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A circle can be inscribed in a trapezoid with bases of lengths 3 and 5, and a circle can also be circumscribed about it. Calculate the area of the pentagon formed by the radii of the inscribed circle that are perpendicular to the legs of the trapezoid, its smaller base, and the corresponding segments of the legs. | \frac{3 \sqrt{15}}{2} | 1/8 |
On a segment of length 1, several smaller segments are colored, and the distance between any two colored points is not equal to 0.1. Prove that the sum of the lengths of the colored segments does not exceed 0.5. | 0.5 | 3/8 |
In acute triangle $ABC$ points $P$ and $Q$ are the feet of the perpendiculars from $C$ to $\overline{AB}$ and from $B$ to $\overline{AC}$, respectively. Line $PQ$ intersects the circumcircle of $\triangle ABC$ in two distinct points, $X$ and $Y$. Suppose $XP=10$, $PQ=25$, and $QY=15$. The value of $AB\cdot AC$ can be written in the form $m\sqrt n$ where $m$ and $n$ are positive integers, and $n$ is not divisible by the square of any prime. Find $m+n$.
| 574 | 1/8 |
Circle \(\omega\) is inscribed in rhombus \(H M_{1} M_{2} T\) so that \(\omega\) is tangent to \(\overline{H M_{1}}\) at \(A\), \(\overline{M_{1} M_{2}}\) at \(I\), \(\overline{M_{2} T}\) at \(M\), and \(\overline{T H}\) at \(E\). Given that the area of \(H M_{1} M_{2} T\) is 1440 and the area of \(E M T\) is 405, find the area of \(A I M E\). | 540 | 1/8 |
Let $ S \equal{} \{1,2,3,\cdots ,280\}$. Find the smallest integer $ n$ such that each $ n$-element subset of $ S$ contains five numbers which are pairwise relatively prime. | 217 | 4/8 |
Given an arithmetic sequence $\{a_n\}$ with the common difference $d$ being an integer, and $a_k=k^2+2$, $a_{2k}=(k+2)^2$, where $k$ is a constant and $k\in \mathbb{N}^*$
$(1)$ Find $k$ and $a_n$
$(2)$ Let $a_1 > 1$, the sum of the first $n$ terms of $\{a_n\}$ is $S_n$, the first term of the geometric sequence $\{b_n\}$ is $l$, the common ratio is $q(q > 0)$, and the sum of the first $n$ terms is $T_n$. If there exists a positive integer $m$, such that $\frac{S_2}{S_m}=T_3$, find $q$． | \frac{\sqrt{13}-1}{2} | 2/8 |
The sum of the numerical coefficients in the binomial $(2a+2b)^7$ is $\boxed{32768}$. | 16384 | 1/8 |
Given the set $M$ consisting of all functions $f(x)$ that satisfy the property: there exist real numbers $a$ and $k$ ($k \neq 0$) such that for all $x$ in the domain of $f$, $f(a+x) = kf(a-x)$. The pair $(a,k)$ is referred to as the "companion pair" of the function $f(x)$.
