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Given triangle \( ABC \), where \( 2h_c = AB \) and \( \angle A = 75^\circ \). Find the measure of angle \( C \). | 75 | 5/8 |
$ABCDE$ is a regular pentagon. What is the degree measure of the acute angle at the intersection of line segments $AC$ and $BD$ ? | 72 | 7/8 |
The set of vertices of a polygon consists of points with coordinates $(x, y)$ where $x$ and $y$ are positive integers, and also satisfy $x \mid 2y + 1$ and $y \mid 2x + 1$. What is the maximum possible area of the polygon? | 20 | 1/8 |
Square $ABCD$ is inscribed in circle $\omega$. An arbitrary point $M$ is chosen on the minor arc $CD$ of circle $\omega$. Inside the square, points $K$ and $L$ are marked such that $KLMD$ is a square. Find $\angle AKD$. | 135 | 1/8 |
Around a circular table, there are exactly 60 chairs. $N$ people sit at the table according to the following rule: every new person must sit next to someone who is already seated. What is the smallest possible value of $N$?
(A) 15
(B) 20
(C) 30
(D) 40
(E) 58 | 20 | 1/8 |
Some positive integers are initially written on a board, where each $2$ of them are different.
Each time we can do the following moves:
(1) If there are 2 numbers (written in the board) in the form $n, n+1$ we can erase them and write down $n-2$ (2) If there are 2 numbers (written in the board) in the form $n, n+4$ we can erase them and write down $n-1$ After some moves, there might appear negative numbers. Find the maximum value of the integer $c$ such that:
Independetly of the starting numbers, each number which appears in any move is greater or equal to $c$ | -3 | 1/8 |
A factory produces a type of instrument. Due to limitations in production capacity and technical level, some defective products are produced. According to experience, the defect rate $p$ of the factory producing this instrument is generally related to the daily output $x$ (pieces) as follows:
$$
P= \begin{cases}
\frac {1}{96-x} & (1\leq x\leq 94, x\in \mathbb{N}) \\
\frac {2}{3} & (x>94, x\in \mathbb{N})
\end{cases}
$$
It is known that for every qualified instrument produced, a profit of $A$ yuan can be made, but for every defective product produced, a loss of $\frac {A}{2}$ yuan will be incurred. The factory wishes to determine an appropriate daily output.
(1) Determine whether producing this instrument can be profitable when the daily output (pieces) exceeds 94 pieces, and explain the reason;
(2) When the daily output $x$ pieces does not exceed 94 pieces, try to express the daily profit $T$ (yuan) of producing this instrument as a function of the daily output $x$ (pieces);
(3) To obtain the maximum profit, how many pieces should the daily output $x$ be? | 84 | 7/8 |
Given a triangle \(ABC\). It is required to divide it into the smallest number of parts so that, by flipping these parts to the other side, the same triangle \(ABC\) can be formed. | 3 | 4/8 |
Given the circle \( \odot O: x^{2}+y^{2}=5 \) and the parabola \( C: y^{2}=2px \) (with \( p > 0 \)), they intersect at point \( A(x_{0}, 2) \). \( AB \) is a diameter of \( \odot O \). A line through point \( B \) intersects the parabola \( C \) at two points \( D \) and \( E \). Find the product of the slopes of the lines \( AD \) and \( AE \). | 2 | 4/8 |
Let $A M O L$ be a quadrilateral with $A M=10, M O=11$, and $O L=12$. Given that the perpendicular bisectors of sides $A M$ and $O L$ intersect at the midpoint of segment $A O$, find the length of side LA. | $\sqrt{77}$ | 5/8 |
From May 1st to May 3rd, the provincial hospital plans to schedule 6 doctors to be on duty, with each person working 1 day and 2 people scheduled per day. Given that doctor A cannot work on the 2nd and doctor B cannot work on the 3rd, how many different scheduling arrangements are possible? | 42 | 7/8 |
Given that for the angles of triangle \(ABC\), \(\sin A + \cos B = \sqrt{2}\) and \(\cos A + \sin B = \sqrt{2}\), find the measure of angle \(C\). | 90 | 7/8 |
Specify the smallest number that ends in 37, has a digit sum of 37, and is divisible by 37. | 99937 | 7/8 |
