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Inside triangle \( ABC \) with angles \(\angle A = 50^\circ\), \(\angle B = 60^\circ\), \(\angle C = 70^\circ\), a point \( M \) is chosen such that \(\angle AMB = 110^\circ\) and \(\angle BMC = 130^\circ\). Find \(\angle MBC\). | 20 | 5/8 |
There are 2011 positive numbers with both their sum and the sum of their reciprocals equal to 2012. Let $x$ be one of these numbers. Find the maximum value of $x + \frac{1}{x}.$ | \frac{8045}{2012} | 2/8 |
Consider all natural numbers whose decimal representations do not contain the digit zero. Prove that the sum of the reciprocals of any number of these numbers does not exceed a certain constant \( C \). | 90 | 3/8 |
Four students take an exam. Three of their scores are $70, 80,$ and $90$. If the average of their four scores is $70$, then what is the remaining score?
$\textbf{(A) }40\qquad\textbf{(B) }50\qquad\textbf{(C) }55\qquad\textbf{(D) }60\qquad \textbf{(E) }70$ | \textbf{(A)}\40 | 1/8 |
In the tetrahedron \( A B C D \),
$$
\begin{array}{l}
AB=1, BC=2\sqrt{6}, CD=5, \\
DA=7, AC=5, BD=7.
\end{array}
$$
Find its volume. | \frac{\sqrt{66}}{2} | 3/8 |
In a school fundraising campaign, $25\%$ of the money donated came from parents. The rest of the money was donated by teachers and students. The ratio of the amount of money donated by teachers to the amount donated by students was $2:3$. What is the ratio of the amount of money donated by parents to the amount donated by students? | 5:9 | 6/8 |
Triangle \(ABC\) has a right angle at \(B\), with \(AB = 3\) and \(BC = 4\). If \(D\) and \(E\) are points on \(AC\) and \(BC\), respectively, such that \(CD = DE = \frac{5}{3}\), find the perimeter of quadrilateral \(ABED\). | 28/3 | 7/8 |
Given an ellipse $\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1 (a > b > 0)$ with its left and right foci being $F_1$ and $F_2$ respectively, and its eccentricity $e = \dfrac{\sqrt{2}}{2}$, the length of the minor axis is $2$.
$(1)$ Find the equation of the ellipse;
$(2)$ Point $A$ is a moving point on the ellipse (not the endpoints of the major axis), the extension line of $AF_2$ intersects the ellipse at point $B$, and the extension line of $AO$ intersects the ellipse at point $C$. Find the maximum value of the area of $\triangle ABC$. | \sqrt{2} | 1/8 |
A regular 201-sided polygon is inscribed inside a circle with center $C$. Triangles are drawn by connecting any three of the 201 vertices of the polygon. How many of these triangles have the point $C$ lying inside the triangle? | 338350 | 2/8 |
Each square of a $(2^n-1) \times (2^n-1)$ board contains either $1$ or $-1$ . Such an arrangement is called *successful* if each number is the product of its neighbors. Find the number of successful arrangements. | 1 | 5/8 |
Given a square grid of size $2023 \times 2023$, with each cell colored in one of $n$ colors. It is known that for any six cells of the same color located in one row, there are no cells of the same color above the leftmost of these six cells and below the rightmost of these six cells. What is the smallest $n$ for which this configuration is possible? | 338 | 1/8 |
Inside the circle $\omega$ are two intersecting circles $\omega_{1}$ and $\omega_{2}$, which intersect at points $K$ and $L$, and are tangent to the circle $\omega$ at points $M$ and $N$. It turns out that the points $K$, $M$, and $N$ are collinear. Find the radius of the circle $\omega$, given that the radii of the circles $\omega_{1}$ and $\omega_{2}$ are 3 and 5, respectively. | 8 | 4/8 |
Let \( A = (2, 0) \) and \( B = (8, 6) \). Let \( P \) be a point on the circle \( x^2 + y^2 = 8x \). Find the smallest possible value of \( AP + BP \). | 6\sqrt{2} | 1/8 |
The internal angle bisector of a triangle from vertex $A$ is parallel to the line $OM$, but not identical to it, where $O$ is the circumcenter and $M$ is the orthocenter. What is the angle at vertex $A$? | 120 | 3/8 |
If \( x^{*} y = xy + 1 \) and \( a = (2 * 4) * 2 \), find \( a \).
If the \( b \)th prime number is \( a \), find \( b \).
