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Let $a_1,a_2,\ldots,a_6$ be a sequence of integers such that for all $1 \le i \le 5$ , $$ a_{i+1}=\frac{a_i}{3} \quad \text{or} \quad a_{i+1}={-}2a_i. $$ Find the number of possible positive values of $a_1+a_2+\cdots+a_6$ less than $1000$ .
*Proposed by **stayhomedomath*** | 142 | 1/8 |
Each of the $2001$ students at a high school studies either Spanish or French, and some study both. The number who study Spanish is between $80$ percent and $85$ percent of the school population, and the number who study French is between $30$ percent and $40$ percent. Let $m$ be the smallest number of students who could study both languages, and let $M$ be the largest number of students who could study both languages. Find $M-m$.
| 298 | 7/8 |
There are three trees planted in the garden: a cherry tree, a plum tree, and an apple tree. The probabilities of the trees thriving are respectively 0.8 and 0.9. What is the probability that:
a) Exactly two trees will thrive;
b) At least two trees will thrive;
c) At least one tree will thrive. | 0.994 | 2/8 |
If $x$ is a number satisfying the equation $\sqrt[3]{x+9}-\sqrt[3]{x-9}=3$, then $x^2$ is between:
$\textbf{(A)}\ 55\text{ and }65\qquad \textbf{(B)}\ 65\text{ and }75\qquad \textbf{(C)}\ 75\text{ and }85\qquad \textbf{(D)}\ 85\text{ and }95\qquad \textbf{(E)}\ 95\text{ and }105$ | \textbf{(C)}\7585 | 1/8 |
In the market supply of light bulbs, products from Factory A account for 70%, while those from Factory B account for 30%. The pass rate for Factory A's products is 95%, and the pass rate for Factory B's products is 80%. What is the probability of purchasing a qualified light bulb manufactured by Factory A? | 0.665 | 2/8 |
Given vectors $\overrightarrow {a}$ and $\overrightarrow {b}$ with magnitudes $|\overrightarrow {a}| = 6\sqrt {3}$ and $|\overrightarrow {b}| = \frac {1}{3}$, and their dot product $\overrightarrow {a} \cdot \overrightarrow {b} = -3$, determine the angle $\theta$ between $\overrightarrow {a}$ and $\overrightarrow {b}$. | \frac{5\pi}{6} | 2/8 |
Consider the graph of $y=f(x)$, which consists of five line segments as described below:
- From $(-5, -4)$ to $(-3, 0)$
- From $(-3, 0)$ to $(-1, -1)$
- From $(-1, -1)$ to $(1, 3)$
- From $(1, 3)$ to $(3, 2)$
- From $(3, 2)$ to $(5, 6)$
What is the sum of the $x$-coordinates of all points where $f(x) = 2.3$? | 4.35 | 1/8 |
Let $\mathcal S$ be a set of $16$ points in the plane, no three collinear. Let $\chi(S)$ denote the number of ways to draw $8$ lines with endpoints in $\mathcal S$ , such that no two drawn segments intersect, even at endpoints. Find the smallest possible value of $\chi(\mathcal S)$ across all such $\mathcal S$ .
*Ankan Bhattacharya* | 1430 | 4/8 |
Prove that the value of the expression \( 333^{555} + 555^{333} \) is divisible by 37. | 37 | 7/8 |
The organizers of a mathematics competition decided to take pictures of 60 participants. It is known that no more than 30 participants can fit in one picture, but any two students must appear in at least one picture together. What is the minimum number of pictures needed to achieve this? | 6 | 7/8 |
A line parallel to the bases of a trapezoid divides it into two similar trapezoids.
Find the segment of this line that is enclosed within the trapezoid, given that the lengths of the bases are \( a \) and \( b \). | \sqrt{} | 7/8 |
Find the remainder when $123456789012$ is divided by $240$. | 132 | 4/8 |
In the rectangular coordinate plane, the number of integer points (i.e., points with both integer x and y coordinates) that satisfy the system of inequalities
$$
\left\{\begin{array}{l}
y \leqslant 3x \\
y \geqslant \frac{1}{3}x \\
x + y \leqslant 100
\end{array}\right.
$$
is ______. | 2551 | 1/8 |
A large rectangle is composed of three identical squares and three identical small rectangles. The perimeter of a square is 24, and the perimeter of a small rectangle is 16. What is the perimeter of the large rectangle?
