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Three balls marked $1,2$ and $3$ are placed in an urn. One ball is drawn, its number is recorded, and then the ball is returned to the urn. This process is repeated and then repeated once more, and each ball is equally likely to be drawn on each occasion. If the sum of the numbers recorded is $6$, what is the probability that the ball numbered $2$ was drawn all three times?
$\textbf{(A)} \ \frac{1}{27} \qquad \textbf{(B)} \ \frac{1}{8} \qquad \textbf{(C)} \ \frac{1}{7} \qquad \textbf{(D)} \ \frac{1}{6} \qquad \textbf{(E)}\ \frac{1}{3}$ | \textbf{(C)}\\frac{1}{7} | 1/8 |
The shaded region below is called a shark's fin falcata, a figure studied by Leonardo da Vinci. It is bounded by the portion of the circle of radius $3$ and center $(0,0)$ that lies in the first quadrant, the portion of the circle with radius $\frac{3}{2}$ and center $(0,\frac{3}{2})$ that lies in the first quadrant, and the line segment from $(0,0)$ to $(3,0)$. What is the area of the shark's fin falcata? | \frac{9\pi}{8} | 3/8 |
For $\{1, 2, 3, \ldots, n\}$ and each of its non-empty subsets a unique alternating sum is defined as follows. Arrange the numbers in the subset in decreasing order and then, beginning with the largest, alternately add and subtract successive numbers. For example, the alternating sum for $\{1, 2, 3, 6,9\}$ is $9-6+3-2+1=5$ and for $\{5\}$ it is simply $5$. Find the sum of all such alternating sums for $n=7$. | 448 | 6/8 |
Among 6 internists and 4 surgeons, there is one chief internist and one chief surgeon. Now, a 5-person medical team is to be formed to provide medical services in rural areas. How many ways are there to select the team under the following conditions?
(1) The team includes 3 internists and 2 surgeons;
(2) The team includes both internists and surgeons;
(3) The team includes at least one chief;
(4) The team includes both a chief and surgeons. | 191 | 1/8 |
How many integers between 10000 and 100000 include the block of digits 178? | 280 | 7/8 |
A set $ S$ of points in space satisfies the property that all pairwise distances between points in $ S$ are distinct. Given that all points in $ S$ have integer coordinates $ (x,y,z)$ where $ 1 \leq x,y, z \leq n,$ show that the number of points in $ S$ is less than $ \min \Big((n \plus{} 2)\sqrt {\frac {n}{3}}, n \sqrt {6}\Big).$ *Dan Schwarz, Romania* | \((n+2)\sqrt{\frac{n}{3}},n\sqrt{6}) | 1/8 |
During what year after 1994 did sales increase the most number of dollars? | 2000 | 2/8 |
Compute the number of positive real numbers $x$ that satisfy $\left(3 \cdot 2^{\left\lfloor\log _{2} x\right\rfloor}-x\right)^{16}=2022 x^{13}$. | 9 | 2/8 |
Determine the greatest common divisor of the following pairs of numbers: $A$ and $C$, and $B$ and $C$.
$$
\begin{aligned}
& A=177^{5}+30621 \cdot 173^{3}-173^{5} \\
& B=173^{5}+30621 \cdot 177^{3}-177^{5} \\
& C=173^{4}+30621^{2}+177^{4}
\end{aligned}
$$ | 30637 | 1/8 |
A stack of $2000$ cards is labelled with the integers from $1$ to $2000,$ with different integers on different cards. The cards in the stack are not in numerical order. The top card is removed from the stack and placed on the table, and the next card is moved to the bottom of the stack. The new top card is removed from the stack and placed on the table, to the right of the card already there, and the next card in the stack is moved to the bottom of the stack. The process - placing the top card to the right of the cards already on the table and moving the next card in the stack to the bottom of the stack - is repeated until all cards are on the table. It is found that, reading from left to right, the labels on the cards are now in ascending order: $1,2,3,\ldots,1999,2000.$ In the original stack of cards, how many cards were above the card labelled 1999? | 927 | 1/8 |
Given a real-valued function \( f \) on the reals such that \( f(x + 19) \leq f(x) + 19 \) and \( f(x + 94) \geq f(x) + 94 \) for all \( x \), show that \( f(x + 1) = f(x) + 1 \) for all \( x \). | f(x+1)=f(x)+1 | 4/8 |
From a set of 32 chess pieces, two pieces are drawn randomly. What is the probability that:
a) 2 dark pieces or 2 pieces of different colors,
b) 1 bishop and 1 pawn or 2 pieces of different colors,
c) 2 different-colored rooks or 2 pieces of the same color but different sizes,
d) 1 king and one knight of the same color, or two pieces of the same color,
e) 2 pieces of the same size or 2 pieces of the same color
are drawn? | \frac{159}{248} | 2/8 |
A set $\mathcal{S}$ of distinct positive integers has the following property: for every integer $x$ in $\mathcal{S},$ the arithmetic mean of the set of values obtained by deleting $x$ from $\mathcal{S}$ is an integer. Given that 1 belongs to $\mathcal{S}$ and that 2002 is the largest element of $\mathcal{S},$ what is the greatest number of elements that $\mathcal{S}$ can have?