If \( c = \left(1 - \frac{1}{2}\right)\left(1 - \frac{1}{3}\right)\left(1 - \frac{1}{4}\right) \cdots \left(1 - \frac{1}{50}\right) \), find \( c \) in the simplest fractional form.
If \( d \) is the area of a square inscribed in a circle of radius 10, find \( d \). | 200 | 6/8 |
Simplify the expression \(\frac{\operatorname{tg}\left(\frac{5}{4} \pi - 4 \alpha\right) \sin^{2}\left(\frac{5}{4} \pi + 4 \alpha\right)}{1 - 2 \cos^{2} 4 \alpha}\). | -\frac{1}{2} | 7/8 |
Given an isosceles right triangle \(ABC\) with a right angle at \(A\), there is a square \(KLMN\) positioned such that points \(K, L, N\) lie on the sides \(AB, BC, AC\) respectively, and point \(M\) is located inside the triangle \(ABC\).
Find the length of segment \(AC\), given that \(AK = 7\) and \(AN = 3\). | 17 | 7/8 |
If for any three distinct numbers $a$, $b$, and $c$ we define $f(a,b,c)=\frac{c+a}{c-b}$, then $f(1,-2,-3)$ is
$\textbf {(A) } -2 \qquad \textbf {(B) } -\frac{2}{5} \qquad \textbf {(C) } -\frac{1}{4} \qquad \textbf {(D) } \frac{2}{5} \qquad \textbf {(E) } 2$ | \textbf{E} | 1/8 |
Find the smallest natural number, which divides $2^{n}+15$ for some natural number $n$ and can be expressed in the form $3x^2-4xy+3y^2$ for some integers $x$ and $y$ . | 23 | 5/8 |
In a table tennis match between player A and player B, the "best of five sets" rule is applied, which means the first player to win three sets wins the match. If the probability of player A winning a set is $\dfrac{2}{3}$, and the probability of player B winning a set is $\dfrac{1}{3}$, then the probability of the match ending with player A winning three sets and losing one set is ______. | \dfrac{8}{27} | 7/8 |
Let vectors $\overrightarrow{a}$, $\overrightarrow{b}$, $\overrightarrow{c}$ satisfy $|\overrightarrow{a}|=|\overrightarrow{b}|=1$, $\overrightarrow{a}\cdot \overrightarrow{b}= \frac{1}{2}$, and $(\overrightarrow{a}- \overrightarrow{c})\cdot(\overrightarrow{b}- \overrightarrow{c})=0$. Then, calculate the maximum value of $|\overrightarrow{c}|$. | \frac{\sqrt{3}+1}{2} | 3/8 |
Let $\{\epsilon_n\}^\infty_{n=1}$ be a sequence of positive reals with $\lim\limits_{n\rightarrow+\infty}\epsilon_n = 0$ . Find \[ \lim\limits_{n\rightarrow\infty}\dfrac{1}{n}\sum\limits^{n}_{k=1}\ln\left(\dfrac{k}{n}+\epsilon_n\right) \] | -1 | 4/8 |
Alice cycled 240 miles in 4 hours, 30 minutes. Then, she cycled another 300 miles in 5 hours, 15 minutes. What was Alice's average speed in miles per hour for her entire journey? | 55.38 | 2/8 |
2011 warehouses are connected by roads such that each warehouse can be reached from any other, possibly by traveling through several roads. Each warehouse contains $x_{1}, \ldots, x_{2011}$ kg of cement, respectively. In one trip, any amount of cement can be transported from one warehouse to another along a connecting road. Ultimately, the plan is to have $y_{1}, \ldots, y_{2011}$ kg of cement at the warehouses, respectively, with the condition that $x_{1} + x_{2} + \ldots + x_{2011} = y_{1} + y_{2} + \ldots + y_{2011}$. What is the minimum number of trips required to execute this plan for any values of $x_{i}$ and $y_{i}$ and any road scheme? | 2010 | 7/8 |
Solve the inequality in integers:
$$
\frac{1}{\sqrt{x-2y+z+1}} + \frac{2}{\sqrt{2x-y+3z-1}} + \frac{3}{\sqrt{3y-3x-4z+3}} > x^2 - 4x + 3
$$ | (3,1,-1) | 1/8 |
On the New Year's table, there are 4 glasses in a row: the first and third contain orange juice, and the second and fourth are empty. While waiting for guests, Valya absent-mindedly and randomly pours juice from one glass to another. Each time, she can take a full glass and pour all its contents into one of the two empty glasses.