The perimeter of a shape is the sum of the lengths of all its sides. | 52 | 3/8 |
Given an isosceles right triangle \(ABC\) with hypotenuse \(AB\). Point \(M\) is the midpoint of side \(BC\). A point \(K\) is chosen on the smaller arc \(AC\) of the circumcircle of triangle \(ABC\). Point \(H\) is the foot of the perpendicular dropped from \(K\) to line \(AB\). Find the angle \(\angle CAK\), given that \(KH = BM\) and lines \(MH\) and \(CK\) are parallel. | 22.5 | 1/8 |
Let $\alpha$ and $\beta$ be reals. Find the least possible value of $(2 \cos \alpha+5 \sin \beta-8)^{2}+(2 \sin \alpha+5 \cos \beta-15)^{2}$. | 100 | 7/8 |
One of the factors of $x^4+2x^2+9$ is:
$\textbf{(A)}\ x^2+3\qquad\textbf{(B)}\ x+1\qquad\textbf{(C)}\ x^2-3\qquad\textbf{(D)}\ x^2-2x-3\qquad\textbf{(E)}\ \text{none of these}$ | (\textbf{E}) | 1/8 |
Problems 8, 9 and 10 use the data found in the accompanying paragraph and figures
Four friends, Art, Roger, Paul and Trisha, bake cookies, and all cookies have the same thickness. The shapes of the cookies differ, as shown.
$\circ$ Art's cookies are trapezoids:
$\circ$ Roger's cookies are rectangles:
$\circ$ Paul's cookies are parallelograms:
$\circ$ Trisha's cookies are triangles:
Each friend uses the same amount of dough, and Art makes exactly $12$ cookies. Art's cookies sell for $60$ cents each. To earn the same amount from a single batch, how much should one of Roger's cookies cost in cents? | 40 | 1/8 |
Given that the sequence $\{a\_n\}$ satisfies $\frac{1}{a_{n+1}} - \frac{1}{a_n} = d (n \in \mathbb{N}^*, d$ is a constant), the sequence $\{\frac{1}{b_n}\}$ is a harmonic sequence, and $b_1 + b_2 + b_3 + ... + b_9 = 90$, find the value of $b_4 + b_6$. | 20 | 3/8 |
12 Smurfs are seated around a round table. Each Smurf dislikes the 2 Smurfs next to them, but does not dislike the other 9 Smurfs. Papa Smurf wants to form a team of 5 Smurfs to rescue Smurfette, who was captured by Gargamel. The team must not include any Smurfs who dislike each other. How many ways are there to form such a team? | 36 | 7/8 |
In the triangle \( \triangle ABC \), the lengths of the sides opposite to angles \( \angle A \), \( \angle B \), and \( \angle C \) are \( a \), \( b \), and \( c \) respectively. Given that:
$$
\begin{array}{l}
\angle C = \max \{\angle A, \angle B, \angle C\}, \\
\sin C = 1 + \cos C \cdot \cos (A-B), \\
\frac{2}{a} + \frac{1}{b} = 1 .
\end{array}
$$
Find the minimum perimeter of the triangle \( \triangle ABC \). | 10 | 1/8 |
Given point $P(2,2)$, and circle $C$: $x^{2}+y^{2}-8y=0$. A moving line $l$ passing through point $P$ intersects circle $C$ at points $A$ and $B$, with the midpoint of segment $AB$ being $M$, and $O$ being the origin.
$(1)$ Find the equation of the trajectory of point $M$;
$(2)$ When $|OP|=|OM|$, find the equation of line $l$ and the area of $\Delta POM$. | \frac{16}{5} | 3/8 |
What is the number of ways in which one can choose $60$ unit squares from a $11 \times 11$ chessboard such that no two chosen squares have a side in common? | 62 | 4/8 |
In triangle $\triangle ABC$, the opposite sides of angles $A$, $B$, and $C$ are $a$, $b$, $c$, and the vectors $\overrightarrow{m}=({\cos C, \cos({\frac{\pi}{2}-B})})$, $\overrightarrow{n}=({\cos({-4\pi+B}), -\sin C})$, and $\overrightarrow{m} \cdot \overrightarrow{n}=-\frac{\sqrt{2}}{2}$. <br/>$(1)$ Find the measure of angle $A$; <br/>$(2)$ If the altitude on side $AC$ is $2$, $a=3$, find the perimeter of $\triangle ABC$. | 5 + 2\sqrt{2} + \sqrt{5} | 1/8 |
Solve for $x$: $0.05x - 0.09(25 - x) = 5.4$. | 54.6428571 | 1/8 |
How many integers from 1 to 1997 have a digit sum that is divisible by 5? | 399 | 2/8 |
Given $(625^{\log_2 250})^{\frac{1}{4}}$, find the value of the expression. | 250 | 1/8 |
What is the smallest natural number \( k \) for which the expression \( 2019 \cdot 2020 \cdot 2021 \cdot 2022 + k \) is a square of a natural number? | 1 | 5/8 |
Given a sequence $\{a_{n}\}$ where $a_{1}=1$ and $a_{n+1}-a_{n}=\left(-1\right)^{n+1}\frac{1}{n(n+2)}$, calculate the sum of the first 40 terms of the sequence $\{\left(-1\right)^{n}a_{n}\}$. | \frac{20}{41} | 3/8 |