| 30 | 2/8 |
Consider the set of points that are inside or within one unit of a rectangular parallelepiped (box) that measures $3$ by $4$ by $5$ units. Given that the volume of this set is $\frac{m + n\pi}{p},$ where $m, n,$ and $p$ are positive integers, and $n$ and $p$ are relatively prime, find $m + n + p.$
| 505 | 7/8 |
Find the value of $x$ if $\log_8 x = 1.75$. | 32\sqrt[4]{2} | 4/8 |
For given integer $n \geq 3$ , set $S =\{p_1, p_2, \cdots, p_m\}$ consists of permutations $p_i$ of $(1, 2, \cdots, n)$ . Suppose that among every three distinct numbers in $\{1, 2, \cdots, n\}$ , one of these number does not lie in between the other two numbers in every permutations $p_i$ ( $1 \leq i \leq m$ ). (For example, in the permutation $(1, 3, 2, 4)$ , $3$ lies in between $1$ and $4$ , and $4$ does not lie in between $1$ and $2$ .) Determine the maximum value of $m$ . | 2^{n-1} | 2/8 |
There are two prime numbers \( p \) such that \( 5p \) can be expressed in the form \( \left\lfloor \frac{n^2}{5} \right\rfloor \) for some positive integer \( n \). What is the sum of these two prime numbers? | 52 | 3/8 |
Let the set \( P_n = \{1, 2, \cdots, n\} \) where \( n \in \mathbf{Z}_{+} \). Define \( f(n) \) as the number of sets \( A \) that satisfy the following conditions:
1. \( A \subseteq P_n \) and \( \bar{A} = P_n \setminus A \);
2. If \( x \in A \), then \( 2x \notin A \);
3. If \( x \in \bar{A} \), then \( 2x \notin \bar{A} \).
Find \( f(2018) \). | 2^{1009} | 2/8 |
Boys were collecting apples. Each boy collected either 10 apples or 10% of the total number of apples collected, and there were both types of boys. What is the minimum number of boys that could have been? | 6 | 7/8 |
Prove that if \( a \) and \( b \) are positive numbers, then
$$
\left(1+\frac{1}{a}\right)\left(1+\frac{1}{b}\right) \geq \frac{8}{1+ab}
$$ | (1+\frac{1}{})(1+\frac{1}{b})\ge\frac{8}{1+} | 6/8 |
Let \( P \) and \( Q \) be two non-constant real polynomials that are relatively prime. Show that there are at most three real numbers \( \lambda \) for which \( P + \lambda Q \) is a square (i.e., a polynomial that is a square of another polynomial). | 3 | 1/8 |
Suppose in a right triangle where angle \( Q \) is at the origin and \( \cos Q = 0.5 \). If the length of \( PQ \) is \( 10 \), what is \( QR \)? | 20 | 3/8 |
The owner of an individual clothing store purchased 30 dresses for $32 each. The selling price of the 30 dresses varies for different customers. Using $47 as the standard price, any excess amount is recorded as positive and any shortfall is recorded as negative. The results are shown in the table below:
| Number Sold | 7 | 6 | 3 | 5 | 4 | 5 |
|-------------|---|---|---|---|---|---|
| Price/$ | +3 | +2 | +1 | 0 | -1 | -2 |
After selling these 30 dresses, how much money did the clothing store earn? | 472 | 1/8 |
Given that \( O \) and \( H \) are the circumcenter and orthocenter, respectively, of an acute triangle \( \triangle ABC \). Points \( M \) and \( N \) lie on \( AB \) and \( AC \) respectively, such that \( AM = AO \) and \( AN = AH \). Let \( R \) be the radius of the circumcircle. Prove that \( MN = R \). | MN=R | 6/8 |
Find the product of the divisors of \(72\). | 72^6 | 3/8 |
Two factories, A and B, collaborated to produce a batch of protective clothing. They started work at the same time. Initially, factory A's production speed was $\frac{1}{3}$ faster than factory B's. Midway, factory B stopped for 1 day to adjust its machinery, after which its production speed doubled. It took a total of 6 days from start to finish, and both factories produced the same amount of protective clothing. How many days did the two factories work after factory B adjusted its machinery? | 3 | 7/8 |
Determine all functions $f: \mathbb{Z}\to\mathbb{Z}$ satisfying \[f\big(f(m)+n\big)+f(m)=f(n)+f(3m)+2014\] for all integers $m$ and $n$ .