The product of the two $102$-digit numbers $404,040,404,...,040,404$ and $707,070,707,...,070,707$ has thousands digit $A$ and units digit $B$. Calculate the sum of $A$ and $B$. | 13 | 2/8 |
If \( f(x) = x^{6} - 2 \sqrt{2006} x^{5} - x^{4} + x^{3} - 2 \sqrt{2007} x^{2} + 2 x - \sqrt{2006} \), then find \( f(\sqrt{2006} + \sqrt{2007}) \). | \sqrt{2007} | 1/8 |
How many numbers with less than four digits (from 0 to 9999) are neither divisible by 3, nor by 5, nor by 7? | 4571 | 5/8 |
Let $p>3$ be a prime and let $a_1,a_2,...,a_{\frac{p-1}{2}}$ be a permutation of $1,2,...,\frac{p-1}{2}$. For which $p$ is it always possible to determine the sequence $a_1,a_2,...,a_{\frac{p-1}{2}}$ if it for all $i,j\in\{1,2,...,\frac{p-1}{2}\}$ with $i\not=j$ the residue of $a_ia_j$ modulo $p$ is known? | p \geq 7 | 1/8 |
We play on a chessboard with sides of length 100. A move consists of choosing a rectangle formed by squares on the chessboard and inverting the colors. What is the smallest number of moves needed to make the entire chessboard completely black? | 100 | 1/8 |
Construct a rational parameterization of the circle \( x^2 + y^2 = 1 \) by drawing lines through the point \((1,0)\). | (\frac{^2-1}{^2+1},\frac{-2t}{^2+1}) | 7/8 |
Find the ratio of the volume of a regular hexagonal pyramid to the volume of a regular triangular pyramid, given that the sides of their bases are equal and their slant heights are twice the length of the sides of the base. | \frac{6 \sqrt{1833}}{47} | 3/8 |
Point $A$ has two tangents drawn to a circle, touching the circle at points $M$ and $N$, respectively. A secant from the same point $A$ intersects the circle at points $B$ and $C$, and intersects the chord $MN$ at point $P$. Given that $AB:BC = 2:3$, find the ratio $AP:PC$. | 4:3 | 1/8 |
Circles with centers \( O_1 \) and \( O_2 \) and equal radii are inscribed in the angles at \( B \) and \( C \) of triangle \( ABC \) respectively. Point \( O \) is the center of the incircle of triangle \( ABC \). These circles touch side \( BC \) at points \( K_1, K_2, \) and \( K \) respectively, with \( BK_1 = 4, CK_2 = 8, \) and \( BC = 18 \).
a) Find the length of segment \( CK \).
b) Suppose the circle with center \( O_1 \) touches side \( AB \) at point \( K_3 \). Find the angle \( ABC \), given that point \( O_1 \) is the center of the circumscribed circle around triangle \( OK_1K_3 \). | 60 | 1/8 |
The circles \(\odot O_{1}\) and \(\odot O_{2}\) intersect at points \(P\) and \(Q\). The chord \(PA\) of \(\odot O_{1}\) is tangent to \(\odot O_{2}\), and the chord \(PB\) of \(\odot O_{2}\) is tangent to \(\odot O_{1}\). Let the circumcenter of \(\triangle PAB\) be \(O\). Prove that \(OQ \perp PQ\). | OQ\perpPQ | 1/8 |
Morteza marked six points on a plane and calculated the areas of all 20 triangles with vertices at these points. Can all of these areas be integers, and can their sum be equal to 2019? | No | 3/8 |
Find the angle $C$ of triangle $ABC$ if vertex $A$ is equidistant from the centers of the excircles that touch sides $AB$ and $BC$. | 90 | 2/8 |
Two identical two-digit numbers were written on the board. One of them had 100 added to the left, and the other had 1 added to the right, resulting in the first number becoming 37 times larger than the second. What were the numbers written on the board? | 27 | 7/8 |
Find all triples $ (x,y,z)$ of real numbers that satisfy the system of equations
\[ \begin{cases}x^3 \equal{} 3x\minus{}12y\plus{}50, \\ y^3 \equal{} 12y\plus{}3z\minus{}2, \\ z^3 \equal{} 27z \plus{} 27x. \end{cases}\]
[i]Razvan Gelca.[/i] | (2, 4, 6) | 2/8 |
Camp Koeller offers exactly three water activities: canoeing, swimming, and fishing. None of the campers is able to do all three of the activities. In total, 15 of the campers go canoeing, 22 go swimming, 12 go fishing, and 9 do not take part in any of these activities. Determine the smallest possible number of campers at Camp Koeller. | 34 | 7/8 |
Inside an isosceles triangle \( ABC \), a point \( K \) is marked such that \( CK = AB = BC \) and \(\angle KAC = 30^\circ\). Find the angle \( AKB \). | 150 | 1/8 |
In the right triangle \(ABC\) with the right angle at vertex \(C\), the angle bisector \(CL\) and the median \(CM\) are drawn.
