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Given $n$ points on a plane $(n>3)$, where no three points are collinear, and any two points may or may not be connected by a line segment. Given that $m$ line segments are drawn, prove that there are at least $\frac{m}{3 n} (4 m - n^2)$ triangles. | \frac{}{3n}(4m-n^2) | 3/8 |
If $a$ and $b$ are positive integers that can each be written as a sum of two squares, then $a b$ is also a sum of two squares. Find the smallest positive integer $c$ such that $c=a b$, where $a=x^{3}+y^{3}$ and $b=x^{3}+y^{3}$ each have solutions in integers $(x, y)$, but $c=x^{3}+y^{3}$ does not. | 4 | 6/8 |
Convert the 2004 fractions \(\frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \cdots, \frac{1}{2005}\) into decimal notation. How many of these fractions result in a purely repeating decimal? | 801 | 1/8 |
How many ordered integer pairs $(x,y)$ ($0 \leq x,y < 31$) are there satisfying $(x^2-18)^2 \equiv y^2 \pmod{31}$? | 60 | 7/8 |
The coordinates of the vertices of isosceles trapezoid $ABCD$ are all integers, with $A=(20,100)$ and $D=(21,107)$. The trapezoid has no horizontal or vertical sides, and $\overline{AB}$ and $\overline{CD}$ are the only parallel sides. The sum of the absolute values of all possible slopes for $\overline{AB}$ is $m/n$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$. | 131 | 1/8 |
Five people are gathered in a meeting. Some pairs of people shakes hands. An ordered triple of people $(A,B,C)$ is a *trio* if one of the following is true:
- A shakes hands with B, and B shakes hands with C, or
- A doesn't shake hands with B, and B doesn't shake hands with C.
If we consider $(A,B,C)$ and $(C,B,A)$ as the same trio, find the minimum possible number of trios. | 10 | 6/8 |
Given a set $M$ of $n$ points on a plane, if every three points in $M$ form the vertices of an equilateral triangle, find the maximum value of $n$. | 3 | 7/8 |
Let \(ABC\) be an acute triangle with incenter \(I\) and circumcenter \(O\). Assume that \(\angle OIA = 90^\circ\). Given that \(AI = 97\) and \(BC = 144\), compute the area of \(\triangle ABC\). | 14040 | 1/8 |
At a physical education class, 27 seventh graders attended, some of whom brought one ball each. Occasionally during the class, a seventh grader would give their ball to another seventh grader who did not have a ball.
At the end of the class, \( N \) seventh graders said: "I received balls less often than I gave them away!" Find the maximum possible value of \( N \) given that nobody lied. | 13 | 2/8 |
Given the sequence $\{a_n\}$, $a_1=1$, $a_2=2$, and $a_{n+2}-a_{n}=1+(-1)^{n}$ $(n\in\mathbb{N}_{+})$, calculate the value of $S_{100}$. | 2600 | 7/8 |
The sequence \(\left\{a_{n}\right\}\) satisfies: \(a_{1}=1\), and for each \(n \in \mathbf{N}^{*}\), \(a_{n}\) and \(a_{n+1}\) are the two roots of the equation \(x^{2}+3nx+b_{n}=0\). Find \(\sum_{k=1}^{20} b_{k}\). | 6385 | 4/8 |
Given a convex hexagon, any two pairs of opposite sides have the following property: the distance between their midpoints is equal to half the sum of their lengths multiplied by $\frac{\sqrt{3}}{2}$. Prove that all the interior angles of the hexagon are equal. (A convex hexagon $A B C D E F$ has 3 pairs of opposite sides: $A B$ and $D E$, $B C$ and $E F$, $C D$ and $F A$). | 120 | 1/8 |
Six ants simultaneously stand on the six vertices of a regular octahedron, with each ant at a different vertex. Simultaneously and independently, each ant moves from its vertex to one of the four adjacent vertices, each with equal probability. What is the probability that no two ants arrive at the same vertex? | \frac{5}{256} | 1/8 |
Three cards, each with a positive integer written on it, are lying face-down on a table. Casey, Stacy, and Tracy are told that
(a) the numbers are all different,
(b) they sum to $13$, and
(c) they are in increasing order, left to right.