*Proposed by Netherlands* | f(n)=2n+1007 | 1/8 |
Let $ABCDEF$ be a regular hexagon. Let $G$, $H$, $I$, $J$, $K$, and $L$ be the midpoints of sides $AB$, $BC$, $CD$, $DE$, $EF$, and $AF$, respectively. The segments $\overline{AH}$, $\overline{BI}$, $\overline{CJ}$, $\overline{DK}$, $\overline{EL}$, and $\overline{FG}$ bound a smaller regular hexagon. Let the ratio of the area of the smaller hexagon to the area of $ABCDEF$ be expressed as a fraction $\frac {m}{n}$ where $m$ and $n$ are relatively prime positive integers. Find $m + n$. | 11 | 5/8 |
In the equation \(2bx + b = 3cx + c\), \(b\) and \(c\) can each take any of the values \(1, 2, 3, 4, 5, 6\). How many cases will there be where the solution to the equation is positive? | 3 | 7/8 |
Let $G_{1} G_{2} G_{3}$ be a triangle with $G_{1} G_{2}=7, G_{2} G_{3}=13$, and $G_{3} G_{1}=15$. Let $G_{4}$ be a point outside triangle $G_{1} G_{2} G_{3}$ so that ray $\overrightarrow{G_{1} G_{4}}$ cuts through the interior of the triangle, $G_{3} G_{4}=G_{4} G_{2}$, and $\angle G_{3} G_{1} G_{4}=30^{\circ}$. Let $G_{3} G_{4}$ and $G_{1} G_{2}$ meet at $G_{5}$. Determine the length of segment $G_{2} G_{5}$. | \frac{169}{23} | 3/8 |
Given a circle \(A_0: (x-1)^2 + (y-1)^2 = 1^2\) on the \(xy\) coordinate plane. In the first quadrant, construct circle \(A_1\) such that it is tangent to both the \(x\) and \(y\) axes and externally tangent to circle \(A_0\). Subsequently, construct circles \(A_i\) (\(i=2,3,4,\cdots\)) such that each circle \(A_i\) is externally tangent to both circle \(A_0\) and the preceding circle \(A_{i-1}\), while also being tangent to the \(x\)-axis. Denote \(L_i\) as the circumference of circle \(A_i\). Given:
\[ L = \lim_{n \to +\infty} \left(L_1 + L_2 + L_3 + \cdots + L_n\right) \]
prove that:
\[ \frac{1}{2} < \frac{L}{2\pi} < \frac{\sqrt{2}}{2}. \] | \frac{1}{2}<\frac{L}{2\pi}<\frac{\sqrt{2}}{2} | 1/8 |
Evan has $10$ cards numbered $1$ through $10$ . He chooses some of the cards and takes the product of the numbers on them. When the product is divided by $3$ , the remainder is $1$ . Find the maximum number of cards he could have chose.