Find the expected number of pourings for the first time when the situation is reversed: the first and third glasses are empty, and the second and fourth glasses are full. | 6 | 3/8 |
We regularly transport goods from city $A$ to city $B$, which is $183 \mathrm{~km}$ away. City $A$ is $33 \mathrm{~km}$ from the river, while city $B$ is built on the riverbank. The cost of transportation per kilometer is half as much on the river as on land. Where should we build the road to minimize transportation costs? | 11\sqrt{3} | 7/8 |
The positive integer \( N \) has six digits in increasing order. For example, 124689 is such a number. However, unlike 124689, three of the digits of \( N \) are 3, 4, and 5, and \( N \) is a multiple of 6. How many possible six-digit integers \( N \) are there? | 3 | 2/8 |
The Evil League of Evil is plotting to poison the city's water supply. They plan to set out from their headquarters at $(5, 1)$ and put poison in two pipes, one along the line $y=x$ and one along the line $x=7$ . However, they need to get the job done quickly before Captain Hammer catches them. What's the shortest distance they can travel to visit both pipes and then return to their headquarters? | 4\sqrt{5} | 1/8 |
Each face of a regular icosahedron has a non-negative integer written on it such that the sum of all 20 numbers is 39. Prove that there exist two different faces sharing a common vertex with the same number written on them.
An icosahedron is a convex polyhedron with 12 vertices, 20 faces, and 30 edges, where all faces are congruent equilateral triangles. Five edges meet at each vertex of the icosahedron. | 2 | 3/8 |
The sides of a triangle are 1 and 2, and the angle between them is $60^{\circ}$. A circle is drawn through the center of the inscribed circle of this triangle and the ends of the third side. Find its radius. | 1 | 7/8 |
Let \( m \) be a positive real number such that for any positive real numbers \( x \) and \( y \), the three line segments with lengths \( x + y \), \( \sqrt{x^2 + y^2 + xy} \), and \( m \sqrt{xy} \) can form the three sides of a triangle. Determine the range of values for \( m \). | (2-\sqrt{3},2+\sqrt{3}) | 3/8 |
What is the area enclosed by the geoboard quadrilateral below?
[asy] unitsize(3mm); defaultpen(linewidth(.8pt)); dotfactor=2; for(int a=0; a<=10; ++a) for(int b=0; b<=10; ++b) { dot((a,b)); }; draw((4,0)--(0,5)--(3,4)--(10,10)--cycle); [/asy]
$\textbf{(A)}\ 15\qquad \textbf{(B)}\ 18\frac12 \qquad \textbf{(C)}\ 22\frac12 \qquad \textbf{(D)}\ 27 \qquad \textbf{(E)}\ 41$ | \textbf{(C)}\22\frac{1}{2} | 1/8 |
Given the vertices of a regular 120-gon \( A_{1}, A_{2}, A_{3}, \ldots, A_{120} \), in how many ways can three vertices be selected to form an obtuse triangle? | 205320 | 1/8 |
Given that \(0\) is the circumcenter of \(\triangle ABC\) and it satisfies \(\overrightarrow{CO} = t \cdot \overrightarrow{CA} + \left(\frac{1}{2} - \frac{3t}{4}\right) \cdot \overrightarrow{CB}\) for \(t \in \mathbb{R} \setminus \{0\}\), and that \(|AB| = 3\), find the maximum area of \(\triangle ABC\). | 9 | 1/8 |
Find all functions \( f: \mathbf{R} \rightarrow \mathbf{R} \) such that for all \( x, y, z \in \mathbf{R} \), the inequality \( f(x+y) + f(y+z) + f(z+x) \geq 3 f(x + 2y + 3z) \) is satisfied. | f(x)= | 5/8 |
A sequence $(a_1, a_2,\ldots, a_k)$ consisting of pairwise distinct squares of an $n\times n$ chessboard is called a *cycle* if $k\geq 4$ and squares $a_i$ and $a_{i+1}$ have a common side for all $i=1,2,\ldots, k$ , where $a_{k+1}=a_1$ . Subset $X$ of this chessboard's squares is *mischievous* if each cycle on it contains at least one square in $X$ .
Determine all real numbers $C$ with the following property: for each integer $n\geq 2$ , on an $n\times n$ chessboard there exists a mischievous subset consisting of at most $Cn^2$ squares. | \frac{1}{3} | 1/8 |
Solve the inequality \(\operatorname{tg} \arccos x \leqslant \sin \operatorname{arctg} x\). | [-\frac{1}{\sqrt[4]{2}},0)\cup[\frac{1}{\sqrt[4]{2}},1] | 5/8 |
Given $|m|=4$, $|n|=3$.
