Found problems: 85335
2016 District Olympiad, 4
Consider the triangle $ ABC $ with $ \angle BAC>60^{\circ } $ and $ \angle BCA>30^{\circ } . $ On the other semiplane than that determined by $ BC $ and $ A $ we have the points $ D $ and $ E $ so that
$$ \angle ABE =\angle CBD =\angle BAE +30^{\circ } =\angle BCD +30^{\circ } =90^{\circ } . $$
Note by $ F,H $ the midpoints of $ AE, $ respectively, $ CD, $ and with $ G $ the intersection of $ AC $ and $ DE. $ Show:
[b]a)[/b] $ EBD\sim ABC $
[b]b)[/b] $ FGH\equiv ABC $
1989 Irish Math Olympiad, 4
Let $a$ be a positive real number and let
$b= \sqrt[3] {a+ \sqrt {a^{2}+1}} + \sqrt[3] {a- \sqrt {a^{2}+1}}$.
Prove that $b$ is a positive integer if, and only if, $a$ is a positive integer of the form $\frac{1}{2} n(n^{2}+3)$, for some positive integer $n$.
2006 Stanford Mathematics Tournament, 6
Ten teams of five runners each compete in a cross-country race. A runner finishing in [i]n[/i]th place contributes [i]n[/i] points to his team, and there are no ties. The team with the lowest score wins. Assuming the first place team does not have the same score as any other team, how many winning scores are possible?
2006 Moldova Team Selection Test, 2
Let $C_1$ be a circle inside the circle $C_2$ and let $P$ in the interior of $C_1$, $Q$ in the exterior of $C_2$. One draws variable lines $l_i$ through $P$, not passing through $Q$. Let $l_i$ intersect $C_1$ in $A_i,B_i$, and let the circumcircle of $QA_iB_i$ intersect $C_2$ in $M_i,N_i$. Show that all lines $M_i,N_i$ are concurrent.
2014-2015 SDML (High School), 5
Beth adds the consecutive positive integers $a$, $b$, $c$, $d$, and $e$, and finds that the sum is a perfect square. She then adds $b$, $c$, and $d$ and finds that this sum is a perfect cube. What is the smallest possible value of $e$?
$\text{(A) }47\qquad\text{(B) }137\qquad\text{(C) }227\qquad\text{(D) }677\qquad\text{(E) }1127$
1998 Polish MO Finals, 3
$S$ is a board containing all unit squares in the $xy$ plane whose vertices have integer coordinates and which lie entirely inside the circle $x^2 + y^2 = 1998^2$. In each square of $S$ is written $+1$. An allowed move is to change the sign of every square in $S$ in a given row, column or diagonal. Can we end up with exactly one $-1$ and $+1$ on the rest squares by a sequence of allowed moves?
2006 Moldova National Olympiad, 12.5
Let $ a_{1},a_{2},...,a_{n} $ be real positive numbers and $ k>m, k,m $ natural numbers. Prove that
$(n-1)(a_{1}^m +a_{2}^m+...+a_{n}^m)\leq\frac{a_{2}^k+a_{3}^k+...+a_{n}^k}{a_{1}^{k-m}}+\frac{a_{1}^k+a_{3}^k+...+a_{n}^k}{a_2^{k-m}}+...+\frac{a_{1}^k+a_{2}^k+...+a_{n-1}^k}{a_{n}^{k-m}} $
2021 Iran Team Selection Test, 3
Prove there exist two relatively prime polynomials $P(x),Q(x)$ having integer coefficients and a real number $u>0$ such that if for positive integers $a,b,c,d$ we have:
$$|\frac{a}{c}-1|^{2021} \le \frac{u}{|d||c|^{1010}}$$
$$| (\frac{a}{c})^{2020}-\frac{b}{d}| \le \frac{u}{|d||c|^{1010}}$$
Then we have :
$$bP(\frac{a}{c})=dQ(\frac{a}{c})$$
(Two polynomials are relatively prime if they don't have a common root)
Proposed by [i]Navid Safaii[/i] and [i]Alireza Haghi[/i]
2022 Kyiv City MO Round 2, Problem 4
Points $D, E, F$ are selected on sides $BC, CA, AB$ correspondingly of triangle $ABC$ with $\angle C = 90^\circ$ such that $\angle DAB = \angle CBE$ and $\angle BEC = \angle AEF$. Show that $DB = DF$.
[i](Proposed by Mykhailo Shtandenko)[/i]
1963 All Russian Mathematical Olympiad, 035
Given a triangle $ABC$. We construct two angle bisectors in the corners $A$ and $B$. Than we construct two lines parallel to those ones through the point $C$. $D$ and $E$ are intersections of those lines with the bisectors. It happens, that $(DE)$ line is parallel to $(AB)$. Prove that the triangle is isosceles.
2000 Putnam, 2
Prove that the expression \[ \dfrac {\text {gcd}(m, n)}{n} \dbinom {n}{m} \] is an integer for all pairs of integers $ n \ge m \ge 1 $.
