problem
stringlengths 10
7.44k
| answer
stringlengths 1
270
| difficulty
stringclasses 8
values |
|---|---|---|
Let $M$ be the midpoint of the base $AC$ of an acute-angled isosceles triangle $ABC$ . Let $N$ be the reflection of $M$ in $BC$ . The line parallel to $AC$ and passing through $N$ meets $AB$ at point $K$ . Determine the value of $\angle AKC$ .
(A.Blinkov)
|
90
|
7/8
|
There are 4 boys and 3 girls lining up for a photo. The number of ways they can line up such that no two boys are adjacent is \_\_\_\_ (answer with a number), and the number of ways they can line up such that no two girls are adjacent is \_\_\_\_ (answer with a number).
|
1440
|
5/8
|
Given the set $\{2,3,5,7,11,13\}$, add one of the numbers twice to another number, and then multiply the result by the third number. What is the smallest possible result?
|
22
|
7/8
|
A circle with center $O$ is tangent to the coordinate axes and to the hypotenuse of the $30^\circ$-$60^\circ$-$90^\circ$ triangle $ABC$ as shown, where $AB=1$. To the nearest hundredth, what is the radius of the circle?
$\textbf{(A)}\ 2.18\qquad\textbf{(B)}\ 2.24\qquad\textbf{(C)}\ 2.31\qquad\textbf{(D)}\ 2.37\qquad\textbf{(E)}\ 2.41$
|
\textbf{(D)}\2.37
|
1/8
|
Given the sets \( M=\{x, xy, \lg(xy)\} \) and \( N=\{0, |x|, y\} \), and that \( M=N \), find the value of \( \left(x+\frac{1}{y}\right)+\left(x^{2}+\frac{1}{y^{2}}\right)+\left(x^{3}+\frac{1}{y^{3}}\right)+\cdots+\left(x^{2001}+\frac{1}{y^{2001}}\right) \).
|
-2
|
5/8
|
If \( x = \frac{1}{4} \), which of the following has the largest value?
(A) \( x \)
(B) \( x^2 \)
(C) \( \frac{1}{2} x \)
(D) \( \frac{1}{x} \)
(E) \( \sqrt{x} \)
|
4
|
1/8
|
Design a set of stamps with the following requirements: The set consists of four stamps of different denominations, with denominations being positive integers. Moreover, for any denomination value among the consecutive integers 1, 2, ..., R, it should be possible to achieve it by appropriately selecting stamps of different denominations and using no more than three stamps. Determine the maximum value of R and provide a corresponding design.
|
14
|
1/8
|
Given the set \( X = \{1, 2, \cdots, 100\} \), determine whether there exist 1111 different subsets of \( X \) such that the intersection of any two (including possibly the same subset) contains a number of elements that is a perfect square. Note: Perfect squares include \( 0, 1, 4, 9, 16, \cdots \).
|
Yes
|
1/8
|
Consider a positive real number $a$ and a positive integer $m$ . The sequence $(x_k)_{k\in \mathbb{Z}^{+}}$ is defined as: $x_1=1$ , $x_2=a$ , $x_{n+2}=\sqrt[m+1]{x_{n+1}^mx_n}$ . $a)$ Prove that the sequence is converging. $b)$ Find $\lim_{n\rightarrow \infty}{x_n}$ .
|
^{\frac{+1}{+2}}
|
7/8
|
In triangle $ABC$, $AB=125$, $AC=117$ and $BC=120$. The angle bisector of angle $A$ intersects $\overline{BC}$ at point $L$, and the angle bisector of angle $B$ intersects $\overline{AC}$ at point $K$. Let $M$ and $N$ be the feet of the perpendiculars from $C$ to $\overline{BK}$ and $\overline{AL}$, respectively. Find $MN$.
|
56
|
1/8
|
Find the maximum number of white dominoes that can be cut from the board shown on the left. A domino is a $1 \times 2$ rectangle.
|
16
|
4/8
|
In an acute triangle \( ABC \), angle \( A \) is \( 35^\circ \). Segments \( BB_1 \) and \( CC_1 \) are altitudes, points \( B_2 \) and \( C_2 \) are the midpoints of sides \( AC \) and \( AB \) respectively. Lines \( B_1C_2 \) and \( C_1B_2 \) intersect at point \( K \). Find the measure (in degrees) of angle \( B_1KB_2 \).
|
75
|
1/8
|
Construct a cross-section of a cube.
Given:
- Line KL, where K is the midpoint of \( D_1 C_1 \), and L is the midpoint of \( C_1 B_1 \), lies in the plane face \( A_1 B_1 C_1 D_1 \).
- This line intersects the extended edges \( A_1 B_1 \) and \( A_1 D_1 \) at points F and E respectively.
- It's easy to calculate that \( D_1 E = \frac{1}{2} A_1 D_1 = \frac{1}{3} A_1 E \) and similarly \( B_1 F = \frac{1}{2} A_1 B_1 = \frac{1}{3} A_1 F \).
- The points E and A lie on the same face. It can be shown that the line AE intersects the edge \( D D_1 \) at point N, dividing this edge in the ratio 2:1.