First, Casey looks at the number on the leftmost card and says, "I don't have enough information to determine the other two numbers." Then Tracy looks at the number on the rightmost card and says, "I don't have enough information to determine the other two numbers." Finally, Stacy looks at the number on the middle card and says, "I don't have enough information to determine the other two numbers." Assume that each person knows that the other two reason perfectly and hears their comments. What number is on the middle card? | 4 | 7/8 |
Let $A B C D$ be a parallelogram with $A B=8, A D=11$, and $\angle B A D=60^{\circ}$. Let $X$ be on segment $C D$ with $C X / X D=1 / 3$ and $Y$ be on segment $A D$ with $A Y / Y D=1 / 2$. Let $Z$ be on segment $A B$ such that $A X, B Y$, and $D Z$ are concurrent. Determine the area of triangle $X Y Z$. | \frac{19 \sqrt{3}}{2} | 7/8 |
If a number is selected at random from the set of all five-digit numbers in which the sum of the digits is equal to 43, what is the probability that this number will be divisible by 11? | \frac{1}{5} | 7/8 |
A shooter hits the following scores in five consecutive shots: 9.7, 9.9, 10.1, 10.2, 10.1. The variance of this set of data is __________. | 0.032 | 7/8 |
Solve the equations:<br/>$(1)x^{2}-4x-1=0$;<br/>$(2)\left(x+3\right)^{2}=x+3$. | -2 | 1/8 |
What is the smallest value of $k$ for which it is possible to mark $k$ cells on a $9 \times 9$ board such that any placement of a three-cell corner touches at least two marked cells? | 56 | 1/8 |
On a plane with a given rectangular Cartesian coordinate system, a square is drawn with vertices at the points \((0 ; 0)\),(0 ; 59)\,(59; 59)\, and \((59 ; 0)\). Find the number of ways to choose two grid nodes inside this square (excluding its boundary) so that at least one of these nodes lies on one of the lines \(y=x\) or \(y=59-x\), but both selected nodes do not lie on any line parallel to either of the coordinate axes. | 370330 | 2/8 |
Let $c > 1$ be a real number. A function $f: [0 ,1 ] \to R$ is called c-friendly if $f(0) = 0, f(1) = 1$ and $|f(x) -f(y)| \le c|x - y|$ for all the numbers $x ,y \in [0,1]$ . Find the maximum of the expression $|f(x) - f(y)|$ for all *c-friendly* functions $f$ and for all the numbers $x,y \in [0,1]$ . | \frac{1}{2} | 5/8 |
John needs to catch a train. The train arrives randomly some time between 2:00 and 3:00, waits for 20 minutes, and then leaves. If John also arrives randomly between 2:00 and 3:00, what is the probability that the train will be there when John arrives? | \frac{5}{18} | 7/8 |
March 9, 2014 is a Sunday. Based on this information, determine which day of the week occurs most frequently in the year 2014. | Wednesday | 2/8 |
Consider the system of inequalities $\left\{\begin{array}{l}x + y > 0, \\ x - y < 0\end{array}\right.$ which represents the region $D$ in the plane. A moving point $P$ within region $D$ satisfies the condition that the product of its distances to the lines $x + y = 0$ and $x - y = 0$ is equal to 2. Let the locus of point $P$ be curve $C$. A line $l$ passing through the point $F(2 \sqrt{2}, 0)$ intersects curve $C$ at points $A$ and $B$. If the circle with diameter $\overline{AB}$ is tangent to the $y$-axis, find the slope of line $l$. | -\sqrt{\sqrt{2}-1} | 1/8 |
A rectangle with sides of length $4$ and $2$ is rolled into the lateral surface of a cylinder. The volume of the cylinder is $\_\_\_\_\_\_\_\_.$ | \frac{4}{\pi} | 1/8 |
There are $9$ cards with the numbers $1, 2, 3, 4, 5, 6, 7, 8$ and $9$ . What is the largest number of these cards can be decomposed in a certain order in a row, so that in any two adjacent cards, one of the numbers is divided by the other?
