problem
stringlengths 10
7.44k
| answer
stringlengths 1
270
| difficulty
stringclasses 8
values |
|---|---|---|
Let $p$ be a prime number. Consider the number
$$
1 + \frac{1}{2} + \ldots + \frac{1}{p-1}
$$
written as an irreducible fraction $\frac{a}{b}$. Prove that, if $p \geq 5$, then $p^{2}$ divides $a$. | p^2divides | 1/8 |
In $\triangle RED$, $\measuredangle DRE=75^{\circ}$ and $\measuredangle RED=45^{\circ}$. $RD=1$. Let $M$ be the midpoint of segment $\overline{RD}$. Point $C$ lies on side $\overline{ED}$ such that $\overline{RC}\perp\overline{EM}$. Extend segment $\overline{DE}$ through $E$ to point $A$ such that $CA=AR$. Then $AE=\frac{a-\sqrt{b}}{c}$, where $a$ and $c$ are relatively prime positive integers, and $b$ is a positive integer. Find $a+b+c$. | 56 | 1/8 |
How many roots does the equation \(\sin x - \log_{10} x = 0\) have? | 3 | 2/8 |
Let \(\{a, b, c, d\}\) be a subset of \(\{1, 2, \ldots, 17\}\). If 17 divides \(a - b + c - d\), then \(\{a, b, c, d\}\) is called a "good subset." Find the number of good subsets. | 476 | 1/8 |
Let \( A_i (i = 1, 2, 3, 4) \) be subsets of the set \( S = \{1, 2, 3, \cdots, 2002\} \). Let \( F \) be the set of all quadruples \( (A_1, A_2, A_3, A_4) \). Find the value of \( \sum_{(A_1, A_2, A_3, A_4) \in \mathbf{F}} \left| A_1 \cup A_2 \cup A_3 \cup A_4 \right| \), where \(|x|\) denotes the number of elements in the set \( x \). | 2^{8004}\times30030 | 1/8 |
Points $K$ and $L$ are taken on the sides $BC$ and $CD$ of a square $ABCD$ so that $\angle AKB = \angle AKL$ . Find $\angle KAL$ . | 45 | 3/8 |
The sum of the first four terms of an arithmetic progression, as well as the sum of the first seven terms, are natural numbers. Furthermore, its first term \(a_1\) satisfies the inequality \(a_1 \leq \frac{2}{3}\). What is the greatest value that \(a_1\) can take? | 9/14 | 5/8 |
Given that the sum of the first $n$ terms of the positive arithmetic geometric sequence $\{a_n\}$ is $S_n$, and $\frac{a_{n+1}}{a_n} < 1$, if $a_3 + a_5 = 20$ and $a_2 \cdot a_6 = 64$, calculate $S_6$. | 126 | 2/8 |
Compute
$$
\sum_{n=2009}^{\infty} \frac{1}{\binom{n}{2009}}.$$
Note that \(\binom{n}{k}\) is defined as \(\frac{n!}{k!(n-k)!}\). | \frac{2009}{2008} | 5/8 |
A rectangular table of size \( x \) cm \( \times 80 \) cm is covered with identical sheets of paper of size 5 cm \( \times 8 \) cm. The first sheet is placed in the bottom-left corner, and each subsequent sheet is placed 1 cm higher and 1 cm to the right of the previous one. The last sheet is adjacent to the top-right corner. What is the length \( x \) in centimeters? | 77 | 7/8 |
Six students taking a test sit in a row of seats with aisles only on the two sides of the row. If they finish the test at random times, what is the probability that some student will have to pass by another student to get to an aisle? | \frac{43}{45} | 1/8 |
Let $a, b, c, p, q, r$ be positive integers with $p, q, r \ge 2$ . Denote
\[Q=\{(x, y, z)\in \mathbb{Z}^3 : 0 \le x \le a, 0 \le y \le b , 0 \le z \le c \}. \]
Initially, some pieces are put on the each point in $Q$ , with a total of $M$ pieces. Then, one can perform the following three types of operations repeatedly:
(1) Remove $p$ pieces on $(x, y, z)$ and place a piece on $(x-1, y, z)$ ;
(2) Remove $q$ pieces on $(x, y, z)$ and place a piece on $(x, y-1, z)$ ;
(3) Remove $r$ pieces on $(x, y, z)$ and place a piece on $(x, y, z-1)$ .
