Found problems: 85335
2005 District Olympiad, 2
Let $ABC$ be a triangle inscribed in a circle of center $O$ and radius $R$. Let $I$ be the incenter of $ABC$, and let $r$ be the inradius of the same triangle, $O\neq I$, and let $G$ be its centroid. Prove that $IG\perp BC$ if and only if $b=c$ or $b+c=3a$.
2003 AIME Problems, 4
In a regular tetrahedron the centers of the four faces are the vertices of a smaller tetrahedron. The ratio of the volume of the smaller tetrahedron to that of the larger is $m/n$, where $m$ and $n$ are relatively prime positive integers. Find $m+n.$
2009 Serbia Team Selection Test, 3
Find the largest natural number $ n$ for which there exist different sets $ S_1,S_2,\ldots,S_n$ such that:
$ 1^\circ$ $ |S_i\cup S_j|\leq 2004$ for each two $ 1\leq i,j\le n$ and
$ 2^\circ$ $ S_i\cup S_j\cup S_k\equal{}\{1,2,\ldots,2008\}$ for each three integers $ 1\le i<j<k\le n$.
2019 Baltic Way, 16
For a positive integer $N$, let $f(N)$ be the number of ordered pairs of positive integers $(a,b)$ such that the number
$$\frac{ab}{a+b}$$
is a divisor of $N$. Prove that $f(N)$ is always a perfect square.
2006 AMC 10, 12
The lines $ x \equal{} \frac 14y \plus{} a$ and $ y \equal{} \frac 14x \plus{} b$ intersect at the point $ (1,2)$. What is $ a \plus{} b$?
$ \textbf{(A) } 0 \qquad \textbf{(B) } \frac 34 \qquad \textbf{(C) } 1 \qquad \textbf{(D) } 2 \qquad \textbf{(E) } \frac 94$
2011 Postal Coaching, 1
Does the sequence \[11, 111, 1111, 11111, \ldots\] contain any fifth power of a positive integer? Justify your answer.
2011 National Olympiad First Round, 10
How many interger tuples $(x,y,z)$ are there satisfying $0\leq x,y,z < 2011$, $xy+yz+zx \equiv 0 \pmod{2011}$, and $x+y+z \equiv 0 \pmod{2011}$ ?
$\textbf{(A)}\ 2010 \qquad\textbf{(B)}\ 2011 \qquad\textbf{(C)}\ 2012 \qquad\textbf{(D)}\ 4021 \qquad\textbf{(E)}\ 4023$
2016 Hanoi Open Mathematics Competitions, 12
In the trapezoid $ABCD, AB // CD$ and the diagonals intersect at $O$. The points $P, Q$ are on $AD, BC$ respectively such that $\angle AP B = \angle CP D$ and $\angle AQB = \angle CQD$. Show that $OP = OQ$.
2020 Romania EGMO TST, P3
Let $ABC$ be an acute scalene triangle. The bisector of the angle $\angle ABC$ intersects the altitude $AD$ at $K$. Let $M$ be the projection of $B$ onto $CK$ and let $N$ be the intersection between $BM$ and $AK$. Let $T$ be a point on $AC$ such that $NT$ is parallel to $DM$. Prove that $BM$ is the bisector of the angle $\angle TBC$.
[i]Melih Üçer, Turkey[/i]
STEMS 2021 Phy Cat B, Q3
[b] Newton's Law of Gravity from Kepler's Laws?[/b]
[list=1]
[*] Planets in the solar system move in elliptic orbits with the sun at one of the foci. [/*]
[*] The line joining the sun and the planet sweeps out equal areas in equal times. [/*]
[*] The period of revolution ($T$) and the length of the semi-major axis $(a$) of the ellipse sit in the relation $T^2/a^3 = constant$. [/*]
[/list]
Now answer the following questions:
[list]
[*] Starting from Newton's Law of Gravitation and Kepler's first law, derive the second and third law. It is possible to derive the first law but that is beyond the scope of this exam. [/*]
[*] For convenience work in the complex (Argand) plane and take the sun to be at the origin $(z=0)$. Show that points on the ellipse may be represented by,
\[ z(\theta) = \frac{a(1-\epsilon^2)}{1+\epsilon\cos\theta}\exp(i\theta) = r(\theta) e^{i\theta}\]
where $a$ is the length of the semi-major axis, $\epsilon$ is the eccentricity of the ellipse and $\theta$ is called the \emph{true anomaly} in celestial mechanics. [/*]
[*] Show that Kepler's second law leads to,
\[ \frac{1}{2}r^2 \dot{\theta} = constant\]
where $r$ and $\theta$ are defined as in part (b) and a dot $(.)$ over a variable denotes its time derivative. What is this constant in terms of the other variables of the problem? [/*]
[*] Using the results of parts (b) and (c) along with Kepler's third law obtain Newton's Law of Gravitation. [/*]
[*] Can the above exercise truly be called a "derivation" of Newton's Law of Gravitation? State your reasons. [/*]
[/list]
2022 MIG, 23
Elax creates a partially filled $4 \times 4$ grid, and is trying to write in positive integers such that any four cells that share no rows and columns always sum to a number $S$. Given that the sum of the numbers in the top row is also $S$, what is the missing cell number?
