares, some of which are colored red. In each $2\times3$ rectangle, there are exactly two red squares. Now consider an arbitrary $9\times11$ rectangle. How many red squares does it contain? (The sides of all considered rectangles go along the grid lines.) | 33 |
54,436 | Find the largest integer $n$ which equals the product of its leading digit and the sum of its digits. | 48 |
62,261 | 18. (6 points) There are 300 balls in total, including white balls and red balls, and 100 boxes. Each box contains 3 balls, with 27 boxes containing 1 white ball, 42 boxes containing 2 or 3 red balls, and the number of boxes containing 3 white balls is the same as the number of boxes containing 3 red balls. Therefore, the total number of white balls is.
balls. | 158 |
50,695 | 5. As shown in the figure, quadrilaterals $A B C D$ and $C E F G$ are two squares. Given that $\frac{S_{\triangle A B I}}{S_{\triangle E F I}}=27$, find $\frac{A B}{E F}$. | 3 |
31,065 | We draw squares on the sides of a right-angled triangle. Calculate the area of the hexagon defined by the outer (not coinciding with the vertices of the triangle) vertices of the squares, if we know the hypotenuse of our right-angled triangle $(c)$, as well as the sum of the legs $(d)$. | ^{2}+^{2} |
51,258 | $3.420 \sin 10^{\circ} \cdot \sin 20^{\circ} \cdot \sin 30^{\circ} \cdot \sin 40^{\circ} \cdot \sin 50^{\circ} \cdot \sin 60^{\circ} \cdot \sin 70^{\circ} \cdot \sin 80^{\circ}=\frac{3}{256} \cdot$ | \frac{3}{256} |
59,169 | 3. Egor wrote a number on the board and encrypted it according to the rules of letter puzzles (different letters correspond to different digits, the same letters correspond to the same digits). The result was the word "GUATEMALA". How many different numbers could Egor have initially written if his number was divisible by 8? | 67200 |
10,387 | 9. Investment funds A, B and C claim that they can earn profits of $200 \%, 300 \%$ and $500 \%$ respectively in one year. Tommy has $\$ 90000$ and plans to invest in these funds. However, he knows that only one of these funds can achieve its claim while the other two will close down. He has thought of an investment plan which can guarantee a profit of at least $\$ n$ in one year. Find the greatest possible value of $n$.
(1 mark)
甲、乙、丙三個投資基金分別聲稱可在一年內賺取 $200 \%$ 、 $300 \%$ 和 $500 \%$ 的利潤。湯美有 90000 元，他打算用這筆錢投資在這些基金上。可是他知道，只有一個基金可以兌現承諾, 其餘兩個則會倒閉。他想到一個投資計劃, 可以確保一年後獲得最少 $n$ 元的淨利潤。求 $n$ 的最大可能值。
(1 分) | 30000 |
13,135 | ## Problem Statement
Calculate the indefinite integral:
$$
\int(5 x-2) e^{3 x} d x
$$ | \frac{(15x - 11)}{9} e^{3x} + C |
27,065 | 5. In an isosceles trapezoid \(ABCD\) with bases \(AD\) and \(BC\), perpendiculars \(BH\) and \(DK\) are drawn from vertices \(B\) and \(D\) to the diagonal \(AC\). It is known that the feet of the perpendiculars lie on the segment \(AC\) and \(AC=20\), \(AK=19\), \(AH=3\). Find the area of trapezoid \(ABCD\).