- Similarly, the line AF intersects the edge \( B_1 B \) at point M, also dividing this edge in the ratio 2:1. Therefore, the cross-section will be the pentagon \( AMLKN \).
The projection of this pentagon onto the lower base will be the pentagon \( ABL_1 K_1 D \), and its area is \(\frac{7}{8}\).
If \( T \) is the midpoint of \( KL \) and \( T_1 \) is the projection of point T onto the lower base, the angle \( TAT_1 \) will be the angle between the cross-section and the lower face of the cube. Then, \(\cos(TAT_1) = \frac{3}{\sqrt{17}} \). Using the formula for the area projection, the area of the cross-section will be \(\frac{7\sqrt{17}}{24}\).
|
\frac{7\sqrt{17}}{24}
|
7/8
|
How many solutions does the equation
\[
\frac{(x-1)(x-2)(x-3)\dotsm(x-200)}{(x-1^2)(x-2^2)(x-3^2)\dotsm(x-10^2)(x-11^3)(x-12^3)\dotsm(x-13^3)}
\]
have for \(x\)?
|
190
|
7/8
|
Of the following sets of data the only one that does not determine the shape of a triangle is:
$\textbf{(A)}\ \text{the ratio of two sides and the inc{}luded angle}\\ \qquad\textbf{(B)}\ \text{the ratios of the three altitudes}\\ \qquad\textbf{(C)}\ \text{the ratios of the three medians}\\ \qquad\textbf{(D)}\ \text{the ratio of the altitude to the corresponding base}\\ \qquad\textbf{(E)}\ \text{two angles}$
|
\textbf{(D)}
|
1/8
|
The Minions need to make jam within the specified time. Kevin can finish the job 4 days earlier if he works alone, while Dave would finish 6 days late if he works alone. If Kevin and Dave work together for 4 days and then Dave completes the remaining work alone, the job is completed exactly on time. How many days would it take for Kevin and Dave to complete the job if they work together?
|
12
|
7/8
|
In the universe of Pi Zone, points are labeled with $2 \times 2$ arrays of positive reals. One can teleport from point $M$ to point $M'$ if $M$ can be obtained from $M'$ by multiplying either a row or column by some positive real. For example, one can teleport from $\left( \begin{array}{cc} 1 & 2 3 & 4 \end{array} \right)$ to $\left( \begin{array}{cc} 1 & 20 3 & 40 \end{array} \right)$ and then to $\left( \begin{array}{cc} 1 & 20 6 & 80 \end{array} \right)$ .
A *tourist attraction* is a point where each of the entries of the associated array is either $1$ , $2$ , $4$ , $8$ or $16$ . A company wishes to build a hotel on each of several points so that at least one hotel is accessible from every tourist attraction by teleporting, possibly multiple times. What is the minimum number of hotels necessary?
*Proposed by Michael Kural*
|
17
|
1/8
|
The square root of a two-digit number is expressed as an infinite decimal fraction, the first four digits of which (including the integer part) are the same. Find this number without using tables.
|
79
|
1/8
|
Given an odd function \( f(x) \) defined on \(\mathbf{R}\) that is symmetric about the line \( x=2 \), and when \( 0<x \leq 2 \), \( f(x)=x+1 \). Calculate \( f(-100) + f(-101) \).
|
2
|
7/8
|
In this Number Wall, you add the numbers next to each other and write the sum in the block directly above the two numbers. Which number will be in the block labeled '$m$'? [asy]
draw((0,0)--(8,0)--(8,2)--(0,2)--cycle);
draw((2,0)--(2,2));
draw((4,0)--(4,2));
draw((6,0)--(6,2));
draw((1,2)--(7,2)--(7,4)--(1,4)--cycle);
draw((3,2)--(3,4));
draw((5,2)--(5,4));
draw((2,4)--(2,6)--(6,6)--(6,4)--cycle);
draw((4,4)--(4,6));
draw((3,6)--(3,8)--(5,8)--(5,6));
label("$m$",(1,1));
label("3",(3,1));
label("9",(5,1));
label("6",(7,1));
label("16",(6,3));
label("54",(4,7));
[/asy]
|
12
|
3/8
|
Print 90,000 five-digit numbers
$$
10000, 10001, \cdots, 99999
$$
on cards, with each card displaying one five-digit number. Some numbers printed on the cards (e.g., 19806 when reversed reads 90861) can be read in two different ways and may cause confusion. How many cards will display numbers that do not cause confusion?
|
89100
|
1/8
|
In a convex quadrilateral $ABCD$, the midpoint of side $AD$ is marked as point $M$. Line segments $BM$ and $AC$ intersect at point $O$. It is known that $\angle ABM = 55^\circ$, $\angle AMB = 70^\circ$, $\angle BOC = 80^\circ$, and $\angle ADC = 60^\circ$. What is the measure of angle $BCA$ in degrees?
|
35
|
1/8
|
Determine the number of solutions to the equation
\[\tan (10 \pi \cos \theta) = \cot (10 \pi \sin \theta)\]
where $\theta \in (0, 2 \pi).$
|
56
|
6/8
|
Rebecca has four resistors, each with resistance 1 ohm. Every minute, she chooses any two resistors with resistance of \(a\) and \(b\) ohms respectively, and combines them into one by one of the following methods:
- Connect them in series, which produces a resistor with resistance of \(a+b\) ohms;
- Connect them in parallel, which produces a resistor with resistance of \(\frac{a b}{a+b}\) ohms;
- Short-circuit one of the two resistors, which produces a resistor with resistance of either \(a\) or \(b\) ohms.
