Found problems: 85335
1988 Putnam, A2
A not uncommon calculus mistake is to believe that the product rule for derivatives says that $(fg)' = f'g'$. If $f(x)=e^{x^2}$, determine, with proof, whether there exists an open interval $(a,b)$ and a nonzero function $g$ defined on $(a,b)$ such that this wrong product rule is true for $x$ in $(a,b)$.
2021 AMC 10 Spring, 9
What is the least possible value of $(xy-1)^2+(x+y)^2$ for real numbers $x$ and $y$?
$\textbf{(A)}\ 0 \qquad\textbf{(B)}\ \frac14 \qquad\textbf{(C)}\ \frac12 \qquad\textbf{(D)}\ 1 \qquad\textbf{(E)}\ 2$
2015 Peru Cono Sur TST, P5
Find the smallest term of the sequence $a_1, a_2, a_3, \ldots$ defined by $a_1=2014^{2015^{2016}}$ and
$$
a_{n+1}=
\begin{cases}
\frac{a_n}{2} & \text{ if } a_n \text{ is even} \\
a_n + 7 & \text{ if } a_n \text{ is odd} \\
\end{cases}
$$
2006 Estonia Math Open Senior Contests, 8
Four points $ A, B, C, D$ are chosen on a circle in such a way that arcs $ AB, BC,$ and $ CD$ are of the same length and the $ arc DA$ is longer than these three. Line $ AD$ and the line tangent to the circle at $ B$ intersect at $ E$. Let $ F$ be the other endpoint of the diameter starting at $ C$ of the circle. Prove that triangle $ DEF$ is equilateral.
2011 Northern Summer Camp Of Mathematics, 2
Find all functions $f: \mathbb N \cup \{0\} \to \mathbb N\cup \{0\}$ such that $f(1)>0$ and
\[f(m^2+3n^2)=(f(m))^2 + 3(f(n))^2 \quad \forall m,n \in \mathbb N\cup \{0\}.\]
2013 May Olympiad, 3
Many distinct points are marked in the plane. A student draws all the segments determined by those points, and then draws a line [i]r[/i] that does not pass through any of the marked points, but cuts exactly $60$ drawn segments. How many segments were not cut by [i]r[/i]? Give all possibilites.
2020 Princeton University Math Competition, B2
Seven students in Princeton Juggling Club are searching for a room to meet in. However, they must stay at least $6$ feet apart from each other, and due to midterms, the only open rooms they can find are circular. In feet, what is the smallest diameter of any circle which can contain seven points, all of which are at least $6$ feet apart from each other?
2015 ISI Entrance Examination, 6
Find all $n\in \mathbb{N} $ so that 7 divides $5^n + 1$
MathLinks Contest 3rd, 2
The sequence $\{x_n\}_{n\ge1}$ is defined by $x_1 = 7$, $x_{n+1} = 2x^2_n - 1$, for all positive integers $n$. Prove that for all positive integers $n$ the number $x_n$ cannot be divisible by $2003$.
2014 Harvard-MIT Mathematics Tournament, 2
There are $10$ people who want to choose a committee of 5 people among them. They do this by first electing a set of $1, 2, 3,$ or $4$ committee leaders, who then choose among the remaining people to complete the 5-person committee. In how many ways can the committee be formed, assuming that people are distinguishable? (Two committees that have the same members but different sets of leaders are considered to be distinct.)
1987 IberoAmerican, 2
In a triangle $ABC$, $M$ and $N$ are the respective midpoints of the sides $AC$ and $AB$, and $P$ is the point of intersection of $BM$ and $CN$. Prove that, if it is possible to inscribe a circle in the quadrilateral $AMPN$, then the triangle $ABC$ is isosceles.
2012 Benelux, 2
Find all quadruples $(a,b,c,d)$ of positive real numbers such that $abcd=1,a^{2012}+2012b=2012c+d^{2012}$ and $2012a+b^{2012}=c^{2012}+2012d$.
2019 Auckland Mathematical Olympiad, 5
$2019$ circles split a plane into a number of parts whose boundaries are arcs of those circles. How many colors are needed to color this geographic map if any two neighboring parts must be coloured with different colours?
1981 Tournament Of Towns, (013) 3
Prove that every real positive number may be represented as a sum of nine numbers whose decimal representation consists of the digits $0$ and $7$.