*Proposed by Evan Chang* | 6 | 7/8 |
How many positive integers less than 900 can be written as a product of two or more consecutive prime numbers? | 14 | 6/8 |
The sum of the (decimal) digits of a natural number $n$ equals $100$ , and the sum of digits of $44n$ equals $800$ . Determine the sum of digits of $3n$ . | 300 | 4/8 |
Given real numbers $b_0, b_1, \dots, b_{2019}$ with $b_{2019} \neq 0$, let $z_1,z_2,\dots,z_{2019}$ be the roots in the complex plane of the polynomial \[ P(z) = \sum_{k=0}^{2019} b_k z^k. \] Let $\mu = (|z_1| + \cdots + |z_{2019}|)/2019$ be the average of the distances from $z_1,z_2,\dots,z_{2019}$ to the origin. Determine the largest constant $M$ such that $\mu \geq M$ for all choices of $b_0,b_1,\dots, b_{2019}$ that satisfy \[ 1 \leq b_0 < b_1 < b_2 < \cdots < b_{2019} \leq 2019. \] | 2019^{-1/2019} | 1/8 |
Given that \( p \) and \( q \) are positive integers such that \( p + q > 2017 \), \( 0 < p < q \leq 2017 \), and \((p, q) = 1\), find the sum of all fractions of the form \(\frac{1}{pq}\). | 1/2 | 1/8 |
How many positive integers $n$ satisfy the following condition:
$(130n)^{50} > n^{100} > 2^{200}\ ?$
$\textbf{(A) } 0\qquad \textbf{(B) } 7\qquad \textbf{(C) } 12\qquad \textbf{(D) } 65\qquad \textbf{(E) } 125$ | \textbf{(E)}125 | 1/8 |
Let $\lfloor x \rfloor$ be the greatest integer less than or equal to $x$. Then the number of real solutions to $4x^2-40\lfloor x \rfloor +51=0$ is
$\mathrm{(A)\ } 0 \qquad \mathrm{(B) \ }1 \qquad \mathrm{(C) \ } 2 \qquad \mathrm{(D) \ } 3 \qquad \mathrm{(E) \ }4$ | (E)\4 | 1/8 |
We have placed a 1 forint coin heads up in every cell of a $3 \times 3$ grid. What is the minimum number of coins that need to be flipped so that there are no three heads or three tails in any row, column, or diagonal? | 4 | 3/8 |
Find the sum of the absolute values of the roots of \( x^{4} - 4x^{3} - 4x^{2} + 16x - 8 = 0 \). | 2+2\sqrt{2}+2\sqrt{3} | 6/8 |
Show that for all \(x, y > 0\), the following inequality holds:
\[ \frac{x}{x^{4} + y^{2}} + \frac{y}{x^{2} + y^{4}} \leq \frac{1}{xy} \] | \frac{x}{x^4+y^2}+\frac{y}{x^2+y^4}\le\frac{1}{xy} | 4/8 |
Bernie has 2020 marbles and 2020 bags labeled $B_{1}, \ldots, B_{2020}$ in which he randomly distributes the marbles (each marble is placed in a random bag independently). If $E$ the expected number of integers $1 \leq i \leq 2020$ such that $B_{i}$ has at least $i$ marbles, compute the closest integer to $1000E$. | 1000 | 2/8 |
Determine the largest number among those $A$ values for which the following statement holds. No matter in what order we write down the first 100 positive integers, there will always be ten consecutive numbers in the sequence whose sum is at least $A$. | 505 | 3/8 |
Moor has $3$ different shirts, labeled $T, E,$ and $A$ . Across $5$ days, the only days Moor can wear shirt $T$ are days $2$ and $5$ . How many different sequences of shirts can Moor wear across these $5$ days? | 72 | 7/8 |
All three-digit natural numbers with the first digit being odd and greater than 1 are written on the board. What is the maximum number of quadratic equations of the form \(a x^{2}+b x+c=0\) that can be formed using these numbers as \(a\), \(b\), and \(c\), each used no more than once, such that all these equations have roots? | 100 | 1/8 |
Given the circle \( O: x^{2}+y^{2}=4 \) and the curve \( C: y=3|x-t| \), and points \( A(m, n) \) and \( B(s, p) \) \((m, n, s, p \in \mathbb{N}^*) \) on the curve \( C \), such that the ratio of the distance from any point on the circle \( O \) to point \( A \) and to point \( B \) is a constant \( k (k>1) \), find the value of \( t \). | \frac{4}{3} | 5/8 |
What is the smallest integer \( n > 0 \) such that for any integer \( m \) in the range \( 1, 2, 3, \ldots, 1992 \), we can always find an integral multiple of \( 1/n \) in the open interval \( (m/1993, (m + 1)/1994) \)? | 3987 | 1/8 |
