problem
stringlengths 10
7.44k
| answer
stringlengths 1
270
| difficulty
stringclasses 8
values |
|---|---|---|
For every natural number $n$ we define \[f(n)=\left\lfloor n+\sqrt{n}+\frac{1}{2}\right\rfloor\] Show that for every integer $k \geq 1$ the equation \[f(f(n))-f(n)=k\] has exactly $2k-1$ solutions. | 2k-1 | 1/8 |
In the diagram, $PQ$ and $RS$ are diameters of a circle with radius 4. If $PQ$ and $RS$ are perpendicular, what is the area of the shaded region?
[asy]
size(120);
import graph;
fill((-1,-1)--(-1,1)--(1,1)--(1,-1)--cycle,mediumgray);
fill(Arc((0,0),sqrt(2),45,135)--cycle,mediumgray);fill(Arc((0,0),sqrt(2),225,315)--cycle,mediumgray);
draw(Circle((0,0),sqrt(2)));
draw((-1,-1)--(1,1)--(1,-1)--(-1,1)--cycle);
label("$P$",(-1,1),NW); label("$R$",(1,1),NE); label("$S$",(-1,-1),SW); label("$Q$",(1,-1),SE);
[/asy] | 16+8\pi | 1/8 |
Let $n \ge 3$ be a positive integer. Find all nonconstant real polynomials $f_1(x), f_2(x), ..., f_n(x)$ such that $f_k(x)f_{k+1}(x) = f_{k+1}(f_{k+2}(x))$ , $1 \le k \le n$ for all real x. [All suffixes are taken modulo $n$ .] | f_k(x)=x^2 | 1/8 |
Given that for reals $a_1,\cdots, a_{2004},$ equation $x^{2006}-2006x^{2005}+a_{2004}x^{2004}+\cdots +a_2x^2+a_1x+1=0$ has $2006$ positive real solution, find the maximum possible value of $a_1.$ | -2006 | 7/8 |
In a spatial quadrilateral \(ABCD\), \(AB = 2\), \(BC = 3\), \(CD = 4\), \(DA = 5\). Find \(\overrightarrow{AC} \cdot \overrightarrow{BD}\). | 7 | 6/8 |
Given the parametric equation of line $l$ as $$\begin{cases} x= \sqrt {3}+t \\ y=7+ \sqrt {3}t\end{cases}$$ ($t$ is the parameter), a coordinate system is established with the origin as the pole and the positive half of the $x$-axis as the polar axis. The polar equation of curve $C$ is $\rho \sqrt {a^{2}\sin^{2}\theta+4\cos^{2}\theta}=2a$ ($a>0$).
1. Find the Cartesian equation of curve $C$.
2. Given point $P(0,4)$, line $l$ intersects curve $C$ at points $M$ and $N$. If $|PM|\cdot|PN|=14$, find the value of $a$. | \frac{2\sqrt{21}}{3} | 4/8 |
When a five-digit number is multiplied by 9, the resulting number consists of the same digits in reverse order. Find the original number. | 10989 | 7/8 |
Max repeatedly throws a fair coin in a hurricane. For each throw, there is a $4 \%$ chance that the coin gets blown away. He records the number of heads $H$ and the number of tails $T$ before the coin is lost. (If the coin is blown away on a toss, no result is recorded for that toss.) What is the expected value of $|H-T|$? | \frac{24}{7} | 2/8 |
Find the derivative.
\[ y=\arcsin \frac{1}{2 x+3}+2 \sqrt{x^{2}+3 x+2}, \quad 2 x+3>0 \] | \frac{4\sqrt{x^2+3x+2}}{2x+3} | 7/8 |
There are 200 students gathered in the gym. Every pair of acquaintances shook hands. It turned out that any two strangers made at least 200 handshakes in total. Prove that there were at least 10,000 handshakes in total. | 10000 | 7/8 |
If the graph of the function $f(x)=\sin \omega x+\sin (\omega x- \frac {\pi}{2})$ ($\omega > 0$) is symmetric about the point $\left( \frac {\pi}{8},0\right)$, and there is a zero point within $\left(- \frac {\pi}{4},0\right)$, determine the minimum value of $\omega$. | 10 | 5/8 |
The probability of snow for each of the next four days is $\frac{3}{4}$. However, if it snows any day before the last day, the probability of snow on the last day increases to $\frac{4}{5}$. What is the probability that it will snow at least once during these four days? Express your answer as a common fraction. | \dfrac{1023}{1280} | 5/8 |
In $\triangle ABC$, $\angle C = 90^\circ$, $\angle B = 30^\circ$, and $AC = 1$. Point $M$ is the midpoint of $AB$. The triangle $\triangle ACM$ is folded along $CM$ so that the distance between points $A$ and $B$ is $\sqrt{2}$. Find the distance from point $A$ to the plane $BCM$. | \frac{\sqrt{6}}{3} | 5/8 |
If a positive integer has eight positive divisors and the sum of these eight positive divisors is 3240, it is called a "good number." For example, 2006 is a good number because the sum of its positive divisors $1, 2, 17, 34, 59, 118, 1003, 2006$ is 3240. Find the smallest good number. | 1614 | 1/8 |
One way to pack a 100 by 100 square with 10000 circles, each of diameter 1, is to put them in 100 rows with 100 circles in each row. If the circles are repacked so that the centers of any three tangent circles form an equilateral triangle, what is the maximum number of additional circles that can be packed? | 1443 | 2/8 |
Given that a normal vector of line $l$ is $\overrightarrow{n}=(\sqrt{3}, -1)$, find the size of the slope angle of line $l$. | \frac{\pi}{3} | 1/8 |
Given two polynomials in the variable \( x \) with integer coefficients. Their product is a polynomial in the variable \( x \) with even coefficients, not all of which are divisible by 4. Prove that in one of the polynomials, all coefficients are even, and in the other, at least one coefficient is odd. | 2 | 4/8 |
Two concentric circles have radii $1$ and $2$. Two points on the outer circle are chosen independently and uniformly at random. What is the probability that the chord joining the two points intersects the inner circle?
