Found problems: 85335
Russian TST 2018, P2
Inside the acute-angled triangle $ABC$, the points $P{}$ and $Q{}$ are chosen so that $\angle ACP = \angle BCQ$ and $\angle CBP =\angle ABQ$. The point $Z{}$ is the projection of $P{}$ onto the line $BC$. The point $Q'$ is symmetric to $Q{}$ with respect to $Z{}$. The points $K{}$ and $L{}$ are chosen on the rays $AB$ and $AC$ respectively, so that $Q'K \parallel QC$ and $Q'L \parallel QB$. Prove that $\angle KPL=\angle BPC$.
2019 Durer Math Competition Finals, 10
In an isosceles, obtuse-angled triangle, the lengths of two internal angle bisectors are in a $2:1$ ratio. Find the obtuse angle of the triangle.
2021 CCA Math Bonanza, L4.4
Let $ABC$ be a triangle with $AB=5$, $BC=7$, $CA=8$, and let $M$ be the midpoint of $BC$. Points $P$ and $Q$ are chosen on the circumcircle of $ABC$ such that $MPQ$ and $ABC$ are similar (with vertices in that order). The product of all different possible areas of $MPQ$ can be written as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Compute $m+n$.
[i]2021 CCA Math Bonanza Lightning Round #4.4[/i]
2021 JHMT HS, 4
There is a unique differentiable function $f$ from $\mathbb{R}$ to $\mathbb{R}$ satisfying $f(x) + (f(x))^3 = x + x^7$ for all real $x.$ The derivative of $f(x)$ at $x = 2$ can be expressed as a common fraction $a/b.$ Compute $a + b.$
1993 Miklós Schweitzer, 4
Let f be a ternary operation on a set of at least four elements for which
(1) $f ( x , x , y ) \equiv f ( x , y , x ) \equiv f( x , y , y ) \equiv x$
(2) $f ( x , y , z ) = f ( y , z , x ) = f ( y , x , z ) \in \{ x , y , z \}$
for pairwise distinct x,y,z.
Prove that f is a nontrivial composition of g such that g is not a composition of f.
(The n-variable operation g is trivial if $g(x_1, ..., x_n) \equiv x_i$ for some i ($1 \leq i \leq n$) )
2009 ISI B.Math Entrance Exam, 1
Let $x,y,z$ be non-zero real numbers. Suppose $\alpha, \beta, \gamma$ are complex numbers such that $|\alpha|=|\beta|=|\gamma|=1$. If $x+y+z=0=\alpha x+\beta y+\gamma z$, then prove that $\alpha =\beta =\gamma$.
2015 Postal Coaching, Problem 1
Let $n \in \mathbb{N}$ be such that $gcd(n, 6) = 1$. Let $a_1 < a_2 < \cdots < a_n$ and $b_1 < b_2 < \cdots < b_n$ be two collection of positive integers such that $a_j + a_k + a_l = b_j + b_k + b_l$ for all integers $1 \le j < k < l \le n$. Prove that $a_j = b_j$ for all $1 \le j \le n$.
1983 National High School Mathematics League, 5
$f(x)=ax^2-c$. If$-4\leq f(1)\leq -1,-z\leq f(2)\leq 5$, then
$\text{(A)}7\leq f(3)\leq26\qquad\text{(B)}-4\leq f(3)\leq15\qquad\text{(C)}-1\leq f(3)\leq23\qquad\text{(D)}-\frac{28}{3}\leq f(3)\leq\frac{35}{3}$
2024 Bulgarian Autumn Math Competition, 11.2
Let $ABC$ be a triangle with $\angle ABC = 60^{\circ}$. Find the angles of the triangle if $\angle BHI = 60^{\circ}$, where $H$ and $I$ are the orthocenter and incenter of $ABC$
2016 Korea National Olympiad, 3
Acute triangle $\triangle ABC$ has area $S$ and perimeter $L$. A point $P$ inside $\triangle ABC$ has $dist(P,BC)=1, dist(P,CA)=1.5, dist(P,AB)=2$. Let $BC \cap AP = D$, $CA \cap BP = E$, $AB \cap CP= F$.
Let $T$ be the area of $\triangle DEF$. Prove the following inequality.
$$ \left( \frac{AD \cdot BE \cdot CF}{T} \right)^2 > 4L^2 + \left( \frac{AB \cdot BC \cdot CA}{24S} \right)^2 $$
2012 Tournament of Towns, 6
We attempt to cover the plane with an infi nite sequence of rectangles, overlapping allowed.
(a) Is the task always possible if the area of the $n$th rectangle is $n^2$ for each $n$?
(b) Is the task always possible if each rectangle is a square, and for any number $N$, there exist squares with total area greater than $N$?
2014 PUMaC Geometry A, 2
Triangle $ABC$ has lengths $AB=20$, $AC=14$, $BC=22$. The median from $B$ intersects $AC$ at $M$ and the angle bisector from $C$ intersects $AB$ at $N$ and the median from $B$ at $P$. Let $\dfrac pq=\dfrac{[AMPN]}{[ABC]}$ for positive integers $p$, $q$ coprime. Note that $[ABC]$ denotes the area of triangle $ABC$. Find $p+q$.