In the right triangular prism \( ABC-A_1B_1C_1 \), \( AB=1 \), \( BC=C_1C=\sqrt{3} \), and \( \angle ABC=90^\circ \). Point \( P \) is a moving point on plane \( ABC \). Find the minimum value of \( A_1P + \frac{1}{2}PC \). | \frac{5}{2} | 1/8 |
What is the smallest natural number $n$ for which there exist natural numbers $x$ and $y$ satisfying the equation a) $x \cdot (x+n) = y^{2}$; b) $x \cdot (x+n) = y^{3}$? | 2 | 3/8 |
Several businessmen decided to start a company and share the profits equally. One of the businessmen was appointed as the director. One day, this director transferred part of the profit from the company's account to their personal account. This amount was three times larger than what each of the others would receive if they had divided the remaining profit equally among themselves. After this, the director left the company. The next director, one of the remaining businessmen, did the same thing as the previous director, and so on. Eventually, the penultimate director transferred to their personal account an amount that was also three times larger than what was left for the last businessman. As a result of these profit distributions, the last businessman received 190 times less money than the first director. How many businessmen started this company? | 19 | 2/8 |
The sequence \(\left(a_{n}\right)\) is defined such that \(a_{n}=n^{2}\) for \(1 \leq n \leq 5\) and for all natural numbers \(n\) the following equality holds: \(a_{n+5} + a_{n+1} = a_{n+4} + a_{n}\). Find \(a_{2015}\). | 17 | 7/8 |
How many numbers from 1 to 100 are divisible by 3, but do not contain the digit 3? | 26 | 4/8 |
Given that the terminal side of angle \\(\alpha\\) passes through the point \\(P(m,2\sqrt{2})\\), \\(\sin \alpha= \frac{2\sqrt{2}}{3}\\) and \\(\alpha\\) is in the second quadrant.
\\((1)\\) Find the value of \\(m\\);
\\((2)\\) If \\(\tan \beta= \sqrt{2}\\), find the value of \\( \frac{\sin \alpha\cos \beta+3\sin \left( \frac{\pi}{2}+\alpha\right)\sin \beta}{\cos (\pi+\alpha)\cos (-\beta)-3\sin \alpha\sin \beta}\\). | \frac{\sqrt{2}}{11} | 7/8 |
Find all triples $(p, q, r)$ of prime numbers for which $4q - 1$ is a prime number and $$ \frac{p + q}{p + r} = r - p $$ holds.
*(Walther Janous)* | (2,3,3) | 3/8 |
Among the digits 0, 1, ..., 9, calculate the number of three-digit numbers that can be formed using repeated digits. | 252 | 1/8 |
There are $2017$ distinct points in the plane. For each pair of these points, construct the midpoint of the segment joining the pair of points. What is the minimum number of distinct midpoints among all possible ways of placing the points? | 2016 | 1/8 |
Let $ \left(a_{n}\right)$ be the sequence of reals defined by $ a_{1}=\frac{1}{4}$ and the recurrence $ a_{n}= \frac{1}{4}(1+a_{n-1})^{2}, n\geq 2$. Find the minimum real $ \lambda$ such that for any non-negative reals $ x_{1},x_{2},\dots,x_{2002}$, it holds
\[ \sum_{k=1}^{2002}A_{k}\leq \lambda a_{2002}, \]
where $ A_{k}= \frac{x_{k}-k}{(x_{k}+\cdots+x_{2002}+\frac{k(k-1)}{2}+1)^{2}}, k\geq 1$. | \frac{1}{2005004} | 1/8 |
In the diagram, the grid is made up of squares. What is the area of the shaded region? [asy]
size(8cm);
// Fill area
fill((0, 0)--(0, 2)--(3, 2)--(3, 3)--(7, 3)--(7, 4)--(12, 4)--cycle, gray(0.75));
defaultpen(1);
// Draw grid
draw((0, 0)--(12, 0));
draw((0, 1)--(12, 1));
draw((0, 2)--(12, 2));
draw((3, 3)--(12, 3));
draw((7, 4)--(12, 4));
draw((0, 0)--(12, 4));
draw((0, 2)--(0, 0));
draw((1, 2)--(1, 0));
draw((2, 2)--(2, 0));
draw((3, 3)--(3, 0));
draw((4, 3)--(4, 0));
draw((5, 3)--(5, 0));
draw((6, 3)--(6, 0));
draw((7, 4)--(7, 0));
draw((8, 4)--(8, 0));
draw((9, 4)--(9, 0));
draw((10, 4)--(10, 0));
draw((11, 4)--(11, 0));
draw((12, 4)--(12, 0));
// Draw lengths
path height = (-0.5, 0)--(-0.5, 2);
path width = (0, -0.5)--(12, -0.5);
path height2 = (12.5, 0)--(12.5, 4);
draw(height); draw(width); draw(height2);
draw((-0.6, 0)--(-0.4, 0));