| 8 | 3/8 |
For a real number $a$ and an integer $n(\geq 2)$ , define $$ S_n (a) = n^a \sum_{k=1}^{n-1} \frac{1}{k^{2019} (n-k)^{2019}} $$ Find every value of $a$ s.t. sequence $\{S_n(a)\}_{n\geq 2}$ converges to a positive real. | 2019 | 6/8 |
Given a triangle $\triangle ABC$ with sides $a$, $b$, $c$ opposite to angles $A$, $B$, $C$ respectively. If $a=2$, $A= \frac{\pi}{3}$, and $\frac{\sqrt{3}}{2} - \sin(B-C) = \sin 2B$, find the area of $\triangle ABC$. | \frac{2\sqrt{3}}{3} | 1/8 |
Let $ABC$ be a triangle where $M$ is the midpoint of $\overline{AC}$, and $\overline{CN}$ is the angle bisector of $\angle{ACB}$ with $N$ on $\overline{AB}$. Let $X$ be the intersection of the median $\overline{BM}$ and the bisector $\overline{CN}$. In addition $\triangle BXN$ is equilateral with $AC=2$. What is $BX^2$? | \frac{10-6\sqrt{2}}{7} | 1/8 |
Two convex polygons with \(M\) and \(N\) sides \((M > N)\) are drawn on a plane. What is the maximum possible number of regions into which they can divide the plane? | 2N+2 | 1/8 |
Consider that for integers from 1 to 1500, $x_1+2=x_2+4=x_3+6=\cdots=x_{1500}+3000=\sum_{n=1}^{1500}x_n + 3001$. Find the value of $\left\lfloor|S|\right\rfloor$, where $S=\sum_{n=1}^{1500}x_n$. | 1500 | 1/8 |
We colored three vertices of a regular icosahedron red. Show that there is at least one vertex of the icosahedron that has at least two red neighbors. | 2 | 1/8 |
The coordinates of the vertices of isosceles trapezoid $ABCD$ are all integers, with $A=(20,100)$ and $D=(21,107)$. The trapezoid has no horizontal or vertical sides, and $\overline{AB}$ and $\overline{CD}$ are the only parallel sides. The sum of the absolute values of all possible slopes for $\overline{AB}$ is $m/n$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.
| 131 | 1/8 |
The area of rhombus \(ABCD\) is 2. The triangle \(ABD\) has an inscribed circle that touches the side \(AB\) at point \(K\). A line \(KL\) is drawn through point \(K\) parallel to the diagonal \(AC\) of the rhombus (the point \(L\) lies on the side \(BC\)). Find the angle \(BAD\) given that the area of triangle \(KLB\) is \(a\). | 2\arcsin\sqrt{} | 1/8 |
Let the set
$$
\begin{array}{l}
M=\{(m, n) \mid m \text{ and } n \text{ are coprime, } m+n>2020, \\
\quad 0<m<n \leqslant 2020\}.
\end{array}
$$
Find the sum \( S=\sum_{(m, n) \in M} \frac{1}{mn} \). | \frac{1}{2} | 1/8 |
Among the numbers $1, 2, 3, \cdots, 50$, if 10 consecutive numbers are selected, what is the probability that exactly 3 of them are prime numbers? | 22/41 | 4/8 |
Given a hyperbola $C_{1}$ defined by $2x^{2}-y^{2}=1$, find the area of the triangle formed by a line parallel to one of the asymptotes of $C_{1}$, the other asymptote, and the x-axis. | \frac{\sqrt{2}}{8} | 1/8 |
The average age of 40 sixth-graders is 12. The average age of 30 of their teachers is 45. What is the average age of all these sixth-graders and their teachers? | 26.14 | 1/8 |
Let \( a, b, c \) be non-negative real numbers such that \( a + b + c = 3 \).