Find the area of triangle \(ABC\) if \(LM = a\) and \(CM = b\). | \frac{b^2(b^2-^2)}{b^2+^2} | 3/8 |
In the arithmetic sequence $\left\{a_{n}\right\}$, if $\frac{a_{11}}{a_{10}} < -1$ and the sum of its first $n$ terms $S_{n}$ has a maximum value, then when $S_{n}$ takes the smallest positive value, $n = (\quad$ ). | 19 | 4/8 |
The ecology club at a school has 30 members: 12 boys and 18 girls. A 4-person committee is to be chosen at random. What is the probability that the committee has at least 1 boy and at least 1 girl? | \dfrac{530}{609} | 7/8 |
What is the maximum value of $\frac{(3^t - 5t)t}{9^t}$ for real values of $t$? | \frac{1}{20} | 3/8 |
One side of a rectangle was tripled, and the other side was halved to form a square. What is the side length of the square if the area of the rectangle is $54 \ \mathrm{m}^{2}$? | 9\, | 1/8 |
For which smallest natural number \( k \) does the expression \( 2016 \cdot 2017 \cdot 2018 \cdot 2019 + k \) become a square of a natural number? | 1 | 7/8 |
Let $N$ be the greatest integer multiple of 8, no two of whose digits are the same. What is the remainder when $N$ is divided by 1000?
| 120 | 2/8 |
Let \( f(x) = |\ln x| \). If the function \( g(x) = f(x) - a x \) has three zero points on the interval \( (0, 4) \), determine the range of values for the real number \( a \). | (\frac{\ln2}{2},\frac{1}{e}) | 3/8 |
How many positive integers \( k \) are there such that
\[
\frac{k}{2013}(a+b) = \mathrm{lcm}(a, b)
\]
has a solution in positive integers \( (a, b) \)? | 1006 | 1/8 |
Given a semicircle with diameter \( AB \), center \( O \), a line intersects the semicircle at points \( C \) and \( D \), and intersects the line \( AB \) at point \( M \) (\(MB < MA, MD < MC\)). Let the second intersection point of the circumcircles of \( \triangle AOC \) and \( \triangle DOB \) be \( K \). Prove that \( \angle M K O \) is a right angle. | \angleMKO=90 | 1/8 |
Mom decides to take Xiaohua on a driving tour to 10 cities during the holidays. After checking the map, Xiaohua is surprised to find that among any three cities, either all three pairs of cities are connected by highways, or exactly one pair of cities is not connected by a highway. How many highways must have been constructed between these 10 cities? (Note: There can be at most one highway between any two cities.) | 40 | 4/8 |
What is the maximum possible area of a triangle if the sides \(a, b, c\) satisfy the following inequalities:
$$
0 < a \leq 1 \leq b \leq 2 \leq c \leq 3
$$ | 1 | 4/8 |
In triangle $PQR$, let the side lengths be $PQ = 7,$ $PR = 8,$ and $QR = 5$. Calculate:
\[\frac{\cos \frac{P - Q}{2}}{\sin \frac{R}{2}} - \frac{\sin \frac{P - Q}{2}}{\cos \frac{R}{2}}.\] | \frac{16}{7} | 7/8 |
For how many positive integers \( n \) less than 200 is \( n^n \) a cube and \( (n+1)^{n+1} \) a square? | 40 | 2/8 |
Let $ABC$ be an acute triangle. Let $H$ and $D$ be points on $[AC]$ and $[BC]$ , respectively, such that $BH \perp AC$ and $HD \perp BC$ . Let $O_1$ be the circumcenter of $\triangle ABH$ , and $O_2$ be the circumcenter of $\triangle BHD$ , and $O_3$ be the circumcenter of $\triangle HDC$ . Find the ratio of area of $\triangle O_1O_2O_3$ and $\triangle ABH$ . | 1/4 | 5/8 |
