Found problems: 85335
Prove that the sum of the lengths of the medians of a triangle is at least three quarters of its perimeter.
For any integer $n+1,\ldots, 2n$ ($n$ is a natural number) consider its greatest odd divisor. Prove that the sum of all these divisors equals $n^2.$
$n$ is a positive integer, $F_n=2^{2^{n}}+1$. Prove that for $n \geq 3$, there exists a prime factor of $F_n$ which is larger than $2^{n+2}(n+1)$.
[u]Set 4[/u]
[b]G10.[/b] Let $ABCD$ be a square with side length $1$. It is folded along a line $\ell$ that divides the square into two pieces with equal area. The minimum possible area of the resulting shape is $A$. Find the integer closest to $100A$.
[b]G11.[/b] The $10$-digit number $\underline{1A2B3C5D6E}$ is a multiple of $99$. Find $A + B + C + D + E$.
[b]G12.[/b] Let $A, B, C, D$ be four points satisfying $AB = 10$ and $AC = BC = AD = BD = CD = 6$. If $V$ is the volume of tetrahedron $ABCD$, then find $V^2$.
[u]Set 5[/u]
[b]G13.[/b] Nate the giant is running a $5000$ meter long race. His first step is $4$ meters, his next step is $6$ meters, and in general, each step is $2$ meters longer than the previous one. Given that his $n$th step will get him across the finish line, find $n$.
[b]G14.[/b] In square $ABCD$ with side length $2$, there exists a point $E$ such that $DA = DE$. Let line $BE$ intersect side $AD$ at $F$ such that $BE = EF$. The area of $ABE$ can be expressed in the form $a -\sqrt{b}$ where $a$ is a positive integer and $b$ is a square-free integer. Find $a + b$.
[b]G15.[/b] Patrick the Beetle is located at $1$ on the number line. He then makes an infinite sequence of moves where each move is either moving $1$, $2$, or $3$ units to the right. The probability that he does reach $6$ at some point in his sequence of moves is $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Find $m + n$.
[u]Set 6[/u]
[b]G16.[/b] Find the smallest positive integer $c$ greater than $1$ for which there do not exist integers $0 \le x, y \le9$ that satisfy $2x + 3y = c$.
[b]G17.[/b] Jaeyong is on the point $(0, 0)$ on the coordinate plane. If Jaeyong is on point $(x, y)$, he can either walk to $(x + 2, y)$, $(x + 1, y + 1)$, or $(x, y + 2)$. Call a walk to $(x + 1, y + 1)$ an Brilliant walk. If Jaeyong cannot have two Brilliant walks in a row, how many ways can he walk to the point $(10, 10)$?
[b]G18.[/b] Deja vu?
Let $ABCD$ be a square with side length $1$. It is folded along a line $\ell$ that divides the square into two pieces with equal area. The maximum possible area of the resulting shape is $B$. Find the integer closest to $100B$.
PS. You should use hide for answers. Sets 1-3 have been posted [url=https://artofproblemsolving.com/community/c3h3131303p28367061]here [/url] and 7-9 [url=https://artofproblemsolving.com/community/c3h3131308p28367095]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
Four ambassadors and one advisor for each of them are to be seated at a round table with $12$ chairs numbered in order from $1$ to $12$. Each ambassador must sit in an even-numbered chair. Each advisor must sit in a chair adjacent to his or her ambassador. There are $N$ ways for the $8$ people to be seated at the table under these conditions. Find the remainder when $N$ is divided by $1000$.
On the plane is a set of at least four points. If any one point from this set is removed, the resulting set has an axis of symmetry. Is it necessarily true that the whole set has an axis of symmetry?
Find the smallest positive integer $n$ or show no such $n$ exists, with the following property: there are infinitely many distinct $n$-tuples of positive rational numbers $(a_1, a_2, \ldots, a_n)$ such that both
$$a_1+a_2+\dots +a_n \quad \text{and} \quad \frac{1}{a_1} + \frac{1}{a_2} + \dots + \frac{1}{a_n}$$
are integers.
Let $\omega$ be the incircle of triangle $ABC$ and $\omega$ touches $BC$ at $D$. $AD$ meets $\omega$ again at $L$. Let $K$ be $A$-excenter, and $M,N$ be the midpoint of $BC,KM$, respectively. Prove that $B,C,N,L$ are concyclic.
Consider the infinite grid of lattice points in $\mathbb{Z}^3$. Little D and Big Z play a game, where Little D first loses a shoe on an unmunched point in the grid. Then, Big Z munches a shoe-free plane perpendicular to one of the coordinate axes. They continue to alternate turns in this fashion, with Little D's goal to lose a shoe on each of $n$ consecutive lattice points on a line parallel to one of the coordinate axes. Determine all $n$ for which Little D can accomplish his goal.
[i]David Yang.[/i]
Let $ O$ be an interior point of the triangle $ ABC$. Prove that there exist positive integers $ p,q$ and $ r$ such that
\[ |p\cdot\overrightarrow{OA} \plus{} q\cdot\overrightarrow{OB} \plus{} r\cdot\overrightarrow{OC}|<\frac{1}{2007}\]
For any real numbers $a$ and $b>0$, define an [i]extension[/i] of an interval $[a-b,a+b] \subseteq \mathbb{R}$ be $[a-2b, a+2b]$. We say that $P_1, P_2, \ldots, P_k$ covers the set $X$ if $X \subseteq P_1 \cup P_2 \cup \ldots \cup P_k$.
