Found problems: 85335
2006 Brazil National Olympiad, 2
Let $n$ be an integer, $n \geq 3$. Let $f(n)$ be the largest number of isosceles triangles whose vertices belong to some set of $n$ points in the plane without three colinear points. Prove that there exists positive real constants $a$ and $b$ such that $an^{2}< f(n) < bn^{2}$ for every integer $n$, $n \geq 3$.
1994 Bundeswettbewerb Mathematik, 4
Let $a,b$ be real numbers ($b\ne 0$) and consider the infinite arithmetic sequence $a, a+b ,a +2b , \ldots.$ Show that this sequence contains an infinite geometric subsequence if and only if $\frac{a}{b}$ is rational.
1909 Eotvos Mathematical Competition, 2
Show that the radian measure of an acute angle is less than the arithmetic mean of its sine and its tangent.
2018 Hanoi Open Mathematics Competitions, 15
There are $n$ distinct straight lines on a plane such that every line intersects exactly $12$ others. Determine all the possible values of $n$.
2022 CCA Math Bonanza, L3.1
Kongol rolls two fair 6-sided die. The probability that one roll is a divisor of the other can be expressed as $\frac{p}{q}$. Determine $p+q$.
[i]2022 CCA Math Bonanza Lightning Round 3.1[/i]
2017 BMT Spring, 8
The numerical value of the following integral $$\int^1_0 (-x^2 + x)^{2017} \lfloor 2017x \rfloor dx$$ can be expressed in the form $a\frac{m!^2}{ n!}$ where a is minimized. Find $a + m + n$.
(Note $\lfloor x\rfloor$ is the largest integer less than or equal to x.)
2009 German National Olympiad, 3
Let $ ABCD$ be a (non-degenerate) quadrangle and $ N$ the intersection of $ AC$ and $ BD$. Denote by $ a,b,c,d$ the length of the altitudes from $ N$ to $ AB,BC,CD,DA$, respectively.
Prove that $ \frac{1}{a}\plus{}\frac{1}{c} \equal{} \frac{1}{b}\plus{}\frac{1}{d}$ if $ ABCD$ has an incircle.
Extension: Prove that the converse is true, too.
[If this has already been posted, I humbly apologize. A quick search turned up nothing.]
2006 Purple Comet Problems, 15
A snowman is built on a level plane by placing a ball radius $6$ on top of a ball radius $8$ on top of a ball radius $10$ as shown. If the average height above the plane of a point in the snowman is $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers, find $m + n$.
[asy]
size(150);
draw(circle((0,0),24));
draw(ellipse((0,0),24,9));
draw(circle((0,-56),32));
draw(ellipse((0,-56),32,12));
draw(circle((0,-128),40));
draw(ellipse((0,-128),40,15));
[/asy]
1991 IMTS, 5
Prove that there are infinitely many positive integers $n$ such that $n \times n \times n$ can not be filled completely with 2 x 2 x 2 and 3 x 3 x 3 solid cubes.
2021 Taiwan TST Round 1, N
For each positive integer $n$, define $V_n=\lfloor 2^n\sqrt{2020}\rfloor+\lfloor 2^n\sqrt{2021}\rfloor$. Prove that, in the sequence $V_1,V_2,\ldots,$ there are infinitely many odd integers, as well as infinitely many even integers.
[i]Remark.[/i] $\lfloor x\rfloor$ is the largest integer that does not exceed the real number $x$.
2002 USAMTS Problems, 4
Let $f(n)$ be the number of ones that occur in the decimal representations of all the numbers from 1 to $n$. For example, this gives $f(8)=1$, $f(9)=1$, $f(10)=2$, $f(11)=4$, and $f(12)=5$. Determine the value of $f(10^{100})$.
2014 Harvard-MIT Mathematics Tournament, 14
Let $ABCD$ be a trapezoid with $AB\parallel CD$ and $\angle D=90^\circ$. Suppose that there is a point $E$ on $CD$ such that $AE=BE$ and that triangles $AED$ and $CEB$ are similar, but not congruent. Given that $\tfrac{CD}{AB}=2014$, find $\tfrac{BC}{AD}$.
2014 ASDAN Math Tournament, 1
Compute the smallest positive integer that is $3$ more than a multiple of $5$, and twice a multiple of $6$.
2021 Federal Competition For Advanced Students, P1, 2
Let $ABC$ denote a triangle. The point $X$ lies on the extension of $AC$ beyond $A$, such that $AX = AB$. Similarly, the point $Y$ lies on the extension of $BC$ beyond $B$ such that $BY = AB$. Prove that the circumcircles of $ACY$ and $BCX$ intersect a second time in a point different from $C$ that lies on the bisector of the angle $\angle BCA$.