1. The focal distance of the parabola $4x^{2}=y$ is \_\_\_\_\_\_\_\_\_\_\_\_
2. The equation of the hyperbola that has the same asymptotes as the hyperbola $\frac{x^{2}}{2} -y^{2}=1$ and passes through $(2,0)$ is \_\_\_\_\_\_\_\_\_\_\_\_
3. In the plane, the distance formula between a point $(x_{0},y_{0})$ and a line $Ax+By+C=0$ is $d= \frac{|Ax_{0}+By_{0}+C|}{\sqrt{A^{2}+B^{2}}}$. By analogy, the distance between the point $(0,1,3)$ and the plane $x+2y+3z+3=0$ is \_\_\_\_\_\_\_\_\_\_\_\_
4. If point $A$ has coordinates $(1,1)$, $F_{1}$ is the lower focus of the ellipse $5y^{2}+9x^{2}=45$, and $P$ is a moving point on the ellipse, then the maximum value of $|PA|+|PF_{1}|$ is $M$, the minimum value is $N$, so $M-N=$ \_\_\_\_\_\_\_\_\_\_\_\_ | 2\sqrt{2} | 4/8 |
Find the possible value of $x + y$ given that $x^3 + 6x^2 + 16x = -15$ and $y^3 + 6y^2 + 16y = -17$. | -4 | 4/8 |
On the side \( AC \) of triangle \( ABC \), a point \( E \) is marked. It is known that the perimeter of triangle \( ABC \) is 25 cm, the perimeter of triangle \( ABE \) is 15 cm, and the perimeter of triangle \( BCE \) is 17 cm. Find the length of segment \( BE \). | 3.5\, | 1/8 |
Angle ABC is a right angle. The diagram shows four quadrilaterals, where three are squares on each side of triangle ABC, and one square is on the hypotenuse. The sum of the areas of all four squares is 500 square centimeters. What is the number of square centimeters in the area of the largest square? | \frac{500}{3} | 6/8 |
Let $a$ be a positive number. Consider the set $S$ of all points whose rectangular coordinates $(x, y)$ satisfy all of the following conditions:
\begin{enumerate}
\item $\frac{a}{2} \le x \le 2a$
\item $\frac{a}{2} \le y \le 2a$
\item $x+y \ge a$
\item $x+a \ge y$
\item $y+a \ge x$
\end{enumerate}
The boundary of set $S$ is a polygon with | 6 | 7/8 |
If $\log_8{3}=p$ and $\log_3{5}=q$, then, in terms of $p$ and $q$, $\log_{10}{5}$ equals | \frac{3pq}{1+3pq} | 7/8 |
\(\{a_{n}\}\) is an arithmetic sequence with \(a_{1} = 1\) and common difference \(d\). \(\{b_{n}\}\) is a geometric sequence with common ratio \(q\). The first three terms of the sequence \(\{a_{n} + b_{n}\}\) are 3, 12, 23. Find \(d + q\). | 9 | 7/8 |
Let $ n\ge 3$ be a natural number. Find all nonconstant polynomials with real coeficcietns $ f_{1}\left(x\right),f_{2}\left(x\right),\ldots,f_{n}\left(x\right)$ , for which
\[ f_{k}\left(x\right)f_{k+ 1}\left(x\right) = f_{k +1}\left(f_{k + 2}\left(x\right)\right), \quad 1\le k\le n,\]
for every real $ x$ (with $ f_{n +1}\left(x\right)\equiv f_{1}\left(x\right)$ and $ f_{n + 2}\left(x\right)\equiv f_{2}\left(x\right)$ ). | f_k(x)=x^2 | 1/8 |
A natural number $k > 1$ is called *good* if there exist natural numbers $$ a_1 < a_2 < \cdots < a_k $$ such that $$ \dfrac{1}{\sqrt{a_1}} + \dfrac{1}{\sqrt{a_2}} + \cdots + \dfrac{1}{\sqrt{a_k}} = 1 $$ .
Let $f(n)$ be the sum of the first $n$ *[good* numbers, $n \geq$ 1. Find the sum of all values of $n$ for which $f(n+5)/f(n)$ is an integer. | 18 | 6/8 |
Let \( f: \mathbb{Z}_{>0} \rightarrow \mathbb{Z} \) be a function with the following properties:
(i) \( f(1) = 0 \),
(ii) \( f(p) = 1 \) for all prime numbers \( p \),
(iii) \( f(x y) = y f(x) + x f(y) \) for all \( x, y \) in \( \mathbb{Z}_{>0} \).