draw((-0.6, 2)--(-0.4, 2));
draw((0, -0.6)--(0, -0.4));
draw((12, -0.6)--(12, -0.4));
draw((12.4, 0)--(12.6, 0));
draw((12.4, 4)--(12.6, 4));
// label lengths
label("$2$", (-0.5, 1), W);
label("$12$", (6, -0.5), S);
label("$4$", (12.5, 2), E);
[/asy] | 14 | 6/8 |
Given the line $y=kx+1$ intersects the parabola $C: x^2=4y$ at points $A$ and $B$, and a line $l$ is parallel to $AB$ and is tangent to the parabola $C$ at point $P$, find the minimum value of the area of triangle $PAB$ minus the length of $AB$. | -\frac{64}{27} | 6/8 |
For each positive integer $n$, define $s(n)$ to equal the sum of the digits of $n$. The number of integers $n$ with $100 \leq n \leq 999$ and $7 \leq s(n) \leq 11$ is $S$. What is the integer formed by the rightmost two digits of $S$? | 24 | 2/8 |
Let $F(0) = 0,$ $F(1) = \frac{3}{2},$ and
\[F(n) = \frac{5}{2} F(n - 1) - F(n - 2)\]for $n \ge 2.$ Find
\[\sum_{n = 0}^\infty \frac{1}{F(2^n)}.\] | 1 | 5/8 |
On standard dice, the total number of pips on each pair of opposite faces is 7. Two standard dice are placed in a stack, so that the total number of pips on the two touching faces is 5. What is the total number of pips on the top and bottom faces of the stack?
A 5
B 6
C 7
D 8
E 9 | 9 | 6/8 |
On a faded piece of paper it is possible to read the following:
\[(x^2 + x + a)(x^{15}- \cdots ) = x^{17} + x^{13} + x^5 - 90x^4 + x - 90.\]
Some parts have got lost, partly the constant term of the first factor of the left side, partly the majority of the summands of the second factor. It would be possible to restore the polynomial forming the other factor, but we restrict ourselves to asking the following question: What is the value of the constant term $a$ ? We assume that all polynomials in the statement have only integer coefficients. | 2 | 1/8 |
Gavrila is in an elevator cabin which is moving downward with a deceleration of 5 m/s². Find the force with which Gavrila presses on the floor. Gavrila's mass is 70 kg, and the acceleration due to gravity is 10 m/s². Give the answer in newtons, rounding to the nearest whole number if necessary. | 350 | 1/8 |
On an $8 \times 8$ chessboard, there are 16 rooks, each placed in a different square. What is the minimum number of pairs of rooks that can attack each other (a pair of rooks can attack each other if they are in the same row or column and there are no other rooks between them)? | 16 | 7/8 |
A cube has exactly six faces and twelve edges. How many vertices does a cube have?
(A) 4
(B) 5
(C) 6
(D) 7
(E) 8 | 8 | 1/8 |
The sequence \(\{x_{n}\}\) is defined as follows: \(x_{1} = \frac{1}{2}\), \(x_{k+1} = x_{k}^{2} + x_{k}\). Find the integer part of the sum \(\frac{1}{x_{1}+1} + \frac{1}{x_{2}+1} + \cdots + \frac{1}{x_{100}+1}\). | 1 | 7/8 |
Let \( A \), \( B \), and \( C \) be three points on the edge of a circular chord such that \( B \) is due west of \( C \) and \( ABC \) is an equilateral triangle whose side is 86 meters long. A boy swam from \( A \) directly toward \( B \). After covering a distance of \( x \) meters, he turned and swam westward, reaching the shore after covering a distance of \( y \) meters. If \( x \) and \( y \) are both positive integers, determine \( y \). | 12 | 1/8 |
With all angles measured in degrees, the product $\prod_{k=1}^{45} \csc^2(2k-1)^\circ=m^n$, where $m$ and $n$ are integers greater than 1. Find $m+n$.
| 91 | 6/8 |
The $52$ cards in a deck are numbered $1, 2, \cdots, 52$. Alex, Blair, Corey, and Dylan each picks a card from the deck without replacement and with each card being equally likely to be picked, The two persons with lower numbered cards from a team, and the two persons with higher numbered cards form another team. Let $p(a)$ be the probability that Alex and Dylan are on the same team, given that Alex picks one of the cards $a$ and $a+9$, and Dylan picks the other of these two cards. The minimum value of $p(a)$ for which $p(a)\ge\frac{1}{2}$ can be written as $\frac{m}{n}$. where $m$ and $n$ are relatively prime positive integers. Find $m+n$.