(Theresia Eisenkölbl)
2023 USAMTS Problems, 5
Let $m$ and $n$ be positive integers. Let $S$ be the set of all points $(x, y)$ with integer
coordinates such that $1 \leq x,y \leq m+n-1$ and $m+1 \leq x +y \leq 2m+n-1.$ Let $L$ be the
set of the $3m+3n-3$ lines parallel to one of $x = 0, y = 0,$ or $x + y = 0$ and passing through
at least one point in $S$. For which pairs $(m, n)$ does there exist a subset $T$ of $S$ such that
every line in $L$ intersects an odd number of elements of $T$?
2020 LMT Fall, B5
Given the following system of equations
$a_1 + a_2 + a_3 = 1$
$a_2 + a_3 + a_4 = 2$
$a_3 + a_4 + a_5 = 3$
$...$
$a_{12} + a_{13} + a_{14} = 12$
$a_{13} + a_{14} + a_1 = 13$
$a_{14 }+ a_1 + a_2 = 14$
find the value of $a_{14}$.
2008 ITest, 46
Let $S$ be the sum of all $x$ in the interval $[0,2\pi)$ that satisfy \[\tan^2 x - 2\tan x\sin x=0.\] Compute $\lfloor10S\rfloor$.
2008 Postal Coaching, 1
Let $ABCD$ be a quadrilateral that can be inscribed in a circle. Denote by $P$ the intersection point of lines $AD$ and $BC$, and by $Q$ the intersection point of lines $AB$ and $DC$. Let $E$ be the fourth vertex of the parallelogram $ABCE$, and $F$ the intersection of lines $CE$ is $PQ$. Prove that the points $D,E, F$, and $Q$ lie on the same circle.
2008 Germany Team Selection Test, 2
Point $ P$ lies on side $ AB$ of a convex quadrilateral $ ABCD$. Let $ \omega$ be the incircle of triangle $ CPD$, and let $ I$ be its incenter. Suppose that $ \omega$ is tangent to the incircles of triangles $ APD$ and $ BPC$ at points $ K$ and $ L$, respectively. Let lines $ AC$ and $ BD$ meet at $ E$, and let lines $ AK$ and $ BL$ meet at $ F$. Prove that points $ E$, $ I$, and $ F$ are collinear.
[i]Author: Waldemar Pompe, Poland[/i]
2006 AMC 10, 22
Elmo makes $ N$ sandwiches for a fundraiser. For each sandwich he uses $ B$ globs of peanut butter at 4 cents per glob and $ J$ blobs of jam at 5 cents per glob. The cost of the peanut butter and jam to make all the sandwiches is $ \$$2.53. Assume that $ B, J,$ and $ N$ are all positive integers with $ N > 1$. What is the cost of the jam Elmo uses to make the sandwiches?
$ \textbf{(A) } \$1.05 \qquad \textbf{(B) } \$1.25 \qquad \textbf{(C) } \$1.45 \qquad \textbf{(D) } \$1.65 \qquad \textbf{(E) } \$1.85$
2007 JBMO Shortlist, 5
The real numbers $x,y,z, m, n$ are positive, such that $m + n \ge 2$. Prove that
$x\sqrt{yz(x + my)(x + nz)} + y\sqrt{xz(y + mx)(y + nz)} + z\sqrt{xy(z + mx)(x + ny) }\le \frac{3(m + n)}{8}
(x + y)(y + z)(z + x)$
1989 Vietnam National Olympiad, 1
Let $ n$ and $ N$ be natural number. Prove that for any $ \alpha$ , $ 0\le\alpha\le N$, and any real $ x$, it holds that \[{ |\sum_ {k=0}^n}\frac{\sin((\alpha+k)x)}{N+k}|\le\min\{(n+1)|x|, \frac{1}{N|\sin\frac{x}{2}|}\}\]
2012 China Team Selection Test, 3
Let $a_1<a_2$ be two given integers. For any integer $n\ge 3$, let $a_n$ be the smallest integer which is larger than $a_{n-1}$ and can be uniquely represented as $a_i+a_j$, where $1\le i<j\le n-1$. Given that there are only a finite number of even numbers in $\{a_n\}$, prove that the sequence $\{a_{n+1}-a_{n}\}$ is eventually periodic, i.e. that there exist positive integers $T,N$ such that for all integers $n>N$, we have
\[a_{T+n+1}-a_{T+n}=a_{n+1}-a_{n}.\]
2019 China Team Selection Test, 5
Determine all functions $f: \mathbb{Q} \to \mathbb{Q}$ such that
$$f(2xy + \frac{1}{2}) + f(x-y) = 4f(x)f(y) + \frac{1}{2}$$
for all $x,y \in \mathbb{Q}$.
2011 AMC 10, 12
Keiko walks once around a track at exactly the same constant speed every day. The sides of the track are straight, and the ends are semicircles. The track has width 6 meters, and it takes her 36 seconds longer to walk around the outside edge of the track than around the inside edge. What is Keiko's speed in meters per second?
$ \textbf{(A)}\ \frac{\pi}{3} \qquad
\textbf{(B)}\ \frac{2\pi}{3} \qquad
\textbf{(C)}\ \pi \qquad
\textbf{(D)}\ \frac{4\pi}{3} \qquad
\textbf{(E)}\ \frac{5\pi}{3} $