Determine the smallest integer \( n \geq 2015 \) that satisfies \( f(n) = n \). | 3125 | 2/8 |
In the quadrilateral $ABCD$ , $AB = BC = CD$ and $\angle BMC = 90^\circ$ , where $M$ is the midpoint of $AD$ . Determine the acute angle between the lines $AC$ and $BD$ . | 30 | 4/8 |
In a convex 13-gon, all diagonals are drawn. They divide it into polygons. Consider the polygon with the largest number of sides among them. What is the greatest number of sides that it can have? | 13 | 5/8 |
Egorov decided to open a savings account to buy a car worth 900,000 rubles. The initial deposit is 300,000 rubles. Every month, Egorov plans to add 15,000 rubles to his account. The bank offers a monthly interest rate of $12\%$ per annum. The interest earned each month is added to the account balance, and the interest for the following month is calculated on the new balance. After how many months will there be enough money in the account to buy the car? | 29 | 6/8 |
Construct a square on one side of an equilateral triangle. On one non-adjacent side of the square, construct a regular pentagon, as shown. On a non-adjacent side of the pentagon, construct a hexagon. Continue to construct regular polygons in the same way, until you construct an octagon. How many sides does the resulting polygon have?
[asy] defaultpen(linewidth(0.6)); pair O=origin, A=(0,1), B=A+1*dir(60), C=(1,1), D=(1,0), E=D+1*dir(-72), F=E+1*dir(-144), G=O+1*dir(-108); draw(O--A--B--C--D--E--F--G--cycle); draw(O--D, dashed); draw(A--C, dashed);[/asy] | 23 | 6/8 |
Let $P(x) = x^2 - 3x - 7$, and let $Q(x)$ and $R(x)$ be two quadratic polynomials also with the coefficient of $x^2$ equal to $1$. David computes each of the three sums $P + Q$, $P + R$, and $Q + R$ and is surprised to find that each pair of these sums has a common root, and these three common roots are distinct. If $Q(0) = 2$, then $R(0) = \frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$. | 71 | 5/8 |
A student's final score on a 150-point test is directly proportional to the time spent studying multiplied by a difficulty factor for the test. The student scored 90 points on a test with a difficulty factor of 1.5 after studying for 2 hours. What score would the student receive on a second test of the same format if they studied for 5 hours and the test has a difficulty factor of 2? | 300 | 2/8 |
In the diagram, $ABC$ is a straight line. What is the value of $y$?
[asy]
draw((-2,0)--(8,0),linewidth(0.7)); draw((8,0)--(5,-5.5)--(0,0),linewidth(0.7));
label("$A$",(-2,0),W); label("$B$",(0,0),N); label("$C$",(8,0),E); label("$D$",(5,-5.5),S);
label("$148^\circ$",(0,0),SW); label("$58^\circ$",(7,0),S);
label("$y^\circ$",(5,-4.5));
[/asy] | 90 | 3/8 |
In a regular tetrahedron \( S-ABC \) with side length \( a \), \( E \) and \( F \) are the midpoints of \( SA \) and \( BC \) respectively. Find the angle between the skew lines \( BE \) and \( SF \). | \arccos(\frac{2}{3}) | 5/8 |
The bases \(AB\) and \(CD\) of trapezoid \(ABCD\) are 55 and 31 respectively, and its diagonals are mutually perpendicular. Find the dot product of vectors \(\overrightarrow{AD}\) and \(\overrightarrow{BC}\). | 1705 | 7/8 |
The line joining $(3,2)$ and $(6,0)$ divides the square shown into two parts. What fraction of the area of the square is above this line? Express your answer as a common fraction.