(1) When $m$ and $n$ have the same sign, find the value of $m-n$.
(2) When $m$ and $n$ have opposite signs, find the value of $m+n$. | -1 | 7/8 |
In the expansion of $(a + b)^n$ there are $n + 1$ dissimilar terms. The number of dissimilar terms in the expansion of $(a + b + c)^{10}$ is:
$\textbf{(A)}\ 11\qquad \textbf{(B)}\ 33\qquad \textbf{(C)}\ 55\qquad \textbf{(D)}\ 66\qquad \textbf{(E)}\ 132$ | \textbf{D} | 1/8 |
A rectangle is divided by two straight lines into four smaller rectangles with perimeters of 6, 10, and 12 (the rectangle with a perimeter of 10 shares sides with the rectangles with perimeters of 6 and 12). Find the perimeter of the fourth rectangle. | 8 | 3/8 |
Given $0\leqslant x\_0 < 1$, for all integers $n > 0$, let $x\_n= \begin{cases} 2x_{n-1}, & 2x_{n-1} < 1 \\ 2x_{n-1}-1, & 2x_{n-1} \geqslant 1 \end{cases}$. Find the number of $x\_0$ that makes $x\_0=x\_6$ true. | 64 | 1/8 |
Calculate the limit
$$
\lim _{x \rightarrow 0}\left(\frac{1+x^{2} 2^{x}}{1+x^{2} 5^{x}}\right)^{1 / \sin ^{3} x}
$$ | \frac{2}{5} | 6/8 |
Find all solutions to $aabb=n^4-6n^3$ , where $a$ and $b$ are non-zero digits, and $n$ is an integer. ( $a$ and $b$ are not necessarily distinct.) | 6655 | 7/8 |
Given that $a > 1$ and $b > 0$, and $a + 2b = 2$, find the minimum value of $\frac{2}{a - 1} + \frac{a}{b}$. | 4(1 + \sqrt{2}) | 1/8 |
In the quadrilateral \(ABCD\), the lengths of the sides \(BC\) and \(CD\) are 2 and 6, respectively. The points of intersection of the medians of triangles \(ABC\), \(BCD\), and \(ACD\) form an equilateral triangle. What is the maximum possible area of quadrilateral \(ABCD\)? If necessary, round the answer to the nearest 0.01. | 29.32 | 1/8 |
There exists a scalar $c$ so that
\[\mathbf{i} \times (\mathbf{v} \times \mathbf{i}) + \mathbf{j} \times (\mathbf{v} \times \mathbf{j}) + \mathbf{k} \times (\mathbf{v} \times \mathbf{k}) = c \mathbf{v}\]for all vectors $\mathbf{v}.$ Find $c.$ | 2 | 3/8 |
There are 100 people in a room with ages $1,2, \ldots, 100$. A pair of people is called cute if each of them is at least seven years older than half the age of the other person in the pair. At most how many pairwise disjoint cute pairs can be formed in this room? | 43 | 2/8 |
Given a positive integer $n \geq 2$, and a set $S$ containing $n$ pairwise distinct real numbers $a_{1}, a_{2}, \ldots, a_{n}$, define $k(S)$ as the number of distinct numbers of the form $a_{i} + 2^{j}$ for $i, j = 1, 2, \ldots, n$. Find the minimum possible value of $k(S)$ for all possible sets $S$. | \frac{n(n+1)}{2} | 1/8 |
Given the parabola C: y² = 3x with focus F, and a line l with slope $\frac{3}{2}$ intersecting C at points A and B, and the x-axis at point P.
(1) If |AF| + |BF| = 4, find the equation of line l;
(2) If $\overrightarrow{AP}$ = 3$\overrightarrow{PB}$, find |AB|. | \frac{4\sqrt{13}}{3} | 6/8 |
Find the minimum of \( |\sin x + \cos x + \tan x + \cot x + \sec x + \csc x| \) for real \( x \). | 2\sqrt{2}-1 | 7/8 |
For all \( m \) and \( n \) satisfying \( 1 \leq n \leq m \leq 5 \), the polar equation
$$
\rho = \frac{1}{1 - C_{m}^{n} \cos \theta}
$$
represents how many different hyperbolas? | 10 | 1/8 |
Let \(a, b, c\) be numbers such that \(0 < c \leq b \leq a\). Prove that
$$
\frac{a^{2}-b^{2}}{c}+\frac{c^{2}-b^{2}}{a}+\frac{a^{2}-c^{2}}{b} \geq 3a-4b+c
$$ | \frac{^2-b^2}{}+\frac{^2-b^2}{}+\frac{^2-^2}{b}\ge3a-4b+ | 1/8 |
Let the sequence of non-negative integers $\left\{a_{n}\right\}$ satisfy:
$$
a_{n} \leqslant n \quad (n \geqslant 1), \quad \text{and} \quad \sum_{k=1}^{n-1} \cos \frac{\pi a_{k}}{n} = 0 \quad (n \geqslant 2).