Find the maximum value of \( S = \left(a^{2} - ab + b^{2}\right)\left(b^{2} - bc + c^{2}\right)\left(c^{2} - ca + a^{2}\right) \). | 12 | 3/8 |
Let \( p \) and \( q \) be positive integers such that
\[
\frac{6}{11} < \frac{p}{q} < \frac{5}{9}
\]
and \( q \) is as small as possible. What is \( p+q \)? | 31 | 6/8 |
The polynomial \( p(x) \equiv x^3 + ax^2 + bx + c \) has three positive real roots. Find a necessary and sufficient condition on \( a \), \( b \), and \( c \) for the roots to be \( \cos A \), \( \cos B \), and \( \cos C \) for some triangle \( ABC \). | ^2-2b-2c=1 | 2/8 |
In a finite sequence of real numbers, the sum of any three consecutive terms is negative, and the sum of any four consecutive terms is positive. Find the maximum number of terms \( r \) in this sequence. | 5 | 3/8 |
Given the equation $x^{2}+a|x|+a^{2}-3=0 \text{ where } a \in \mathbf{R}$ has a unique real solution, find the value of $a$. | \sqrt{3} | 7/8 |
Determine the maximal possible length of the sequence of consecutive integers which are expressible in the form $ x^3\plus{}2y^2$ , with $ x, y$ being integers. | 5 | 2/8 |
Distribute 6 volunteers into 4 groups, with each group having at least 1 and at most 2 people, and assign them to four different exhibition areas of the fifth Asia-Europe Expo. The number of different allocation schemes is ______ (answer with a number). | 1080 | 5/8 |
The square \( STUV \) is formed by a square bounded by 4 equal rectangles. The perimeter of each rectangle is \( 40 \text{ cm} \). What is the area, in \( \text{cm}^2 \), of the square \( STUV \)?
(a) 400
(b) 200
(c) 160
(d) 100
(e) 80
| 400 | 1/8 |
On a very long narrow highway where overtaking is impossible, $n$ cars are driving in random order, each at its preferred speed. If a fast car catches up to a slow car, the fast car has to slow down and drive at the same speed as the slow car. As a result, the cars form groups. Find the expected number of "lonely" cars, i.e., groups consisting of a single car. | 1 | 1/8 |
The recruits stood in a row one after another, all facing the same direction. Among them were three brothers: Peter, Nikolai, and Denis. There were 50 people in front of Peter, 100 in front of Nikolai, and 170 in front of Denis. At the command "About-Face!", everyone turned to face the opposite direction. It turned out that in front of one of the brothers now stood four times as many people as in front of another. How many recruits, including the brothers, could there be? List all possible options. | 211 | 7/8 |
In an acute-angled triangle \(ABC\), the altitude \(AD = a\) and the altitude \(CE = b\). The acute angle between \(AD\) and \(CE\) is \(\alpha\). Find \(AC\). | \frac{\sqrt{^{2}+b^{2}-2ab\cos\alpha}}{\sin\alpha} | 1/8 |
Show that the equation
\[x(x+1)(x+2)\dots (x+2020)-1=0\]
has exactly one positive solution $x_0$ , and prove that this solution $x_0$ satisfies
\[\frac{1}{2020!+10}<x_0<\frac{1}{2020!+6}.\] | \frac{1}{2020!+10}<x_0<\frac{1}{2020!+6} | 2/8 |
A decorative window is made up of a rectangle with semicircles at either end. The ratio of $AD$ to $AB$ is $3:2$. And $AB$ is 30 inches. What is the ratio of the area of the rectangle to the combined area of the semicircles? | 6:\pi | 2/8 |
At the vertices of the grid of a \$4 \times 5\$ rectangle, there are 30 bulbs, all of which are initially off. In one move, it is allowed to draw any line that does not touch any bulbs (the sizes of the bulbs can be ignored, assuming them to be points) such that on one side of the line none of the bulbs are lit, and then turn on all the bulbs on that side of the line. Each move must light at least one bulb. Is it possible to light all the bulbs in exactly four moves? | Yes | 2/8 |
Given a positive integer $n (\geqslant 2)$, find the minimum value of $|X|$ such that for any $n$ two-element subsets $B_{1}, B_{2}, \cdots, B_{n}$ of the set $X$, there exists a subset $Y$ of $X$ satisfying:
(1) $|Y| = n$;
(2) For $i = 1, 2, \cdots, n$, we have $\left|Y \cap B_{i}\right| \leqslant 1$.