Suppose that after three minutes, Rebecca has a single resistor with resistance \(R\) ohms. How many possible values are there for \(R\)?
|
15
|
2/8
|
Given that the circumference of a sector is $20\,cm$ and its area is $9\,cm^2$, find the radian measure of the central angle of the sector.
|
\frac{2}{9}
|
5/8
|
Points \( A \) and \( B \) lie on a circle with center \( O \) and radius 6, and point \( C \) is equidistant from points \( A, B, \) and \( O \). Another circle with center \( Q \) and radius 8 is circumscribed around triangle \( A C O \). Find \( B Q \).
|
10
|
4/8
|
Each of the $10$ dwarfs either always tells the truth or always lies. It is known that each of them loves exactly one type of ice cream: vanilla, chocolate or fruit. First, Snow White asked those who like the vanilla ice cream to raise their hands, and everyone raised their hands, then those who like chocolate ice cream - and half of the dwarves raised their hands, then those who like the fruit ice cream - and only one dwarf raised his hand. How many of the gnomes are truthful?
|
4
|
7/8
|
Compute the perimeter of the triangle that has area $3-\sqrt{3}$ and angles $45^\circ$ , $60^\circ$ , and $75^\circ$ .
|
3\sqrt{2} + 2\sqrt{3} - \sqrt{6}
|
1/8
|
The members of a distinguished committee were choosing a president, and each member gave one vote to one of the 27 candidates. For each candidate, the exact percentage of votes the candidate got was smaller by at least 1 than the number of votes for that candidate. What was the smallest possible number of members of the committee?
|
134
|
3/8
|
If $\left\{a_{n}\right\}$ is an increasing sequence and for any positive integer $n$, $a_{n} = n^{2} + \lambda n$ always holds, then the value range of the real number $\lambda$ is ______ .
|
(-3,+\infty)
|
3/8
|
A ferry boat shuttles tourists to an island every hour starting at 10 AM until its last trip, which starts at 3 PM. One day the boat captain notes that on the 10 AM trip there were 100 tourists on the ferry boat, and that on each successive trip, the number of tourists was 1 fewer than on the previous trip. How many tourists did the ferry take to the island that day?
$\textbf{(A)}\ 585 \qquad \textbf{(B)}\ 594 \qquad \textbf{(C)}\ 672 \qquad \textbf{(D)}\ 679 \qquad \textbf{(E)}\ 694$
|
585\\textbf{(A)}
|
1/8
|
In the triangular prism $S-ABC$, the lateral edge $SC$ is equal in length to the base edge $AB$, and forms a $60^{\circ}$ angle with the base plane $ABC$. The vertices $A$, $B$, and $C$ as well as the midpoints of the lateral edges of the prism all lie on a sphere with radius 1. Prove that the center of this sphere lies on the edge $AB$, and determine the height of the prism.
|
\sqrt{3}
|
1/8
|
An angle is inscribed with a circle of radius \( R \), and the length of the chord connecting the points of tangency is \( a \). Two tangents are drawn parallel to this chord, forming a trapezoid. Find the area of this trapezoid.
|
\frac{8R^3}{}
|
1/8
|
Let $a, b, c, d$ be real numbers such that $a^{2}+b^{2}+c^{2}+d^{2}=1$. Determine the minimum value of $(a-b)(b-c)(c-d)(d-a)$ and determine all values of $(a, b, c, d)$ such that the minimum value is achieved.
|
-\frac{1}{8}
|
3/8
|
The integer $y$ has 24 positive factors. The numbers 20 and 35 are factors of $y$. What is the smallest possible value of $y$?
|
1120
|
1/8
|
Let $A$ be the set of positive integers that have no prime factors other than $2$, $3$, or $5$. The infinite sum \[\frac{1}{1} + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \frac{1}{5} + \frac{1}{6} + \frac{1}{8} + \frac{1}{9} + \frac{1}{10} + \frac{1}{12} + \frac{1}{15} + \frac{1}{16} + \frac{1}{18} + \frac{1}{20} + \cdots\]of the reciprocals of the elements of $A$ can be expressed as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. What is $m+n$?
$\textbf{(A) } 16 \qquad \textbf{(B) } 17 \qquad \textbf{(C) } 19 \qquad \textbf{(D) } 23 \qquad \textbf{(E) } 36$
|
\textbf{(C)}19
|
1/8
|
Consider polynomials $P$ of degree $2015$ , all of whose coefficients are in the set $\{0,1,\dots,2010\}$ . Call such a polynomial *good* if for every integer $m$ , one of the numbers $P(m)-20$ , $P(m)-15$ , $P(m)-1234$ is divisible by $2011$ , and there exist integers $m_{20}, m_{15}, m_{1234}$ such that $P(m_{20})-20, P(m_{15})-15, P(m_{1234})-1234$ are all multiples of $2011$ . Let $N$ be the number of good polynomials. Find the remainder when $N$ is divided by $1000$ .