[asy]
draw((-2,0)--(7,0),linewidth(1),Arrows);
draw((0,-1)--(0,4),linewidth(1),Arrows);
draw((1,.25)--(1,-.25),linewidth(1));
draw((2,.25)--(2,-.25),linewidth(1));
draw((3,.25)--(3,-.25),linewidth(1));
draw((4,.25)--(4,-.25),linewidth(1));
draw((5,.25)--(5,-.25),linewidth(1));
draw((6,.25)--(6,-.25),linewidth(1));
draw((.25,1)--(-.25,1),linewidth(1));
draw((.25,2)--(-.25,2),linewidth(1));
draw((.25,3)--(-.25,3),linewidth(1));
draw((3,0)--(6,0)--(6,3)--(3,3)--(3,0)--cycle,linewidth(2));
label("$y$",(0,4),N);
label("$x$",(7,0),E);
label("$(3,0)$",(3,0),S);
label("$(6,3)$",(6,3),N);
[/asy] | \frac{2}{3} | 7/8 |
Compute
\[\cos^6 0^\circ + \cos^6 1^\circ + \cos^6 2^\circ + \dots + \cos^6 90^\circ.\] | \frac{229}{8} | 5/8 |
Given a parallelepiped $A B C D A_{1} B_{1} C_{1} D_{1}$, the point $X$ is chosen on the edge $A_{1} D_{1}$ and the point $Y$ is chosen on the edge $B C$. It is known that $A_{1} X=5$, $B Y=3$, and $B_{1} C_{1}=14$. The plane $C_{1} X Y$ intersects the ray $D A$ at point $Z$. Find $D Z$. | 20 | 7/8 |
In bag A, there are 3 white balls and 2 red balls, while in bag B, there are 2 white balls and 4 red balls. If a bag is randomly chosen first, and then 2 balls are randomly drawn from that bag, the probability that the second ball drawn is white given that the first ball drawn is red is ______. | \frac{17}{32} | 7/8 |
In $\triangle ABC$, the sides opposite to angles A, B, and C are denoted as $a$, $b$, and $c$ respectively. Given that $a= \sqrt{2}$, $b=2$, and $\sin B - \cos B = \sqrt{2}$, find the measure of angle $A$. | \frac{\pi}{6} | 1/8 |
The positive reals $ x,y$ and $ z$ are satisfying the relation $ x \plus{} y \plus{} z\geq 1$ . Prove that:
$ \frac {x\sqrt {x}}{y \plus{} z} \plus{} \frac {y\sqrt {y}}{z \plus{} x} \plus{} \frac {z\sqrt {z}}{x \plus{} y}\geq \frac {\sqrt {3}}{2}$
*Proposer*:**Baltag Valeriu** | \frac{\sqrt{3}}{2} | 4/8 |
Given that all faces of the tetrahedron P-ABC are right triangles, and the longest edge PC equals $2\sqrt{3}$, the surface area of the circumscribed sphere of this tetrahedron is \_\_\_\_\_\_. | 12\pi | 1/8 |
Let \(ABCD\) be a quadrilateral with \(BC = CD = DA = 1\), \(\angle DAB = 135^\circ\), and \(\angle ABC = 75^\circ\). Find \(AB\). | \frac{\sqrt{6}-\sqrt{2}}{2} | 1/8 |
In the rectangular coordinate system xOy, the parametric equation of line l is $$\begin{cases} \overset{x=t}{y=1+t}\end{cases}$$ (t is the parameter), line m is parallel to line l and passes through the coordinate origin, and the parametric equation of circle C is $$\begin{cases} \overset{x=1+cos\phi }{y=2+sin\phi }\end{cases}$$ (φ is the parameter). Establish a polar coordinate system with the coordinate origin as the pole and the positive semi-axis of the x-axis as the polar axis.
1. Find the polar coordinate equations of line m and circle C.
2. Suppose line m and circle C intersect at points A and B. Find the perimeter of △ABC. | 2+ \sqrt {2} | 7/8 |
The real number sequence \(a_0, a_1, a_2, \cdots, a_n, \cdots\) satisfies the following equation: \(a_0 = a\), where \(a\) is a real number,
\[ a_n = \frac{a_{n-1} \sqrt{3} + 1}{\sqrt{3} - a_{n-1}}, \quad n \in \mathbf{N} \]
Find \( a_{1994} \). | \frac{\sqrt{3}}{1-\sqrt{3}} | 4/8 |
In the quadrilateral \(ABCD\), \(AB\) is parallel to \(CD\) and \(AD\) is parallel to \(BC\). Prove that there exists a point from which the distances to the lines containing the sides of the quadrilateral are proportional to these sides. | 0 | 1/8 |
The vertices of a regular hexagon are labeled $\cos (\theta), \cos (2 \theta), \ldots, \cos (6 \theta)$. For every pair of vertices, Bob draws a blue line through the vertices if one of these functions can be expressed as a polynomial function of the other (that holds for all real $\theta$ ), and otherwise Roberta draws a red line through the vertices. In the resulting graph, how many triangles whose vertices lie on the hexagon have at least one red and at least one blue edge? | 14 | 3/8 |
Find all surjective functions $f:\mathbb{N}\to\mathbb{N}$ such that for all positive integers $a$ and $b$ , exactly one of the following equations is true:
\begin{align*}
f(a)&=f(b), <br />
f(a+b)&=\min\{f(a),f(b)\}.