Find the smallest positive integer $M$ such that one can always perform a sequence of operations, making a piece placed on $(0,0,0)$ , no matter how the pieces are distributed initially. | p^^^ | 6/8 |
Kolya drew 10 line segments and marked all their intersection points in red. After counting the red points, he noticed the following property: on each segment, there are exactly three red points.
a) Provide an example of the arrangement of 10 line segments with this property.
b) What can be the maximum number of red points for 10 segments with this property? | 15 | 6/8 |
Let $a$ , $b$ , $c$ , $d$ be complex numbers satisfying
\begin{align*}
5 &= a+b+c+d
125 &= (5-a)^4 + (5-b)^4 + (5-c)^4 + (5-d)^4
1205 &= (a+b)^4 + (b+c)^4 + (c+d)^4 + (d+a)^4 + (a+c)^4 + (b+d)^4
25 &= a^4+b^4+c^4+d^4
\end{align*}
Compute $abcd$ .
*Proposed by Evan Chen* | 70 | 1/8 |
How many multiples of 15 are between 15 and 305? | 20 | 5/8 |
Steph Curry is playing the following game and he wins if he has exactly 5 points at some time. Flip a fair coin. If heads, shoot a 3-point shot which is worth 3 points. If tails, shoot a free throw which is worth 1 point. He makes \(\frac{1}{2}\) of his 3-point shots and all of his free throws. Find the probability he will win the game. (Note he keeps flipping the coin until he has exactly 5 or goes over 5 points). | \frac{140}{243} | 3/8 |
Let the arithmetic sequence $\left\{a_{n}\right\}(n \geqslant 1)$ contain the terms 1 and $\sqrt{2}$. Prove that no three terms in $\left\{a_{n}\right\}$ form a geometric sequence. | 3 | 6/8 |
The ratio of the short side of a certain rectangle to the long side is equal to the ratio of the long side to the diagonal. What is the square of the ratio of the short side to the long side of this rectangle?
$\textbf{(A)}\ \frac{\sqrt{3}-1}{2}\qquad\textbf{(B)}\ \frac{1}{2}\qquad\textbf{(C)}\ \frac{\sqrt{5}-1}{2} \qquad\textbf{(D)}\ \frac{\sqrt{2}}{2} \qquad\textbf{(E)}\ \frac{\sqrt{6}-1}{2}$ | \textbf{(C)}\\frac{\sqrt{5}-1}{2} | 1/8 |
Given that the terminal side of angle $\alpha$ passes through point $P(-4a, 3a) (a \neq 0)$, find the value of $\sin \alpha + \cos \alpha - \tan \alpha$. | \frac{19}{20} | 6/8 |
Stephanie enjoys swimming. She goes for a swim on a particular date if, and only if, the day, month (where January is replaced by '01' through to December by '12') and year are all of the same parity (that is they are all odd, or all are even). On how many days will she go for a swim in the two-year period between January 1st of one year and December 31st of the following year inclusive? | 183 | 1/8 |
In the circular sector \( OAB \), the central angle of which is \( 45^{\circ} \), a rectangle \( KMRT \) is inscribed. Side \( KM \) of the rectangle lies along the radius \( OA \), vertex \( P \) is on the arc \( AB \), and vertex \( T \) is on the radius \( OB \). The side \( KT \) is 3 units longer than the side \( KM \). The area of the rectangle \( KMRT \) is 18. Find the radius of the circle. | 3\sqrt{13} | 3/8 |
Professor Severus Snape brewed three potions, each in a volume of 600 ml. The first potion makes the drinker intelligent, the second makes them beautiful, and the third makes them strong. To have the effect of the potion, it is sufficient to drink at least 30 ml of each potion. Severus Snape intended to drink his potions, but he was called away by the headmaster and left the signed potions in large jars on his table. Taking advantage of his absence, Harry, Hermione, and Ron approached the table with the potions and began to taste them.
The first to try the potions was Hermione: she approached the first jar of the intelligence potion and drank half of it, then poured the remaining potion into the second jar of the beauty potion, mixed the contents thoroughly, and drank half of it. Then it was Harry's turn: he drank half of the third jar of the strength potion, then poured the remainder into the second jar, thoroughly mixed everything in the jar, and drank half of it. Now all the contents were left in the second jar, which Ron ended up with. What percentage of the contents of this jar does Ron need to drink to ensure that each of the three potions will have an effect on him? | 40 | 7/8 |
Find all real numbers $x,y,z$ so that
\begin{align*}
x^2 y + y^2 z + z^2 &= 0 \\
z^3 + z^2 y + z y^3 + x^2 y &= \frac{1}{4}(x^4 + y^4).