In a certain triangle, the difference of two sides: $b-c$ is twice the distance of the angle bisector $f_{\alpha}$, originating from the common endpoint $A$ of the two sides, from the altitude foot $M$. What is the angle $\alpha$ enclosed by the two sides? | 60 | 1/8 |
A simple graph \( G \) on 2020 vertices has its edges colored red and green. It turns out that any monochromatic cycle has even length. Given this information, what is the maximum number of edges \( G \) could have? | 1530150 | 2/8 |
Given that the probability that a ball is tossed into bin k is 3^(-k) for k = 1,2,3,..., find the probability that the blue ball is tossed into a higher-numbered bin than the yellow ball. | \frac{7}{16} | 1/8 |
In how many ways can four black balls, four white balls, and four blue balls be distributed into six different boxes? | 2000376 | 3/8 |
Let \( C \) be a circle with two diameters intersecting at an angle of 30 degrees. A circle \( S \) is tangent to both diameters and to \( C \), and has radius 1. Find the largest possible radius of \( C \). | 1+\sqrt{6}+\sqrt{2} | 1/8 |
In the diagram, points \(B\), \(C\), and \(D\) lie on a line. Also, \(\angle ABC = 90^\circ\) and \(\angle ACD = 150^\circ\). The value of \(x\) is: | 60 | 2/8 |
Let $AB$ be a diameter of a circle centered at $O$. Let $E$ be a point on the circle, and let the tangent at $B$ intersect the tangent at $E$ and $AE$ at $C$ and $D$, respectively. If $\angle BAE = 43^\circ$, find $\angle CED$, in degrees.
[asy]
import graph;
unitsize(2 cm);
pair O, A, B, C, D, E;
O = (0,0);
A = (0,1);
B = (0,-1);
E = dir(-6);
D = extension(A,E,B,B + rotate(90)*(B));
C = extension(E,E + rotate(90)*(E),B,B + rotate(90)*(B));
draw(Circle(O,1));
draw(B--A--D--cycle);
draw(B--E--C);
label("$A$", A, N);
label("$B$", B, S);
label("$C$", C, S);
label("$D$", D, SE);
label("$E$", E, dir(0));
dot("$O$", O, W);
[/asy] | 47^\circ | 1/8 |
Xiao Hua needs to attend an event at the Youth Palace at 2 PM, but his watch gains 4 minutes every hour. He reset his watch at 10 AM. When Xiao Hua arrives at the Youth Palace according to his watch at 2 PM, how many minutes early is he actually? | 16 | 6/8 |
A herd of elephants. Springs are bubbling at the bottom of the lake. A herd of 183 elephants could drink it dry in one day, and a herd of 37 elephants could do so in 5 days. How many days will it take for 1 elephant to drink the lake dry? | 365 | 7/8 |
There are 15 consecutive numbers written on the board. Can it be that the sum of any three consecutive numbers is positive, while the sum of any four consecutive numbers is negative? | No | 5/8 |
What is the smallest prime whose digits sum to \(28\)? | 1999 | 7/8 |
Let $z_1=18+83i,~z_2=18+39i,$ and $z_3=78+99i,$ where $i=\sqrt{-1}.$ Let $z$ be the unique complex number with the properties that $\frac{z_3-z_1}{z_2-z_1}~\cdot~\frac{z-z_2}{z-z_3}$ is a real number and the imaginary part of $z$ is the greatest possible. Find the real part of $z$. | 56 | 2/8 |
Bob is coloring lattice points in the coordinate plane. Find the number of ways Bob can color five points in $\{(x, y) \mid 1 \leq x, y \leq 5\}$ blue such that the distance between any two blue points is not an integer. | 80 | 1/8 |
Jerry buys a bottle of 150 pills. Using a standard 12 hour clock, he sees that the clock reads exactly 12 when he takes the first pill. If he takes one pill every five hours, what hour will the clock read when he takes the last pill in
the bottle? | 1 | 7/8 |
The height of a triangle, equal to 2, divides the angle of the triangle in the ratio 2:1, and the base of the triangle into parts, the smaller of which is equal to 1. Find the area of the triangle. | 11/3 | 5/8 |
Robert has 4 indistinguishable gold coins and 4 indistinguishable silver coins. Each coin has an engraving of one face on one side, but not on the other. He wants to stack the eight coins on a table into a single stack so that no two adjacent coins are face to face. Find the number of possible distinguishable arrangements of the 8 coins.