$\textbf{(A)}\ \frac{1}{6}\qquad \textbf{(B)}\ \frac{1}{4}\qquad \textbf{(C)}\ \frac{2-\sqrt{2}}{2}\qquad \textbf{(D)}\ \frac{1}{3}\qquad \textbf{(E)}\ \frac{1}{2}\qquad$ | \textbf{(D)}\:\frac{1}{3} | 1/8 |
n and r are positive integers. Find the smallest k for which we can construct r subsets \( A_1, A_2, \ldots, A_r \) of \( \{0, 1, 2, \ldots, n-1\} \) each with k elements such that each integer \( 0 \le m < n \) can be written as a sum of one element from each of the r subsets. | \lceiln^{1/r}\rceil | 3/8 |
Given that \( P \) is the vertex of a right circular cone with an isosceles right triangle as its axial cross-section, \( PA \) is a generatrix of the cone, and \( B \) is a point on the base of the cone. Find the maximum value of \(\frac{PA + AB}{PB}\). | \sqrt{4 + 2\sqrt{2}} | 1/8 |
In a $7 \times 7$ grid, 19 squares are colored red. A row or column is considered to be red if it contains at least 4 red squares. What is the maximum number of red rows and columns in this grid? | 8 | 2/8 |
Four people are sitting around a round table, with identical coins placed in front of each person. Everyone flips their coin simultaneously. If the coin lands heads up, the person stands up; if it lands tails up, the person remains seated. Calculate the probability that no two adjacent people stand up. | \frac{7}{16} | 7/8 |
Define the function \( f(k) \) as follows:
\[ f(k)=\begin{cases}
0, & k \text{ is a perfect square} \\
\left\lfloor \frac{1}{\{\sqrt{k}\}} \right\rfloor, & k \text{ is not a perfect square}
\end{cases} \]
Compute \( \sum_{k=1}^{240} f(k) \). | 768 | 1/8 |
Suppose $a, b$, and $c$ are distinct positive integers such that $\sqrt{a \sqrt{b \sqrt{c}}}$ is an integer. Compute the least possible value of $a+b+c$. | 7 | 3/8 |
Each page number of a 488-page book is printed one time in the book. The first page is page 1 and the last page is page 488. When printing all of the page numbers, how many more 4's are printed than 8's? | 90 | 5/8 |
$(1)$ When throwing two uniformly weighted dice, let $A=$"the first time an odd number appears," $B=$"the sum of the two dice is a multiple of $3$." Determine whether events $A$ and $B$ are independent, and explain the reason;<br/>$(2)$ Athletes A and B are participating in a shooting assessment test. Each person has two chances to shoot. If at least one shot hits the target in the two attempts, the assessment is considered passed. It is known that the probability of A hitting the target is $0.7$, and the probability of B hitting the target is $0.6$. A and B's shooting do not affect each other. Find the probability that exactly one of them passes the assessment. | 0.2212 | 5/8 |
Define the sequence \(\left\{a_{n}\right\}\) by:
\[ a_{1} = \frac{1}{2}, \quad a_{n+1} = \frac{a_{n}^{2}}{a_{n}^{2} - 2a_{n} + 2} \quad (n = 1, 2, \ldots). \]
Prove that for every positive integer \(n \) (\(n \geqslant 2\)), the following inequality holds:
\[ a_{n} + \frac{1}{2} a_{n-1} + \frac{1}{2^{2}} a_{n-2} + \cdots + \frac{1}{2^{n-1}} a_{1} < \frac{1}{2^{n-1}}. \] | a_{n}+\frac{1}{2}a_{n-1}+\frac{1}{2^{2}}a_{n-2}+\cdots+\frac{1}{2^{n-1}}a_{1}<\frac{1}{2^{n-1}} | 1/8 |
Find the value of \( \cos (\angle OBC + \angle OCB) \) in triangle \( \triangle ABC \), where angle \( \angle A \) is an obtuse angle, \( O \) is the orthocenter, and \( AO = BC \). | -\frac{\sqrt{2}}{2} | 6/8 |
Let $a_1,a_2,\ldots,a_n$ be a permutation of the numbers $1,2,\ldots,n$ , with $n\geq 2$ . Determine the largest possible value of the sum \[ S(n)=|a_2-a_1|+ |a_3-a_2| + \cdots + |a_n-a_{n-1}| . \]
*Romania* | \lfloor\frac{n^2}{2}\rfloor-1 | 2/8 |
A *site* is any point $(x, y)$ in the plane such that $x$ and $y$ are both positive integers less than or equal to 20.