\end{align*} | (0, 0, 0) | 1/8 |
Find all polynomials $W$ with real coefficients possessing the following property: if $x+y$ is a rational number, then $W(x)+W(y)$ is rational. | W(x)=ax+b | 7/8 |
Given a right triangle $PQR$ with $\angle PQR = 90^\circ$, suppose $\cos Q = 0.6$ and $PQ = 15$. What is the length of $QR$? | 25 | 1/8 |
Given the ellipse $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1 (a>b>0)$ and the line $x+y=1$ intersect at points $M$ and $N$, and $OM \perp ON$ (where $O$ is the origin), when the eccentricity $e \in\left[-\frac{3}{3}, \overline{2}\right]$, find the range of values for the length of the major axis of the ellipse. | [\sqrt{5},\sqrt{6}] | 1/8 |
In a coordinate system, a circle with radius $7$ and center is on the y-axis placed inside the parabola with equation $y = x^2$ , so that it just touches the parabola in two points. Determine the coordinate set for the center of the circle.
| (0,\frac{197}{4}) | 7/8 |
For a finite sequence \( B = (b_1, b_2, \dots, b_{50}) \) of numbers, the Cesaro sum is defined as
\[
\frac{S_1 + \cdots + S_{50}}{50},
\]
where \( S_k = b_1 + \cdots + b_k \) and \( 1 \leq k \leq 50 \).
If the Cesaro sum of the 50-term sequence \( (b_1, \dots, b_{50}) \) is 500, what is the Cesaro sum of the 51-term sequence \( (2, b_1, \dots, b_{50}) \)? | 492 | 1/8 |
Let \( a_{1}, a_{2}, \cdots, a_{21} \) be a permutation of \( 1, 2, \cdots, 21 \) such that
$$
\left|a_{20}-a_{21}\right| \geqslant\left|a_{19}-a_{21}\right| \geqslant\left|a_{18}-a_{21}\right| \geqslant \cdots \geqslant\left|a_{1}-a_{21}\right|.
$$
The number of such permutations is \[\qquad\]. | 3070 | 2/8 |
Given that the perimeter of triangle \( \triangle ABC \) is 1, and \(\sin 2A + \sin 2B = 4 \sin A \sin B\):
(1) Prove that \( \triangle ABC \) is a right triangle.
(2) Find the maximum area of \( \triangle ABC \). | \frac{3-2\sqrt{2}}{4} | 2/8 |
In a football tournament, eight teams played, and each team played once with every other team. Teams that score fifteen or more points advance to the next round. A win gives 3 points, a draw gives 1 point, and a loss gives 0 points. What is the maximum number of teams that can advance to the next round? | 5 | 5/8 |
Let $ABCD$ be a quadrilateral with $\overline{AB}\parallel\overline{CD}$ , $AB=16$ , $CD=12$ , and $BC<AD$ . A circle with diameter $12$ is inside of $ABCD$ and tangent to all four sides. Find $BC$ . | 13 | 5/8 |
Please write an irrational number that is smaller than $3$. | \sqrt{2} | 7/8 |
In how many ways can the following selections be made from a full deck of 52 cards:
a) 4 cards with different suits and different ranks?