| 263 | 7/8 |
How many triples \((x, y, z) \in \mathbb{R}^{3}\) satisfy the following system of equations:
\[
\begin{array}{l}
x = 2018 - 2019 \cdot \operatorname{sign}(y + z) \\
y = 2018 - 2019 \cdot \operatorname{sign}(z + x) \\
z = 2018 - 2019 \cdot \operatorname{sign}(x + y)
\end{array}
\]
where for any real number \(\alpha\), \(\operatorname{sign}(\alpha)\) is defined as:
\[
\operatorname{sign}(\alpha) =
\begin{cases}
+1 & \text{if } \alpha > 0 \\
0 & \text{if } \alpha = 0 \\
-1 & \text{if } \alpha < 0
\end{cases}
\] | 3 | 4/8 |
A biologist wishes to estimate the fish population in a protected area. Initially, on March 1st, she captures and tags 80 fish, then releases them back into the water. Four months later, on July 1st, she captures another 90 fish for a follow-up study, finding that 4 of these are tagged. For her estimation, she assumes that 30% of these fish have left the area by July 1st due to various environmental factors, and that an additional 50% of the fish in the July sample weren't in the area on March 1st due to new arrivals. How many fish does she estimate were in the area on March 1st? | 900 | 7/8 |
A permutation of \{1,2, \ldots, 7\} is chosen uniformly at random. A partition of the permutation into contiguous blocks is correct if, when each block is sorted independently, the entire permutation becomes sorted. For example, the permutation $(3,4,2,1,6,5,7)$ can be partitioned correctly into the blocks $[3,4,2,1]$ and $[6,5,7]$, since when these blocks are sorted, the permutation becomes $(1,2,3,4,5,6,7)$. Find the expected value of the maximum number of blocks into which the permutation can be partitioned correctly. | \frac{151}{105} | 1/8 |
Find the smallest possible value of \(x+y\) where \(x, y \geq 1\) and \(x\) and \(y\) are integers that satisfy \(x^{2}-29y^{2}=1\). | 11621 | 2/8 |
Suppose $n$ and $r$ are nonnegative integers such that no number of the form $n^2+r-k(k+1) \text{ }(k\in\mathbb{N})$ equals to $-1$ or a positive composite number. Show that $4n^2+4r+1$ is $1$ , $9$ , or prime. | 1,9, | 1/8 |
From vertex $A$ of an equilateral triangle $ABC$ , a ray $Ax$ intersects $BC$ at point $D$ . Let $E$ be a point on $Ax$ such that $BA =BE$ . Calculate $\angle AEC$ . | 30 | 4/8 |
As shown in the diagram, there is a sequence of curves \(P_{0}, P_{1}, P_{2}, \cdots\). It is given that \(P_{0}\) is an equilateral triangle with an area of 1. Each \(P_{k+1}\) is obtained from \(P_{k}\) by performing the following operations: each side of \(P_{k}\) is divided into three equal parts, an equilateral triangle is constructed outwards on the middle segment of each side, and the middle segments are then removed (\(k=0,1,2, \cdots\)). Let \(S_{n}\) denote the area enclosed by the curve \(P_{n}\).
1. Find a general formula for the sequence \(\{S_{n}\}\).
2. Evaluate \(\lim _{n \rightarrow \infty} S_{n}\). | \frac{8}{5} | 7/8 |
Let \(S\) be the set of all nonzero real numbers. Let \(f : S \to S\) be a function such that
\[f(x) + f(y) = cf(xyf(x + y))\]
for all \(x, y \in S\) such that \(x + y \neq 0\) and for some nonzero constant \(c\). Determine all possible functions \(f\) that satisfy this equation and calculate \(f(5)\). | \frac{1}{5} | 2/8 |
$(1)$ $f(n)$ is a function defined on the set of positive integers, satisfying:<br/>① When $n$ is a positive integer, $f(f(n))=4n+9$;<br/>② When $k$ is a non-negative integer, $f(2^{k})=2^{k+1}+3$. Find the value of $f(1789)$.<br/>$(2)$ The function $f$ is defined on the set of ordered pairs of positive integers, and satisfies the following properties:<br/>① $f(x,x)=x$;<br/>② $f(x,y)=f(y,x)$;<br/>③ $(x+y)f(x,y)=yf(x,x+y)$. Find $f(14,52)$. | 364 | 7/8 |
How many paths are there from the starting point $C$ to the end point $D$, if every step must be up or to the right in a grid of 8 columns and 7 rows? | 6435 | 1/8 |
Let \( a \) and \( b \) be two real numbers. We set \( s = a + b \) and \( p = ab \). Express \( a^3 + b^3 \) in terms of \( s \) and \( p \) only. | ^3-3sp | 7/8 |
Let $l$ be a tangent line at the point $T\left(t,\ \frac{t^2}{2}\right)\ (t>0)$ on the curve $C: y=\frac{x^2}{2}.$ A circle $S$ touches the curve $C$ and the $x$ axis.
