Found problems: 85335
The five sides and five diagonals of a regular pentagon are drawn on a piece of paper. Two people play a game, in which they take turns to colour one of these ten line segments. The first player colours line segments blue, while the second player colours line segments red. A player cannot colour a line segment that has already been coloured. A player wins if they are the first to create a triangle in their own colour, whose three vertices are also vertices of the regular pentagon. The game is declared a draw if all ten line segments have been coloured without a player winning. Determine whether the first player, the second player, or neither player can force a win.
Let $n \geq 3$ be an integer. Let $t_1$, $t_2$, ..., $t_n$ be positive real numbers such that \[n^2 + 1 > \left( t_1 + t_2 + \cdots + t_n \right) \left( \frac{1}{t_1} + \frac{1}{t_2} + \cdots + \frac{1}{t_n} \right).\] Show that $t_i$, $t_j$, $t_k$ are side lengths of a triangle for all $i$, $j$, $k$ with $1 \leq i < j < k \leq n$.
Let $ABCD$ be a parallelogram which is not a rectangle and $E$ be the point in its plane such that $AE \perp AB$ and $CE \perp CB$. Prove that $\angle DEA = \angle CEB$.
Define the numbers $a_0, a_1, \ldots, a_n$ in the following way:
\[ a_0 = \frac{1}{2}, \quad a_{k+1} = a_k + \frac{a^2_k}{n} \quad (n > 1, k = 0,1, \ldots, n-1). \]
Prove that \[ 1 - \frac{1}{n} < a_n < 1.\]
Let the [i]taxicab distance[/i] between points $(x_1,y_1)$ and $(x_2,y_2)$ in the coordinate plane is given by $|x_1-x_2|+|y_1-y_2|$. For how many points $P$ with integer coordinates is the taxicab distance between $P$ and the origin less than or equal to $20$?
$\textbf{(A) }441\qquad\textbf{(B) }761\qquad\textbf{(C) }841\qquad\textbf{(D) }921\qquad\textbf{(E) }924$
A divisor $d$ of a positive integer $n$ is said to be a [i]close[/i] divisor of $n$ if $\sqrt{n}<d<2\sqrt{n}$. Does there exist a positive integer with exactly $2020$ close divisors?
Suppose that $x,y$ and $\frac{x^2+y^2+6}{xy}$ are positive integers . Prove that $\frac{x^2+y^2+6}{xy}$ is a perfect cube.
Show that
\[-2 \le \cos \theta\left(\sin \theta + \sqrt{\sin ^2 \theta +3}\right) \le 2\]
for all values of $\theta$.
Find the maximum number of subsets from $\left \{ 1,...,n \right \}$ such that for any two of them like $A,B$ if $A\subset B$ then $\left | B-A \right |\geq 3$. (Here $\left | X \right |$ is the number of elements of the set $X$.)
Suppose $a$, $b$, $c$, and $d$ are positive real numbers that satisfy the system of equations \begin{align*}(a+b)(c+d)&=143,\\(a+c)(b+d)&=150,\\(a+d)(b+c)&=169.\end{align*} Compute the smallest possible value of $a^2+b^2+c^2+d^2$.
In triangle $ABC$, the angle bisectors $AD$, $BE$ and $CF$ are drawn, intersecting at point $I$. The perpendicular bisector of the segment $AD$ intersects lines $BE$ and $CF$ at points $M$ and $N$, respectively. Prove that points $A$, $I$, $M$ and $ N$ lie on the same circle.
Let the regular hexagon $ABCDEF$ be inscribed in a circle with center $O$, $N$ be such a point Let $E-N-C$, $M$ a point such that $A- M-C$ and $R$ a point on the circumference, such that $D-N- R$. If $\angle EFR = 90^o$, $\frac{AM}{AC}=\frac{CN}{EC}$ and $AC=\sqrt3$, calculate $AM$.