Initially, each of the 400 sites is unoccupied. Amy and Ben take turns placing stones with Amy going first. On her turn, Amy places a new red stone on an unoccupied site such that the distance between any two sites occupied by red stones is not equal to $\sqrt{5}$ . On his turn, Ben places a new blue stone on any unoccupied site. (A site occupied by a blue stone is allowed to be at any distance from any other occupied site.) They stop as soon as a player cannot place a stone.
Find the greatest $K$ such that Amy can ensure that she places at least $K$ red stones, no matter how Ben places his blue stones.
*Proposed by Gurgen Asatryan, Armenia* | 100 | 7/8 |
Find all polynomials $P(x)$ with integer coefficients such that $P(P(n) + n)$ is a prime number for infinitely many integers $n$ . | P(x)=p | 3/8 |
Given positive integers \( x, y, z \) satisfying \( xyz = (22-x)(22-y)(22-z) \) and \( x + y + z < 44 \). Let \( M \) and \( N \) represent the maximum and minimum values of \( x^2 + y^2 + z^2 \) respectively. Find \( M + N \). | 926 | 2/8 |
A $5 \times 5$ grid of squares is filled with integers. Call a rectangle corner-odd if its sides are grid lines and the sum of the integers in its four corners is an odd number. What is the maximum possible number of corner-odd rectangles within the grid? | 60 | 1/8 |
Three cones are placed on a table, standing on their bases and touching each other. The radii of their bases are \(2r\), \(3r\), and \(10r\). A truncated cone (frustum) is placed on the table with its smaller base downward, sharing a slant height with each of the other cones. Find \(r\), if the radius of the smaller base of the truncated cone is 15. | 29 | 6/8 |
Find the smallest positive integer $n$ such that for any $n$ mutually coprime integers greater than 1 and not exceeding 2009, there is at least one prime number among them. | 15 | 4/8 |
Xiao Li drove from location A to location B. Two hours after departure, the car broke down at location C, and it took 40 minutes to repair. After the repair, the speed was only 75% of the normal speed, resulting in arrival at location B being 2 hours later than planned. If the car had instead broken down at location D, which is 72 kilometers past location C, with the same repair time of 40 minutes and the speed after the repair still being 75% of the normal speed, then the arrival at location B would be only 1.5 hours later than planned. Determine the total distance in kilometers between location A and location B. | 288 | 7/8 |
An HMMT party has $m$ MIT students and $h$ Harvard students for some positive integers $m$ and $h$, For every pair of people at the party, they are either friends or enemies. If every MIT student has 16 MIT friends and 8 Harvard friends, and every Harvard student has 7 MIT enemies and 10 Harvard enemies, compute how many pairs of friends there are at the party. | 342 | 1/8 |
Let \( a \) and \( b \) be positive real numbers such that \(\frac{1}{a}+\frac{1}{b} \leqslant 2 \sqrt{2}\) and \((a-b)^{2}=4(a b)^{3}\). Find \(\log_{a} b\). | -1 | 3/8 |
Three concentric circles have radii $3,$ $4,$ and $5.$ An equilateral triangle with one vertex on each circle has side length $s.$ The largest possible area of the triangle can be written as $a + \tfrac{b}{c} \sqrt{d},$ where $a,$ $b,$ $c,$ and $d$ are positive integers, $b$ and $c$ are relatively prime, and $d$ is not divisible by the square of any prime. Find $a+b+c+d.$ | 41 | 1/8 |
In a sequence of numbers \(1, 4, 7, 10, \cdots, 697, 700\), the first number is 1, and each subsequent number is equal to the previous number plus 3, until 700. Calculate the number of trailing zeroes in the product of all these numbers. For example, the number of trailing zeroes in 12003000 is 3. | 60 | 6/8 |
Given that \( f(x) \) is a function defined on the set of real numbers, and \( f(x+2)[1 - f(x)] = 1 + f(x) \). If \( f(1) = 2 + \sqrt{3} \), find the value of \( f(1949) \). | \sqrt{3}-2 | 1/8 |
Adding two dots above the decimal 0.142857 makes it a repeating decimal. The 2020th digit after the decimal point is 5. What is the repeating cycle? $\quad$ . | 142857 | 1/8 |
The distance between locations A and B is 135 kilometers. Two cars, a large one and a small one, travel from A to B. The large car departs 4 hours earlier than the small car, but the small car arrives 30 minutes earlier than the large car. The speed ratio of the small car to the large car is 5:2. Find the speeds of both cars. | 18 | 7/8 |
Using the digits 1 to 6 to form the equation shown below, where different letters represent different digits, the two-digit number represented by $\overline{A B}$ is what?