b) 6 cards such that all four suits are represented among them? | 8682544 | 6/8 |
In triangle $ABC$, angle $B$ equals $120^\circ$, and $AB = 2 BC$. The perpendicular bisector of side $AB$ intersects $AC$ at point $D$. Find the ratio $CD: DA$. | 3:2 | 5/8 |
Call a set of integers "spacy" if it contains no more than one out of any three consecutive integers. How many subsets of $\{1, 2, 3, \dots, 15\}$, including the empty set, are spacy? | 406 | 4/8 |
The cells of a \(5 \times 5\) grid are each colored red, white, or blue. Sam starts at the bottom-left cell of the grid and walks to the top-right cell by taking steps one cell either up or to the right. Thus, he passes through 9 cells on his path, including the start and end cells. Compute the number of colorings for which Sam is guaranteed to pass through a total of exactly 3 red cells, exactly 3 white cells, and exactly 3 blue cells no matter which route he takes. | 1680 | 1/8 |
A function \( f: \mathbb{N} \rightarrow \mathbb{N} \) has the property that for all positive integers \( m \) and \( n \), exactly one of the \( f(n) \) numbers
\[
f(m+1), f(m+2), \ldots, f(m+f(n))
\]
is divisible by \( n \). Prove that \( f(n)=n \) for infinitely many positive integers \( n \). | f(n)=nfor\inftyinitelymanypositiveintegersn | 1/8 |
Real numbers $x$, $y$, and $z$ are chosen independently and at random from the interval $[0,n]$ for some positive integer $n$. The probability that no two of $x$, $y$, and $z$ are within 1 unit of each other is greater than $\frac {1}{2}$. What is the smallest possible value of $n$? | 10 | 4/8 |
Let the function \( f(x) \) for any real number \( x \) satisfy: \( f(2-x) = f(2+x) \) and \( f(7-x) = f(7+x) \), and \( f(0) = 0 \). Let \( M \) be the number of roots of \( f(x) = 0 \) in the interval \([-1000, 1000]\). Find the minimum value of \( M \). | 401 | 6/8 |
The sum of the first $n$ terms of the sequence $\{a_n\}$ is $S_n=n^2+n+1$, and $b_n=(-1)^n(a_n-2)$ $(n\in\mathbb{N}^*)$, then the sum of the first $50$ terms of the sequence $\{b_n\}$ is $\_\_\_\_\_\_\_$. | 49 | 7/8 |
For a given positive integer \( k \), let \( f_{1}(k) \) represent the square of the sum of the digits of \( k \), and define \( f_{n+1}(k) = f_{1}\left(f_{n}(k)\right) \) for \( n \geq 1 \). Find the value of \( f_{2005}\left(2^{2006}\right) \). | 169 | 3/8 |
Let $C$ and $C^{\prime}$ be two externally tangent circles with centers $O$ and $O^{\prime}$ and radii 1 and 2, respectively. From $O$, a tangent is drawn to $C^{\prime}$ with the point of tangency at $P^{\prime}$, and from $O^{\prime}$, a tangent is drawn to $C$ with the point of tangency at $P$, both tangents being in the same half-plane relative to the line passing through $O$ and $O^{\prime}$. Find the area of the triangle $O X O^{\prime}$, where $X$ is the intersection point of $O^{\prime} P$ and $O P^{\prime}$. | \frac{4\sqrt{2} - \sqrt{5}}{3} | 7/8 |
Given the ellipse $\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1$ $(a > b > 0)$ with eccentricity $e = \dfrac{\sqrt{6}}{3}$, and the distance between the left focus and one endpoint of the minor axis is $\sqrt{3}$.
$(I)$ Find the standard equation of the ellipse;
$(II)$ Given the fixed point $E(-1, 0)$, if the line $y = kx + 2$ intersects the ellipse at points $A$ and $B$. Is there a real number $k$ such that the circle with diameter $AB$ passes through point $E$? Please explain your reasoning. | \dfrac{7}{6} | 7/8 |
Given that plane vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ are non-zero vectors, $|\overrightarrow{a}|=2$, and $\overrightarrow{a} \bot (\overrightarrow{a}+2\overrightarrow{b})$, calculate the projection of vector $\overrightarrow{b}$ in the direction of vector $\overrightarrow{a}$. | -1 | 5/8 |
The function f is defined recursively by f(1)=f(2)=1 and f(n)=f(n-1)-f(n-2)+n for all integers n ≥ 3. Find the value of f(2018). | 2017 | 7/8 |
Assuming that a knight in chess can move $n$ squares horizontally and 1 square vertically (or 1 square horizontally and $n$ squares vertically) with each step, and given that the knight is placed on an infinite chessboard, determine for which natural numbers $n$ the knight can move to any specified square, and for which $n$ it cannot. | n | 2/8 |
Find the number of five-digit numbers in decimal notation that contain at least one digit 8. | 37512 | 7/8 |
When $\frac{1}{2222}$ is expressed as a decimal, what is the sum of the first 60 digits after the decimal point? | 108 | 4/8 |
A deck of $2n$ cards numbered from $1$ to $2n$ is shuffled and n cards are dealt to $A$ and $B$ . $A$ and $B$ alternately discard a card face up, starting with $A$ . The game when the sum of the discards is first divisible by $2n + 1$ , and the last person to discard wins. What is the probability that $A$ wins if neither player makes a mistake? | 0 | 4/8 |
On the side $AB$ of triangle $ABC$, point $K$ is marked. Segment $CK$ intersects the median $AM$ of the triangle at point $P$. It turns out that $AK = AP$.