(10 points) | 120 |
7,935 | A geometric sequence $(a_n)$ has $a_1=\sin x$ $a_2=\cos x$ , and $a_3= \tan x$ for some real number $x$ . For what value of $n$ does $a_n=1+\cos x$ | 8 |
20,519 | How many positive integers $x$ with $200 \leq x \leq 600$ have exactly one digit that is a prime number? | 156 |
24,535 | Given \(\sin \alpha + \sin (\alpha + \beta) + \cos (\alpha + \beta) = \sqrt{3}\), where \(\beta \in \left[\frac{\pi}{4}, \pi\right]\), find the value of \(\beta\). | \dfrac{\pi}{4} |
12,032 |
Let \( n = 34000 \). Among the vertices of a regular \( n \)-gon \( A_{1} A_{2} \ldots A_{n} \), the vertices \( A_{i} \) are painted red if the index \( i \) is a power of 2, i.e., \( i = 1, 2, 4, 8, 16, \ldots \). In how many ways can you select 400 vertices of this \( n \)-gon such that they form a regular 400-gon and none of them are red? | 77 |
1,202 | 2. Write down the first 15 consecutive odd numbers in a row. Divide the numbers in this row into groups so that the first group contains one number, the second group contains the next two numbers, the third group contains three numbers, and so on. Find the sum of the numbers in each group and indicate the common pattern for all groups. | n^3 |
34,179 | 3. On the hundredth year of his reign, the Immortal Treasurer decided to start issuing new coins. In this year, he put into circulation an unlimited supply of coins with a value of $2^{100}-1$, the following year - with a value of $2^{101}-1$, and so on. As soon as the value of the next new coin can be exactly matched using the previously issued new coins, the Treasurer will be removed from office. In which year of his reign will this happen? (I. Bogdanov) | 200 |
30,749 | 46th Putnam 1985 Problem A3 x is a real. Define a i 0 = x/2 i , a i j+1 = a i j 2 + 2 a i j . What is lim n→∞ a n n ? Solution | e^x-1 |
30,495 | In the tetrahedron $A-BCD$, $AB=CD=5$, $AC=BD=\sqrt{34}$, $AD=BC=\sqrt{41}$, then the radius of the circumscribed sphere of the tetrahedron $A-BCD$ is | \frac{5\sqrt{2}}{2} |
18,872 | In a convex hexagon, two random diagonals are chosen independently of each other. Find the probability that these diagonals intersect inside the hexagon (inside means not at a vertex).
# | \dfrac{5}{12} |
11,480 | Given a triangle \(ABC\), let \(D\) be the point on the ray \(BA\) such that \(BD = BA + AC\). If \(K\) and \(M\) are points on the sides \(BA\) and \(BC\), respectively, such that the triangles \(BDM\) and \(BCK\) have the same areas, prove that \(\angle BKM = \frac{1}{2} \angle BAC\). | \angle BKM = \frac{1}{2} \angle BAC |
27,105 | 13. Let the intersection locus of two perpendicular tangents to the ellipse $\frac{x^{2}}{5}+\frac{y^{2}}{4}=1$ be $C$. Suppose the tangents $P A, P B$ to curve $C$ intersect at point $P$, and are tangent to $C$ at points $A, B$ respectively. Find the minimum value of $\overrightarrow{P A} \cdot \overrightarrow{P B}$. | 18\sqrt{2}-27 |
55,169 | 3. Fill the numbers $1,2,3, \ldots, 9,10$ into 10 circles that form a rectangle, such that the sum of the numbers on each side of the rectangle is equal. The maximum sum is ( ). | 22 |
10,246 | Let \( A \) be a natural number greater than 1, and \( B \) be a natural divisor of \( A^2 + 1 \). Prove that if \( B - A > 0 \), then \( B - A > \sqrt{A} \). | B - A > \sqrt{A} |
30,067 | 1. Let $a, b$ and $c$ be non-zero real numbers and
$$
a+\frac{b}{c}=b+\frac{c}{a}=c+\frac{a}{b}=1
$$
Calculate the value of the expression $a b+b c+c a$. | 0 |
30,909 | Two players, Blake and Ruby, play the following game on an infinite grid of unit squares, all initially colored white. The players take turns starting with Blake. On Blake's turn, Blake selects one white unit square and colors it blue. On Ruby's turn, Ruby selects two white unit squares and colors them red. The players alternate until Blake decides to end the game. At this point, Blake gets a score, given by the number of unit squares in the largest (in terms of area) simple polygon containing only blue unit squares.
What is the largest score Blake can guarantee? | 4 |
64,494 | ## Task 4 - 110614
Two places $A$ and $B$ are connected by a $999 \mathrm{~km}$ long road.
At intervals of $1 \mathrm{~km}$, milestones are placed along this road, each marked on both sides such that one side indicates the distance from $A$ and the other side the distance from $B$ in km. For example, the stone at the exit of $A$ bears the inscription 0 and 999, and the stone at the entrance of $B$ bears the inscription 999 and 0.