$$
\overline{A B} \times (\overline{C D} - E) + F = 2021
$$ | 32 | 7/8 |
This problem involves four different natural numbers written by a teacher. For each pair of these numbers, Peter found their greatest common divisor. He obtained six numbers: 1, 2, 3, 4, 5, and $N$, where $N > 5$. What is the smallest possible value of $N$? | 14 | 1/8 |
Given the line \( L: x + y - 9 = 0 \) and the circle \( M: 2x^2 + 2y^2 - 8x - 8y - 1 = 0 \), point \( A \) is on line \( L \), and points \( B \) and \( C \) are on circle \( M \). In triangle \( \triangle ABC \), \(\angle BAC = 45^\circ\) and \( AB \) passes through the center of circle \( M \). Then, the range of the x-coordinate of point \( A \) is \(\quad\) | [3,6] | 1/8 |
Let $x, y, n$ be positive integers with $n>1$. How many ordered triples $(x, y, n)$ of solutions are there to the equation $x^{n}-y^{n}=2^{100}$ ? | 49 | 1/8 |
Let $ABCD$ be a rectangle and let $\overline{DM}$ be a segment perpendicular to the plane of $ABCD$. Suppose that $\overline{DM}$ has integer length, and the lengths of $\overline{MA},\overline{MC},$ and $\overline{MB}$ are consecutive odd positive integers (in this order). What is the volume of pyramid $MABCD?$
$\textbf{(A) }24\sqrt5 \qquad \textbf{(B) }60 \qquad \textbf{(C) }28\sqrt5\qquad \textbf{(D) }66 \qquad \textbf{(E) }8\sqrt{70}$ | \textbf{(A)}24\sqrt{5} | 1/8 |
1. Given that $α$ and $β$ are acute angles, and $\cos α= \frac{4}{5}$, $\cos (α+β)=- \frac{16}{65}$, find the value of $\cos β$.
2. Given that $0 < β < \frac{π}{4} < α < \frac{3}{4}π$, $\cos ( \frac{π}{4}-α)= \frac{3}{5}$, $\sin ( \frac{3π}{4}+β)= \frac{5}{13}$, find the value of $\sin (α+β)$. | \frac{56}{65} | 6/8 |
An arithmetic sequence \(\{a_{n}\}\) with \(a_1 > 0\) has a sum of the first \(n\) terms denoted by \(S_n\). Given that \(S_9 > 0\) and \(S_{10} < 0\), for which value of \(n\) is \(S_n\) maximized? | 5 | 7/8 |
Given an integer $n \geq 2$, determine the largest constant $C_{n}$ such that for any real numbers $a_{1}, a_{2}, \cdots, a_{n}$ satisfying $a_{1} + a_{2} + \cdots + a_{n} = 0$, the following inequality holds:
\[
\sum_{i=1}^{n} \min \{a_{i}, a_{i+1}\} \leq C_{n} \cdot \min_{1 \leq i \leq n} a_{i},
\]
where it is understood that $a_{n+1}=a_{1}$. | \frac{n}{n-1} | 1/8 |
Consider a convex pentagon $ABCDE$. Let $P_A, P_B, P_C, P_D,$ and $P_E$ denote the centroids of triangles $BCDE, ACDE, ABDE, ABCD,$ and $ABCE$, respectively. Compute $\frac{[P_A P_B P_C P_D P_E]}{[ABCDE]}$. | \frac{1}{16} | 3/8 |
$\mathbb{N}$ is the set of positive integers. Determine all functions $f:\mathbb{N}\to\mathbb{N}$ such that for every pair $(m,n)\in\mathbb{N}^2$ we have that:
\[f(m)+f(n) \ | \ m+n .\] | f(n)=n | 3/8 |
Find the number of different monic quadratic polynomials (i.e., with the leading coefficient equal to 1) with integer coefficients such that they have two different roots which are powers of 5 with natural exponents, and their coefficients do not exceed in absolute value $125^{48}$. | 5112 | 5/8 |
The cost of five water bottles is \ $13, rounded to the nearest dollar, and the cost of six water bottles is \$ 16, also rounded to the nearest dollar. If all water bottles cost the same integer number of cents, compute the number of possible values for the cost of a water bottle.