Find the ratio $BK: PM$. | 2 | 4/8 |
While Travis is having fun on cubes, Sherry is hopping in the same manner on an octahedron. An octahedron has six vertices and eight regular triangular faces. After five minutes, how likely is Sherry to be one edge away from where she started? | \frac{11}{16} | 5/8 |
Given the ellipse \(\Gamma\):
\[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \quad (a > b > 0) \]
Let \(A\) and \(B\) be the left and top vertices of the ellipse \(\Gamma\), respectively. Let \(P\) be a point in the fourth quadrant on the ellipse \(\Gamma\). The line segment \(PA\) intersects the y-axis at point \(C\), and the line segment \(PB\) intersects the x-axis at point \(D\). Find the maximum area of the triangle \(\triangle PCD\). | \frac{(\sqrt{2}-1)}{2} | 1/8 |
Compute $\sqrt[3]{466560000}$. | 360 | 1/8 |
In an extended hexagonal lattice, each point is still one unit from its nearest neighbor. The lattice is now composed of two concentric hexagons where the outer hexagon has sides twice the length of the inner hexagon. All vertices are connected to their nearest neighbors. How many equilateral triangles have all three vertices in this extended lattice? | 20 | 1/8 |
A right triangle is inscribed in the ellipse given by the equation $x^2 + 9y^2 = 9$. One vertex of the triangle is at the point $(0,1)$, and one leg of the triangle is fully contained within the x-axis. Find the squared length of the hypotenuse of the inscribed right triangle, expressed as the ratio $\frac{m}{n}$ with $m$ and $n$ coprime integers, and give the value of $m+n$. | 11 | 1/8 |
A list of positive integers is called good if the maximum element of the list appears exactly once. A sublist is a list formed by one or more consecutive elements of a list. For example, the list $10,34,34,22,30,22$ the sublist $22,30,22$ is good and $10,34,34,22$ is not. A list is very good if all its sublists are good. Find the minimum value of $k$ such that there exists a very good list of length $2019$ with $k$ different values on it. | 11 | 2/8 |
Two mothers with their children want to sit on a bench with 4 places. In how many ways can they sit so that each mother sits next to her child? Each mother is walking with one child. | 8 | 7/8 |
A paper equilateral triangle of side length 2 on a table has vertices labeled \(A\), \(B\), and \(C\). Let \(M\) be the point on the sheet of paper halfway between \(A\) and \(C\). Over time, point \(M\) is lifted upwards, folding the triangle along segment \(BM\), while \(A\), \(B\), and \(C\) remain on the table. This continues until \(A\) and \(C\) touch. Find the maximum volume of tetrahedron \(ABCM\) at any time during this process. | \frac{\sqrt{3}}{6} | 1/8 |
Point \( O \) is the circumcenter of the acute-angled triangle \( ABC \). The circumcircle \(\omega\) of triangle \(AOC\) intersects the sides \(AB\) and \(BC\) again at points \(E\) and \(F\). It is found that the line \(EF\) bisects the area of triangle \(ABC\). Find angle \(B\). | 45 | 1/8 |
Brian writes down four integers $w > x > y > z$ whose sum is $44$. The pairwise positive differences of these numbers are $1, 3, 4, 5, 6,$ and $9$. What is the sum of the possible values for $w$? | 31 | 6/8 |
A club consists initially of 20 total members, which includes eight leaders. Each year, all the current leaders leave the club, and each remaining member recruits three new members. Afterwards, eight new leaders are elected from outside. How many total members will the club have after 4 years? | 980 | 3/8 |
Suppose positive real numbers \( x, y, z \) satisfy \( x y z = 1 \). Find the maximum value of \( f(x, y, z) = (1 - yz + z)(1 - zx + x)(1 - xy + y) \) and the corresponding values of \( x, y, z \). | 1 | 3/8 |