*Proposed by Yang Liu*
|
460
|
2/8
|
Three segments, not lying in the same plane, intersect at one point and are divided into halves by this point. Prove that there are exactly two tetrahedrons in which these segments connect the midpoints of opposite edges.
|
2
|
6/8
|
Find the fraction with the smallest denominator in the open interval \(\left(\frac{47}{245}, \frac{34}{177}\right)\).
|
\frac{19}{99}
|
1/8
|
How many diagonals within a regular nine-sided polygon span an odd number of vertices between their endpoints?
|
18
|
5/8
|
In a certain area, there are 100,000 households, among which there are 99,000 ordinary households and 1,000 high-income households. A simple random sampling method is used to select 990 households from the ordinary households and 100 households from the high-income households for a survey. It was found that a total of 120 households own 3 or more sets of housing, among which there are 40 ordinary households and 80 high-income households. Based on these data and combining your statistical knowledge, what do you think is a reasonable estimate of the proportion of households in the area that own 3 or more sets of housing?
|
4.8\%
|
7/8
|
Let $2000 < N < 2100$ be an integer. Suppose the last day of year $N$ is a Tuesday while the first day of year $N+2$ is a Friday. The fourth Sunday of year $N+3$ is the $m$ th day of January. What is $m$ ?
*Based on a proposal by Neelabh Deka*
|
23
|
4/8
|
Let the function \( f(x) = x^2 + ax + b \) where \( a \) and \( b \) are real constants. It is known that the inequality \( |f(x)| \leqslant |2x^2 + 4x - 30| \) holds for any real number \( x \). A sequence \( \{a_n\} \) and \( \{b_n\} \) is defined as: \( a_1 = \frac{1}{2} \), \( 2a_n = f(a_{n-1}) + 15 \) for \( n = 2, 3, 4, \ldots \), and \( b_n = \frac{1}{2+a_n} \) for \( n = 1, 2, 3, \ldots \). Let \( S_n \) be the sum of the first \( n \) terms of the sequence \( \{b_n\} \) and \( T_n \) be the product of the first \( n \) terms of the sequence \( \{b_n\} \).
(1) Prove that \( a = 2 \) and \( b = -15 \).
(2) Prove that for any positive integer \( n \), \( 2^{n+1} T_n + S_n \) is a constant value.
|
2
|
1/8
|
Let $\triangle ABC$ be an acute isosceles triangle with circumcircle $\omega$. The tangents to $\omega$ at vertices $B$ and $C$ intersect at point $T$. Let $Z$ be the projection of $T$ onto $BC$. Assume $BT = CT = 20$, $BC = 24$, and $TZ^2 + 2BZ \cdot CZ = 478$. Find $BZ \cdot CZ$.
|
144
|
1/8
|
Let $p$ be a prime. It is given that there exists a unique nonconstant function $\chi:\{1,2,\ldots, p-1\}\to\{-1,1\}$ such that $\chi(1) = 1$ and $\chi(mn) = \chi(m)\chi(n)$ for all $m, n \not\equiv 0 \pmod p$ (here the product $mn$ is taken mod $p$ ). For how many positive primes $p$ less than $100$ is it true that \[\sum_{a=1}^{p-1}a^{\chi(a)}\equiv 0\pmod p?\] Here as usual $a^{-1}$ denotes multiplicative inverse.
*Proposed by David Altizio*
|
24
|
5/8
|
All edges (including the sides of the base) of a triangular pyramid are equal. Find the ratio of the radius of the sphere inscribed in the pyramid to its height.
|
1:4
|
1/8
|
Natural numbers $1, 2, 3, 4, \cdots$ are arranged in order according to the direction indicated by arrows, and they turn at positions of numbers like $2, 3, 5, 7, 10, \cdots$ and so on.
(1) If 2 is counted as the first turn, then the number at the 45th turn is $\qquad$ .
(2) From 1978 to 2010, the natural numbers that are exactly at turning positions are $\qquad$ .
|
1981
|
2/8
|
Thirty-nine students from seven classes came up with 60 problems, with students of the same class coming up with the same number of problems (not equal to zero), and students from different classes coming up with a different number of problems. How many students came up with one problem each?
|
33
|
5/8
|
Given $\{1,a, \frac{b}{a}\}=\{0,a^2,a+b\}$, calculate the value of $a^{2005}+b^{2005}$.
|
-1
|
5/8
|
It is given isosceles triangle $ABC$ ( $AB=AC$ ) such that $\angle BAC=108^{\circ}$ . Angle bisector of angle $\angle ABC$ intersects side $AC$ in point $D$ , and point $E$ is on side $BC$ such that $BE=AE$ . If $AE=m$ , find $ED$
|
m
|
2/8
|
$(MON 1)$ Find the number of five-digit numbers with the following properties: there are two pairs of digits such that digits from each pair are equal and are next to each other, digits from different pairs are different, and the remaining digit (which does not belong to any of the pairs) is different from the other digits.
|
1944
|
4/8
|
Chords \(AB\) and \(CD\) of a circle with center \(O\) both have a length of 5. The extensions of segments \(BA\) and \(CD\) beyond points \(A\) and \(D\) intersect at point \(P\), where \(DP=13\). The line \(PO\) intersects segment \(AC\) at point \(L\). Find the ratio \(AL:LC\).