\end{align*}
*Remarks:* $\mathbb{N}$ denotes the set of all positive integers. A function $f:X\to Y$ is said to be surjective if for every $y\in Y$ there exists $x\in X$ such that $f(x)=y$ . | f(n)=\nu_2(n)+1 | 1/8 |
The function $y=f(x)$ is an even function with the smallest positive period of $4$, and when $x \in [-2,0]$, $f(x)=2x+1$. If there exist $x\_1$, $x\_2$, $…x\_n$ satisfying $0 \leqslant x\_1 < x\_2 < … < x\_n$, and $|f(x\_1)-f(x\_2)|+|f(x\_2)-f(x\_1)|+…+|f(x\_{n-1}-f(x\_n))|=2016$, then the minimum value of $n+x\_n$ is \_\_\_\_\_\_. | 1513 | 7/8 |
A rectangular piece of paper has integer side lengths. The paper is folded so that a pair of diagonally opposite vertices coincide, and it is found that the crease is of length 65. Find a possible value of the perimeter of the paper. | 408 | 2/8 |
If, in the expansion of \((a + 2b)^n\), there exist three consecutive terms whose binomial coefficients form an arithmetic progression, then the largest three-digit positive integer \(n\) is _____. | 959 | 4/8 |
Define a function \( D(n) \) on the set of positive integers such that:
(i) \( D(1)=0 \),
(ii) \( D(p)=1 \) when \( p \) is a prime number,
(iii) For any two positive integers \( u \) and \( v \),
\[ D(uv) = uD(v) + vD(u). \]
1. Prove that these three conditions are consistent (i.e., not contradictory) and uniquely determine the function \( D(n) \). Derive a formula for \( \frac{D(n)}{n} \) assuming \( n=p_{1}^{a_{1}} p_{2}^{a_{2}} \cdots p_{k}^{a_{k}} \), where \( p_{1}, p_{2}, \cdots, p_{k} \) are distinct primes.
2. Find \( n \) such that \( D(n)=n \).
3. Define \( D^{1}(n)=D(n) \) and \( D^{k+1}(n)=D\left(D^{k}(n)\right) \) for \( k=1,2, \cdots \). Find the limit of \( D^{m}(63) \) as \( m \rightarrow \infty \).
(Note: This is from the 10th Putnam Mathematics Competition, 1950) | +\infty | 1/8 |
Quadrilateral \(ABCD\) is inscribed in a circle with diameter \(AC\). Points \(K\) and \(M\) are the projections of vertices \(A\) and \(C\) respectively onto the line \(BD\). Through point \(K\) a line parallel to \(BC\) is drawn, intersecting \(AC\) at point \(P\). Prove that the angle \(KPM\) is a right angle. | \angleKPM=90 | 1/8 |
Find the sum of all positive integers $B$ such that $(111)_B=(aabbcc)_6$ , where $a,b,c$ represent distinct base $6$ digits, $a\neq 0$ . | 237 | 1/8 |
Given the function $f(x)=\cos (x- \frac {π}{4})-\sin (x- \frac {π}{4}).$
(I) Determine the evenness or oddness of the function $f(x)$ and provide a proof;
(II) If $θ$ is an angle in the first quadrant and $f(θ+ \frac {π}{3})= \frac { \sqrt {2}}{3}$, find the value of $\cos (2θ+ \frac {π}{6})$. | \frac {4 \sqrt {2}}{9} | 7/8 |
Three of the roots of the equation $x^4 -px^3 +qx^2 -rx+s = 0$ are $\tan A, \tan B$ , and $\tan C$ , where $A, B$ , and $C$ are angles of a triangle. Determine the fourth root as a function only of $p, q, r$ , and $s.$ | \frac{r-p}{-1} | 2/8 |
Use Horner's Rule to find the value of $v_2$ when the polynomial $f(x) = x^5 + 4x^4 + x^2 + 20x + 16$ is evaluated at $x = -2$. | -4 | 1/8 |
Knights, who always tell the truth, and liars, who always lie, live on an island. One day, 100 residents of this island lined up, and each of them said one of the following phrases:
- "To the left of me there are as many liars as knights."
- "To the left of me there is 1 more liar than knights."
- "To the left of me there are 2 more liars than knights."
- "To the left of me there are 99 more liars than knights."