1. Determine whether the function $f(x) = x^2$ belongs to set $M$ and explain your reasoning.
2. If $f(x) = \sin x \in M$, find all companion pairs $(a,k)$ for the function $f(x)$.
3. If $(1,1)$ and $(2,-1)$ are both companion pairs of the function $f(x)$, where $f(x) = \cos(\frac{\pi}{2}x)$ for $1 \leq x < 2$ and $f(x) = 0$ for $x=2$. Find all zeros of the function $y=f(x)$ when $2014 \leq x \leq 2016$. | 2016 | 2/8 |
Given the number $200 \ldots 002$ (100 zeros). We need to replace two of the zeros with non-zero digits such that the resulting number is divisible by 66. How many ways can this be done? | 27100 | 4/8 |
Consider $n \geq 2$ distinct points in the plane $A_1,A_2,...,A_n$ . Color the midpoints of the segments determined by each pair of points in red. What is the minimum number of distinct red points? | 2n-3 | 2/8 |
Find the least positive integer $n$ such that $$\frac 1{\sin 30^\circ\sin 31^\circ}+\frac 1{\sin 32^\circ\sin 33^\circ}+\cdots+\frac 1{\sin 88^\circ\sin 89^\circ}+\cos 89^\circ=\frac 1{\sin n^\circ}.$$ | n = 1 | 1/8 |
A solid in the shape of a right circular cone is 4 inches tall and its base has a 3-inch radius. The entire surface of the cone, including its base, is painted. A plane parallel to the base of the cone divides the cone into two solids, a smaller cone-shaped solid $C$ and a frustum-shaped solid $F,$ in such a way that the ratio between the areas of the painted surfaces of $C$ and $F$ and the ratio between the volumes of $C$ and $F$ are both equal to $k$. Given that $k=\frac m n,$ where $m$ and $n$ are relatively prime positive integers, find $m+n.$
| 512 | 7/8 |
You have infinitely many boxes, and you randomly put 3 balls into them. The boxes are labeled $1,2, \ldots$. Each ball has probability $1 / 2^{n}$ of being put into box $n$. The balls are placed independently of each other. What is the probability that some box will contain at least 2 balls? | 5 / 7 | 4/8 |
Elbert and Yaiza each draw 10 cards from a 20-card deck with cards numbered 1, 2, 3, ..., 20. Starting with the player who has the card numbered 1, the players take turns placing down the lowest-numbered card from their hand that is greater than every card previously placed. When a player cannot place a card, they lose and the game ends. Given that Yaiza lost and 5 cards were placed in total, compute the number of ways the cards could have been initially distributed. (The order of cards in a player's hand does not matter.) | 324 | 1/8 |
A sphere with radius $R$ is tangent to one base of a truncated cone and also tangent to its lateral surface along a circle that is the circumference of the other base of the cone. Find the volume of the body consisting of the cone and the sphere, given that the surface area of this body is $S$. | \frac{SR}{3} | 1/8 |
Consider an arithmetic sequence $\left\{a_{n}\right\}$ with the sum of the first $n$ terms denoted as $S_{n}$. It is given that $a_{3} = 12$, $S_{12} > 0$, and $S_{13} < 0$.
(1) Determine the range of values for the common difference $d$.
(2) Identify which of $S_{1}, S_{2}, \cdots, S_{12}$ is the largest value and provide a justification. | S_6 | 7/8 |
In the sequence $\left\{x_{n}\right\}$, where $x_{1}=a$, $x_{2}=b$, and $x_{n}=\frac{x_{n-1}+x_{n-2}}{2}$, prove that $\lim _{n \rightarrow \infty} x_{n}$ exists and determine its value. | \frac{2b}{3} | 7/8 |
In the geometric sequence ${a_n}$ where $q=2$, if the sum of the series $a_2 + a_5 + \dots + a_{98} = 22$, calculate the sum of the first 99 terms of the sequence $S_{99}$. | 77 | 7/8 |
A line drawn through the circumcenter and orthocenter of triangle \(ABC\) cuts off equal segments \(C A_{1}\) and \(C B_{1}\) from its sides \(CA\) and \(CB\). Prove that angle \(C\) is equal to \(60^\circ\). | 60 | 1/8 |
Let \( A, B, C \) be distinct points on the parabola \( y = x^2 \). \( R \) is the radius of the circumcircle of triangle \( \triangle ABC \). Find the range of values for \( R \). | (\frac{1}{2},+\infty) | 1/8 |
Let \( n \) be a positive integer greater than 2. A function \( f: \mathbf{R}^{2} \rightarrow \mathbf{R} \) satisfies the following condition: For any regular \( n \)-gon \( A_{1} A_{2} \cdots A_{n} \), we have \(\sum_{i=1}^{n} f(A_{i}) = 0 \). Prove that \( f \equiv 0 \). | f\equiv0 | 1/8 |
In the expansion of $(2x +3y)^{20}$ , find the number of coefficients divisible by $144$ .
*Proposed by Hannah Shen* | 15 | 1/8 |
A round-robin tennis tournament is organized where each player is supposed to play every other player exactly once. However, the tournament is scheduled to have one rest day during which no matches will be played. If there are 10 players in the tournament, and the tournament was originally scheduled for 9 days, but one day is now a rest day, how many matches will be completed? | 40 | 7/8 |
Real numbers \(a\), \(b\), and \(c\) and positive number \(\lambda\) make \(f(x) = x^3 + ax^2 + b x + c\) have three real roots \(x_1\), \(x_2\), \(x_3\), such that:
(1) \(x_2 - x_1 = \lambda\);
(2) \(x_3 > \frac{1}{2}(x_1 + x_2)\).