|
13/18
|
1/8
|
The three-digit number 999 has a special property: It is divisible by 27, and its digit sum is also divisible by 27. The four-digit number 5778 also has this property, as it is divisible by 27 and its digit sum is also divisible by 27. How many four-digit numbers have this property?
|
75
|
1/8
|
A man and his faithful dog simultaneously started moving along the perimeter of a block from point \( A \) at time \( t_{0} = 0 \) minutes. The man moved at a constant speed clockwise, while the dog ran at a constant speed counterclockwise. It is known that the first time they met was \( t_{1} = 2 \) minutes after starting, and this meeting occurred at point \( B \). Given that they continued moving after this, each in their original direction and at their original speed, determine the next time they will both be at point \( B \) simultaneously. Note that \( A B = C D = 100 \) meters and \( B C = A D = 200 \) meters.
|
14
|
7/8
|
There are 99 children standing in a circle, each initially holding a ball. Every minute, each child with a ball throws their ball to one of their two neighbors. If a child receives two balls, one of the balls is irrevocably lost. What is the minimum amount of time after which only one ball can remain with the children?
|
98
|
1/8
|
Given \( x, y, z \in \mathbf{R}^{+} \) and \( x^{3} + y^{3} + z^{3} = 1 \), prove that \( \frac{x^{2}}{1 - x^{2}} + \frac{y^{2}}{1 - y^{2}} + \frac{z^{2}}{1 - z^{2}} \geqslant \frac{3 \sqrt{3}}{2} \).
|
\frac{3\sqrt{3}}{2}
|
1/8
|
The product of the digits of a four-digit number is 810. If none of the digits is repeated, what is the sum of the digits?
(A) 18
(B) 19
(C) 23
(D) 25
(E) 22
|
23
|
6/8
|
Compute the unique positive integer $m$ such that
\[1 \cdot 2^1 + 2 \cdot 2^2 + 3 \cdot 2^3 + \dots + m \cdot 2^m = 2^{m + 8}.\]
|
129
|
1/8
|
There are \( n \) real numbers written on a blackboard. It is allowed to erase any two numbers, say \( a \) and \( b \), and write another number \(\frac{1}{4}(a+b)\) on the board. This operation is performed \( n-1 \) times, resulting in only one number left on the blackboard. Initially, all \( n \) numbers on the board are 1. Prove that the final number remaining on the blackboard is not less than \(\frac{1}{n}\).
|
\frac{1}{n}
|
3/8
|
Find the least positive integer \(M\) for which there exist a positive integer \(n\) and polynomials \(P_1(x)\), \(P_2(x)\), \(\ldots\), \(P_n(x)\) with integer coefficients satisfying \[Mx=P_1(x)^3+P_2(x)^3+\cdots+P_n(x)^3.\]
*Proposed by Karthik Vedula*
|
6
|
1/8
|
It is known that the ellipse $C_1$ and the parabola $C_2$ have a common focus $F(1,0)$. The center of $C_1$ and the vertex of $C_2$ are both at the origin. A line $l$ passes through point $M(4,0)$ and intersects the parabola $C_2$ at points $A$ and $B$ (with point $A$ in the fourth quadrant).
1. If $|MB| = 4|AM|$, find the equation of line $l$.
2. If the symmetric point $P$ of the origin $O$ with respect to line $l$ lies on the parabola $C_2$, and line $l$ intersects the ellipse $C_1$ at common points, find the minimum length of the major axis of the ellipse $C_1$.
|
\sqrt{34}
|
4/8
|
Define functions $f$ and $g$ as nonconstant, differentiable, real-valued functions on $R$ . If $f(x+y)=f(x)f(y)-g(x)g(y)$ , $g(x+y)=f(x)g(y)+g(x)f(y)$ , and $f'(0)=0$ , prove that $\left(f(x)\right)^2+\left(g(x)\right)^2=1$ for all $x$ .
|
(f(x))^2+((x))^2=1
|
1/8
|
Let $a$ , $b$ and $c$ be real numbers such that $abc(a+b)(b+c)(c+a)\neq0$ and $(a+b+c)\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right)=\frac{1007}{1008}$ Prove that $\frac{ab}{(a+c)(b+c)}+\frac{bc}{(b+a)(c+a)}+\frac{ca}{(c+b)(a+b)}=2017$
|
2017
|
3/8
|
The extension of the altitude $BH$ of triangle $ABC$ intersects its circumcircle at point $D$ (points $B$ and $D$ are on opposite sides of line $AC$). The degree measures of arcs $AD$ and $CD$ that do not contain point $B$ are $120^{\circ}$ and $90^{\circ}$, respectively. Determine the ratio in which segment $BD$ is divided by side $AC$.
|
1:\sqrt{3}
|
1/8
|
On a board, there are $N \geq 9$ distinct non-negative numbers less than one. It turns out that for any eight distinct numbers from the board, there exists a ninth number on the board, different from them, such that the sum of these nine numbers is an integer. For which $N$ is this possible?
|
9
|
2/8
|
The ratio of the number of games won to the number of games lost (no ties) by the Middle School Middies is $11/4$. To the nearest whole percent, what percent of its games did the team lose?