Here, $|A|$ represents the number of elements in the finite set $A$. | 2n-1 | 4/8 |
In a rectangle \(ABCD\) with an area of 1 (including its boundary), there are 5 points such that no three points are collinear. Determine the minimum number of triangles, formed by these 5 points as vertices, that have an area of at most \(\frac{1}{4}\). | 2 | 1/8 |
Calculate the value of the expression \( \sqrt[3]{11 + 4 \sqrt[3]{14 + 10 \sqrt[3]{17 + 18 \sqrt[3]{(\ldots)}}}} \). | 3 | 5/8 |
Given \(\triangle ABC\). On ray \(BA\), take a point \(D\) such that \(|BD| = |BA| + |AC|\). Let \(K\) and \(M\) be two points on rays \(BA\) and \(BC\) respectively, such that the area of \(\triangle BDM\) is equal to the area of \(\triangle BCK\). Find \(\angle BKM\) if \(\angle BAC = \alpha\). | \frac{\alpha}{2} | 4/8 |
Every day at noon, a mail steamer leaves from Le Havre to New York, and at the same time, another steamer from the same company leaves New York for Le Havre. Each of these steamers takes exactly seven days to complete their journey, and they travel the same route.
How many steamers from the same company will a steamer traveling from Le Havre to New York meet on its way? | 15 | 1/8 |
Misha and Masha had the same multi-digit whole number written in their notebooks, which ended with 9876. Masha placed a plus sign between the third and fourth digits from the right, while Misha placed a plus sign between the fourth and fifth digits from the right. To the students' surprise, both resulting sums turned out to be the same. What was the original number written by the students? List all possible answers and prove that there are no other solutions. | 9999876 | 2/8 |
Person A and Person B started working on the same day. The company policy states that Person A works for 3 days and then rests for 1 day, while Person B works for 7 days and then rests for 3 consecutive days. How many days do Person A and Person B rest on the same day within the first 1000 days? | 100 | 7/8 |
Given the function \( f(x)=\left(1-x^{3}\right)^{-1 / 3} \), find \( f(f(f \ldots f(2018) \ldots)) \) where the function \( f \) is applied 2019 times. | 2018 | 7/8 |
Show that \(x^4 - 1993x^3 + (1993 + n)x^2 - 11x + n = 0\) has at most one integer root if \(n\) is an integer. | 1 | 2/8 |
For which values of the parameter \( a \) does the equation \( f(x) = p(x) \) have one solution, if \( f(x) = \left|\frac{2 x^{3}-5 x^{2}-2 x+5}{(1.5 x-3)^{2}-(0.5 x-2)^{2}}\right| \) and \( p(x) = |2 x+5|+a \)? If there is more than one value for the parameter, indicate the sum of these values. | -10 | 3/8 |
Let $g:[0,1]\rightarrow \mathbb{R}$ be a continuous function and let $f_{n}:[0,1]\rightarrow \mathbb{R}$ be a
sequence of functions defined by $f_{0}(x)=g(x)$ and $$ f_{n+1}(x)=\frac{1}{x}\int_{0}^{x}f_{n}(t)dt. $$ Determine $\lim_{n\to \infty}f_{n}(x)$ for every $x\in (0,1]$ . | (0) | 2/8 |
Find the least positive integer $N$ such that the set of $1000$ consecutive integers beginning with $1000\cdot N$ contains no square of an integer. | 282 | 1/8 |
In a math class, the teacher gave the students 2 questions. Scoring is as follows: getting one question correct earns 10 points, partially correct earns 5 points, and completely wrong or unanswered earns 0 points. After grading, the teacher found that there were students with each possible score, and for each score, there were 5 students who had exactly the same score distribution for both questions. How many students are in this class? | 45 | 4/8 |
In the Cartesian coordinate system, the line \( l \) passing through the origin \( O \) intersects the curve \( y = e^{x-1} \) at two different points \( A \) and \( B \). Lines parallel to the \( y \)-axis are drawn through \( A \) and \( B \), intersecting the curve \( y = \ln x \) at points \( C \) and \( D \), respectively. The slope of the line \( CD \) is \(\qquad\). | 1 | 7/8 |
In $\triangle ABC$, $\sin ^{2}A-\sin ^{2}C=(\sin A-\sin B)\sin B$, then angle $C$ equals to $\dfrac {\pi}{6}$. | \dfrac {\pi}{3} | 2/8 |
Given that \( f(x) \) is an odd function defined on \(\mathbf{R} \), with \( f(1) = 1 \), and for any \( x < 0 \), it holds that
$$
f\left( \frac{x}{x-1} \right) = x f(x).