| 630 | 1/8 |
Simão needs to find a number that is the code to the Treasure Chest, which is hidden in the table.
| 5 | 9 | 4 | 9 | 4 | 1 |
|---|---|---|---|---|---|
| 6 | 3 | 7 | 3 | 4 | 8 |
| 8 | 2 | 4 | 2 | 5 | 5 |
| 7 | 4 | 5 | 7 | 5 | 2 |
| 2 | 7 | 6 | 1 | 2 | 8 |
| 5 | 2 | 3 | 6 | 7 | 1 |
To discover the code, he needs to form groups of 3 digits that are in successive cells, either horizontally or vertically, and whose sum is 14. Once these groups are identified, the code is the sum of the numbers that do not appear in these groups. What is this code? | 29 | 1/8 |
Let \( \omega^3 = 1 \), where \( \omega \neq 1 \). Show that \( z_1 \), \( z_2 \), and \(-\omega z_1 - \omega^2 z_2\) are the vertices of an equilateral triangle. | z_1,z_2,-\omegaz_1-\omega^2z_2 | 1/8 |
Find the minimum point of the function $f(x)=x+2\cos x$ on the interval $[0, \pi]$. | \dfrac{5\pi}{6} | 6/8 |
Consider the following sequence $$ (a_n)_{n=1}^{\infty}=(1,1,2,1,2,3,1,2,3,4,1,2,3,4,5,1,\dots) $$ Find all pairs $(\alpha, \beta)$ of positive real numbers such that $\lim_{n\to \infty}\frac{\displaystyle\sum_{k=1}^n a_k}{n^{\alpha}}=\beta$ .
(Proposed by Tomas Barta, Charles University, Prague) | (\frac{3}{2},\frac{\sqrt{2}}{3}) | 7/8 |
There is a positive real number $x$ not equal to either $\tfrac{1}{20}$ or $\tfrac{1}{2}$ such that\[\log_{20x} (22x)=\log_{2x} (202x).\]The value $\log_{20x} (22x)$ can be written as $\log_{10} (\tfrac{m}{n})$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$. | 112 | 7/8 |
Let \( P \in \mathbf{C}[X] \) be such that for all \( z \in \mathbf{C} \) with \( |z| = 1 \), \( P(z) \) is real.
Show that \( P \) is constant. | P | 5/8 |
Find all functions \( f \) from \(\mathbb{R}_{+}^{*}\) to \(\mathbb{R}_{+}^{*}\) that satisfy the following condition:
\[
\forall x, y \in \mathbb{R}_{+}^{*}, \quad f(x+f(y))=f(x+y)+f(y)
\] | f(x)=2x | 1/8 |
Given point \( A \) has coordinates \( (0,3) \), and points \( B \) and \( C \) are on the circle \( O: x^{2}+y^{2}=25 \), satisfying \(\angle B A C=90^{\circ}\). Find the maximum area of \(\triangle A B C\). | \frac{25+3\sqrt{41}}{2} | 1/8 |
Given 50 feet of fencing, where 5 feet is used for a gate that does not contribute to the enclosure area, what is the greatest possible number of square feet in the area of a rectangular pen enclosed by the remaining fencing? | 126.5625 | 4/8 |
Find the biggest natural number $m$ that has the following property: among any five 500-element subsets of $\{ 1,2,\dots, 1000\}$ there exist two sets, whose intersection contains at least $m$ numbers. | 200 | 5/8 |
Let the sequence $(a_{n})$ be defined by $a_{1} = t$ and $a_{n+1} = 4a_{n}(1 - a_{n})$ for $n \geq 1$ . How many possible values of t are there, if $a_{1998} = 0$ ? | 2^{1996}+1 | 1/8 |
On the refrigerator, MATHCOUNTS is spelled out with 10 magnets, one letter per magnet. Two vowels and three consonants fall off and are put away in a bag. If the Ts are indistinguishable, how many distinct possible collections of letters could be put in the bag? | 75 | 7/8 |
From the numbers $1, 2, \cdots, 1000$, choose $k$ numbers such that any three of the chosen numbers can form the side lengths of a triangle. Find the minimum value of $k$. | 16 | 1/8 |
Assume real numbers $a_i,b_i\,(i=0,1,\cdots,2n)$ satisfy the following conditions:
(1) for $i=0,1,\cdots,2n-1$ , we have $a_i+a_{i+1}\geq 0$ ;
(2) for $j=0,1,\cdots,n-1$ , we have $a_{2j+1}\leq 0$ ;
(2) for any integer $p,q$ , $0\leq p\leq q\leq n$ , we have $\sum_{k=2p}^{2q}b_k>0$ .