Determine the number of these stones whose inscriptions use at most two different digits (e.g., 722 and 277)! | 40 |
52,155 | 29th Putnam 1968 Problem B1 The random variables X, Y can each take a finite number of integer values. They are not necessarily independent. Express prob( min(X, Y) = k) in terms of p 1 = prob( X = k), p 2 = prob(Y = k) and p 3 = prob( max(X, Y) = k). | p_1+p_2-p_3 |
54,893 | 59. As shown in the figure, in the right triangle $\triangle OAB$, $\angle AOB=30^{\circ}, AB=2$. If the right triangle $\triangle OAB$ is rotated $90^{\circ}$ clockwise around point $O$ to get the right triangle $\triangle OCD$, then the area swept by $AB$ is $\qquad$. | \pi |
18,763 | The vertices of a regular $n$-sided polygon inscribed in a circle of radius $r$ are $A_{1}, A_{2}, \ldots, A_{n}$, and $P$ is an internal point of the polygon. Prove that $P A_{1}+P A_{2}+\ldots+P A_{n} \geqq n \cdot r$. | PA_1 + PA_2 + \ldots + PA_n \geq n \cdot r |
50,463 | Problem 2. Vasya solved problems for 10 days - at least one problem each day. Each day (except the first), if the weather was cloudy, he solved one more problem than the previous day, and if it was sunny, one less problem. In the first 9 days, Vasya solved 13 problems. What was the weather like on the tenth day? [5 points]
(B. Frenkin) | Cloudy |
27,849 | Example 4 Let real numbers $x_{1}, x_{2}, \cdots, x_{1991}$ satisfy the condition
$$
\sum_{i=1}^{1990}\left|x_{i}-x_{i+1}\right|=1991 \text {. }
$$
and $y_{k}=\frac{1}{k} \sum_{i=1}^{k} x_{i}(k=1,2, \cdots, 1991)$. Find the maximum value of $\sum_{i=1}^{1990}\left|y_{i}-y_{i+1}\right|$.
(25th All-Soviet Union Mathematical Olympiad) | 1990 |
33,374 | 8. Variant 1.
Given a parallelogram $A B C D$. Let $B P$ and $C Q$ be the perpendiculars dropped from vertices $B$ and $C$ to diagonals $A C$ and $B D$ respectively (point $P$ lies on segment $A C$, and point $Q$ lies on segment $B D$). Find the ratio $\frac{10 B D}{A C}$, if $\frac{A P}{A C}=\frac{4}{9}$ and $\frac{D Q}{D B}=\frac{28}{81}$. | 6 |
24,794 | If $\triangle ABC$ satisfies that $\cot A, \cot B, \cot C$ form an arithmetic sequence, then the maximum value of $\angle B$ is ______. | \dfrac{\pi}{3} |
65,549 | The blank squares in the figure must be filled with numbers in such a way that each number, starting from the second row, is equal to the sum of the two adjacent numbers in the immediately preceding row. For example, the number in the first cell of the second row is 11, because $11=5+6$. What number will appear in the square marked with $\times$?
(a) 4
(b) 6
(c) 9
(d) 15
(e) 10
 | 10 |
51,069 | 1. The range of the function $f(x)=\sin ^{4} x \tan x+\cos ^{4} x \cot x$ is | (-\infty,-\frac{1}{2}]\cup[\frac{1}{2},+\infty) |
69,003 | 119. Find the probability that event $A$ will occur exactly 70 times in 243 trials, if the probability of this event occurring in each trial is 0.25. | 0.0231 |
52,665 | (3) Let the function $f(x)=a|x|+\frac{b}{x}$ (where $a, b$ are constants), and $f(-2)=0$, $f(x)$ has two intervals of monotonic increase. Find the relationship between $a$ and $b$ that satisfies the above conditions. | 4a |
56,575 | $4 \cdot 34$ An integer has exactly 4 prime factors, the sum of whose squares is 476. Find this integer.
(China National Training Team Practice Question, 1990) | 1989 |
8,174 | You are given 10 numbers - one one and nine zeros. You are allowed to select two numbers and replace each of them with their arithmetic mean.
What is the smallest number that can end up in the place of the one? | \dfrac{1}{512} |
9,523 | The extension of the median $A M$ of triangle $A B C$ intersects its circumscribed circle at point $D$. Find $B C$ if $A C = D C = 1$. | \sqrt{2} |
20,698 | Three, there are $n$ people, it is known that any two of them make at most one phone call, and any $n-2$ of them have the same total number of phone calls, which is $3^{k}$ times, where $k$ is a natural number. Find all possible values of $n$.