$$
Find all possible values of $a_{2021}$. | 2021 | 5/8 |
Given that \(\theta\) is an angle in the third quadrant, and \(\sin^{4}\theta + \cos^{4}\theta = \frac{5}{9}\), determine the value of \(\sin 2\theta\). | -\frac{2\sqrt{2}}{3} | 1/8 |
Given that a floor is tiled in a similar pattern with a $4 \times 4$ unit repeated pattern and each of the four corners looks like the scaled down version of the original, determine the fraction of the tiled floor made up of darker tiles, assuming symmetry and pattern are preserved. | \frac{1}{2} | 1/8 |
Consider the following algorithm.
Step 0. Set \( n = m \).
Step 1. If \( n \) is even, divide \( n \) by two. If \( n \) is odd, increase \( n \) by one.
Step 2. If \( n > 1 \), go to Step 1. If \( n = 1 \), end the algorithm. How many natural numbers \( m \) exist such that Step 1 will be performed exactly 15 times when this algorithm is executed? | 610 | 1/8 |
Grandpa is twice as strong as Grandma, Grandma is three times as strong as Granddaughter, Granddaughter is four times as strong as Doggie, Doggie is five times as strong as Cat, and Cat is six times as strong as Mouse. Grandpa, Grandma, Granddaughter, Doggie, and Cat together with Mouse can pull up the Turnip, but without Mouse they can't. How many Mice are needed so that they can pull up the Turnip on their own? | 1237 | 7/8 |
Let $S$ be the set of integers $n > 1$ for which $\tfrac1n = 0.d_1d_2d_3d_4\ldots$, an infinite decimal that has the property that $d_i = d_{i+12}$ for all positive integers $i$. Given that $9901$ is prime, how many positive integers are in $S$? (The $d_i$ are digits.)
| 255 | 4/8 |
On a plane, 6 lines intersect pairwise, but only three pass through the same point. Find the number of non-overlapping line segments intercepted. | 21 | 1/8 |
Suppose that one of every 500 people in a certain population has a particular disease, which displays no symptoms. A blood test is available for screening for this disease. For a person who has this disease, the test always turns out positive. For a person who does not have the disease, however, there is a $2\%$ false positive rate--in other words, for such people, $98\%$ of the time the test will turn out negative, but $2\%$ of the time the test will turn out positive and will incorrectly indicate that the person has the disease. Let $p$ be the probability that a person who is chosen at random from this population and gets a positive test result actually has the disease. Which of the following is closest to $p$?
$\textbf{(A)}\ \frac{1}{98}\qquad\textbf{(B)}\ \frac{1}{9}\qquad\textbf{(C)}\ \frac{1}{11}\qquad\textbf{(D)}\ \frac{49}{99}\qquad\textbf{(E)}\ \frac{98}{99}$ | \textbf{(C)}\\frac{1}{11} | 1/8 |
Given a sequence $\{a_{n}\}$ with the sum of the first $n$ terms denoted as $S_{n}$, $a_{1}=3$, $\frac{{S}_{n+1}}{{S}_{n}}=\frac{{3}^{n+1}-1}{{3}^{n}-1}$, $n\in N^{*}$.
$(1)$ Find $S_{2}$, $S_{3}$, and the general formula for $\{a_{n}\}$;
$(2)$ Let $b_n=\frac{a_{n+1}}{(a_n-1)(a_{n+1}-1)}$, the sum of the first $n$ terms of the sequence $\{b_{n}\}$ is denoted as $T_{n}$. If $T_{n}\leqslant \lambda (a_{n}-1)$ holds for all $n\in N^{*}$, find the minimum value of $\lambda$. | \frac{9}{32} | 7/8 |
There are a total of 10040 numbers written on a blackboard, including 2006 of the number 1, 2007 of the number 2, 2008 of the number 3, 2009 of the number 4, and 2010 of the number 5. Each operation consists of erasing 4 different numbers and writing the 5th number (for example, erasing one 1, one 2, one 3, and one 4, and writing one 5; or erasing one 2, one 3, one 4, and one 5, and writing one 1, etc.). If after a finite number of operations, there are exactly two numbers left on the blackboard, what is the product of these two numbers?