(E Turkevich)
2008 HMNT, 1
Find the minimum of $x^2 - 2x$ over all real numbers $x.$
1998 Belarus Team Selection Test, 2
In town $ A,$ there are $ n$ girls and $ n$ boys, and each girl knows each boy. In town $ B,$ there are $ n$ girls $ g_1, g_2, \ldots, g_n$ and $ 2n \minus{} 1$ boys $ b_1, b_2, \ldots, b_{2n\minus{}1}.$ The girl $ g_i,$ $ i \equal{} 1, 2, \ldots, n,$ knows the boys $ b_1, b_2, \ldots, b_{2i\minus{}1},$ and no others. For all $ r \equal{} 1, 2, \ldots, n,$ denote by $ A(r),B(r)$ the number of different ways in which $ r$ girls from town $ A,$ respectively town $ B,$ can dance with $ r$ boys from their own town, forming $ r$ pairs, each girl with a boy she knows. Prove that $ A(r) \equal{} B(r)$ for each $ r \equal{} 1, 2, \ldots, n.$
2007 Indonesia TST, 2
Let $a > 3$ be an odd integer. Show that for every positive integer $n$ the number $a^{2^n}- 1$ has at least $n + 1$ distinct prime divisors.
VMEO III 2006 Shortlist, N10
The notation $\phi (n)$ is the number of positive integers smaller than $n$ and coprime with $n$, $\pi (n)$ is the number of primes that do not exceed $n$. Prove that for any natural number $n > 1$, we have
$$\phi (n) \ge \frac{\pi (n)}{2}$$
2012 India Regional Mathematical Olympiad, 6
Let $a$ and $b$ be real numbers such that $a \ne 0$. Prove that not all the roots of $ax^4 + bx^3 + x^2 + x + 1 = 0$ can be real.
2014 Indonesia MO Shortlist, C5
Determine all pairs of natural numbers $(m, r)$ with $2014 \ge m \ge r \ge 1$ that fulfill
$\binom{2014}{m}+\binom{m}{r}=\binom{2014}{r}+\binom{2014-r}{m-r} $
2019 JBMO Shortlist, G4
Triangle $ABC$ is such that $AB < AC$. The perpendicular bisector of side $BC$ intersects lines $AB$ and $AC$ at points $P$ and $Q$, respectively. Let $H$ be the orthocentre of triangle $ABC$, and let $M$ and $N$ be the midpoints of segments $BC$ and $PQ$, respectively. Prove that lines $HM$ and $AN$ meet on the circumcircle of $ABC$.
MathLinks Contest 6th, 2.3
Let $\sigma : \{1, 2, . . . , n\} \to \{1, 2, . . . , n\}$ be a bijective mapping. Let $S_n$ be the set of all such mappings and let $d_k(\sigma) = |\sigma(k) - \sigma(k + 1)|$, for all $k \in \{1, 2, ..., n\}$, where $\sigma (n + 1) = \sigma (1)$. Also let $d(\sigma) = \min \{d_k(\sigma) | 1 \le k \le n\}$. Find $\max_{\sigma \in S_n} d(\sigma)$.
2022 Korea -Final Round, P3
A function $g \colon \mathbb{R} \to \mathbb{R}$ is given such that its range is a finite set. Find all functions $f \colon \mathbb{R} \to \mathbb{R}$ that satisfies $$2f(x+g(y))=f(2g(x)+y)+f(x+3g(y))$$ for all $x, y \in \mathbb{R}$.
2018 PUMaC Live Round, 8.3
If $a$ and $b$ are positive integers such that $3\sqrt{2+\sqrt{2+\sqrt{3}}}=a\cos\frac{\pi}{b}$, find $a+b$.
2019 Romania National Olympiad, 4
Let $n \geq 3$ and $a_1,a_2,...,a_n$ be complex numbers different from $0$ with $|a_i| < 1$ for all $i \in \{1,2,...,n-1 \}.$ If the coefficients of $f = \prod_{i=1}^n (X-a_i)$ are integers, prove that
$\textbf{a)}$ The numbers $a_1,a_2,...,a_n$ are distinct.
$\textbf{b)}$ If $a_j^2 = a_ia_k,$ then $i=j=k.$