$\text{(A)}\ 24\% \qquad \text{(B)}\ 27\% \qquad \text{(C)}\ 36\% \qquad \text{(D)}\ 45\% \qquad \text{(E)}\ 73\%$
|
(B)\27
|
1/8
|
For which values of the parameter \( a \) does the equation
$$
5^{x^{2}-6 a x+9 a^{2}}=a x^{2}-6 a^{2} x+9 a^{3}+a^{2}-6 a+6
$$
have exactly one solution?
|
1
|
2/8
|
Given vectors $\overrightarrow{a}=(1,\sin x)$, $\overrightarrow{b}=(\sin x,-1)$, $\overrightarrow{c}=(1,\cos x)$, where $x\in(0,\pi)$.
(Ⅰ) If $(\overrightarrow{a}+ \overrightarrow{b})\nparallel \overrightarrow{c}$, find $x$;
(Ⅱ) In $\triangle ABC$, the sides opposite to angles $A$, $B$, $C$ are $a$, $b$, $c$ respectively, and $B$ is the $x$ found in (Ⅰ), $2\sin^2B+2\sin^2C-2\sin^2A=\sin B\sin C$, find the value of $\sin \left(C- \frac{\pi}{3}\right)$.
|
\frac{1-3\sqrt{5}}{8}
|
3/8
|
Given a natural number \( a \), let \( S(a) \) represent the sum of its digits (for example, \( S(123) = 1 + 2 + 3 = 6 \) ). If a natural number \( n \) has all distinct digits, and \( S(3n) = 3S(n) \), what is the maximum value of \( n \)?
|
3210
|
6/8
|
In a right triangle \( \triangle A M N \), given \( \angle B A C=60^{\circ} \). Let \( O_{1} \) and \( O_{2} \) be the centers of circles, and \( P \) and \( R \) be the points of tangency of these circles with side \( BC \). In the right triangle \( O_{1}O_{2}Q \) with right angle at \( Q \) and point \( Q \in O_{2}R \), we have \( O_{1}O_{2}^{2} = O_{1}Q^{2} + QO_{2}^{2} \), \( (16)^{2} = O_{1}Q^{2} + (8)^{2} \).
$$
P R = O_{1}Q = 8 \sqrt{3}, \quad \angle O_{2} O_{1} Q = 30^{\circ}
$$
Given \( \angle O_{1} O_{2} Q = 60^{\circ} \).
$$
\begin{aligned}
& \angle R O_{2} C = \frac{1}{2} \angle R O_{2} E = \frac{1}{2}\left(360^{\circ}-90^{\circ}-2 \cdot 60^{\circ}\right) = 75^{\circ}, \\
& \angle O_{2} C R = 15^{\circ}, \quad \angle A C B = 30^{\circ}, \quad \angle A B C = 90^{\circ}, \\
& R C = 12 \operatorname{ctg} 15^{\circ} = 12 \sqrt{\frac{1+\cos 30^{\circ}}{1-\cos 30^{\circ}}} = 12 \sqrt{\frac{2+\sqrt{3}}{2-\sqrt{3}}} = 12(2+\sqrt{3}) = 24+12 \sqrt{3}, \\
& B C = B P + P R + R C = 4 + 8 \sqrt{3} + 24 + 12 \sqrt{3} = 28 + 20 \sqrt{3}, \\
& A C = B C / \sin 60^{\circ} = (120 + 56 \sqrt{3}) / 3, \\
& A N = A C - C E - N E = (120 + 56 \sqrt{3} - 72 - 36 \sqrt{3} - 36) / 3 = (12 + 20 \sqrt{3}) / 3, \\
& M N = A N / \operatorname{tg} 30^{\circ} = (12 \sqrt{3} + 60) / 3 = 4 \sqrt{3} + 20.
\end{aligned}
$$
Calculate the area of triangle \( \triangle A M N \):
\( S = A N \cdot M N / 2 = (12 + 20 \sqrt{3})(4 \sqrt{3} + 20) / 6 = (224 \sqrt{3} + 240) / 3 \).
|
\frac{224\sqrt{3}+240}{3}
|
6/8
|
Two numbers \( x \) and \( y \) satisfy the equation \( 26x^2 + 23xy - 3y^2 - 19 = 0 \) and are respectively the sixth and eleventh terms of a decreasing arithmetic progression consisting of integers. Find the common difference of this progression.
|
-3
|
7/8
|
A tournament among 2021 ranked teams is played over 2020 rounds. In each round, two teams are selected uniformly at random among all remaining teams to play against each other. The better-ranked team always wins, and the worse-ranked team is eliminated. Let \( p \) be the probability that the second-best ranked team is eliminated in the last round. Compute \( \lfloor 2021 p \rfloor \).
|
674
|
3/8
|
For how many ordered pairs of positive integers $(x, y)$ with $x < y$ is the harmonic mean of $x$ and $y$ equal to $12^{10}$?
|
409
|
4/8
|
Four friends, One, Two, Five, and Ten, are located on one side of a dark tunnel and have only one flashlight. It takes one minute for person One to walk through the tunnel, two minutes for Two, five minutes for Five, and ten minutes for Ten. The tunnel is narrow, and at most two people can walk at the same time with the flashlight. Whenever two people walk together, they walk at the speed of the slower one. Show that all four friends can go from one side of the tunnel to the other in 17 minutes.