| 8 | 1/8 |
Let $\mathrm{P}$ be a point on the side $\mathrm{BC}$ of a triangle $\mathrm{ABC}$. The parallel through $\mathrm{P}$ to $\mathrm{AB}$ intersects side $\mathrm{AC}$ at point $\mathrm{Q}$, and the parallel through $\mathrm{P}$ to $\mathrm{AC}$ intersects side $\mathrm{AB}$ at point $\mathrm{R}$. The ratio of the areas of triangles $\mathrm{RBP}$ and $\mathrm{QPC}$ is $\mathrm{k}^{2}$.
Determine the ratio of the areas of triangles $\mathrm{ARQ}$ and $\mathrm{ABC}$. | \frac{k}{(1+k)^2} | 3/8 |
The integers from 1 to \( k \) are concatenated to form the integer \( N = 123456789101112 \ldots \). Determine the smallest integer value of \( k > 2019 \) such that \( N \) is divisible by 9. | 2024 | 3/8 |
Solve the following equation in the set of real numbers:
\[ 8^{x}+27^{x}+2 \cdot 30^{x}+54^{x}+60^{x}=12^{x}+18^{x}+20^{x}+24^{x}+45^{x}+90^{x} \] | -1,0,1 | 1/8 |
Determine the exact value of the series
\[\frac{1}{5 + 1} + \frac{2}{5^2 + 1} + \frac{4}{5^4 + 1} + \frac{8}{5^8 + 1} + \frac{16}{5^{16} + 1} + \dotsb.\] | \frac{1}{4} | 6/8 |
Line $\ell$ passes through $A$ and into the interior of the equilateral triangle $ABC$ . $D$ and $E$ are the orthogonal projections of $B$ and $C$ onto $\ell$ respectively. If $DE=1$ and $2BD=CE$ , then the area of $ABC$ can be expressed as $m\sqrt n$ , where $m$ and $n$ are positive integers and $n$ is not divisible by the square of any prime. Determine $m+n$ .
[asy]
import olympiad;
size(250);
defaultpen(linewidth(0.7)+fontsize(11pt));
real r = 31, t = -10;
pair A = origin, B = dir(r-60), C = dir(r);
pair X = -0.8 * dir(t), Y = 2 * dir(t);
pair D = foot(B,X,Y), E = foot(C,X,Y);
draw(A--B--C--A^^X--Y^^B--D^^C--E);
label(" $A$ ",A,S);
label(" $B$ ",B,S);
label(" $C$ ",C,N);
label(" $D$ ",D,dir(B--D));
label(" $E$ ",E,dir(C--E));
[/asy] | 10 | 5/8 |
Given the function $f(x) = x^2 - (2t + 1)x + t \ln x$ where $t \in \mathbb{R}$,
(1) If $t = 1$, find the extreme values of $f(x)$.
(2) Let $g(x) = (1 - t)x$, and suppose there exists an $x_0 \in [1, e]$ such that $f(x_0) \geq g(x_0)$ holds. Find the maximum value of the real number $t$. | \frac{e(e - 2)}{e - 1} | 2/8 |
A point is randomly dropped on the interval $[5 ; 7]$ and let $k$ be the resulting value. Find the probability that the roots of the equation $\left(k^{2}-3 k-4\right) x^{2}+(3 k-7) x+2=0$ satisfy the condition $x_{1} \leq 2 x_{2}$. | 1/3 | 4/8 |
Let ABC be an acute, non-isosceles triangle with orthocenter H, with M being the midpoint of side AB, and w being the angle bisector of angle ACB. Let S be the intersection point of the perpendicular bisector of side AB with w, and F be the foot of the perpendicular dropped from H to w.
Prove that segments MS and MF are of equal length.