*Proposed by Eugene Chen* | 11 | 7/8 |
On the continuation of side \( BC \) of parallelogram \( ABCD \), a point \( F \) is taken beyond point \( C \). Segment \( AF \) intersects diagonal \( BD \) at point \( E \) and side \( CD \) at point \( G \), where \( GF=3 \) and \( AE \) is 1 more than \( EG \). What part of the area of parallelogram \( ABCD \) is the area of triangle \( ADE \)? | \frac{1}{6} | 3/8 |
Anders is solving a math problem, and he encounters the expression $\sqrt{15!}$. He attempts to simplify this radical by expressing it as $a \sqrt{b}$ where $a$ and $b$ are positive integers. The sum of all possible distinct values of $ab$ can be expressed in the form $q \cdot 15!$ for some rational number $q$. Find $q$. | 4 | 7/8 |
Each corner cube is removed from this $3\text{ cm}\times 3\text{ cm}\times 3\text{ cm}$ cube. The surface area of the remaining figure is
[asy]
draw((2.7,3.99)--(0,3)--(0,0));
draw((3.7,3.99)--(1,3)--(1,0));
draw((4.7,3.99)--(2,3)--(2,0));
draw((5.7,3.99)--(3,3)--(3,0));
draw((0,0)--(3,0)--(5.7,0.99));
draw((0,1)--(3,1)--(5.7,1.99));
draw((0,2)--(3,2)--(5.7,2.99));
draw((0,3)--(3,3)--(5.7,3.99));
draw((0,3)--(3,3)--(3,0));
draw((0.9,3.33)--(3.9,3.33)--(3.9,0.33));
draw((1.8,3.66)--(4.8,3.66)--(4.8,0.66));
draw((2.7,3.99)--(5.7,3.99)--(5.7,0.99));
[/asy] | 54 | 7/8 |
There are 90 children in a chess club. During a session, they were divided into 30 groups of 3 people each, and in each group, everyone played one game with everyone else. No other games were played. A total of 30 "boy vs. boy" games and 14 "girl vs. girl" games were played. How many "mixed" groups were there, i.e., groups that included both boys and girls? | 23 | 7/8 |
A semicircle with diameter $d$ is contained in a square whose sides have length 8. Given the maximum value of $d$ is $m - \sqrt{n},$ find $m+n.$ | 544 | 2/8 |
Three fair six-sided dice are rolled. What is the probability that the values shown on two of the dice sum to the value shown on the remaining die?
$\textbf{(A)}\ \dfrac16\qquad\textbf{(B)}\ \dfrac{13}{72}\qquad\textbf{(C)}\ \dfrac7{36}\qquad\textbf{(D)}\ \dfrac5{24}\qquad\textbf{(E)}\ \dfrac29$ | \textbf{(D)}\frac{5}{24} | 1/8 |
The number of edges of a convex polyhedron is 99. What is the maximum number of edges a plane, not passing through its vertices, can intersect? | 66 | 1/8 |
Consider a string of $n$ $7$'s, $7777\cdots77,$ into which $+$ signs are inserted to produce an arithmetic expression. For example, $7+77+777+7+7=875$ could be obtained from eight $7$'s in this way. For how many values of $n$ is it possible to insert $+$ signs so that the resulting expression has value $7000$? | 108 | 1/8 |
Two circles of radius \( r \) are externally tangent to each other and internally tangent to the ellipse \( x^2 + 4y^2 = 8 \). Find \( r \). | \frac{\sqrt{6}}{2} | 7/8 |
For each real number $x$, let $\textbf{[}x\textbf{]}$ be the largest integer not exceeding $x$
(i.e., the integer $n$ such that $n\le x<n+1$). Which of the following statements is (are) true?