1997 Irish Math Olympiad, 1
Given a positive integer $ n$, denote by $ \sigma (n)$ the sum of all positive divisors of $ n$. We say that $ n$ is $ abundant$ if $ \sigma (n)>2n.$ (For example, $ 12$ is abundant since $ \sigma (12)\equal{}28>2 \cdot 12$.) Let $ a,b$ be positive integers and suppose that $ a$ is abundant. Prove that $ ab$ is abundant.
2012 India IMO Training Camp, 2
Find the least positive integer that cannot be represented as $\frac{2^a-2^b}{2^c-2^d}$ for some positive integers $a, b, c, d$.
2015 District Olympiad, 2
For every real number $ a, $ define the set $ A_a=\left\{ n\in\{ 0\}\cup\mathbb{N}\bigg|\sqrt{n^2+an}\in\{ 0\}\cup\mathbb{N}\right\} . $
[b]a)[/b] Show the equivalence: $ \# A_a\in\mathbb{N}\iff a\neq 0, $ where $ \# B $ is the cardinal of $ B. $
[b]b)[/b] Determine $ \max A_{40} . $
2006 Harvard-MIT Mathematics Tournament, 2
Compute $\displaystyle\lim_{x\to 0}\dfrac{e^{x\cos x}-1-x}{\sin(x^2)}.$
1978 Vietnam National Olympiad, 6
Given a rectangular parallelepiped $ABCDA'B'C'D'$ with the bases $ABCD, A'B'C'D'$, the edges $AA',BB', CC',DD'$ and $AB = a,AD = b,AA' = c$. Show that there exists a triangle with the sides equal to the distances from $A,A',D$ to the diagonal $BD'$ of the parallelepiped. Denote those distances by $m_1,m_2,m_3$. Find the relationship between $a, b, c,m_1,m_2,m_3$.
Putnam 1939, A2
Let $C$ be the curve $y = x^3$ (where $x$ takes all real values). The tangent at $A$ meets the curve again at $B.$ Prove that the gradient at $B$ is $4$ times the gradient at $A.$
2010 National Olympiad First Round, 3
How many real pairs $(x,y)$ are there such that
\[
x^2+2y = 2xy \\
x^3+x^2y = y^2
\]
$ \textbf{(A)}\ 3
\qquad\textbf{(B)}\ 2
\qquad\textbf{(C)}\ 1
\qquad\textbf{(D)}\ 0
\qquad\textbf{(E)}\ \text{None}
$
2002 Switzerland Team Selection Test, 10
Given an integer $m\ge 2$, find the smallest integer $k > m$ such that for any partition of the set $\{m,m + 1,..,k\}$ into two classes $A$ and $B$ at least one of the classes contains three numbers $a,b,c$ (not necessarily distinct) such that $a^b = c$.
2025 CMIMC Algebra/NT, 2
I plotted the graphs $y=(x-0)^2, y=(x-5)^2, \ldots, y=(x-45)^2.$ I also draw a line $y=k,$ and notice that it intersects the parabolas at exactly $19$ distinct points. What is $k$?
1995 Chile National Olympiad, 7
In a semicircle of radius $4$ three circles are inscribed, as indicated in the figure. Larger circles have radii $ R_1 $ and $ R_2 $, and the larger circle has radius $ r $.
a) Prove that $ \dfrac {1} {\sqrt{r}} = \dfrac {1} {\sqrt{R_1}} + \dfrac {1} {\sqrt{R_2}} $
b) Prove that $ R_1 + R_2 \le 8 (\sqrt{2} -1) $
c) Prove that $ r \le \sqrt{2} -1 $
[img]https://cdn.artofproblemsolving.com/attachments/0/9/aaaa65d1f4da4883973751e1363df804b9944c.jpg[/img]
2016 HMNT, 4
Meghal is playing a game with $2016$ rounds $1, 2, ..., 201$6. In round $n$, two rectangular double-sided mirrors are arranged such that they share a common edge and the angle between the faces is $\frac{2\pi}{n+2}$. Meghal shoots a laser at these mirrors and her score for the round is the number of points on the two mirrors at which the laser beam touches a mirror. What is the maximum possible score Meghal could have after she finishes the game?
2001 India IMO Training Camp, 2
Let $p > 3$ be a prime. For each $k\in \{1,2, \ldots , p-1\}$, define $x_k$ to be the unique integer in $\{1, \ldots, p-1\}$ such that $kx_k\equiv 1 \pmod{p}$ and set $kx_k = 1+ pn_k$. Prove that :
\[\sum_{k=1}^{p-1}kn_k \equiv \frac{p-1}{2} \pmod{p}\]
2017 Princeton University Math Competition, A4/B6
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 $m+n$?
1996 Austrian-Polish Competition, 1
Let $k \ge 1$ be a positive integer. Prove that there exist exactly $3^{k-1}$ natural numbers $n$ with the following properties:
(i) $n$ has exactly $k$ digits (in decimal representation),
(ii) all the digits of $n$ are odd,
(iii) $n$ is divisible by $5$,
(iv) the number $m = n/5$ has $k$ odd digits