Prove that $\sum_{i=0}^{2n}(-1)^i a_i b_i\geq 0$ , and determine when the equality holds. | \sum_{i=0}^{2n}(-1)^ia_ib_i\ge0 | 2/8 |
Given real numbers \( x \) and \( y \) satisfy \( x^2 + y^2 = 20 \). Find the maximum value of \( xy + 8x + y \). | 42 | 1/8 |
Let \( M \) and \( N \) be two points on the Thales' circle of segment \( AB \), distinct from \( A \) and \( B \). Let \( C \) be the midpoint of segment \( NA \), and \( D \) be the midpoint of segment \( NB \). The circle is intersected at the point \( E \) a second time by the line \( MC \), and at point \( F \) by the line \( MD \). What is the value of the expression
\[
MC \cdot CE + MD \cdot DF
\]
if \( AB = 2 \) units? | 1 | 6/8 |
Find all binary operations $\diamondsuit: \mathbb R_{>0}\times \mathbb R_{>0}\to \mathbb R_{>0}$ (meaning $\diamondsuit$ takes pairs of positive real numbers to positive real numbers) such that for any real numbers $a, b, c > 0$ ,
- the equation $a\,\diamondsuit\, (b\,\diamondsuit \,c) = (a\,\diamondsuit \,b)\cdot c$ holds; and
- if $a\ge 1$ then $a\,\diamondsuit\, a\ge 1$ .
*Evan Chen* | \diamondsuit\frac{}{b} | 3/8 |
How many positive, three-digit integers contain at least one $4$ as a digit but do not contain a $6$ as a digit? | 200 | 7/8 |
In an equilateral triangle $\triangle PRS$, if $QS=QT$ and $\angle QTS=40^\circ$, what is the value of $x$? | 80 | 1/8 |
A quadratic function $f(x) = x^2 - ax + b$ has one root in the interval $[-1, 1]$ and another root in the interval $[1, 2]$. Find the minimum value of $a - 2b$.
A. 0
B. $-1$
C. -2
D. 1 | -1 | 1/8 |
Can you find two numbers such that the difference of their squares is a cube, and the difference of their cubes is a square? What are the two smallest numbers with this property? | 106 | 1/8 |
Let \( f(x) = x^2 + px + q \). It is known that the inequality \( |f(x)| > \frac{1}{2} \) has no solutions on the interval \([4, 6]\). Find \( \underbrace{f(f(\ldots f}_{2017}\left(\frac{9 - \sqrt{19}}{2}\right)) \ldots) \). If necessary, round the answer to two decimal places. | 6.68 | 1/8 |
Let $ABC$ be an equilateral triangle. Suppose that the points on sides $AB$, $BC$, and $AC$, including vertices $A$, $B$, and $C$, are divided arbitrarily into two disjoint subsets. Then at least one of the subsets contains the vertex of a right triangle. | 1 | 2/8 |
Ana and Luíza train every day for the Big Race that will take place at the end of the year at school, each running at the same speed. The training starts at point $A$ and ends at point $B$, which are $3000 \mathrm{~m}$ apart. They start at the same time, but when Luíza finishes the race, Ana still has $120 \mathrm{~m}$ to reach point $B$. Yesterday, Luíza gave Ana a chance: "We will start at the same time, but I will start some meters before point $A$ so that we arrive together." How many meters before point $A$ should Luíza start? | 125 | 4/8 |
Let $R$ be the set of all real numbers. Find all functions $f: R \rightarrow R$ such that for all $x, y \in R$, the following equation holds:
$$
f\left(x^{2}+f(y)\right)=y+(f(x))^{2}.