$\textbf{I. [}x+1\textbf{]}=\textbf{[}x\textbf{]}+1\text{ for all }x \\ \textbf{II. [}x+y\textbf{]}=\textbf{[}x\textbf{]}+\textbf{[}y\textbf{]}\text{ for all }x\text{ and }y \\ \textbf{III. [}xy\textbf{]}=\textbf{[}x\textbf{]}\textbf{[}y\textbf{]}\text{ for all }x\text{ and }y$
$\textbf{(A) }\text{none}\qquad \textbf{(B) }\textbf{I }\text{only}\qquad \textbf{(C) }\textbf{I}\text{ and }\textbf{II}\text{ only}\qquad \textbf{(D) }\textbf{III }\text{only}\qquad \textbf{(E) }\text{all}$ | \textbf{(B)}\textbf{I} | 1/8 |
In triangle $ABC$, angle $ACB$ is 50 degrees, and angle $CBA$ is 70 degrees. Let $D$ be the foot of the perpendicular from $A$ to $BC$, $O$ the center of the circle circumscribed about triangle $ABC$, and $E$ the other end of the diameter which goes through $A$. Find the angle $DAE$, in degrees.
[asy]
unitsize(1.5 cm);
pair A, B, C, D, E, O;
A = dir(90);
B = dir(90 + 100);
C = dir(90 - 140);
D = (A + reflect(B,C)*(A))/2;
E = -A;
O = (0,0);
draw(Circle(O,1));
draw(A--B--C--cycle);
draw(A--D);
draw(A--E,dashed);
label("$A$", A, N);
label("$B$", B, W);
label("$C$", C, SE);
label("$D$", D, SW);
label("$E$", E, S);
dot("$O$", O, dir(0));
[/asy] | 20^\circ | 3/8 |
For $n \in \mathbb{N}$ , consider non-negative valued functions $f$ on $\{1,2, \cdots , n\}$ satisfying $f(i) \geqslant f(j)$ for $i>j$ and $\sum_{i=1}^{n} (i+ f(i))=2023.$ Choose $n$ such that $\sum_{i=1}^{n} f(i)$ is at least. How many such functions exist in that case? | 15 | 1/8 |
In a plane Cartesian coordinate system, let \( O \) be the origin and \( \Gamma : y^{2} = 2px \) (\( p > 0 \)) be a parabola with focus \( F \) and directrix \( l \). A chord \( AB \) of the parabola \( \Gamma \) passes through the focus \( F \) and is not perpendicular to the \( x \)-axis.
1. Prove that the circle with \( AB \) as its diameter is tangent to the directrix \( l \).
2. Suppose the tangency point is \( P \). The line \( OP \) intersects the parabola \( \Gamma \) at a second point \( Q \). Find the minimum value of the area of triangle \( ABQ \). | \frac{3\sqrt{3}}{4}p^2 | 2/8 |
Let \[\begin{aligned} a &= \sqrt{2}+\sqrt{3}+\sqrt{6}, \\ b &= -\sqrt{2}+\sqrt{3}+\sqrt{6}, \\ c&= \sqrt{2}-\sqrt{3}+\sqrt{6}, \\ d&=-\sqrt{2}-\sqrt{3}+\sqrt{6}. \end{aligned}\]Evaluate $\left(\frac1a + \frac1b + \frac1c + \frac1d\right)^2.$ | \frac{96}{529} | 6/8 |
In the base of a regular triangular prism is a triangle \( ABC \) with a side length of \( a \). Points \( A_{1} \), \( B_{1} \), and \( C_{1} \) are taken on the lateral edges, and their distances from the base plane are \( \frac{a}{2} \), \( a \), and \( \frac{3a}{2} \) respectively. Find the angle between the planes \( ABC \) and \( A_{1} B_{1} C_{1} \). | 45 | 7/8 |
Given the equation \(2x^4 + ax^3 + 9x^2 + ax + 2 = 0\) with real coefficients, all four roots are complex numbers, and their magnitudes are not equal to 1. Determine the range of values for \(a\). | (-2\sqrt{10},2\sqrt{10}) | 2/8 |
Given the function \( f(x) = \ln (ax + 1) + \frac{1-x}{1+x}, \, x \geq 0, \, a > 0 \):
1. If \( f(x) \) has an extremum at \( x = 1 \), find the value of \( a \).
2. If \( f(x) \geq \ln 2 \) is always true, find the range of \( a \). | [1,+\infty) | 1/8 |
Let \( c_{n}=11 \ldots 1 \) be a number in which the decimal representation contains \( n \) ones. Then \( c_{n+1}=10 \cdot c_{n}+1 \). Therefore:
\[ c_{n+1}^{2}=100 \cdot c_{n}^{2} + 22 \ldots 2 \cdot 10 + 1 \]
For example,
\( c_{2}^{2}=11^{2}=(10 \cdot 1+1)^{2}=100+2 \cdot 10+1=121 \),
\( c_{3}^{2} = 111^{2} = 100 \cdot 11^{2} + 220 + 1 = 12100 + 220 + 1 = 12321 \),
\( c_{4}^{2} = 1111^{2} = 100 \cdot 111^{2} + 2220 + 1 = 1232100 + 2220 + 1 = 1234321 \), etc.