---
The translation maintains the original text's line breaks and format. | 5 |
10,868 | Let $P$ be any point in the first quadrant on the ellipse $C: \frac{x^{2}}{5} + y^{2} = 1$. $F_{1}$ and $F_{2}$ are the left and right foci of the ellipse, respectively. The lines $P F_{1}$ and $P F_{2}$ intersect the ellipse $C$ at points $M$ and $N$, respectively. Given that $\overrightarrow{P F_{1}} = \lambda_{1} \overrightarrow{F_{1} M}$ and $\overrightarrow{P F_{2}} = \lambda_{2} \overrightarrow{F_{2} N}$, find the coordinates of point $P$ such that the slope of the line segment $MN$ is $-\frac{1}{9}$. | \left( \dfrac{5\sqrt{6}}{6}, \dfrac{\sqrt{6}}{6} \right) |
31,901 | The 38th question, the sequence $\left\{x_{n}\right\}$ satisfies $x_{1}=1$, and for any $n \in Z^{+}$, it holds that $x_{n+1}=x_{n}+3 \sqrt{x_{n}}+\frac{n}{\sqrt{x_{n}}}$, try to find the value of $\lim _{n \rightarrow+\infty} \frac{n^{2}}{x_{n}}$. | \frac{4}{9} |
12,092 | 11.1. Find all solutions of the system of equations in real numbers:
$$
\left\{\begin{array}{c}
x y+z+t=1 \\
y z+t+x=3 \\
z t+x+y=-1 \\
t x+y+z=1
\end{array}\right.
$$ | (1, 0, -1, 2) |
27,004 | 2. Four cars $A, B, C$, and $D$ start simultaneously from the same point on a circular track. $A$ and $B$ drive clockwise, while $C$ and $D$ drive counterclockwise. All cars move at constant (but pairwise different) speeds. Exactly 7 minutes after the start of the race, $A$ meets $C$ for the first time, and at the same moment, $B$ meets $D$ for the first time. After another 46 minutes, $A$ and $B$ meet for the first time. After how much time from the start of the race will all the cars meet for the first time? | 371 |
65,378 | 99***. A rod is broken into two parts at a randomly chosen point; then the larger of the two resulting parts is again broken into two parts at a randomly chosen point. What is the probability that a triangle can be formed from the three resulting pieces? | 2\ln2-1 |
68,390 | 9. (10 points) In the figure, $AB$ is the diameter of circle $O$, 6 cm long, and square $BCDE$ has one vertex $E$ on the circumference of the circle, $\angle ABE=45^{\circ}$. The difference between the area of the non-shaded part of circle $O$ and the area of the non-shaded part of square $BCDE$ is $\qquad$ square centimeters (take $\pi=3.14$) | 10.26 |
52,696 | 4. (8 points) As shown in the figure, semi-circular lace is attached around a square with a side length of 10 decimeters. The number of circular paper pieces needed is ( ).
A. 8
B. 40
C. 60
D. 80 | 40 |
67,804 | [list]
[*] A power grid with the shape of a $3\times 3$ lattice with $16$ nodes (vertices of the lattice) joined by wires (along the sides of squares. It may have happened that some of the wires have burned out. In one test technician can choose any two nodes and check if electrical current circulates between them (i.e there is a chain of intact wires joining the chosen nodes) . Technicial knows that current will circulate from any node to another node. What is the least number of tests required to demonstrate this?
[*] Previous problem for the grid of $5\times 5$ lattice.[/list] | 35 |
57,256 | [Theorem on the lengths of a tangent and a secant; the product of the entire secant and its external segment. Auxiliary similar triangles [Mean proportional in a right triangle
From point $A$, two tangents (with points of tangency $M$ and $N$) and a secant intersecting the circle at points $B$ and $C$, and the chord $MN$ at point $P$, are drawn, with $AB: BC=2: 3$. Find $AP: PC$.
# | 4:3 |
25,374 | 1. [5 points] Point $D$ lies on side $A C$ of triangle $A B C$. The circle with diameter $B D$ intersects sides $A B$ and $B C$ at points $P$ and $T$ respectively. Points $M$ and $N$ are the midpoints of segments $A D$ and $C D$ respectively. It is known that $P M \| T N$.
a) Find the angle $A B C$.
b) Suppose additionally that $M P=\frac{1}{2}, N T=2, B D=\sqrt{3}$. Find the area of triangle $A B C$. | \frac{5\sqrt{13}}{3\sqrt{2}} |
65,374 | 15. Given a set $S$ satisfying $|S|=10$. Let $A_{1}, A_{2}, \cdots, A_{k}$ be non-empty subsets of $S$, and the intersection of any two subsets contains at most two elements. Find the maximum value of $k$.