Denote by $(x(t),\ y(t))$ the center of the circle $S$ . Find $\lim_{r\rightarrow +0}\int_r^1 x(t)y(t)\ dt.$ | \frac{\sqrt{2}+1}{30} | 1/8 |
A point $P$ is chosen uniformly at random inside a square of side length 2. If $P_{1}, P_{2}, P_{3}$, and $P_{4}$ are the reflections of $P$ over each of the four sides of the square, find the expected value of the area of quadrilateral $P_{1} P_{2} P_{3} P_{4}$. | 8 | 1/8 |
A quadrangular prism \(A B C D A_{1} B_{1} C_{1} D_{1}\) has a rhombus \(A B C D\) at its base, where \(A C = 4\) and \(\angle D B C = 30^{\circ}\). A sphere passes through the vertices \(D, A, B, B_{1}, C_{1}, D_{1}\).
a) Find the area of the circle obtained by intersecting the sphere with the plane passing through points \(B, C\), and \(D\).
b) Find the angle \(A_{1} C D\).
c) It is additionally given that the radius of the sphere is 5. Find the volume of the prism. | 48\sqrt{3} | 1/8 |
The base of the pyramid is a rectangle with an area of $S$. Two of the lateral faces are perpendicular to the plane of the base, while the other two are inclined to it at angles of 30° and $60^{\circ}$. Find the volume of the pyramid. | \frac{S\sqrt{S}}{3} | 5/8 |
Let $f(n) = \frac{x_1 + x_2 + \cdots + x_n}{n}$, where $n$ is a positive integer. If $x_k = (-1)^k, k = 1, 2, \cdots, n$, the set of possible values of $f(n)$ is: | $\{0, -\frac{1}{n}\}$ | 1/8 |
In a convex polygon with an odd number of vertices \(2n + 1\), two diagonals are chosen independently at random. Find the probability that these diagonals intersect inside the polygon. | \frac{n(2n-1)}{3(2n^2-n-2)} | 7/8 |
Let line \( l: y = kx + m \) (where \( k \) and \( m \) are integers) intersect the ellipse \( \frac{x^2}{16} + \frac{y^2}{12} = 1 \) at two distinct points \( A \) and \( B \), and intersect the hyperbola \( \frac{x^2}{4} - \frac{y^2}{12} = 1 \) at two distinct points \( C \) and \( D \). The vector sum \( \overrightarrow{AC} + \overrightarrow{BD} = \overrightarrow{0} \). The total number of such lines is _____. | 9 | 3/8 |
Given the set $M=\{x|2x^{2}-3x-2=0\}$ and the set $N=\{x|ax=1\}$. If $N \subset M$, what is the value of $a$? | \frac{1}{2} | 1/8 |
Two swimmers, at opposite ends of a $90$-foot pool, start to swim the length of the pool,
one at the rate of $3$ feet per second, the other at $2$ feet per second.
They swim back and forth for $12$ minutes. Allowing no loss of times at the turns, find the number of times they pass each other.
$\textbf{(A)}\ 24\qquad \textbf{(B)}\ 21\qquad \textbf{(C)}\ 20\qquad \textbf{(D)}\ 19\qquad \textbf{(E)}\ 18$ | \textbf{(C)}\20 | 1/8 |
Find the greatest real $k$ such that, for every tetrahedron $ABCD$ of volume $V$ , the product of areas of faces $ABC,ABD$ and $ACD$ is at least $kV^2$ . | 9/2 | 1/8 |
Let $a$ and $b$ be positive integers such that $ab \neq 1$. Determine all possible integer values of $f(a, b)=\frac{a^{2}+b^{2}+ab}{ab-1}$. | 7 | 1/8 |
The vertices of a $3 \times 1 \times 1$ rectangular prism are $A, B, C, D, E, F, G$, and $H$ so that $A E, B F$, $C G$, and $D H$ are edges of length 3. Point $I$ and point $J$ are on $A E$ so that $A I=I J=J E=1$. Similarly, points $K$ and $L$ are on $B F$ so that $B K=K L=L F=1$, points $M$ and $N$ are on $C G$ so that $C M=M N=N G=1$, and points $O$ and $P$ are on $D H$ so that $D O=O P=P H=1$. For every pair of the 16 points $A$ through $P$, Maria computes the distance between them and lists the 120 distances. How many of these 120 distances are equal to $\sqrt{2}$? | 32 | 1/8 |
In a \(7 \times 7\) table, some cells are black while the remaining ones are white. In each white cell, the total number of black cells located with it in the same row or column is written; nothing is written in the black cells. What is the maximum possible sum of the numbers in the entire table? | 168 | 5/8 |
Given positive numbers $x$ and $y$ satisfying $x^2+y^2=1$, find the maximum value of $\frac {1}{x}+ \frac {1}{y}$. | 2\sqrt{2} | 2/8 |
Let point \( A \) (not on the y-axis) be a moving point on the line \( x - 2y + 13 = 0 \). From point \( A \), draw two tangents to the parabola \( y^2 = 8x \), with \( M \) and \( N \) being the points of tangency. Lines \( AM \) and \( AN \) intersect the y-axis at points \( B \) and \( C \), respectively. Prove:
1. The line \( MN \) always passes through a fixed point.
2. The circumcircle of triangle \( ABC \) always passes through a fixed point, and find the minimum radius of this circle. | \frac{3\sqrt{5}}{2} | 1/8 |
Points with coordinates $(1,1),(5,1)$ and $(1,7)$ are three vertices of a rectangle. What are the coordinates of the fourth vertex of the rectangle? | (5,7) | 7/8 |
Given a chessboard consisting of $16 \times 16$ small squares, with alternating black and white colors, define an operation $A$ as follows: change the color of all squares in the $i$-th row and all squares in the $j$-th column to their opposite color $(1 \leqslant i, j \leqslant 16)$. We call $A$ an operation on the small square located at the $i$-th row and $j$-th column.
Can we change all the squares on the chessboard to the same color through a certain number of operations? Prove your conclusion. | Yes | 3/8 |
The diagram below shows \( \triangle ABC \), which is isosceles with \( AB = AC \) and \( \angle A = 20^\circ \). The point \( D \) lies on \( AC \) such that \( AD = BC \). The segment \( BD \) is constructed as shown. Determine \( \angle ABD \) in degrees. | 10 | 6/8 |
Consider those functions $f$ that satisfy $f(x+5)+f(x-5) = f(x)$ for all real $x$. Find the least common positive period $p$ for all such functions. | 30 | 6/8 |
Let $s_1, s_2, s_3$ be the three roots of $x^3 + x^2 +\frac92x + 9$ . $$ \prod_{i=1}^{3}(4s^4_i + 81) $$ can be written as $2^a3^b5^c$ . Find $a + b + c$ . | 16 | 3/8 |
One standard balloon can lift a basket with contents weighing not more than 80 kg. Two standard balloons can lift the same basket with contents weighing not more than 180 kg. What is the weight, in kg, of the basket? | 20 | 5/8 |
In the sequence \( a_{0}, a_{1}, a_{2}, \cdots, a_{n} \) where \( a_{0}=\frac{1}{2} \) and \( a_{k+1}=a_{k}+\frac{1}{n} a_{k}^{2} \) for \( k=0, 1, 2, \cdots, n-1 \), prove that \( 1-\frac{1}{n}<a_{n}<1 \). | 1-\frac{1}{n}<a_n<1 | 3/8 |
Find the minimum value of $n (n > 0)$ such that the function \\(f(x)= \begin{vmatrix} \sqrt {3} & \sin x \\\\ 1 & \cos x\\end{vmatrix} \\) when shifted $n$ units to the left becomes an even function. | \frac{5\pi}{6} | 4/8 |
Let $P$ be a convex polygon with area 1. Show that there exists a rectangle with area 2 that contains it. Can this result be improved? | 2 | 5/8 |
Let $ x \equal{} \sqrt[3]{\frac{4}{25}}\,$ . There is a unique value of $ y$ such that $ 0 < y < x$ and $ x^x \equal{} y^y$ . What is the value of $ y$ ? Express your answer in the form $ \sqrt[c]{\frac{a}{b}}\,$ , where $ a$ and $ b$ are relatively prime positive integers and $ c$ is a prime number. | \sqrt[3]{\frac{32}{3125}} | 6/8 |
The price of Type A remote control car is 46.5 yuan, and the price of Type B remote control car is 54.5 yuan. Lele has 120 yuan. If he buys both types of remote control cars, will he have enough money? If so, how much money will he have left after the purchase? | 19 | 5/8 |
Define a power cycle to be a set $S$ consisting of the nonnegative integer powers of an integer $a$, i.e. $S=\left\{1, a, a^{2}, \ldots\right\}$ for some integer $a$. What is the minimum number of power cycles required such that given any odd integer $n$, there exists some integer $k$ in one of the power cycles such that $n \equiv k$ $(\bmod 1024) ?$ | 10 | 1/8 |
For any finite sequence of positive integers \pi, let $S(\pi)$ be the number of strictly increasing subsequences in \pi with length 2 or more. For example, in the sequence $\pi=\{3,1,2,4\}$, there are five increasing sub-sequences: $\{3,4\},\{1,2\},\{1,4\},\{2,4\}$, and $\{1,2,4\}$, so $S(\pi)=5$. In an eight-player game of Fish, Joy is dealt six cards of distinct values, which she puts in a random order \pi from left to right in her hand. Determine $\sum_{\pi} S(\pi)$ where the sum is taken over all possible orders \pi of the card values. | 8287 | 7/8 |
A prime number $ q $ is called***'Kowai'***number if $ q = p^2 + 10$ where $q$ , $p$ , $p^2-2$ , $p^2-8$ , $p^3+6$ are prime numbers. WE know that, at least one ***'Kowai'*** number can be found. Find the summation of all ***'Kowai'*** numbers.