We observe that in all listed numbers \( c_{2}^{2}, c_{3}^{2}, c_{4}^{2} \), the digit with respect to which these numbers are symmetric (2 in the case of \( c_{2}^{2}, 3 \) in the case of \( c_{3}^{2}, 4 \) in the case of \( c_{4}^{2} \)) coincides with the number of ones in the number that was squared.
The given number \( c=123456787654321 \) is also symmetric with respect to the digit 8, which suggests that it might be the square of the number \( c_{8} = 11111111 \). This can be verified by performing multiplication by columns or using the recursive relation. | 11111111 | 1/8 |
What is the sum of every third odd number between $100$ and $300$? | 6800 | 6/8 |
Two lines passing through the origin and perpendicular to each other intersect the parabola $y^{2} = 4p(x + p)$ ($p > 0$) at points $A$, $B$, and $C$, $D$. When is $|AB| + |CD|$ minimized? | 16p | 7/8 |
Assume the quartic $x^{4}-a x^{3}+b x^{2}-a x+d=0$ has four real roots $\frac{1}{2} \leq x_{1}, x_{2}, x_{3}, x_{4} \leq 2$. Find the maximum possible value of $\frac{\left(x_{1}+x_{2}\right)\left(x_{1}+x_{3}\right) x_{4}}{\left(x_{4}+x_{2}\right)\left(x_{4}+x_{3}\right) x_{1}}$ (over all valid choices of $\left.a, b, d\right)$. | \frac{5}{4} | 1/8 |
A triangle has vertices $A(0,0)$, $B(12,0)$, and $C(8,10)$. The probability that a randomly chosen point inside the triangle is closer to vertex $B$ than to either vertex $A$ or vertex $C$ can be written as $\frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Find $p+q$. | 409 | 1/8 |
Find all positive integer solutions to \( a^3 - b^3 - c^3 = 3abc \) and \( a^2 = 2(a + b + c) \). | (4,3,1) | 6/8 |
How far apart are the midpoints of two skew edges of a regular octahedron with edge length $a$? | \frac{\sqrt{3}}{2} | 5/8 |
In every row of a grid $100 \times n$ is written a permutation of the numbers $1,2 \ldots, 100$ . In one move you can choose a row and swap two non-adjacent numbers with difference $1$ . Find the largest possible $n$ , such that at any moment, no matter the operations made, no two rows may have the same permutations. | 2^{99} | 2/8 |
Xiaopang uses two stopwatches to measure the speed of a train. He finds that the train takes 40 seconds to pass through a 660-meter bridge and takes 10 seconds to pass by him at the same speed. Based on the data provided by Xiaopang, calculate the length of the train in meters. | 220 | 6/8 |
In the plane rectangular coordinate system $xOy$, the parametric equations of the line $l$ are $\left\{\begin{array}{l}x=2t+1,\\ y=2t\end{array}\right.$ (where $t$ is a parameter). Taking the coordinate origin $O$ as the pole and the non-negative half-axis of the $x$-axis as the polar axis, the polar coordinate equation of the curve $C$ is $\rho ^{2}-4\rho \sin \theta +3=0$.
$(1)$ Find the rectangular coordinate equation of the line $l$ and the general equation of the curve $C$;
$(2)$ A tangent line to the curve $C$ passes through a point $A$ on the line $l$, and the point of tangency is $B$. Find the minimum value of $|AB|$. | \frac{\sqrt{14}}{2} | 7/8 |
It is given that \( k \) is a positive integer not exceeding 99. There are no natural numbers \( x \) and \( y \) such that \( x^{2} - k y^{2} = 8 \). Find the difference between the maximum and minimum possible values of \( k \). | 96 | 2/8 |
The projections of a plane convex polygon onto the $OX$ axis, the bisector of the 1st and 3rd coordinate angles, the $OY$ axis, and the bisector of the 2nd and 4th coordinate angles are respectively $4, 3\sqrt{2}, 5,$ and $4\sqrt{2}$. The area of the polygon is denoted by $S$. Prove that $S \geq 10$. | S\ge10 | 1/8 |
The calculator's keyboard has digits from 0 to 9 and symbols of two operations. Initially, the display shows the number 0. Any keys can be pressed. The calculator performs operations in the sequence of key presses. If an operation symbol is pressed several times in a row, the calculator will remember only the last press. The absent-minded Scientist pressed very many buttons in a random sequence. Find the approximate probability that the result of the resulting sequence of operations is an odd number. | 1/3 | 3/8 |
The average of \(a, b\) and \(c\) is 16. The average of \(c, d\) and \(e\) is 26. The average of \(a, b, c, d\), and \(e\) is 20. The value of \(c\) is:
(A) 10
(B) 20
(C) 21
(D) 26
(E) 30 | 26 | 1/8 |
There is a wooden stick 240 cm long. First, starting from the left end, a line is drawn every 7 cm. Then, starting from the right end, a line is drawn every 6 cm. The stick is cut at each marked line. How many of the resulting smaller sticks are 3 cm long? | 12 | 1/8 |
Petrov booked an apartment in a newly built house, which has five identical entrances. Initially, the entrances were numbered from left to right, and Petrov's apartment number was 636. Later, the developer changed the numbering to the opposite direction (right to left, as shown in the diagram). Then, Petrov's apartment number became 242. How many apartments are in the building? (The numbering of apartments within each entrance has not changed.) | 985 | 7/8 |
Amina and Bert alternate turns tossing a fair coin. Amina goes first and each player takes three turns. The first player to toss a tail wins. If neither Amina nor Bert tosses a tail, then neither wins. What is the probability that Amina wins? | \frac{21}{32} | 7/8 |
Let $P(x) = x^3 + 8x^2 - x + 3$ and let the roots of $P$ be $a, b,$ and $c.$ The roots of a monic polynomial $Q(x)$ are $ab - c^2, ac - b^2, bc - a^2.$ Find $Q(-1).$ | 1536 | 4/8 |
Let $n$ and $k$ be integers, $1\le k\le n$ . Find an integer $b$ and a set $A$ of $n$ integers satisfying the following conditions:
(i) No product of $k-1$ distinct elements of $A$ is divisible by $b$ .