Prove that there exists an integer $M$ with the following property: for every finite subset $A \subseteq \mathbb{R}$, there exists a subset $B \subseteq A$ with at most $M$ numbers, so that for every $100$ closed intervals that covers $B$, their extensions covers $A$.
Given a Pacman of radius $1$, and mouth opening angle $90^\circ$, what is the largest (circular) pellet it can eat? The pellet must lie entirely outside the yellow portion and entirely inside the circumcircle of the Pacman. Let the radius be equal to $a\sqrt b+c$. where $b$ is square free. Find $a+b+c$.
An $m\times n\times p$ rectangular box has half the volume of an $(m+2)\times(n+2)\times(p+2)$ rectangular box, where $m, n,$ and $p$ are integers, and $m\le n\le p.$ What is the largest possible value of $p$?
Let $x, y$, and $z$ be real numbers satisfying $x + y + z = xyz.$ Prove that
\[x(1 - y^2)(1 - z^2) + y(1 -z^2)(1 - x^2) + z(1 - x^2)(1 - y^2) = 4xyz.\]
Let $I$ be the incentre of acute-angled triangle $ABC$. Let the incircle meet $BC, CA$, and $AB$ at $D, E$, and $F,$ respectively. Let line $EF$ intersect the circumcircle of the triangle at $P$ and $Q$, such that $F$ lies between $E$ and $P$. Prove that $\angle DPA + \angle AQD =\angle QIP$.
(Slovakia)
Determine all four-digit numbers $\overline{abcd} $ such that
$(a + b)(a + c)(a + d)(b + c)(b + d)(c + d) =\overline{abcd} $:
Two farmers agree that pigs are worth $ \$300$ and that goats are worth $ \$210$. When one farmer owes the other money, he pays the debt in pigs or goats, with ``change'' received in the form of goats or pigs as necessary. (For example, a $ \$390$ debt could be paid with two pigs, with one goat received in change.) What is the amount of the smallest positive debt that can be resolved in this way?
$ \textbf{(A) } \$5\qquad \textbf{(B) } \$10\qquad \textbf{(C) } \$30\qquad \textbf{(D) } \$90\qquad \textbf{(E) } \$210$
If $n$ is a positive integer such that $8n+1$ is a perfect square, then
$\textbf{(A)}~n\text{ must be odd}$
$\textbf{(B)}~n\text{ cannot be a perfect square}$
$\textbf{(C)}~n\text{ cannot be a perfect square}$
$\textbf{(D)}~\text{None of the above}$
How many triangles formed by three vertices of a regular $17$-gon are obtuse?
$\text{(A) }156\qquad\text{(B) }204\qquad\text{(C) }357\qquad\text{(D) }476\qquad\text{(E) }524$
Find all functions from $\mathbb{N}\cup\{0\}\to\mathbb{N}\cup\{0\}$ such that $f(m^2+mf(n))=mf(m+n)$, for all $m,n\in \mathbb{N}\cup\{0\}$.
Let $\triangle ABC$ be a right triangle with right angle $\angle B$. Suppose the angle bisector $l$ of $B$ divides the hypotenuse $AC$ into two segments of length $\sqrt{3}-1$ and $\sqrt{3}+1$. What is the measure of the smaller angle between $l$ and $AC$, in radians?
A prime number $p > 2$ and $x,y \in \left\{ 1,2,\ldots, \frac{p-1}{2} \right\}$ are given. Prove that if
$x\left( p-x\right)y\left( p-y\right)$ is a perfect square, then $x = y$.
$ A$ can do a piece of work in $ 9$ days. $ B$ is $ 50\%$ more efficient than $ A$. The number of days it takes $ B$ to do the same piece of work is:
$ \textbf{(A)}\ 13\frac {1}{2} \qquad\textbf{(B)}\ 4\frac {1}{2} \qquad\textbf{(C)}\ 6 \qquad\textbf{(D)}\ 3 \qquad\textbf{(E)}\ \text{none of these answers}$
Let $ABC$ be an acute-angled triangle. Let $D$ and $E$ be the midpoints of sides $\overline{AB}$ and $\overline{AC}$ respectively. Let $F$ be the point such that $D$ is the midpoint of $\overline{EF}$. Let $\Gamma$ be the circumcircle of triangle $FDB$. Let $G$ be a point on the segment $\overline{CD}$ such that the midpoint of $\overline{BG}$ lies on $\Gamma$. Let $H$ be the second intersection of $\Gamma$ and $FC$. Show that the quadrilateral $BHGC$ is cyclic. \\ \\ [i]Proposed by Art Waeterschoot.[/i]
Let $S$ be a nonempty finite set, and $\mathcal {F}$ be a collection of subsets of $S$ such that the following conditions are met:
(i) $\mathcal {F}$ $\setminus$ {$S$} $\neq$ $\emptyset$ ;
(ii) if $F_1, F_2 \in \mathcal {F}$, then $F_1 \cap F_2 \in \mathcal {F}$ and $F_1 \cup F_2 \in \mathcal {F}$.
Prove that there exists $a \in S$ which belongs to at most half of the elements of $\mathcal {F}$.