|
17
|
7/8
|
Given $\log _{2} 3=a$ and $\log _{3} 7=b$, find the value of $\log _{42} 56$.
|
\frac{+3}{+1}
|
3/8
|
The positive reals \( x \) and \( y \) satisfy \( x^2 + y^3 \geq x^3 + y^4 \). Show that \( x^3 + y^3 \leq 2 \).
|
2
|
4/8
|
A square is contained in a cube when all of its points are in the faces or in the interior of the cube. Determine the biggest $\ell > 0$ such that there exists a square of side $\ell$ contained in a cube with edge $1$ .
|
\frac{\sqrt{6}}{2}
|
1/8
|
One hundred musicians are planning to organize a festival with several concerts. In each concert, while some of the one hundred musicians play on stage, the others remain in the audience assisting to the players. What is the least number of concerts so that each of the musicians has the chance to listen to each and every one of the other musicians on stage?
|
9
|
2/8
|
Consider sequences that consist entirely of $A$'s and $B$'s and that have the property that every run of consecutive $A$'s has even length, and every run of consecutive $B$'s has odd length. Examples of such sequences are $AA$, $B$, and $AABAA$, while $BBAB$ is not such a sequence. How many such sequences have length 14?
|
172
|
1/8
|
Person A and person B are playing a game with the following rules: In odd-numbered rounds, the probability of A winning is $\frac{3}{4}$; in even-numbered rounds, the probability of B winning is $\frac{3}{4}$. There are no ties in any round. The game ends when one person has won 2 more rounds than the other. Determine the expected number of rounds played when the game ends.
|
\frac{16}{3}
|
3/8
|
For the quadrilateral \(ABCD\), it is known that \(AB = BD\), \(\angle ABD = \angle DBC\), and \(\angle BCD = 90^\circ\). On segment \(BC\), point \(E\) is marked such that \(AD = DE\). What is the length of segment \(BD\) if it is known that \(BE = 7\) and \(EC = 5\)?
|
17
|
3/8
|
An object in the plane moves from the origin and takes a ten-step path, where at each step the object may move one unit to the right, one unit to the left, one unit up, or one unit down. How many different points could be the final point?
|
221
|
1/8
|
The rational numbers $x$ and $y$, when written in lowest terms, have denominators 60 and 70 , respectively. What is the smallest possible denominator of $x+y$ ?
|
84
|
7/8
|
A sequence of numbers is arranged in the following pattern: \(1, 2, 3, 2, 3, 4, 3, 4, 5, 4, 5, 6, \cdots\). Starting from the first number on the left, find the sum of the first 99 numbers.
|
1782
|
6/8
|
Complex numbers $z_1,$ $z_2,$ and $z_3$ are zeros of a polynomial $Q(z) = z^3 + pz + s,$ where $|z_1|^2 + |z_2|^2 + |z_3|^2 = 300$. The points corresponding to $z_1,$ $z_2,$ and $z_3$ in the complex plane are the vertices of a right triangle with the right angle at $z_3$. Find the square of the hypotenuse of this triangle.
|
450
|
7/8
|
Calculate the area enclosed by the Bernoulli lemniscate \( r^{2} = a^{2} \cos 2\varphi \).
|
^2
|
5/8
|
In triangle $DEF$, $DE = 8$, $EF = 6$, and $FD = 10$.
[asy]
defaultpen(1);
pair D=(0,0), E=(0,6), F=(8,0);
draw(D--E--F--cycle);
label("\(D\)",D,SW);
label("\(E\)",E,N);
label("\(F\)",F,SE);
[/asy]
Point $Q$ is arbitrarily placed inside triangle $DEF$. What is the probability that $Q$ lies closer to $D$ than to either $E$ or $F$?
|
\frac{1}{4}
|
1/8
|
It is given regular $n$ -sided polygon, $n \geq 6$ . How many triangles they are inside the polygon such that all of their sides are formed by diagonals of polygon and their vertices are vertices of polygon?
|
\frac{n(n-4)(n-5)}{6}
|
5/8
|
Let \( S = \{1, 2, 3, \ldots, 9, 10\} \). A non-empty subset of \( S \) is considered "Good" if the number of even integers in the subset is more than or equal to the number of odd integers in the same subset. For example, the subsets \( \{4,8\}, \{3,4,7,8\} \) and \( \{1,3,6,8,10\} \) are "Good". How many subsets of \( S \) are "Good"?