Find the maximum value of \(\frac{2 a^3 + 27 c - 9 a b}{\lambda^3}\). | \frac{3\sqrt{3}}{2} | 3/8 |
Calculate the limit of the numerical sequence:
$$\lim _{n \rightarrow \infty}\left(\frac{7 n^{2}+18 n-15}{7 n^{2}+11 n+15}\right)^{n+2}$$ | e | 7/8 |
Given that the regular price for one backpack is $60, and Maria receives a 20% discount on the second backpack and a 30% discount on the third backpack, calculate the percentage of the $180 regular price she saved. | 16.67\% | 5/8 |
The value of $ 21!$ is $ 51{,}090{,}942{,}171{,}abc{,}440{,}000$ , where $ a$ , $ b$ , and $ c$ are digits. What is the value of $ 100a \plus{} 10b \plus{} c$ ? | 709 | 2/8 |
Suppose that \((a_1, b_1), (a_2, b_2), \ldots, (a_{100}, b_{100})\) are distinct ordered pairs of nonnegative integers. Let \(N\) denote the number of pairs of integers \((i, j)\) satisfying \(1 \leq i < j \leq 100\) and \(\left|a_i b_j - a_j b_i\right| = 1\). Determine the largest possible value of \(N\) over all possible choices of the 100 ordered pairs. | 197 | 3/8 |
A box contains 5 white balls and 5 black balls. I draw them out of the box, one at a time. What is the probability that all of my draws alternate colors, starting and ending with the same color? | \frac{1}{126} | 7/8 |
In the triangle $\triangle ABC$, let $G$ be the centroid, and let $I$ be the center of the inscribed circle. Let $\alpha$ and $\beta$ be the angles at the vertices $A$ and $B$, respectively. Suppose that the segment $IG$ is parallel to $AB$ and that $\beta = 2 \tan^{-1} (1/3)$. Find $\alpha$. | \frac{\pi}{2} | 5/8 |
Three congruent isosceles triangles $DAO$, $AOB$, and $OBC$ have $AD=AO=OB=BC=12$ and $AB=DO=OC=16$. These triangles are arranged to form trapezoid $ABCD$. Point $P$ is on side $AB$ so that $OP$ is perpendicular to $AB$. Points $X$ and $Y$ are the midpoints of $AD$ and $BC$, respectively. When $X$ and $Y$ are joined, the trapezoid is divided into two smaller trapezoids. Find the ratio of the area of trapezoid $ABYX$ to the area of trapezoid $XYCD$ in simplified form and find $p+q$, where the ratio is $p:q$. | 12 | 4/8 |
Calculate the definite integral:
$$
\int_{1}^{8} \frac{5 \sqrt{x+24}}{(x+24)^{2} \cdot \sqrt{x}} \, dx
$$ | \frac{1}{8} | 7/8 |
On an island, there are knights, liars, and followers; each one knows who is who among them. All 2018 island inhabitants were lined up and each was asked to answer "Yes" or "No" to the question: "Are there more knights than liars on the island?" The inhabitants answered one by one in such a way that the others could hear. Knights told the truth, liars lied. Each follower answered the same as the majority of those who had answered before them, and if the number of "Yes" and "No" answers was equal, they could give either answer. It turned out that there were exactly 1009 "Yes" answers. What is the maximum number of followers that could be among the island inhabitants? | 1009 | 1/8 |
Given the integers \( 1, 2, 3, \ldots, 40 \), find the greatest possible sum of the positive differences between the integers in twenty pairs, where the positive difference is either 1 or 3. | 58 | 1/8 |
Let an integer $ n > 3$ . Denote the set $ T\equal{}\{1,2, \ldots,n\}.$ A subset S of T is called *wanting set* if S has the property: There exists a positive integer $ c$ which is not greater than $ \frac {n}{2}$ such that $ |s_1 \minus{} s_2|\ne c$ for every pairs of arbitrary elements $ s_1,s_2\in S$ . How many does a *wanting set* have at most are there ? | \lfloor\frac{2n}{3}\rfloor | 1/8 |
In the country of Wonderland, an election campaign is being held for the title of the best tea lover, with the candidates being the Mad Hatter, March Hare, and Dormouse. According to a poll, $20\%$ of the residents plan to vote for the Mad Hatter, $25\%$ for the March Hare, and $30\%$ for the Dormouse. The remaining residents are undecided. Determine the smallest percentage of the undecided voters that the Mad Hatter needs to secure to guarantee not losing to both the March Hare and the Dormouse (regardless of how the undecided votes are allocated), given that each undecided voter will vote for one of the candidates. The winner is determined by a simple majority. Justify your answer. | 70 | 1/8 |
Let $S$ be the set of points whose coordinates $x,$ $y,$ and $z$ are integers that satisfy $0\le x\le2,$ $0\le y\le3,$ and $0\le z\le4.$ Two distinct points are randomly chosen from $S.$ The probability that the midpoint of the segment they determine also belongs to $S$ is $m/n,$ where $m$ and $n$ are relatively prime positive integers. Find $m + n.$
| 200 | 7/8 |
Find the maximum value of the function
$$
f(x) = \sqrt{3} \sin 2x + 2 \sin x + 4 \sqrt{3} \cos x.