untranslated part:
将上面的文本翻译成英文，请保留源文本的换行和格式，直接输出翻译结果。
Note: The last sentence is a note for the translation task and is not part of the text to be translated. It has been translated here for clarity. | 175 |
12,401 | At the school reunion, 45 people attended. It turned out that any two of them who have the same number of acquaintances among the attendees are not acquainted with each other. What is the maximum number of pairs of acquaintances that could be among the attendees? | 870 |
11,715 | Find the largest positive integer $N$ so that the number of integers in the set $\{1,2,\dots,N\}$ which are divisible by 3 is equal to the number of integers which are divisible by 5 or 7 (or both). | 65 |
12,527 | Prove: $\frac{1}{m+1} C_{n}^{0}-\frac{1}{m+2} C_{n}^{1}+\frac{1}{m+3} C_{n}^{2}-\cdots+$ $(-1)^{n} \frac{1}{m+n+1} C_{n}^{n}=\frac{1}{(m+n+1) C_{m+n}^{n}}$, where $n \in \mathrm{N}$. | \frac{1}{(m+n+1) \binom{m+n}{n}} |
23,020 | We call a pair $(a,b)$ of positive integers, $a<391$ , *pupusa* if $$ \textup{lcm}(a,b)>\textup{lcm}(a,391) $$ Find the minimum value of $b$ across all *pupusa* pairs.
Fun Fact: OMCC 2017 was held in El Salvador. *Pupusa* is their national dish. It is a corn tortilla filled with cheese, meat, etc. | 18 |
61,093 | Problem 7.4. The figure shows 5 lines intersecting at one point. One of the resulting angles is $34^{\circ}$. How many degrees is the sum of the four angles shaded in gray?
 | 146 |
9,062 | Prove that \( \frac{1}{\frac{1}{a} + \frac{1}{b}} + \frac{1}{\frac{1}{c} + \frac{1}{d}} \leq \frac{1}{\frac{1}{a+c} + \frac{1}{b+d}} \) for positive reals \( a, b, c, \) and \( d \). | \frac{1}{\frac{1}{a} + \frac{1}{b}} + \frac{1}{\frac{1}{c} + \frac{1}{d}} \leq \frac{1}{\frac{1}{a+c} + \frac{1}{b+d}} |
32,205 | 5. Given the equation about $x$: $(2 a-1) \sin x+(2-a) \sin 2 x=\sin 3 x$, if all solutions of the equation in the range $x \in[0,2 \pi]$ form an arithmetic sequence in ascending order, then the range of the real number $a$ is $\qquad$ . | (-\infty,-2]\bigcup{0}\bigcup[2,+\infty) |
63,253 | 63. A, B, and C are playing a round-robin table tennis tournament, with the rule that the winner gets 2 points, the loser gets 0 points, and in case of a draw, each gets 1 point. After the tournament, there are $\qquad$ possible score situations for the three players. | 19 |
63,107 | ## Task 4 - 280914
Three workpieces $W_{1}, W_{2}, W_{3}$ go through an assembly line with four processing machines $M_{1}, M_{2}, M_{3}, M_{4}$. Each workpiece must pass through the machines in the order $M_{1}, M_{2}, M_{3}, M_{4}$, and the order of the three workpieces must be the same at each machine.
The processing times of the workpieces on the individual machines are (in hours) given in the following table:
| | $M_{1}$ | $M_{2}$ | $M_{3}$ | $M_{4}$ |
| :--- | :---: | :---: | :---: | :---: |
| $W_{1}$ | 4 | 1 | 2 | 1.5 |
| $W_{2}$ | 2 | 2.5 | 1 | 0.5 |
| $W_{3}$ | 2 | 3.5 | 1 | 1 |
Two workpieces can never be processed simultaneously on the same machine. The times for changing the workpieces at the machines are so small that they can be neglected.
Give an order of the three workpieces for passing through the assembly line such that the total time (the time from the entry of the first workpiece into machine $M_{1}$ to the exit of the last workpiece from machine $M_{4}$) is as small as possible! Show that the order you give with its total time is better than any other order! | W_{3},W_{1},W_{2} |
25,502 | C2. Queenie and Horst play a game on a $20 \times 20$ chessboard. In the beginning the board is empty. In every turn, Horst places a black knight on an empty square in such a way that his new knight does not attack any previous knights. Then Queenie places a white queen on an empty square. The game gets finished when somebody cannot move.
Find the maximal positive $K$ such that, regardless of the strategy of Queenie, Horst can put at least $K$ knights on the board. | 100 |
56,862 | Three, (10 points) A quadratic trinomial $x^{2}+p x+q$ with coefficients $p$ and $q$ as integers, and roots as irrational numbers $\alpha_{1}, \alpha_{2}$, is called an irrational quadratic trinomial. Find the minimum value of the sum of the absolute values of the roots for all irrational quadratic trinomials. | \sqrt{5} |
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