$$
Find the value of \( \sum_{i=1}^{50} f\left( \frac{1}{i} \right) f\left( \frac{1}{101-i} \right) \). | \frac{2^{98}}{99!} | 3/8 |
Compute the smallest positive integer $a$ for which $$ \sqrt{a +\sqrt{a +...}} - \frac{1}{a +\frac{1}{a+...}}> 7 $$ | 43 | 4/8 |
Points $K$, $L$, $M$, and $N$ lie in the plane of the square $ABCD$ so that $AKB$, $BLC$, $CMD$, and $DNA$ are equilateral triangles. If $ABCD$ has an area of 16, find the area of $KLMN$. Express your answer in simplest radical form.
[asy]
pair K,L,M,I,A,B,C,D;
D=(0,0);
C=(10,0);
B=(10,10);
A=(0,10);
I=(-8.7,5);
L=(18.7,5);
M=(5,-8.7);
K=(5,18.7);
draw(A--B--C--D--cycle,linewidth(0.7));
draw(A--D--I--cycle,linewidth(0.7));
draw(B--L--C--cycle,linewidth(0.7));
draw(A--B--K--cycle,linewidth(0.7));
draw(D--C--M--cycle,linewidth(0.7));
draw(K--L--M--I--cycle,linewidth(0.7));
label("$A$",A,SE);
label("$B$",B,SW);
label("$C$",C,NW);
label("$D$",D,NE);
label("$K$",K,N);
label("$L$",L,E);
label("$M$",M,S);
label("$N$",I,W);
[/asy] | 32 + 16\sqrt{3} | 2/8 |
Given real numbers \(x_{1}, x_{2}, \cdots, x_{1997}\) satisfying the following conditions:
1. \(-\frac{1}{\sqrt{3}} \leq x_{i} \leq \sqrt{3}\) for \(i = 1, 2, \cdots, 1997\).
2. \(x_{1} + x_{2} + \cdots + x_{1997} = -318 \sqrt{3}\).
Find the maximum value of \(x_{1}^{12} + x_{2}^{12} + \cdots + x_{1997}^{12}\). | 189548 | 3/8 |
\[ y = x + \cos(2x) \] in the interval \((0, \pi / 4)\). | \frac{\pi}{12} + \frac{\sqrt{3}}{2} | 6/8 |
Let n be the smallest positive integer such that n is divisible by 20, n^2 is a perfect square, and n^3 is a perfect fifth power. Find the value of n. | 3200000 | 1/8 |
Find the number of pairs $(a,b)$ of natural nunbers such that $b$ is a 3-digit number, $a+1$ divides $b-1$ and $b$ divides $a^{2} + a + 2$ . | 16 | 1/8 |
Jarris the triangle is playing in the \((x, y)\) plane. Let his maximum \(y\) coordinate be \(k\). Given that he has side lengths 6, 8, and 10 and that no part of him is below the \(x\)-axis, find the minimum possible value of \(k\). | 24/5 | 6/8 |
Based on a city's rules, the buildings of a street may not have more than $9$ stories. Moreover, if the number of stories of two buildings is the same, no matter how far they are from each other, there must be a building with a higher number of stories between them. What is the maximum number of buildings that can be built on one side of a street in this city? | 511 | 1/8 |
In the unit cube $A B C D-A_{1} B_{1} C_{1} D_{1}$, points $E, F, G$ are the midpoints of edges $A A_{1}, C_{1} D_{1}$, and $D_{1} A_{1}$, respectively. Find the distance from point $B_{1}$ to the plane $E F G$. | \frac{\sqrt{3}}{2} | 7/8 |
$ABCD$ is a convex quadrilateral such that $AB=2$, $BC=3$, $CD=7$, and $AD=6$. It also has an incircle. Given that $\angle ABC$ is right, determine the radius of this incircle. | \frac{1+\sqrt{13}}{3} | 7/8 |