$$ | \frac{17}{2} | 1/8 |
Let \(ABCD\) be an isosceles trapezoid such that \(AD = BC\), \(AB = 3\), and \(CD = 8\). Let \(E\) be a point in the plane such that \(BC = EC\) and \(AE \perp EC\). Compute \(AE\). | 2\sqrt{6} | 6/8 |
Given $f(x) = 2\sqrt{3}\sin x \cos x + 2\cos^2x - 1$,
(1) Find the maximum value of $f(x)$, as well as the set of values of $x$ for which $f(x)$ attains its maximum value;
(2) In $\triangle ABC$, if $a$, $b$, and $c$ are the lengths of sides opposite the angles $A$, $B$, and $C$ respectively, with $a=1$, $b=\sqrt{3}$, and $f(A) = 2$, determine the angle $C$. | \frac{\pi}{2} | 7/8 |
Given $f(x)= \begin{cases} 2a-(x+ \frac {4}{x}),x < a\\x- \frac {4}{x},x\geqslant a\\end{cases}$.
(1) When $a=1$, if $f(x)=3$, then $x=$ \_\_\_\_\_\_;
(2) When $a\leqslant -1$, if $f(x)=3$ has three distinct real roots that form an arithmetic sequence, then $a=$ \_\_\_\_\_\_. | - \frac {11}{6} | 6/8 |
The area of trapezoid $ABCD$ with bases $AD$ and $BC$ ($AD > BC$) is 48, and the area of triangle $AOB$, where $O$ is the intersection point of the diagonals of the trapezoid, is 9. Find the ratio of the bases of the trapezoid $AD: BC$. | 3 | 1/8 |
Joey wrote a system of equations on a blackboard, where each of the equations was of the form $a + b = c$ or $a \cdot b = c$ for some variables or integers $a, b, c$ . Then Sean came to the board and erased all of the plus signs and multiplication signs, so that the board reads: $x\,\,\,\, z = 15$ $x\,\,\,\, y = 12$ $x\,\,\,\, x = 36$ If $x, y, z$ are integer solutions to the original system, find the sum of all possible values of $100x+10y+z$ . | 2037 | 6/8 |
Let the polynomial \( P(x) = x^n + a_1 x^{n-1} + \cdots + a_{n-1} x + a_n \) have complex roots \( x_1, x_2, \cdots, x_n \). Define \( \alpha = \frac{1}{n} \sum_{k=1}^{n} x_k \) and \( \beta^2 = \frac{1}{n} \sum_{k=1}^{n} |x_k|^2 \), with \( \beta^2 < 1 + |\alpha|^2 \). If a complex number \( x_0 \) satisfies \( |\alpha - x_0|^2 < 1 - \beta^2 + |\alpha|^2 \), prove that \( |P(x_0)| < 1 \). | |P(x_0)|<1 | 2/8 |
Pictured below is a rectangular array of dots with the bottom left point labelled \( A \). In how many ways can two points in the array be chosen so that they, together with point \( A \), form a triangle with positive area?
[Image of a rectangular array of dots with point \( A \) at the bottom left.]
