Found problems: 85335
2004 Junior Balkan Team Selection Tests - Romania, 3
Let $p, q, r$ be primes and let $n$ be a positive integer such that $p^n + q^n = r^2$. Prove that $n = 1$.
Laurentiu Panaitopol
2005 Iran MO (3rd Round), 2
Suppose $\{x_n\}$ is a decreasing sequence that $\displaystyle\lim_{n \rightarrow\infty}x_n=0$. Prove that $\sum(-1)^nx_n$ is convergent
2012 Junior Balkan Team Selection Tests - Moldova, 1
Let $ 1\leq a,b,c,d,e,f,g,h,k \leq 9 $ and $ a,b,c,d,e,f,g,h,k $ are different integers, find the minimum value of the expression $ E = a*b*c+d*e*f+g*h*k $ and prove that it is minimum.
2021-IMOC, N4
There are $m \geq 3$ positive integers, not necessarily distinct, that are arranged in a circle so that any positive integer divides the sum of its neighbours.
Show that if there is exactly one $1$, then for any positive integer $n$, there are at most $\phi(n)$ copies
of $n$.
[i]Proposed By- (usjl, adapted from 2014 Taiwan TST)[/i]
2016 BMT Spring, 7
Find the coefficient of $x^2$ in the following polynomial
$$(1 -x)^2(1 + 2x)^2(1 - 3x)^2... (1 -11x)^2.$$
2010 Romania National Olympiad, 4
Let $f:[-1,1]\to\mathbb{R}$ be a continuous function having finite derivative at $0$, and
\[I(h)=\int^h_{-h}f(x)\text{ d}x,\ h\in [0,1].\]
Prove that
a) there exists $M>0$ such that $|I(h)-2f(0)h|\le Mh^2$, for any $h\in [0,1]$.
b) the sequence $(a_n)_{n\ge 1}$, defined by $a_n=\sum_{k=1}^n\sqrt{k}|I(1/k)|$, is convergent if and only if $f(0)=0$.
[i]Calin Popescu[/i]
2007 ITest, 10
My grandparents are Arthur, Bertha, Christoph, and Dolores. My oldest grandparent is only $4$ years older than my youngest grandparent. Each grandfather is two years older than his wife. If Bertha is younger than Dolores, what is the difference between Bertha's age and the mean of my grandparents’ ages?
$\textbf{(A) }0\hspace{14em}\textbf{(B) }1\hspace{14em}\textbf{(C) }2$
$\textbf{(D) }3\hspace{14em}\textbf{(E) }4\hspace{14em}\textbf{(F) }5$
$\textbf{(G) }6\hspace{14em}\textbf{(H) }7\hspace{14em}\textbf{(I) }8$
$\textbf{(J) }2007$
1974 Putnam, A2
A circle stands in a plane perpendicular to the ground and a point $A$ lies in this plane exterior to the circle and higher than its bottom. A particle starting from rest at $A$ slides without friction down an inclined straight line until it reaches the circle. Which straight line allows descent in the shortest time?
1968 Miklós Schweitzer, 10
Let $ h$ be a triangle of perimeter $ 1$, and let $ H$ be a triangle of perimeter $ \lambda$ homothetic to $ h$. Let $ h_1,h_2,...$ be translates of $ h$ such that , for all $ i$, $ h_i$ is different from $ h_{i\plus{}2}$ and touches $ H$ and $ h_{i\plus{}1}$ (that is, intersects without overlapping). For which values of $ \lambda$ can these triangles be chosen so that the sequence $ h_1,h_2,...$ is periodic? If $ \lambda \geq 1$ is such a value, then determine the number of different triangles in a periodic
chain $ h_1,h_2,...$ and also the number of times such a chain goes around the triangle $ H$.
[i]L. Fejes-Toth[/i]
1983 Poland - Second Round, 3
The point $ P $ lies inside the triangle $ ABC $, with $ \measuredangle PAC = \measuredangle PBC $. The points $ L $ and $ M $ are the projections $ P $ onto the lines $ BC $ and $ CA $, respectively, $ D $ is the midpoint of the segment $ AB $. Prove that $ DL = DM $.
2020 BMT Fall, 8
By default, iPhone passcodes consist of four base-$10$ digits. However, Freya decided to be unconventional and use hexadecimal (base-$16$) digits instead of base-$10$ digits! (Recall that $10_{16} = 16_{10}$.) She sets her passcode such that exactly two of the hexadecimal digits are prime. How many possible passcodes could she have set?
2013 IMC, 1
Let $\displaystyle{z}$ be a complex number with $\displaystyle{\left| {z + 1} \right| > 2}$. Prove that $\displaystyle{\left| {{z^3} + 1} \right| > 1}$.
[i]Proposed by Walther Janous and Gerhard Kirchner, Innsbruck.[/i]
2016 Gulf Math Olympiad, 4
4. Suppose that four people A, B, C and D decide to play games of tennis
doubles. They might first play the team A and B against the team C
and D. Next A and C might play B and D. Finally A and D might play
B and C. The advantage of this arrangement is that two conditions are
satisfied.
(a) Each player is on the same team as each other player exactly once.
(b) Each player is on the opposing team to each other player exactly
twice.
Is it possible to arrange a collection of tennis matches satisfying both
condition (a) and condition (b) in the following circumstances?
(i) There are five players.
(ii) There are seven players.
(iii) There are nine players.