| 59 | 6/8 |
The Wolf and Ivan Tsarevich are 20 versts away from a source of living water, and the Wolf is taking Ivan Tsarevich there at a speed of 3 versts per hour. To revive Ivan Tsarevich, one liter of water is needed, which flows from the source at a rate of half a liter per hour. At the source, there is a Raven with unlimited carrying capacity; it must gather the water, after which it will fly towards the Wolf and Ivan Tsarevich at a speed of 6 versts per hour, spilling a quarter liter of water every hour. After how many hours will it be possible to revive Ivan Tsarevich? | 4 | 5/8 |
Let \( a, b, c \) be positive real numbers such that \( a b c = 1 \). Prove that:
\[
\frac{1}{a^{3}(b+c)}+\frac{1}{b^{3}(c+a)}+\frac{1}{c^{3}(a+b)} \geq \frac{3}{2}.
\] | \frac{3}{2} | 7/8 |
In the Cartesian coordinate system $(xOy)$, a pole is established at the origin $O$ with the non-negative semi-axis of the $x$-axis as the polar axis, forming a polar coordinate system. Given that the equation of line $l$ is $4ρ\cos θ-ρ\sin θ-25=0$, and the curve $W$ is defined by the parametric equations $x=2t, y=t^{2}-1$.
1. Find the Cartesian equation of line $l$ and the general equation of curve $W$.
2. If point $P$ is on line $l$, and point $Q$ is on curve $W$, find the minimum value of $|PQ|$. | \frac{8\sqrt{17}}{17} | 6/8 |
A deck consists of six red cards and six green cards, each with labels $A$, $B$, $C$, $D$, $E$ corresponding to each color. Two cards are dealt from this deck. A winning pair consists of cards that either share the same color or the same label. Calculate the probability of drawing a winning pair.
A) $\frac{1}{2}$
B) $\frac{10}{33}$
C) $\frac{30}{66}$
D) $\frac{35}{66}$
E) $\frac{40}{66}$ | \frac{35}{66} | 1/8 |
What is the largest four-digit negative integer congruent to $1 \pmod{17}$? | -1002 | 7/8 |
Given two vectors $\overrightarrow {a}$ and $\overrightarrow {b}$ with an angle of $\frac {2\pi}{3}$ between them, $|\overrightarrow {a}|=2$, $|\overrightarrow {b}|=3$, let $\overrightarrow {m}=3\overrightarrow {a}-2\overrightarrow {b}$ and $\overrightarrow {n}=2\overrightarrow {a}+k\overrightarrow {b}$:
1. If $\overrightarrow {m} \perp \overrightarrow {n}$, find the value of the real number $k$;
2. Discuss whether there exists a real number $k$ such that $\overrightarrow {m} \| \overrightarrow {n}$, and explain the reasoning. | \frac{4}{3} | 1/8 |
Let $\Delta$ denote the set of all triangles in a plane. Consider the function $f: \Delta\to(0,\infty)$ defined by $f(ABC) = \min \left( \dfrac ba, \dfrac cb \right)$ , for any triangle $ABC$ with $BC=a\leq CA=b\leq AB = c$ . Find the set of values of $f$ . | [1,\frac{\sqrt{5}+1}{2}) | 1/8 |
In the diagram, square $ABCD$ has sides of length $4,$ and $\triangle ABE$ is equilateral. Line segments $BE$ and $AC$ intersect at $P.$ Point $Q$ is on $BC$ so that $PQ$ is perpendicular to $BC$ and $PQ=x.$ [asy]
pair A, B, C, D, E, P, Q;
A=(0,0);
B=(4,0);
C=(4,-4);
D=(0,-4);
E=(2,-3.464);
P=(2.535,-2.535);
Q=(4,-2.535);
draw(A--B--C--D--A--E--B);
draw(A--C);
draw(P--Q, dashed);
label("A", A, NW);
label("B", B, NE);
label("C", C, SE);
label("D", D, SW);
label("E", E, S);
label("P", P, W);
label("Q", Q, dir(0));
label("$x$", (P+Q)/2, N);
label("4", (A+B)/2, N);
[/asy] Determine the measure of angle $BPC.$ | 105^\circ | 4/8 |
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