(Karl Czakler) | MS=MF | 1/8 |
What is the maximum number of non-intersecting diagonals that can be drawn in a convex $n$-gon (diagonals that share a common vertex are allowed)? | n-3 | 7/8 |
Veronica put on five rings: one on her little finger, one on her middle finger, and three on her ring finger. In how many different orders can she take them all off one by one? | 20 | 6/8 |
Given \( x \in (0,1) \) and \( \frac{1}{x} \notin \mathbf{Z} \), define
\[
a_{n}=\frac{n x}{(1-x)(1-2 x) \cdots(1-n x)} \quad \text{for} \quad n=1,2, \cdots
\]
A number \( x \) is called a "good number" if and only if it makes the sequence \(\{a_{n}\}\) satisfy
\[
a_{1}+a_{2}+\cdots+a_{10} > -1 \quad \text{and} \quad a_{1} a_{2} \cdots a_{10} > 0
\]
Find the sum of the lengths of all intervals on the number line that represent all such "good numbers". | 61/210 | 1/8 |
Prove that if $m$ and $n$ are integers such that
$$
(\sqrt{2}-1)^{n} \geq \sqrt{m}-\sqrt{m-1}
$$
then $m \geq n^{2}$. | \gen^{2} | 3/8 |
Let $P(x) = 3\sqrt[3]{x}$, and $Q(x) = x^3$. Determine $P(Q(P(Q(P(Q(4))))))$. | 108 | 7/8 |
Given the function $f(x)=x^{3}- \frac {3}{2}x^{2}+ \frac {3}{4}x+ \frac {1}{8}$, find the value of $\sum\limits_{k=1}^{2016}f( \frac {k}{2017})$. | 504 | 6/8 |
Let $B$ be the set of all binary integers that can be written using exactly $5$ zeros and $8$ ones where leading zeros are allowed. If all possible subtractions are performed in which one element of $B$ is subtracted from another, find the number of times the answer $1$ is obtained. | 330 | 6/8 |
For a positive integer $n$ we denote by $u(n)$ the largest prime number less than or equal to $n$ , and with $v(n)$ the smallest prime number larger than $n$ . Prove that \[ \frac 1 {u(2)v(2)} + \frac 1{u(3)v(3)} + \cdots + \frac 1{ u(2010)v(2010)} = \frac 12 - \frac 1{2011}. \] | \frac{1}{2}-\frac{1}{2011} | 7/8 |
Given the ellipse \(\frac{x^{2}}{9} + \frac{y^{2}}{5} = 1\), the right focus is \(F\), and \(P\) is a point on the ellipse. Point \(A\) is at \((0, 2 \sqrt{3})\). When the perimeter of \(\triangle APF\) is maximized, what is the area of \(\triangle APF\)? | \frac{21\sqrt{3}}{4} | 1/8 |
Xiaopang arranges the 50 integers from 1 to 50 in ascending order without any spaces in between. Then, he inserts a "+" sign between each pair of adjacent digits, resulting in an addition expression: \(1+2+3+4+5+6+7+8+9+1+0+1+1+\cdots+4+9+5+0\). Please calculate the sum of this addition expression. The result is ________. | 330 | 6/8 |
In triangle \(ABC\), angle \(B\) is a right angle, and the medians \(AD\) and \(BE\) are mutually perpendicular. Find angle \(C\). | \arctan(\frac{1}{\sqrt{2}}) | 3/8 |
Two spheres with equal mass and charge are initially at a distance of \( l \) from each other when they are released without an initial velocity. After \( t \) seconds, the distance between them doubles. How much time will it take for the distance between the spheres to double if they are initially released from a distance of \( 3l \)? Air resistance can be neglected. | 3\sqrt{3} | 1/8 |
The letter T is formed by placing a $2\:\text{inch} \times 6\:\text{inch}$ rectangle vertically and a $2\:\text{inch} \times 4\:\text{inch}$ rectangle horizontally across the top center of the vertical rectangle. What is the perimeter of the T, in inches? | 24 | 4/8 |
Let $p$ be an odd prime number, and let $\mathbb{F}_p$ denote the field of integers modulo $p$. Let $\mathbb{F}_p[x]$ be the ring of polynomials over $\mathbb{F}_p$, and let $q(x) \in \mathbb{F}_p[x]$ be given by \[ q(x) = \sum_{k=1}^{p-1} a_k x^k, \] where \[ a_k = k^{(p-1)/2} \mod{p}. \] Find the greatest nonnegative integer $n$ such that $(x-1)^n$ divides $q(x)$ in $\mathbb{F}_p[x]$. | \frac{p-1}{2} | 1/8 |
Consider three collinear points \(B\), \(C\), and \(D\) such that \(C\) is between \(B\) and \(D\). Let \(A\) be a point not on the line \(BD\) such that \(AB = AC = CD\).