|
637
|
7/8
|
Of the following
(1) $a(x-y)=ax-ay$
(2) $a^{x-y}=a^x-a^y$
(3) $\log (x-y)=\log x-\log y$
(4) $\frac{\log x}{\log y}=\log{x}-\log{y}$
(5) $a(xy)=ax \cdot ay$
$\textbf{(A)}\text{Only 1 and 4 are true}\qquad\\\textbf{(B)}\ \text{Only 1 and 5 are true}\qquad\\\textbf{(C)}\ \text{Only 1 and 3 are true}\qquad\\\textbf{(D)}\ \text{Only 1 and 2 are true}\qquad\\\textbf{(E)}\ \text{Only 1 is true}$
|
\textbf{(E)}
|
1/8
|
Fully factor the following expression: $2x^2-8$
|
(2) (x+2) (x-2)
|
1/8
|
Acute triangle $ABP$ , where $AB > BP$ , has altitudes $BH$ , $PQ$ , and $AS$ . Let $C$ denote the intersection of lines $QS$ and $AP$ , and let $L$ denote the intersection of lines $HS$ and $BC$ . If $HS = SL$ and $HL$ is perpendicular to $BC$ , find the value of $\frac{SL}{SC}$ .
|
\frac{1}{3}
|
1/8
|
To actively create a "demonstration school for the prevention and control of myopia in children and adolescents in the city" and cultivate students' good eye habits, a certain school conducted a competition on correct eye knowledge this semester. $20$ student answer sheets were randomly selected and their scores (denoted by $x$, unit: points) were recorded as follows:
$86\ \ 82\ \ 90\ \ 99\ \ 98\ \ 96\ \ 90\ \ 100\ \ 89\ \ 83$
$87\ \ 88\ \ 81\ \ 90\ \ 93\ \ 100\ \ 96\ \ 100\ \ 92\ \ 100$
Organize the data:
<table>
<tr>
<td width="139" align="center">$80\leqslant x \lt 85$</td>
<td width="139" align="center">$85\leqslant x \lt 90$</td>
<td width="139" align="center">$90\leqslant x \lt 95$</td>
<td width="139" align="center">$95\leqslant x\leqslant 100$</td>
</tr>
<tr>
<td align="center">$3$</td>
<td align="center">$4$</td>
<td align="center">$a$</td>
<td align="center">$8$</td>
</tr>
</table>
Analyze the data:
<table>
<tr>
<td width="138" align="center">Mean</td>
<td width="138" align="center">Median</td>
<td width="138" align="center">Mode</td>
</tr>
<tr>
<td align="center">$92$</td>
<td align="center">$b$</td>
<td align="center">$c$</td>
</tr>
</table>
Based on the above information, answer the following questions:
$(1)$ Write down the values of $a$, $b$, and $c$ directly from the table;
$(2)$ $2700$ students from the school participated in the knowledge competition. Estimate the number of people with scores not less than $90$ points;
$(3)$ Choose one quantity from the median and mode, and explain its significance in this question.
|
1755
|
5/8
|
The crafty rabbit and the foolish fox made an agreement: every time the fox crosses the bridge in front of the rabbit's house, the rabbit would double the fox's money. However, each time the fox crosses the bridge, he has to pay the rabbit a toll of 40 cents. Hearing that his money would double each time he crossed the bridge, the fox was very happy. However, after crossing the bridge three times, he discovered that all his money went to the rabbit. How much money did the fox initially have?
|
35
|
7/8
|
In a certain city, the rules for selecting license plate numbers online are as follows: The last five characters of the plate must include two English letters (with the letters "I" and "O" not allowed), and the last character must be a number. How many possible combinations meet these requirements?
|
3456000
|
1/8
|
Given a positive number \( r \), let the set \( T = \left\{(x, y) \mid x, y \in \mathbb{R}, \text{ and } x^{2} + (y-7)^{2} \leq r^{2} \right\} \). This set \( T \) is a subset of the set \( S = \{(x, y) \mid x, y \in \mathbb{R}, \text{ and for any } \theta \in \mathbb{R}, \ \cos 2\theta + x \cos \theta + y \geq 0\} \). Determine the maximum value of \( r \).
|
4\sqrt{2}
|
5/8
|
Given that $α \in \left( \frac{π}{2}, π \right)$ and $3\cos 2α = \sin \left( \frac{π}{4} - α \right)$, find the value of $\sin 2α$.
|
-\frac{17}{18}
|
7/8
|
Irja and Valtteri are tossing coins. They take turns, Irja starting. Each of them has a pebble which reside on opposite vertices of a square at the start. If a player gets heads, she or he moves her or his pebble on opposite vertex. Otherwise the player in turn moves her or his pebble to an adjacent vertex so that Irja proceeds in positive and Valtteri in negative direction. The winner is the one who can move his pebble to the vertex where opponent's pebble lies. What is the probability that Irja wins the game?
|
\frac{4}{7}
|
1/8
|
Find the smallest positive number $\alpha$ such that there exists a positive number $\beta$ satisfying the inequality
$$
\sqrt{1+x}+\sqrt{1-x} \leqslant 2-\frac{x^{\alpha}}{\beta}
$$
for $0 \leqslant x \leqslant 1$.
|
2
|
4/8
|
Let $ABC$ be a triangle with $\angle BCA=90^{\circ}$ , and let $D$ be the foot of the altitude from $C$ . Let $X$ be a point in the interior of the segment $CD$ . Let $K$ be the point on the segment $AX$ such that $BK=BC$ . Similarly, let $L$ be the point on the segment $BX$ such that $AL=AC$ . Let $M$ be the point of intersection of $AL$ and $BK$ .
Show that $MK=ML$ .
*Proposed by Josef Tkadlec, Czech Republic*
|
MK=ML
|
1/8
|
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.