Notation: $A-B-C$ means than points $A,B,C$ are collinear in that order i.e. $ B$ lies between $ A$ and $C$.
How many pairs $(m,n)$ of integers satisfy the equation $m+n=mn$?
$\textbf{(A) }1\qquad\textbf{(B) }2\qquad\textbf{(C) }3\qquad\textbf{(D) }4\qquad \textbf{(E) }\text{more than }4$
Let \( ABC \) be an acute-angled scalene triangle with circumcenter \( O \). Denote by \( M \), \( N \), and \( P \) the midpoints of sides \( BC \), \( CA \), and \( AB \), respectively. Let \( \omega \) be the circle passing through \( A \) and tangent to \( OM \) at \( O \). The circle \( \omega \) intersects \( AB \) and \( AC \) at points \( E \) and \( F \), respectively (where \( E \) and \( F \) are distinct from \( A \)). Let \( I \) be the midpoint of segment \( EF \), and let \( K \) be the intersection of lines \( EF \) and \( NP \). Prove that \( AO = 2IK \) and that triangle \( IMO \) is isosceles.
An equilateral triangle is tiled with $n^2$ smaller congruent equilateral triangles such that there are $n$ smaller triangles along each of the sides of the original triangle. For each of the small equilateral triangles, we randomly choose a vertex $V$ of the triangle and draw an arc with that vertex as center connecting the midpoints of the two sides of the small triangle with $V$ as an endpoint. Find, with proof, the expected value of the number of full circles formed, in terms of $n.$
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Does there exist a convex pentagon from which a similar pentagon can be cut off by a straight line?
(S Tokarev)
If real numbers $x$, $y$, and $z$ satisfy $x^2-yz = 1$ and $y^2-xz = 4$ such that $|x+y+z|$ is minimized, then $z^2-xy$ can be expressed in the form $\sqrt{a}-b$ where $a$ and $b$ are positive integers. Find $a+b$.
[i]Proposed by Andy Xu[/i]
For a positive integer that doesn’t end in $0$, define its reverse to be the number formed by reversing its digits. For instance, the reverse of $102304$ is $403201$. In terms of $n \geq 1$, how many numbers when added to its reverse give $10^{n}-1$, the number consisting of $n$ nines?
Find the minimum of $\sum\limits_{k=0}^{40} \left(x+\frac{k}{2}\right)^2$ where $x$ is a real numbers
Find the number of ordered pairs $(a,b)$ such that $a,b \in \{10,11,\cdots,29,30\}$ and \\
$\hspace{1cm}$ $GCD(a,b)+LCM(a,b)=a+b$.
A1,A2,...,Am are subsets of X and we have |Ai|=mk (m,k natural numbers)
prove that we can separate X into k sets such that every set has at least one member of each Ai.
Let $ABC$ be an isosceles triangle with $AB=BC$. A trisector of $\angle B$ meets $AC$ at $D$. If $AB,AC$ and $BD$ are integers and $AB-BD$ $=$ $3$, find $AC$.
Compute $g(2005)$ where $g$ is a function defined on the natural numbers that has the following properties:
i) $g(1) = 0$
ii) $g(nm) = g(n) + g(m) + g(n)g(m)$ for any pair of integers $n, m$.
iii) $g(n^2 + 1) = (g(n) + 1)^2$ for every integer $n$.
Show that the system of equations
\begin{align*}
\lfloor x\rfloor^2+\lfloor y\rfloor &=0, \\
3x+y &=2,
\end{align*}
has infinitely many solutions and all these solutions satisfy bounds
\begin{align*}
0<\ &x <4, \\
-9\le\ &y\le 1.
\end{align*}
Let $ABCD$ be a convex quadrilateral with $AB=2, AD=7,$ and $CD=3$ such that the bisectors of acute angles $\angle{DAB}$ and $\angle{ADC}$ intersect at the midpoint of $\overline{BC}.$ Find the square of the area of $ABCD.$