Two players, \(A\) and \(B\), play rock-paper-scissors continuously until player \(A\) wins 2 consecutive games. Suppose each player is equally likely to use each hand sign in every game. What is the expected number of games they will play? | 12 | 6/8 |
Calculate the definite integral:
$$
\int_{0}^{2 \pi} \sin^{6} x \cos^{2} x \, dx
$$ | \frac{5\pi}{64} | 6/8 |
The equilateral triangle has sides of \(2x\) and \(x+15\) as shown. Find the perimeter of the triangle in terms of \(x\). | 90 | 5/8 |
Given a polynomial with integer coefficients,
\[16x^5 + b_4x^4 + b_3x^3 + b_2x^2 + b_1x + 24 = 0,\]
find the number of different possible rational roots of this polynomial. | 32 | 6/8 |
Let be the set $ \mathcal{C} =\left\{ f:[0,1]\longrightarrow\mathbb{R}\left| \exists f''\bigg|_{[0,1]} \right.\quad\exists x_1,x_2\in [0,1]\quad x_1\neq x_2\wedge \left( f\left(
x_1 \right) = f\left( x_2 \right) =0\vee f\left(
x_1 \right) = f'\left( x_1 \right) = 0\right) \wedge f''<1 \right\} , $ and $ f^*\in\mathcal{C} $ such that $ \int_0^1\left| f^*(x) \right| dx =\sup_{f\in\mathcal{C}} \int_0^1\left| f(x) \right| dx . $ Find $ \int_0^1\left| f^*(x) \right| dx $ and describe $ f^*. $ | 1/12 | 1/8 |
The ratio of $w$ to $x$ is $4:3$, the ratio of $y$ to $z$ is $3:2$, and the ratio of $z$ to $x$ is $1:6$. What is the ratio of $w$ to $y?$
$\textbf{(A)}\ 4:3 \qquad\textbf{(B)}\ 3:2 \qquad\textbf{(C)}\ 8:3 \qquad\textbf{(D)}\ 4:1 \qquad\textbf{(E)}\ 16:3$ | \textbf{(E)}\16:3 | 1/8 |
Let $d_1 = a^2 + 2^a + a \cdot 2^{(a+1)/2} + a^3$ and $d_2 = a^2 + 2^a - a \cdot 2^{(a+1)/2} + a^3$. If $1 \le a \le 300$, how many integral values of $a$ are there such that $d_1 \cdot d_2$ is a multiple of $3$? | 100 | 4/8 |
I had been planning to work for 20 hours a week for 12 weeks this summer to earn $\$3000$ to buy a used car. Unfortunately, I got sick for the first two weeks of the summer and didn't work any hours. How many hours a week will I have to work for the rest of the summer if I still want to buy the car? | 24 | 7/8 |
On the sides $AB$ and $AC$ of triangle $ABC$ lie points $K$ and $L$, respectively, such that $AK: KB = 4:7$ and $AL: LC = 3:2$. The line $KL$ intersects the extension of side $BC$ at point $M$. Find the ratio $CM: BC$. | 8 : 13 | 1/8 |
The bisectors $\mathrm{AD}$ and $\mathrm{BE}$ of triangle $\mathrm{ABC}$ intersect at point I. It turns out that the area of triangle ABI is equal to the area of quadrilateral CDIE. Find $A B$, given that $C A=9$ and $C B=4$. | 6 | 4/8 |
The three row sums and the three column sums of the array \[
\left[\begin{matrix}4 & 9 & 2\\ 8 & 1 & 6\\ 3 & 5 & 7\end{matrix}\right]
\]
are the same. What is the least number of entries that must be altered to make all six sums different from one another? | 4 | 1/8 |
Any five points are taken inside or on a rectangle with dimensions 2 by 1. Let b be the smallest possible number with the property that it is always possible to select one pair of points from these five such that the distance between them is equal to or less than b. What is b? | \frac{\sqrt{5}}{2} | 3/8 |
Given that \( a_{1} < a_{2} < \cdots < a_{9} \) are positive integers such that the sums created from (at least one, at most nine different elements) are all unique, prove that \( a_{9} > 100 \). | a_9>100 | 7/8 |
Consider a three-person game involving the following three types of fair six-sided dice. - Dice of type $A$ have faces labelled $2,2,4,4,9,9$. - Dice of type $B$ have faces labelled $1,1,6,6,8,8$. - Dice of type $C$ have faces labelled $3,3,5,5,7,7$. All three players simultaneously choose a die (more than one person can choose the same type of die, and the players don't know one another's choices) and roll it. Then the score of a player $P$ is the number of players whose roll is less than $P$ 's roll (and hence is either 0,1 , or 2 ). Assuming all three players play optimally, what is the expected score of a particular player? | \frac{8}{9} | 1/8 |