Given three equilateral triangles \(\triangle A_{1} B_{1} C_{1}\), \(\triangle A_{2} B_{2} C_{2}\), and \(\triangle A_{3} B_{3} C_{3}\) with vertices arranged in reverse order. Let the centroids of \(\triangle A_{1} A_{2} A_{3}\), \(\triangle B_{1} B_{2} B_{3}\), and \(\triangle C_{1} C_{2} C_{3}\) be \(M_{1}\), \(M_{2}\), and \(M_{3}\) respectively. Prove that \(\triangle M_{1} M_{2} M_{3}\) is an equilateral triangle. | \triangleM_1M_2M_3 | 2/8 |
A group of 12 friends decides to form a committee of 5. Calculate the number of different committees that can be formed. Additionally, if there are 4 friends who refuse to work together, how many committees can be formed without any of these 4 friends? | 56 | 7/8 |
Given that $A_k = \frac {k(k - 1)}2\cos\frac {k(k - 1)\pi}2,$ find $|A_{19} + A_{20} + \cdots + A_{98}|.$ | 40 | 5/8 |
Susan wants to determine the average and median number of candies in a carton. She buys 9 cartons of candies, opens them, and counts the number of candies in each one. She finds that the cartons contain 5, 7, 8, 10, 12, 14, 16, 18, and 20 candies. What are the average and median number of candies per carton? | 12 | 7/8 |
A dog from point \( A \) chased a fox, which was 30 meters away from the dog in point \( B \). The dog's jump is 2 meters, and the fox's jump is 1 meter. The dog makes 2 jumps while the fox makes 3 jumps. At what distance from point \( A \) will the dog catch the fox? | 120 | 6/8 |
What is the maximum number of sides of a convex polygon that can be divided into right triangles with acute angles measuring 30 and 60 degrees? | 12 | 1/8 |
Let \( A, B, D, E, F, C \) be six points on a circle in that order, with \( AB = AC \). The line \( AD \) intersects \( BE \) at point \( P \). The line \( AF \) intersects \( CE \) at point \( R \). The line \( BF \) intersects \( CD \) at point \( Q \). The line \( AD \) intersects \( BF \) at point \( S \). The line \( AF \) intersects \( CD \) at point \( T \). Point \( K \) is on segment \( ST \) such that \(\angle SKQ = \angle ACE\). Prove that \(\frac{SK}{KT} = \frac{PQ}{QR}\). | \frac{SK}{KT}=\frac{PQ}{QR} | 1/8 |
The side of the base and the height of a regular quadrilateral pyramid are equal to \( a \). Find the radius of the inscribed sphere. | \frac{\sqrt{5}-1)}{4} | 7/8 |
Let $P$ be the intersection point of the diagonals of quadrilateral $ABCD$, $M$ be the intersection point of the lines connecting the midpoints of its opposite sides, $O$ be the intersection point of the perpendicular bisectors of the diagonals, and $H$ be the intersection point of the lines connecting the orthocenters of triangles $APD$ and $BPC$, $APB$ and $CPD$. Prove that $M$ is the midpoint of $OH$. | MisthemidpointofOH | 1/8 |
A sequence $(a_n)$ of real numbers is defined by $a_0=1$, $a_1=2015$ and for all $n\geq1$, we have
$$a_{n+1}=\frac{n-1}{n+1}a_n-\frac{n-2}{n^2+n}a_{n-1}.$$
Calculate the value of $\frac{a_1}{a_2}-\frac{a_2}{a_3}+\frac{a_3}{a_4}-\frac{a_4}{a_5}+\ldots+\frac{a_{2013}}{a_{2014}}-\frac{a_{2014}}{a_{2015}}$. | 3021 | 6/8 |
A square $ABCD$ with side length $1$ is inscribed in a circle. A smaller square lies in the circle with two vertices lying on segment $AB$ and the other two vertices lying on minor arc $AB$ . Compute the area of the smaller square.