(a) If \(\angle BAC = 36^\circ\), then verify that
\[ \frac{1}{CD} - \frac{1}{BD} = \frac{1}{CD + BD} \]
(b) Now, suppose that
\[ \frac{1}{CD} - \frac{1}{BD} = \frac{1}{CD + BD} \]
Verify that \(\angle BAC = 36^\circ\). | 36 | 6/8 |
Five numbers form an increasing arithmetic progression. The sum of their cubes is zero, and the sum of their squares is 70. Find the smallest of these numbers. | -2\sqrt{7} | 7/8 |
Compute the number of nonempty subsets $S \subseteq\{-10,-9,-8, . . . , 8, 9, 10\}$ that satisfy $$ |S| +\ min(S) \cdot \max (S) = 0. $$ | 335 | 2/8 |
Let \( S(n) \) be the sum of the digits in the decimal representation of the number \( n \). Find \( S\left(S\left(S\left(S\left(2017^{2017}\right)\right)\right)\right) \). | 1 | 6/8 |
If $x=1$ is an extremum point of the function $f(x)=(x^{2}+ax-1)e^{x-1}$, determine the maximum value of $f(x)$. | 5e^{-3} | 1/8 |
Let $F_{1}$ and $F_{2}$ be the two foci of an ellipse $C$. Let $AB$ be a chord of the ellipse passing through the point $F_{2}$. In the triangle $\triangle F_{1}AB$, the lengths are given as follows:
$$
|F_{1}A| = 3, \; |AB| = 4, \; |BF_{1}| = 5.
$$
Find $\tan \angle F_{2}F_{1}B$. | \frac{1}{7} | 6/8 |
The cube below has sides of length 5 feet. If a cylindrical section of radius 1 foot is removed from the solid at an angle of $45^\circ$ to the top face, what is the total remaining volume of the cube? Express your answer in cubic feet in terms of $\pi$. | 125 - 5\sqrt{2}\pi | 3/8 |
A school selects 4 teachers from 8 to teach in 4 remote areas at the same time (one person per area), where teacher A and teacher B cannot go together, and teacher A and teacher C can only go together or not go at all. The total number of different dispatch plans is ___. | 600 | 2/8 |
Elizabetta wants to write the integers 1 to 9 in the regions of the shape shown, with one integer in each region. She wants the product of the integers in any two regions that have a common edge to be not more than 15. In how many ways can she do this? | 16 | 1/8 |
A boy, standing on a horizontal surface of the ground at a distance $L=5$ m from a vertical wall of a house, kicked a ball lying in front of him on the ground. The ball flew at an angle $\alpha=45^{\circ}$ to the horizontal and, after an elastic collision with the wall, fell back to the same place where it initially lay.
1) Find the flight time of the ball.
2) At what height from the ground did the ball bounce off the wall?
Assume the acceleration due to gravity is $10 \mathrm{~m} / \mathrm{c}^{2}$. | 2.5 | 4/8 |
The circles OAB, OBC, and OCA have equal radius \( r \). Show that the circle ABC also has radius \( r \). | r | 6/8 |
The operation $*$ is defined by
\[a * b = \frac{a - b}{1 - ab}.\]Compute
\[1 * (2 * (3 * (\dotsb (999 * 1000) \dotsb))).\] | 1 | 3/8 |
Given \( x^{2} + y^{2} = 25 \), find the maximum value of the function \( f(x, y) = \sqrt{8 y - 6 x + 50} + \sqrt{8 y + 6 x + 50} \). | 6\sqrt{10} | 4/8 |
Given the sequence $\left\{a_{n}\right\}$ with its sum of the first $n$ terms $S_{n}$ satisfying $2 S_{n}-n a_{n}=n$ for $n \in \mathbf{N}^{*}$, and $a_{2}=3$:
1. Find the general term formula for the sequence $\left\{a_{n}\right\}$.
2. Let $b_{n}=\frac{1}{a_{n} \sqrt{a_{n+1}}+a_{n+1} \sqrt{a_{n}}}$ and $T_{n}$ be the sum of the first $n$ terms of the sequence $\left\{b_{n}\right\}$. Determine the smallest positive integer $n$ such that $T_{n}>\frac{9}{20}$. | 50 | 7/8 |
If $a$ and $b$ are nonzero real numbers such that $\left| a \right| \ne \left| b \right|$ , compute the value of the expression
\[
\left( \frac{b^2}{a^2} + \frac{a^2}{b^2} - 2 \right) \times
\left( \frac{a + b}{b - a} + \frac{b - a}{a + b} \right) \times
\left(
\frac{\frac{1}{a^2} + \frac{1}{b^2}}{\frac{1}{b^2} - \frac{1}{a^2}}
- \frac{\frac{1}{b^2} - \frac{1}{a^2}}{\frac{1}{a^2} + \frac{1}{b^2}}
\right).
\] | -8 | 7/8 |
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