MOAA Gunga Bowls, 2021.10
We say that an ordered pair $(a,b)$ of positive integers with $a>b$ is square-ish if both $a+b$ and $a-b$ are perfect squares. For example, $(17,8)$ is square-ish because $17+8=25$ and $17-8=9$ are both perfect squares. How many square-ish pairs $(a,b)$ with $a+b<100$ are there?
[i]Proposed by Nathan Xiong[/i]
1998 AMC 8, 15
Problems $15, 16$, and $17$ all refer to the following:
In the very center of the Irenic Sea lie the beautiful Nisos Isles. In $1998$ the number of people on these islands is only 200, but the population triples every $25$ years. Queen Irene has decreed that there must be at least $1.5$ square miles for every person living in the Isles. The total area of the Nisos Isles is $24,900$ square miles.
15. Estimate the population of Nisos in the year $2050$.
$ \text{(A)}\ 600\qquad\text{(B)}\ 800\qquad\text{(C)}\ 1000\qquad\text{(D)}\ 2000\qquad\text{(E)}\ 3000 $
1995 IberoAmerican, 1
Find all the possible values of the sum of the digits of all the perfect squares.
[Commented by djimenez]
[b]Comment: [/b]I would rewrite it as follows:
Let $f: \mathbb{N}\rightarrow \mathbb{N}$ such that $f(n)$ is the sum of all the digits of the number $n^2$. Find the image of $f$ (where, by image it is understood the set of all $x$ such that exists an $n$ with $f(n)=x$).
2000 Harvard-MIT Mathematics Tournament, 26
What are the last $3$ digits of $1!+2!+\cdots +100!$
2024 China Team Selection Test, 23
$P(z)=a_nz^n+\dots+a_1z+z_0$, with $a_n\neq 0$ is a polynomial with complex coefficients, such that when $|z|=1$, $|P(z)|\leq 1$. Prove that for any $0\leq k\leq n-1$, $|a_k|\leq 1-|a_n|^2$.
[i]Proposed by Yijun Yao[/i]
1983 IMO Longlists, 50
Is it possible to choose $1983$ distinct positive integers, all less than or equal to $10^5$, no three of which are consecutive terms of an arithmetic progression?
2024 ITAMO, 2
We are given a unit square in the plane. A point $M$ in the plane is called [i]median [/i]if there exists points $P$ and $Q$ on the boundary of the square such that $PQ$ has length one and $M$ is the midpoint of $PQ$.
Determine the geometric locus of all median points.
2005 Estonia Team Selection Test, 1
On a plane, a line $\ell$ and two circles $c_1$ and $c_2$ of different radii are given such that $\ell$ touches both circles at point $P$. Point $M \ne P$ on $\ell$ is chosen so that the angle $Q_1MQ_2$ is as large as possible where $Q_1$ and $Q_2$ are the tangency points of the tangent lines drawn from $M$ to $c_i$ and $c_2$, respectively, differing from $\ell$ . Find $\angle PMQ_1 + \angle PMQ_2$·
2007 Harvard-MIT Mathematics Tournament, 4
Find the real number $\alpha$ such that the curve $f(x)=e^x$ is tangent to the curve $g(x)=\alpha x^2$.
1989 IberoAmerican, 3
Let $a,b$ and $c$ be the side lengths of a triangle. Prove that:
\[\frac{a-b}{a+b}+\frac{b-c}{b+c}+\frac{c-a}{c+a}<\frac{1}{16}\]
2018 Bundeswettbewerb Mathematik, 2
Consider all functions $f:\mathbb{R} \to \mathbb{R}$ satisfying $f(1-f(x))=x$ for all $x \in \mathbb{R}$.
a) By giving a concrete example, show that such a function exists.
b) For each such function define the sum
\[S_f=f(-2017)+f(-2016)+\dots+f(-1)+f(0)+f(1)+\dots+f(2017)+f(2018).\]
Determine all possible values of $S_f$.
2019 IMO Shortlist, N2
Find all triples $(a, b, c)$ of positive integers such that $a^3 + b^3 + c^3 = (abc)^2$.
2013 AMC 10, 16
In $\triangle ABC$, medians $\overline{AD}$ and $\overline{CE}$ intersect at $P$, $PE=1.5$, $PD=2$, and $DE=2.5$. What is the area of $AEDC?$
[asy]
unitsize(75);
pathpen = black; pointpen=black;
pair A = MP("A", D((0,0)), dir(200));
pair B = MP("B", D((2,0)), dir(-20));
pair C = MP("C", D((1/2,1)), dir(100));
pair D = MP("D", D(midpoint(B--C)), dir(30));
pair E = MP("E", D(midpoint(A--B)), dir(-90));
pair P = MP("P", D(IP(A--D, C--E)), dir(150)*2.013);
draw(A--B--C--cycle);
draw(A--D--E--C);
[/asy]
$\textbf{(A)}\ 13 \qquad
\textbf{(B)}\ 13.5 \qquad
\textbf{(C)}\ 14 \qquad
\textbf{(D)}\ 14.5 \qquad
\textbf{(E)}\ 15 $