(Note: Include the actual image if providing this to an end-user, as visual context is critical for solving the problem). | 256 | 1/8 |
Let \( Q \) be the set of some permutations of the numbers \( 1, 2, \ldots, 100 \) such that for any \( 1 \leq a, b \leq 100 \) with \( a \neq b \), there is at most one \( \sigma \in Q \) where \( b \) immediately follows \( a \) in \( \sigma \). Find the maximum number of elements in the set \( Q \). | 100 | 7/8 |
Find all natural values of \( n \) for which
$$
\cos \frac{2 \pi}{9}+\cos \frac{4 \pi}{9}+\cdots+\cos \frac{2 \pi n}{9}=\cos \frac{\pi}{9}, \text { and } \log _{2}^{2} n+45<\log _{2} 8 n^{13}
$$
Record the sum of the obtained values of \( n \) as the answer. | 644 | 4/8 |
All edges of a tetrahedron are \(12 \, \text{cm}\). Can it be placed in a box shaped as a rectangular parallelepiped with sides \(9 \, \text{cm}\), \(13 \, \text{cm}\), and \(15 \, \text{cm}\)? | Yes | 4/8 |
On a table, there are 100 piles of pebbles, with 1, 2, ..., up to 100 pebbles in each pile respectively. In one step, you can reduce any number of selected piles, provided that you remove the same number of pebbles from each selected pile.
What is the minimum number of steps needed to remove all the pebbles from the table? | 7 | 5/8 |
Given that F is the right focus of the ellipse $\frac{x^{2}}{a^{2}}+ \frac{y^{2}}{b^{2}}=1(a>b>0)$, and A is one endpoint of the ellipse's minor axis. If F is the trisection point of the chord of the ellipse that passes through AF, calculate the eccentricity of the ellipse. | \frac{\sqrt{3}}{3} | 7/8 |
Joe has $1729$ randomly oriented and randomly arranged unit cubes, which are initially unpainted. He makes two cubes of sidelengths $9$ and $10$ or of sidelengths $1$ and $12$ (randomly chosen). These cubes are dipped into white paint. Then two more cubes of sidelengths $1$ and $12$ or $9$ and $10$ are formed from the same unit cubes, again randomly oriented and randomly arranged, and dipped into paint. Joe continues this process until every side of every unit cube is painted. After how many times of doing this is the expected number of painted faces closest to half of the total? | 7 | 2/8 |
Given $(a+i)i=b+ai$, solve for $|a+bi|$. | \sqrt{2} | 1/8 |
Let $a,$ $b,$ $c,$ $d$ be real numbers such that
\[\frac{(a - b)(c - d)}{(b - c)(d - a)} = \frac{3}{7}.\] Find the product of all possible values of
\[\frac{(a - c)(b - d)}{(a - b)(c - d)}.\] | -\frac{4}{3} | 4/8 |
The real numbers $x$ , $y$ , $z$ , and $t$ satisfy the following equation:
\[2x^2 + 4xy + 3y^2 - 2xz -2 yz + z^2 + 1 = t + \sqrt{y + z - t} \]
Find 100 times the maximum possible value for $t$ . | 125 | 6/8 |
The function defined on the set of real numbers, \(f(x)\), satisfies \(f(x-1) = \frac{1 + f(x+1)}{1 - f(x+1)}\). Find the value of \(f(1) \cdot f(2) \cdot f(3) \cdots f(2008) + 2008\). | 2009 | 7/8 |
The minimum possible sum of the three dimensions of a rectangular box with a volume of 3003 in^3 is what value? | 45 | 2/8 |
Solve the inequality \( 202 \sqrt{x^{3} - 3x - \frac{3}{x} + \frac{1}{x^{3}} + 5} \leq 0 \). | \frac{-1-\sqrt{21}\\sqrt{2\sqrt{21}+6}}{4} | 1/8 |
In how many ways can a barrel with a capacity of 10 liters be emptied using two containers with capacities of 1 liter and 2 liters? | 89 | 5/8 |
A grid sheet of size \(5 \times 7\) was cut into \(2 \times 2\) squares, L-shaped pieces covering 3 cells, and \(1 \times 3\) strips. How many \(2 \times 2\) squares could be obtained? | 5 | 3/8 |
Evaluate the integral $\int_{0}^{\frac{\pi}{2}} \sin^{2} \frac{x}{2} dx =$ \_\_\_\_\_\_. | \frac{\pi}{4} - \frac{1}{2} | 7/8 |
The integer $m$ is the largest positive multiple of $18$ such that every digit of $m$ is either $9$ or $0$. Compute $\frac{m}{18}$. | 555 | 1/8 |
Let ($a_1$, $a_2$, ... $a_{10}$) be a list of the first 10 positive integers such that for each $2\le$ $i$ $\le10$ either $a_i + 1$ or $a_i-1$ or both appear somewhere before $a_i$ in the list. How many such lists are there?