$$ | f(x)=x | 2/8 |
Find the millionth digit after the decimal point in the decimal representation of the fraction \( \frac{3}{41} \). | 7 | 5/8 |
Given a 2x3 rectangle with six unit squares, the lower left corner at the origin, find the value of $c$ such that a slanted line extending from $(c,0)$ to $(4,4)$ divides the entire region into two regions of equal area. | \frac{5}{2} | 1/8 |
Complex numbers \(a, b, c\) form an equilateral triangle with side length 18 in the complex plane. If \(|a+b+c|=36\), find \(|bc + ca + ab|\). | 432 | 4/8 |
Given a set of paired data $(18,24)$, $(13,34)$, $(10,38)$, $(-1,m)$, the regression equation for these data is $y=-2x+59.5$. Find the correlation coefficient $r=$______(rounded to $0.001$). | -0.998 | 4/8 |
The constant term in the expansion of (1+x)(e^(-2x)-e^x)^9. | 84 | 3/8 |
The midpoints of the sides of a regular hexagon $ABCDEF$ are joined to form a smaller hexagon. What fraction of the area of $ABCDEF$ is enclosed by the smaller hexagon? [asy]
import olympiad; import geometry; size(150); defaultpen(linewidth(0.8)); dotfactor=4;
pair[] bigHexagon = new pair[6];
bigHexagon[0] = dir(0);
pair[] smallHexagon = new pair[6];
smallHexagon[0] = (dir(0) + dir(60))/2;
for(int i = 1; i <= 7; ++i){
bigHexagon[i] = dir(60*i);
draw(bigHexagon[i]--bigHexagon[i - 1]);
smallHexagon[i] = (bigHexagon[i] + bigHexagon[i - 1])/2;
draw(smallHexagon[i]--smallHexagon[i - 1]);
}
dot(Label("$A$",align=dir(0)),dir(0)); dot(Label("$B$",align=dir(60)),dir(60)); dot(Label("$C$",align=dir(120)),dir(120)); dot(Label("$D$",align=dir(180)),dir(180)); dot(Label("$E$",align=dir(240)),dir(240)); dot(Label("$F$",align=dir(300)),dir(300));
[/asy] | \frac{3}{4} | 6/8 |
In the Cartesian coordinate system $xoy$, point $P(0, \sqrt{3})$ is given. The parametric equation of curve $C$ is $\begin{cases} x = \sqrt{2} \cos \varphi \\ y = 2 \sin \varphi \end{cases}$ (where $\varphi$ is the parameter). A polar coordinate system is established with the origin as the pole and the positive half-axis of $x$ as the polar axis. The polar equation of line $l$ is $\rho = \frac{\sqrt{3}}{2\cos(\theta - \frac{\pi}{6})}$.
(Ⅰ) Determine the positional relationship between point $P$ and line $l$, and explain the reason;
(Ⅱ) Suppose line $l$ intersects curve $C$ at two points $A$ and $B$, calculate the value of $\frac{1}{|PA|} + \frac{1}{|PB|}$. | \sqrt{14} | 5/8 |
Let $x=\frac{\sum\limits_{n=1}^{44} \cos n^\circ}{\sum\limits_{n=1}^{44} \sin n^\circ}$. What is the greatest integer that does not exceed $100x$? | 241 | 7/8 |
Consider the natural implementation of computing Fibonacci numbers:
\begin{tabular}{l}
1: \textbf{FUNCTION} $\text{FIB}(n)$ :
2: $\qquad$ \textbf{IF} $n = 0$ \textbf{OR} $n = 1$ \textbf{RETURN} 1
3: $\qquad$ \textbf{RETURN} $\text{FIB}(n-1) + \text{FIB}(n-2)$ \end{tabular}
When $\text{FIB}(10)$ is evaluated, how many recursive calls to $\text{FIB}$ occur? | 176 | 3/8 |
Find the number of ways in which the letters in "HMMTHMMT" can be rearranged so that each letter is adjacent to another copy of the same letter. For example, "MMMMTTHH" satisfies this property, but "HHTMMMTM" does not. | 12 | 1/8 |
Place 7 different-colored goldfish into 3 glass tanks numbered 1, 2, and 3, such that the number of fish in each tank is not less than the number on its label. Determine the number of different ways to do this. | 455 | 5/8 |
Given an ellipse \(\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1\) (where \(a > b > 0\)) with the left focal point \(F\), and a point \(P(x_{0}, y_{0})\) on the ellipse where \(x_{0} > 0\). A tangent to the circle \(x^{2} + y^{2} = b^{2}\) is drawn at point \(P\), intersecting the ellipse at a second point \(Q\). Let \(I\) be the incenter of triangle \(PFQ\), and \(\angle PFQ = 2\alpha\). Then \( |FI|\cos \alpha = \) ___. | a | 1/8 |
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