[asy]
size(100);
add(grid(4,4));
label("$11$", (0.5,1.5));
label("$10$", (0.5,2.5));
label("?", (0.5,3.5));
label("$8$", (1.5,3.5));
label("$7$", (2.5,2.5));
label("$4$", (3.5,0.5));
label("$9$", (3.5,1.5));
[/asy]
$\textbf{(A) }4\qquad\textbf{(B) }6\qquad\textbf{(C) }9\qquad\textbf{(D) }10\qquad\textbf{(E) }12$
1992 Mexico National Olympiad, 4
Show that $1 + 11^{11} + 111^{111} + 1111^{1111} +...+ 1111111111^{1111111111}$ is divisible by $100$.
2019 Mediterranean Mathematics Olympiad, 2
Let $m_1<m_2<\cdots<m_s$ be a sequence of $s\ge2$ positive integers, none of which can be written as the sum of (two or more) distinct other numbers in the sequence. For every integer $r$ with $1\le r<s$, prove that
\[ r\cdot m_r+m_s ~\ge~ (r+1)(s-1). \]
(Proposed by Gerhard Woeginger, Austria)
2016 Germany Team Selection Test, 2
The positive integers $a_1,a_2, \dots, a_n$ are aligned clockwise in a circular line with $n \geq 5$. Let $a_0=a_n$ and $a_{n+1}=a_1$. For each $i \in \{1,2,\dots,n \}$ the quotient \[ q_i=\frac{a_{i-1}+a_{i+1}}{a_i} \] is an integer. Prove \[ 2n \leq q_1+q_2+\dots+q_n < 3n. \]
2022 International Zhautykov Olympiad, 1
Non-zero polynomials $P(x)$, $Q(x)$, and $R(x)$ with real coefficients satisfy the identities
$$ P(x) + Q(x) + R(x) = P(Q(x)) + Q(R(x)) + R(P(x)) = 0. $$
Prove that the degrees of the three polynomials are all even.
2022 Portugal MO, 2
Let $P$ be a point on a circle $C_1$ and let $C_2$ be a circle with center $P$ that intersects $C_1$ at two points Q and R. The circle $C_3$, with center $Q$ and which passes through $R$, intersects $C_2$ at another point S, as in figure. Shows that $QS$ is tangent to $C_1$.
[img]https://cdn.artofproblemsolving.com/attachments/7/5/f48d414c68c33c4efaf4d6c8bebcf6f1fad4ba.png[/img]
2018 Harvard-MIT Mathematics Tournament, 5
Is it possible for the projection of the set of points $(x, y, z)$ with $0 \leq x, y, z \leq 1$ onto some two-dimensional plane to be a simple convex pentagon?
2023 Balkan MO Shortlist, C1
Joe and Penny play a game. Initially there are $5000$ stones in a pile, and the two players remove stones from the pile by making a sequence of moves. On the $k$-th move, any number of stones between $1$ and $k$ inclusive may be removed. Joe makes the odd-numbered moves and Penny makes the even-numbered moves. The player who removes the very last stone is the winner. Who wins if both players play perfectly?
1993 Cono Sur Olympiad, 3
Find the number of elements that a set $B$ can have, contained in $(1, 2, ... , n)$, according to the following property: For any elements $a$ and $b$ on $B$ ($a \ne b$), $(a-b) \not| (a+b)$.
2014 Hanoi Open Mathematics Competitions, 14
Let be given $a < b < c$ and $f(x) =\frac{c(x - a)(x - b)}{(c - a)(c - b)}+\frac{a(x - b)(x - c)}{(a - b)(a -c)}+\frac{b(x -c)(x - a)}{(b - c)(b - a)}$.
Determine $f(2014)$.
2024 HMNT, 4
Albert writes down all of the multiples of $9$ between $9$ and $999,$ inclusive. Compute the sum of the digits he wrote.
1992 All Soviet Union Mathematical Olympiad, 568
A cinema has its seats arranged in $n$ rows $\times m$ columns. It sold mn tickets but sold some seats more than once. The usher managed to allocate seats so that every ticket holder was in the correct row or column. Show that he could have allocated seats so that every ticket holder was in the correct row or column and at least one person was in the correct seat. What is the maximum $k$ such that he could have always put every ticket holder in the correct row or column and at least $k$ people in the correct seat?
1996 Spain Mathematical Olympiad, 6
A regular pentagon is constructed externally on each side of a regular pentagon of side $1$. The figure is then folded and the two edges of the external pentagons meeting at each vertex of the original pentagon are glued together. Find the volume of water that can be poured into the obtained container.
2020/2021 Tournament of Towns, P1
Is it possible that a product of 9 consecutive positive integers is equal to a sum of 9 consecutive (not necessarily the same) positive integers?
[i]Boris Frenkin[/i]
2022 AMC 12/AHSME, 9
The sequence $a_0,a_1,a_2,\cdots$ is a strictly increasing arithmetic sequence of positive integers such that
\[2^{a_7}=2^{27} \cdot a_7.\]
What is the minimum possible value of $a_2$?
$\textbf{(A)}8~\textbf{(B)}12~\textbf{(C)}16~\textbf{(D)}17~\textbf{(E)}22$