$\textbf{(A)}\ \ 120\qquad\textbf{(B)}\ 512\qquad\textbf{(C)}\ \ 1024\qquad\textbf{(D)}\ 181,440\qquad\textbf{(E)}\ \ 362,880$ | \textbf{(B)}\512 | 1/8 |
Let \( n \) be the product of the first 10 primes, and let
\[ S = \sum_{xy \mid n} \varphi(x) \cdot y, \]
where \( \varphi(x) \) denotes the number of positive integers less than or equal to \(x\) that are relatively prime to \( x \), and the sum is taken over ordered pairs \((x, y)\) of positive integers for which \( xy \) divides \( n \). Compute \(\frac{S}{n}\). | 1024 | 7/8 |
Two players play a game where they are each given 10 indistinguishable units that must be distributed across three locations. (Units cannot be split.) At each location, a player wins at that location if the number of units they placed there is at least 2 more than the units of the other player. If both players distribute their units randomly (i.e., there is an equal probability of them distributing their units for any attainable distribution across the 3 locations), the probability that at least one location is won by one of the players can be expressed as \(\frac{a}{b}\), where \(a, b\) are relatively prime positive integers. Compute \(100a + b\). | 1011 | 6/8 |
Let $ n $ be a natural number. How many numbers of the form $ \pm 1\pm 2\pm 3\pm\cdots\pm n $ are there? | \frac{n(n+1)}{2}+1 | 4/8 |
Wang Wei walks from location A to location B, while Zhang Ming rides a bicycle from location B to location A. They meet on the road after half an hour. After reaching location A, Zhang Ming immediately returns to location B and catches up with Wang Wei 20 minutes after their first meeting. After arriving at location B, Zhang Ming turns back, and they meet for the third time $\qquad$ minutes after their second meeting. | 40 | 1/8 |
Given that for all \( x \in [-1, 1] \), \( x^{3} - a x + 1 \geqslant 0 \), find the range of the real number \( a \). | [0,\frac{3\sqrt[3]{2}}{2}] | 3/8 |
Place each of the digits 4, 5, 6, and 7 in exactly one square to make the smallest possible product. The grid placement is the same as described before. | 2622 | 7/8 |
The organizing committee of the sports meeting needs to select four volunteers from Xiao Zhang, Xiao Zhao, Xiao Li, Xiao Luo, and Xiao Wang to take on four different tasks: translation, tour guide, etiquette, and driver. If Xiao Zhang and Xiao Zhao can only take on the first two tasks, while the other three can take on any of the four tasks, then the total number of different dispatch plans is \_\_\_\_\_\_ (The result should be expressed in numbers). | 36 | 2/8 |
A circle is drawn through two adjacent vertices of a square such that the length of the tangent to it from the third vertex of the square is three times the side length of the square. Find the area of the circle if the side length of the square is \(a\). | \frac{65\pi^2}{4} | 2/8 |
A natural number undergoes the following operation: the rightmost digit of its decimal representation is discarded, and then the number obtained after discarding is added to twice the discarded digit. For example, $157 \mapsto 15 + 2 \times 7 = 29$, $5 \mapsto 0 + 2 \times 5 = 10$. A natural number is called ‘good’ if after repeatedly applying this operation, the resulting number stops changing. Find the smallest such good number. | 19 | 7/8 |
A book has a total of 100 pages, numbered sequentially from 1, 2, 3, 4…100. The digit “2” appears in the page numbers a total of \_\_\_\_\_\_ times. | 20 | 7/8 |