Let the sequence \(\{a_{n}\}\) have 10 terms, where \(a_{i} \in \{1, -1\}\) for \(i = 1, 2, \ldots, 10\), and for any \(k \leqslant 9\), the following conditions hold:
\[ \left|\sum_{i=1}^{k} a_{i}\right| \leq 2 \]
\[ \sum_{i=1}^{10} a_{i} = 0 \]
How many sequences \(\{a_{n}\}\) satisfy these conditions? | 162 | 6/8 |
In $\triangle ABC$, $2\sin ^{2} \frac{A}{2}= \sqrt{3}\sin A$, $\sin (B-C)=2\cos B\sin C$, find the value of $\frac{AC}{AB}$ . | \frac{1+\sqrt{13}}{2} | 7/8 |
A piece of wire is spirally wound around a cylindrical tube, forming 10 turns. The length of the tube is 9 cm, and the circumference of the tube is 4 cm. The ends of the spiral lie on the same generating line of the cylinder. Find the length of the wire. | 41\, | 1/8 |
Given the ellipse \(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1\) (\(a > b > 0\)), where the right focus is \(F\) and the right directrix \(l\) intersects the x-axis at point \(N\). A line \(PM\), perpendicular to directrix \(l\), is drawn from a point \(P\) on the ellipse with the foot \(M\). It is known that \(PN\) bisects \(\angle FPM\) and the quadrilateral \(OFMP\) is a parallelogram. Prove that \(e > \frac{2}{3}\). | \frac{2}{3} | 1/8 |
Let event $A$ be "Point $M(x,y)$ satisfies $x^{2}+y^{2}\leqslant a(a > 0)$", and event $B$ be "Point $M(x,y)$ satisfies $\begin{cases} & x-y+1\geqslant 0 \\ & 5x-2y-4\leqslant 0 \\ & 2x+y+2\geqslant 0 \end{cases}$. If $P(B|A)=1$, then find the maximum value of the real number $a$. | \dfrac{1}{2} | 7/8 |
In the non-convex quadrilateral $ABCD$ shown below, $\angle BCD$ is a right angle, $AB=12$, $BC=4$, $CD=3$, and $AD=13$. What is the area of quadrilateral $ABCD$? | 36 | 2/8 |
The values of the quadratic polynomial \( ax^{2} + bx + c \) are negative for all \( x \). Prove that \(\frac{b}{a} < \frac{c}{a} + 1\). | \frac{b}{}<\frac{}{}+1 | 7/8 |
In the acute triangle \( \triangle ABC \), the circumradius \( R = 1 \), and the sides opposite to angles \( A, B, \) and \( C \) are \( a, b, \) and \( c \) respectively. Prove that:
$$
\frac{a}{1-\sin A}+\frac{b}{1-\sin B}+\frac{c}{1-\sin C} \geqslant 18+12 \sqrt{3}
$$ | 18+12\sqrt{3} | 7/8 |
An ant walks from the bottom left corner of a \( 10 \times 10 \) square grid to the diagonally-opposite corner, always walking along grid lines and taking as short a route as possible. Let \( N(k) \) be the number of different paths that the ant could follow if it makes exactly \( k \) turns. Find \( N(6) - N(5) \). | 3456 | 4/8 |
For what values of the parameter \( a \) will the minimum value of the function
$$
f(x)=|7x - 3a + 8| + |5x + 4a - 6| + |x - a - 8| - 24
$$
be the smallest? | \frac{82}{43} | 1/8 |
If the sets of real numbers
$$
A=\{2x, 3y\} \text{ and } B=\{6, xy\}
$$
have exactly one common element, then the product of all elements in $A \cup B$ is ____. | 0 | 6/8 |
One of Euler's conjectures was disproved in then 1960s by three American mathematicians when they showed there was a positive integer $ n$ such that \[133^5 \plus{} 110^5 \plus{} 84^5 \plus{} 27^5 \equal{} n^5.\] Find the value of $ n$ . | 144 | 6/8 |
First, select $n$ numbers from the set $ \{1, 2, \cdots, 2020\} $. Then, from these $n$ numbers, choose any two numbers $a$ and $b$, such that $a$ does not divide $b$. Find the maximum value of $n$. | 1010 | 7/8 |
In triangle \( ABC \), \( AB = 33 \), \( AC = 21 \), and \( BC = m \), where \( m \) is a positive integer. If point \( D \) can be found on \( AB \) and point \( E \) can be found on \( AC \) such that \( AD = DE = EC = n \), where \( n \) is a positive integer, what must the value of \( m \) be? | 30 | 5/8 |
Vanya wrote the number 1 on the board and then several more numbers. Each time Vanya writes the next number, Mitya calculates the median of the numbers on the board and writes it down in his notebook. At a certain point, the numbers in Mitya's notebook are: $1, 2, 3, 2.5, 3, 2.5, 2, 2, 2, 2.5$.