2015 Princeton University Math Competition, A2/B3
What is the sum of all positive integers $n$ such that $\text{lcm}(2n, n^2) = 14n - 24$?
2015 Bosnia And Herzegovina - Regional Olympiad, 2
For positive integer $n$, find all pairs of coprime integers $p$ and $q$ such that $p+q^2=(n^2+1)p^2+q$
2005 Hungary-Israel Binational, 1
Does there exist a sequence of $2005$ consecutive positive integers that
contains exactly $25$ prime numbers?
1999 National Olympiad First Round, 35
Flights are arranged between 13 countries. For $ k\ge 2$, the sequence $ A_{1} ,A_{2} ,\ldots A_{k}$ is said to a cycle if there exist a flight from $ A_{1}$ to $ A_{2}$, from $ A_{2}$ to $ A_{3}$, $ \ldots$, from $ A_{k \minus{} 1}$ to $ A_{k}$, and from $ A_{k}$ to $ A_{1}$. What is the smallest possible number of flights such that how the flights are arranged, there exist a cycle?
$\textbf{(A)}\ 14 \qquad\textbf{(B)}\ 53 \qquad\textbf{(C)}\ 66 \qquad\textbf{(D)}\ 79 \qquad\textbf{(E)}\ 156$
2020 USMCA, 29
Let $ABC$ be a triangle with circumcircle $\Gamma$ and let $D$ be the midpoint of minor arc $BC$. Let $E, F$ be on $\Gamma$ such that $DE \bot AC$ and $DF \bot AB$. Lines $BE$ and $DF$ meet at $G$, and lines $CF$ and $DE$ meet at $H$. Given that $AB = 8, AC = 10$, and $\angle BAC = 60^\circ$, find the area of $BCHG$.
[i] Note: this is a modified version of Premier #2 [/i]
2021 CHMMC Winter (2021-22), 7
Let $ABC$ be a triangle with $AB = 5$, $BC = 6$, and $CA = 7$. Denote $\Gamma$ the incircle of $ABC$, let $I$ be the center of $\Gamma$ . The circumcircle of $BIC$ intersects $\Gamma$ at $X_1$ and $X_2$. The circumcircle of $CIA$ intersects $\Gamma$ at $Y_1$ and $Y_2$. The circumcircle of $AIB$ intersects $\Gamma$ at $Z_1$ and $Z_2$. The area of the triangle determined by $\overline{X_1X_2}$, $\overline{Y_1Y_2}$, and $\overline{Z_1Z_2}$ equals $\frac{m \sqrt{p}}{n}$ for positive integers $m, n$, and $p$, where $m$ and$ n$ are relatively prime and $p$ is squarefree.
Compute $m+n+ p$.
2010 Contests, 4
Solid camphor is insoluble in water but is soluble in vegetable oil. The best explanation for this behavior is that camphor is a(n)
${ \textbf{(A)}\ \text{Ionic solid} \qquad\textbf{(B)}\ \text{Metallic solid} \qquad\textbf{(C)}\ \text{Molecular solid} \qquad\textbf{(D)}\ \text{Network solid} } $
1999 IMO Shortlist, 8
Given a triangle $ABC$. The points $A$, $B$, $C$ divide the circumcircle $\Omega$ of the triangle $ABC$ into three arcs $BC$, $CA$, $AB$. Let $X$ be a variable point on the arc $AB$, and let $O_{1}$ and $O_{2}$ be the incenters of the triangles $CAX$ and $CBX$. Prove that the circumcircle of the triangle $XO_{1}O_{2}$ intersects the circle $\Omega$ in a fixed point.
2004 AMC 8, 10
Handy Aaron helped a neighbor $1\frac{1}{4}$ hours on Monday, $50$ minutes on Tuesday, from $8:20$ to $10:45$ on Wednesday morning, and a half-hour on Friday. He is paid $\$3$ per hour. How much did he earn for the week?
$\textbf{(A)}\ 8\qquad
\textbf{(B)}\ 9\qquad
\textbf{(C)}\ 10\qquad
\textbf{(D)}\ 12\qquad
\textbf{(E)}\ 15$
1988 IMO Shortlist, 12
In a triangle $ ABC,$ choose any points $ K \in BC, L \in AC, M \in AB, N \in LM, R \in MK$ and $ F \in KL.$ If $ E_1, E_2, E_3, E_4, E_5, E_6$ and $ E$ denote the areas of the triangles $ AMR, CKR, BKF, ALF, BNM, CLN$ and $ ABC$ respectively, show that
\[ E \geq 8 \cdot \sqrt [6]{E_1 E_2 E_3 E_4 E_5 E_6}.
\]
2015 JBMO Shortlist, 2
The point ${P}$ is outside the circle ${\Omega}$. Two tangent lines, passing from the point ${P}$ touch the circle ${\Omega}$ at the points ${A}$ and ${B}$. The median${AM \left(M\in BP\right)}$ intersects the circle ${\Omega}$ at the point ${C}$ and the line ${PC}$ intersects again the circle ${\Omega}$ at the point ${D}$. Prove that the lines ${AD}$ and ${BP}$ are parallel.
(Moldova)
2012 Saint Petersburg Mathematical Olympiad, 5
$n \geq k$ -two natural numbers. $S$ -such natural, that have not less than $n$ divisors. All divisors of $S$ are written
in descending order. What minimal number of divisors can have number from $k$-th place ?