Let \( ABC \) be a triangle such that \( AB = AC = 182 \) and \( BC = 140 \). Let \( X_1 \) lie on \( AC \) such that \( CX_1 = 130 \). Let the line through \( X_1 \) perpendicular to \( BX_1 \) at \( X_1 \) meet \( AB \) at \( X_2 \). Define \( X_2, X_3, \ldots \) as follows: for \( n \) odd and \( n \geq 1 \), let \( X_{n+1} \) be the intersection of \( AB \) with the perpendicular to \( X_{n-1} X_n \) through \( X_n \); for \( n \) even and \( n \geq 2 \), let \( X_{n+1} \) be the intersection of \( AC \) with the perpendicular to \( X_{n-1} X_n \) through \( X_n \). Find \( BX_1 + X_1X_2 + X_2X_3 + \ldots \) | \frac{1106}{5} | 1/8 |
A stacking of circles in the plane consists of a base, or some number of unit circles centered on the $x$-axis in a row without overlap or gaps, and circles above the $x$-axis that must be tangent to two circles below them (so that if the ends of the base were secured and gravity were applied from below, then nothing would move). How many stackings of circles in the plane have 4 circles in the base? | 14 | 1/8 |
A class of 54 students in the fifth grade took a group photo. The fixed price is 24.5 yuan for 4 photos. Additional prints cost 2.3 yuan each. If every student in the class wants one photo, how much money in total needs to be paid? | 139.5 | 4/8 |
For a given increasing function \( f: \mathbb{R} \rightarrow \mathbb{R} \), define the function \( g(x, y) \) as follows:
\[ g(x, y) = \frac{f(x+y) - f(x)}{f(x) - f(x-y)}, \quad x \in \mathbb{R}, \; y > 0. \]
Suppose that for \( x=0 \) and all \( y > 0 \), and for \( x \neq 0 \) and all \( y \in (0, |x|] \), the following inequality holds:
\[ 2^{-1} < g(x, y) < 2. \]
Prove that for all \( x \in \mathbb{R} \) and \( y > 0 \), the following inequality holds:
\[ 14^{-1} < g(x, y) < 14. \] | \frac{1}{14}<(x,y)<14 | 1/8 |
Show that \(\sin^n(2x) + (\sin^n(x) - \cos^n(x))^2 \leq 1\). | \sin^n(2x)+(\sin^n(x)-\cos^n(x))^2\le1 | 1/8 |
What is the distance from Boguli to Bolifoyn? | 10 | 1/8 |
A parabola is inscribed in an equilateral triangle \( ABC \) of side length 1 such that \( AC \) and \( BC \) are tangent to the parabola at \( A \) and \( B \), respectively. Find the area between \( AB \) and the parabola. | \frac{\sqrt{3}}{6} | 7/8 |
What is the maximum number of checkers that can be placed on an $8 \times 8$ board so that each one is being attacked? | 32 | 1/8 |
What is the probability of having $2$ adjacent white balls or $2$ adjacent blue balls in a random arrangement of $3$ red, $2$ white and $2$ blue balls? | $\dfrac{10}{21}$ | 2/8 |
Sector $OAB$ is a quarter of a circle of radius 3 cm. A circle is drawn inside this sector, tangent at three points as shown. What is the number of centimeters in the radius of the inscribed circle? Express your answer in simplest radical form. [asy]
import olympiad; import geometry; size(100); defaultpen(linewidth(0.8));
draw(Arc(origin,3,90,180));
draw((-3,0)--(origin)--(0,3));
real x = 3/(1 + sqrt(2));
draw(Circle((-x,x),x)); label("$B$",(0,3),N); label("$A$",(-3,0),W);
label("$O$",(0,0),SE); label("3 cm",(0,0)--(-3,0),S);
[/asy] | 3\sqrt{2}-3 | 6/8 |
The sequence \( a_n \) is defined as follows: \( a_1 = 1 \), \( a_{n+1} = a_n + \frac{1}{a_n} \) for \( n \geq 1 \). Prove that \( a_{100} > 14 \). | a_{100}>14 | 6/8 |
In a high school with $500$ students, $40\%$ of the seniors play a musical instrument, while $30\%$ of the non-seniors do not play a musical instrument. In all, $46.8\%$ of the students do not play a musical instrument. How many non-seniors play a musical instrument?