a) What number was written on the board fourth?
b) What number was written on the board eighth? | 2 | 2/8 |
In the following diagram (not to scale), $A$ , $B$ , $C$ , $D$ are four consecutive vertices of an 18-sided regular polygon with center $O$ . Let $P$ be the midpoint of $AC$ and $Q$ be the midpoint of $DO$ . Find $\angle OPQ$ in degrees.
[asy]
pathpen = rgb(0,0,0.6)+linewidth(0.7); pointpen = black+linewidth(3); pointfontpen = fontsize(10); pen dd = rgb(0,0,0.6)+ linewidth(0.7) + linetype("4 4"); real n = 10, start = 360/n*6-15;
pair O=(0,0), A=dir(start), B=dir(start+360/n), C=dir(start+2*360/n), D=dir(start+3*360/n), P=(A+C)/2, Q=(O+D)/2; D(D("O",O,NE)--D("A",A,W)--D("B",B,SW)--D("C",C,S)--D("D",D,SE)--O--D("P",P,1.6*dir(95))--D("Q",Q,NE)); D(A--C); D(A--(A+dir(start-360/n))/2, dd); D(D--(D+dir(start+4*360/n))/2, dd);
[/asy] | 30 | 5/8 |
What is the minimum force required to press a cube with a volume of $10 \, \text{cm}^{3}$, floating in water, so that it is completely submerged? The density of the material of the cube is $400 \, \text{kg/m}^{3}$, and the density of water is $1000 \, \text{kg/m}^{3}$. Give the answer in SI units. Assume the acceleration due to gravity is $10 \, \text{m/s}^{2}$. | 0.06\, | 1/8 |
Let \( \triangle ABC \) be a triangle. The incircle of \( \triangle ABC \) touches the sides \( AB \) and \( AC \) at the points \( Z \) and \( Y \), respectively. Let \( G \) be the point where the lines \( BY \) and \( CZ \) meet, and let \( R \) and \( S \) be points such that the two quadrilaterals \( BCYR \) and \( BCSZ \) are parallelograms.
Prove that \( GR = GS \). | GR=GS | 1/8 |
A rectangle has side lengths in the ratio of \(2:5\). If all of its sides are extended by \(9 \text{ cm}\), the resulting rectangle has side lengths in the ratio \(3:7\).
What will be the ratio of the side lengths of the rectangle that results from extending all of its sides by an additional \(9 \text{ cm}\)? | 5:11 | 3/8 |
There are 100 houses in a row on a street. A painter comes and paints every house red. Then, another painter comes and paints every third house (starting with house number 3) blue. Another painter comes and paints every fifth house red (even if it is already red), then another painter paints every seventh house blue, and so forth, alternating between red and blue, until 50 painters have been by. After this is finished, how many houses will be red? | 52 | 1/8 |
Prove that for positive \( x, y, z \), the following inequality holds:
$$
(x + 8y + 2z)(x + 2y + z)(x + 4y + 4z) \geq 256xyz
$$ | (x+8y+2z)(x+2y+z)(x+4y+4z)\ge256xyz | 1/8 |
Determine the smallest possible value of $$ |2^m - 181^n|, $$ where $m$ and $n$ are positive integers.
| 7 | 7/8 |
A trapezoid \(ABCD\) with bases \(AD\) and \(BC\) is such that angle \(ABD\) is right and \(BC + CD = AD\). Find the ratio of the bases \(AD: BC\). | 2 | 2/8 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.