It is known that each phrase was said by exactly one person. What is the minimum number of liars that can be among these 100 residents? | 50 | 1/8 |
Suppose that $m$ and $n$ are positive integers with $m<n$ such that the interval $[m, n)$ contains more multiples of 2021 than multiples of 2000. Compute the maximum possible value of $n-m$. | 191999 | 1/8 |
In the decimal representation of the even number \( M \), only the digits \( 0, 2, 4, 5, 7, \) and \( 9 \) participate, and digits may repeat. It is known that the sum of the digits of the number \( 2M \) is 31, and the sum of the digits of the number \( M / 2 \) is 28. What values can the sum of the digits of the number \( M \) have? List all possible answers. | 29 | 1/8 |
Let positive integers $a$, $b$, $c$, and $d$ be randomly and independently selected with replacement from the set $\{1, 2, 3,\dots, 2000\}$. Find the probability that the expression $ab+bc+cd+d+1$ is divisible by $4$. | \frac{1}{4} | 1/8 |
In a chorus performance, there are 6 female singers (including 1 lead singer) and 2 male singers arranged in two rows.
(1) If there are 4 people per row, how many different arrangements are possible?
(2) If the lead singer stands in the front row and the male singers stand in the back row, with again 4 people per row, how many different arrangements are possible? | 5760 | 7/8 |
Let positive real numbers \(a, b, c\) satisfy \(a+b+c=1\). Prove that \(10\left(a^{3}+b^{3}+c^{3}\right)-9\left(a^{5}+b^{5}+c^{5}\right) \geqslant 1\). | 1 | 4/8 |
The line $12x+5y=60$ forms a triangle with the coordinate axes. What is the sum of the lengths of the altitudes of this triangle?
$\textbf{(A)}\; 20 \qquad\textbf{(B)}\; \dfrac{360}{17} \qquad\textbf{(C)}\; \dfrac{107}{5} \qquad\textbf{(D)}\; \dfrac{43}{2} \qquad\textbf{(E)}\; \dfrac{281}{13}$ | \textbf{(E)}\;\frac{281}{13} | 1/8 |
The cells of a 2001×2001 chessboard are colored in a checkerboard pattern in black and white, such that the corner cells are black. For each pair of differently colored cells, a vector is drawn from the center of the black cell to the center of the white cell. Prove that the sum of the drawn vectors equals 0. | 0 | 6/8 |
Let the sequence $a_{1}, a_{2}, \cdots$ be defined recursively as follows: $a_{n}=11a_{n-1}-n$ . If all terms of the sequence are positive, the smallest possible value of $a_{1}$ can be written as $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. What is $m+n$ ? | 121 | 7/8 |
Given $$\sqrt {2 \frac {2}{3}}=2 \sqrt { \frac {2}{3}}$$, $$\sqrt {3 \frac {3}{8}}=3 \sqrt { \frac {3}{8}}$$, $$\sqrt {4 \frac {4}{15}}=4 \sqrt { \frac {4}{15}}$$, ..., if $$\sqrt {6 \frac {a}{t}}=6 \sqrt { \frac {a}{t}}$$ (where $a$, $t$∈$R^*$), then $a=$ \_\_\_\_\_\_ , $t=$ \_\_\_\_\_\_ . | 35 | 6/8 |
Given a parallelogram $ABCD$ , let $\mathcal{P}$ be a plane such that the distance from vertex $A$ to $\mathcal{P}$ is $49$ , the distance from vertex $B$ to $\mathcal{P}$ is $25$ , and the distance from vertex $C$ to $\mathcal{P}$ is $36$ . Find the sum of all possible distances from vertex $D$ to $\mathcal{P}$ .
*Proposed by **HrishiP*** | 220 | 6/8 |
How many ways are there to insert plus signs (+) between the digits of 1111111111111111 (fifteen 1's) so that the result will be a multiple of 30? | 2002 | 6/8 |
Let the integer part and decimal part of $2+\sqrt{6}$ be $x$ and $y$ respectively. Find the values of $x$, $y$, and the square root of $x-1$. | \sqrt{3} | 7/8 |
In a circle, two perpendicular chords $KM$ and $LN$ are drawn. It is known that lines $KL$ and $MN$ are parallel, and two sides of the quadrilateral $KLMN$ are equal to 2. Find the radius of the circle. | \sqrt{2} | 4/8 |
All old Mother Hubbard had in her cupboard was a Giant Bear chocolate bar. She gave each of her children one-twelfth of the chocolate bar. One third of the bar was left. How many children did she have?
A) 6
B) 8
C) 12
D) 15
E) 18 | 8 | 1/8 |
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