$\textbf{(A) } 66 \qquad\textbf{(B) } 154 \qquad\textbf{(C) } 186 \qquad\textbf{(D) } 220 \qquad\textbf{(E) } 266$ | \textbf{(B)}154 | 1/8 |
If two lines \( l \) and \( m \) have equations \( y = -2x + 8 \), and \( y = -3x + 9 \), what is the probability that a point randomly selected in the 1st quadrant and below \( l \) will fall between \( l \) and \( m \)? Express your answer as a decimal to the nearest hundredth. | 0.16 | 2/8 |
Julio has two cylindrical candles with different heights and diameters. The two candles burn wax at the same uniform rate. The first candle lasts 6 hours, while the second candle lasts 8 hours. He lights both candles at the same time and three hours later both candles are the same height. What is the ratio of their original heights? | 5:4 | 1/8 |
Find \( \cot^{-1}(1) + \cot^{-1}(3) + \ldots + \cot^{-1}(n^2 + n + 1) + \ldots \), where \( \cot^{-1}(m) \) is taken to be the value in the range \( (0, \frac{\pi}{2}] \). | \frac{\pi}{2} | 5/8 |
Given the set \( A = \{0, 1, 2, \cdots, 9\} \), and a collection of non-empty subsets \( B_1, B_2, \cdots, B_k \) of \( A \) such that for \( i \neq j \), \( B_i \cap B_j \) has at most two elements, determine the maximum value of \( k \). | 175 | 3/8 |
Evaluate $\sqrt[3]{1+27} \cdot \sqrt[3]{1+\sqrt[3]{27}}$. | \sqrt[3]{112} | 1/8 |
In certain cells of a rectangular board of size 101 by 99, there is a turtle in each cell. Every minute, each turtle simultaneously crawls to an adjacent cell of the board, sharing a side with the one they are currently in. Each subsequent move is made perpendicular to the previous one: if the previous move was horizontal - left or right, then the next one will be vertical - up or down, and vice versa. What is the maximum number of turtles that can move indefinitely on the board such that at each moment no cell contains more than one turtle? | 9800 | 1/8 |
A laptop is originally priced at $\$1200$. It is on sale for $15\%$ off. John applies two additional coupons: one gives $10\%$ off the discounted price, and another gives $5\%$ off the subsequent price. What single percent discount would give the same final price as these three successive discounts? | 27.325\% | 4/8 |
The vertices of $\triangle ABC$ are $A = (0,0)\,$, $B = (0,420)\,$, and $C = (560,0)\,$. The six faces of a die are labeled with two $A\,$'s, two $B\,$'s, and two $C\,$'s. Point $P_1 = (k,m)\,$ is chosen in the interior of $\triangle ABC$, and points $P_2\,$, $P_3\,$, $P_4, \dots$ are generated by rolling the die repeatedly and applying the rule: If the die shows label $L\,$, where $L \in \{A, B, C\}$, and $P_n\,$ is the most recently obtained point, then $P_{n + 1}^{}$ is the midpoint of $\overline{P_n L}$. Given that $P_7 = (14,92)\,$, what is $k + m\,$? | 344 | 6/8 |
Consider the integer\[N = 9 + 99 + 999 + 9999 + \cdots + \underbrace{99\ldots 99}_\text{321 digits}.\]Find the sum of the digits of $N$.
| 342 | 3/8 |
Determine the area of the triangle bounded by the axes and the curve $y = (x-5)^2 (x+3)$. | 300 | 5/8 |
There are $n$ teams that need to hold a double round-robin tournament (each pair of teams competes twice, with each team having one match at their home field). Each team can play multiple away games in a week (from Sunday to Saturday). However, if a team has a home game in a particular week, they cannot have any away games scheduled in that week. Determine the maximum value of $n$ if all games can be completed within 4 weeks.
(Note: A match held at team A's field between teams A and B is considered a home game for team A and an away game for team B.) | 6 | 1/8 |
The curve \( x^{2} + 4(y - a)^{2} = 4 \) intersects the curve \( x^{2} = 4y \). Find the range of values for \( a \). | [-1,\frac{5}{4}) | 1/8 |
A circle $ \Gamma$ is inscribed in a quadrilateral $ ABCD$ . If $ \angle A\equal{}\angle B\equal{}120^{\circ}, \angle D\equal{}90^{\circ}$ and $ BC\equal{}1$ , find, with proof, the length of $ AD$ . | \frac{\sqrt{3}-1}{2} | 5/8 |
Joe's quiz scores were 88, 92, 95, 81, and 90, and then he took one more quiz and scored 87. What was his mean score after all six quizzes? | 88.83 | 6/8 |
Given a constant function on the interval $(0,1)$, $f(x)$, which satisfies: when $x \notin \mathbf{Q}$, $f(x)=0$; and when $x=\frac{p}{q}$ (with $p, q$ being integers, $(p, q)=1, 0<p<q$), $f(x)=\frac{p+1}{q}$. Determine the maximum value of $f(x)$ on the interval $\left(\frac{7}{8}, \frac{8}{9}\right)$. | $\frac{16}{17}$ | 4/8 |
Medians $\overline{DP}$ and $\overline{EQ}$ of $\triangle DEF$ intersect at an angle of $60^\circ$. If $DP = 21$ and $EQ = 27$, determine the length of side $DE$. | 2\sqrt{67} | 6/8 |
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