| \frac{1}{25} | 2/8 |
In the triangle \( ABC \), \(\angle B = 90^\circ\), \(\angle C = 20^\circ\), \( D \) and \( E \) are points on \( BC \) such that \(\angle ADC =140^\circ\) and \(\angle AEC =150^\circ\). Suppose \( AD=10 \). Find \( BD \cdot CE \). | 50 | 7/8 |
$ABCD$ is a rectangle (see the accompanying diagram) with $P$ any point on $\overline{AB}$. $\overline{PS} \perp \overline{BD}$ and $\overline{PR} \perp \overline{AC}$. $\overline{AF} \perp \overline{BD}$ and $\overline{PQ} \perp \overline{AF}$. Then $PR + PS$ is equal to: | $AF$ | 6/8 |
The medians \( A M \) and \( B E \) of triangle \( A B C \) intersect at point \( O \). Points \( O, M, E, C \) lie on the same circle. Find \( A B \) if \( B E = A M = 3 \). | 2\sqrt{3} | 4/8 |
Nine people sit down for dinner where there are three choices of meals. Three people order the beef meal, three order the chicken meal, and three order the fish meal. The waiter serves the nine meals in random order. Find the number of ways in which the waiter could serve the meal types to the nine people so that exactly one person receives the type of meal ordered by that person. | 216 | 1/8 |
Three different numbers are chosen at random from the list \(1, 3, 5, 7, 9, 11, 13, 15, 17, 19\). The probability that one of them is the mean of the other two is \(p\). What is the value of \(\frac{120}{p}\) ? | 720 | 6/8 |
A collection of 8 cubes consists of one cube with edge-length $k$ for each integer $k, 1 \le k \le 8.$ A tower is to be built using all 8 cubes according to the rules:
Any cube may be the bottom cube in the tower.
The cube immediately on top of a cube with edge-length $k$ must have edge-length at most $k+2.$
Let $T$ be the number of different towers than can be constructed. What is the remainder when $T$ is divided by 1000?
| 458 | 1/8 |
A sequence of integers that is monotonically increasing, where the first term is odd, the second term is even, the third term is odd, the fourth term is even, and so on, is called an alternating sequence. The empty set is also considered an alternating sequence. Let \( A(n) \) represent the number of alternating sequences where all integers are taken from the set \(\{1, 2, \cdots, n\}\). It is clear that \(A(1) = 2\) and \(A(2) = 3\). Find \(A(20)\) and prove your conclusion. | 17711 | 1/8 |
Do there exist polynomials \( a(x) \), \( b(x) \), \( c(y) \), \( d(y) \) such that \( 1 + xy + x^2 y^2 \equiv a(x)c(y) + b(x)d(y) \)? | No | 1/8 |
The 16 squares on a piece of paper are numbered as shown in the diagram. While lying on a table, the paper is folded in half four times in the following sequence:
(1) fold the top half over the bottom half
(2) fold the bottom half over the top half
(3) fold the right half over the left half
(4) fold the left half over the right half.
Which numbered square is on top after step 4? | 9 | 1/8 |
Let \( XYZ \) be an acute-angled triangle. Let \( s \) be the side length of the square which has two adjacent vertices on side \( YZ \), one vertex on side \( XY \), and one vertex on side \( XZ \). Let \( h \) be the distance from \( X \) to the side \( YZ \) and \( b \) be the distance from \( Y \) to \( Z \).
(a) If the vertices have coordinates \( X=(2,4), Y=(0,0) \), and \( Z=(4,0) \), find \( b, h \), and \( s \).
(b) Given the height \( h=3 \) and \( s=2 \), find the base \( b \).
(c) If the area of the square is 2017, determine the minimum area of triangle \( XYZ \). | 4034 | 6/8 |
If \( a b c + c b a = 1069 \), then there are \(\qquad\) such \(\underline{a b c}\). | 8 | 5/8 |
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