(ii) Every product of $k$ distinct elements of $A$ is divisible by $b$ .
(iii) For all distinct $a,a'$ in $A$ , $a$ does not divide $a'$ . | 2^k | 1/8 |
Let $F$ be the focus of the parabola $C: y^2=4x$, point $A$ lies on $C$, and point $B(3,0)$. If $|AF|=|BF|$, then calculate the distance of point $A$ from point $B$. | 2\sqrt{2} | 7/8 |
$a,b,c$ are positive numbers such that $ a^2 + b^2 + c^2 = 2abc + 1 $ . Find the maximum value of
\[ (a-2bc)(b-2ca)(c-2ab) \] | 1/8 | 1/8 |
Six chairs are placed in a row. Find the number of ways 3 people can sit randomly in these chairs such that no two people sit next to each other. | 24 | 6/8 |
In an experiment, a scientific constant $C$ is determined to be $2.43865$ with an error of at most $\pm 0.00312$.
The experimenter wishes to announce a value for $C$ in which every digit is significant.
That is, whatever $C$ is, the announced value must be the correct result when $C$ is rounded to that number of digits.
The most accurate value the experimenter can announce for $C$ is | 2.44 | 7/8 |
Let $c_0,c_1>0$ . And suppose the sequence $\{c_n\}_{n\ge 0}$ satisfies
\[ c_{n+1}=\sqrt{c_n}+\sqrt{c_{n-1}}\quad \text{for} \;n\ge 1 \]
Prove that $\lim_{n\to \infty}c_n$ exists and find its value.
*Proposed by Sadovnichy-Grigorian-Konyagin* | 4 | 4/8 |
Let \( n \) be a positive integer,
\[
S_{n}=\left\{\left(a_{1}, a_{2}, \cdots, a_{2^{n}}\right) \mid a_{i}=0,1\right\}.
\]
For \( a, b \in S_{n} \),
\[
a=\left(a_{1}, a_{2}, \cdots, a_{2^{n}}\right), b=\left(b_{1}, b_{2}, \cdots, b_{2^{n}}\right),
\]
define \( d(a, b)=\sum_{i=1}^{2^{n}}\left|a_{i}-b_{i}\right| \).
If for any \( a, b \in A, a \neq b \), we have \( d(a, b) \geq 2^{n-1} \), then \( A \subseteq S_{n} \) is called a "good subset".
Find the maximum value of \( |A| \). | 2^{n+1} | 7/8 |
For any positive integer \( n \), let \( a_{n} \) denote the number of triangles with integer side lengths whose longest side is \( 2n \).
(1) Find an expression for \( a_{n} \) in terms of \( n \);
(2) Given the sequence \( \{b_{n}\} \) satisfies
$$
\sum_{k=1}^{n}(-1)^{n-k} \binom{n}{k} b_{k}=a_{n}\quad (n \in \mathbf{Z}_{+}),
$$
find the number of positive integers \( n \) such that \( b_{n} \leq 2019 a_{n} \). | 12 | 1/8 |
Let \( p, q, r, s \) be distinct primes such that \( pq - rs \) is divisible by 30. Find the minimum possible value of \( p + q + r + s \). | 54 | 1/8 |
Compute
\[\sum_{j = 0}^\infty \sum_{k = 0}^\infty 2^{-3k - j - (k + j)^2}.\] | \frac{4}{3} | 7/8 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.