Found problems: 85335
2018 USAJMO, 4
Triangle $ABC$ is inscribed in a circle of radius 2 with $\angle ABC \geq 90^\circ$, and $x$ is a real number satisfying the equation $x^4 + ax^3 + bx^2 + cx + 1 = 0$, where $a=BC$, $b=CA$, $c=AB$. Find all possible values of $x$.
2013 Chile National Olympiad, 4
Consider a function f defined on the positive integers that meets the following conditions: $$f(1) = 1 \, , \,\, f(2n) = 2f(n) \, , \,\, nf(2n + 1) = (2n + 1)(f(n) + n) $$ for all $n \ge 1$.
a) Prove that $f(n)$ is an integer for all $n$.
b) Find all positive integers $m$ less than $2013$ that satisfy the equation $f(m) = 2m$.
PEN D Problems, 10
Let $p$ be a prime number of the form $4k+1$. Suppose that $2p+1$ is prime. Show that there is no $k \in \mathbb{N}$ with $k<2p$ and $2^k \equiv 1 \; \pmod{2p+1}$.
2005 AMC 12/AHSME, 20
Let $ a,b,c,d,e,f,g$ and $ h$ be distinct elements in the set
\[ \{ \minus{} 7, \minus{} 5, \minus{} 3, \minus{} 2,2,4,6,13\}.
\]What is the minimum possible value of
\[ (a \plus{} b \plus{} c \plus{} d)^2 \plus{} (e \plus{} f \plus{} g \plus{} h)^2
\]$ \textbf{(A)}\ 30\qquad
\textbf{(B)}\ 32\qquad
\textbf{(C)}\ 34\qquad
\textbf{(D)}\ 40\qquad
\textbf{(E)}\ 50$
2006 IMO Shortlist, 1
A sequence of real numbers $ a_{0},\ a_{1},\ a_{2},\dots$ is defined by the formula
\[ a_{i \plus{} 1} \equal{} \left\lfloor a_{i}\right\rfloor\cdot \left\langle a_{i}\right\rangle\qquad\text{for}\quad i\geq 0;
\]here $a_0$ is an arbitrary real number, $\lfloor a_i\rfloor$ denotes the greatest integer not exceeding $a_i$, and $\left\langle a_i\right\rangle=a_i-\lfloor a_i\rfloor$. Prove that $a_i=a_{i+2}$ for $i$ sufficiently large.
[i]Proposed by Harmel Nestra, Estionia[/i]
1995 APMO, 4
Let $C$ be a circle with radius $R$ and centre $O$, and $S$ a fixed point in the interior of $C$. Let $AA'$ and $BB'$ be perpendicular chords through $S$. Consider the rectangles $SAMB$, $SBN'A'$, $SA'M'B'$, and $SB'NA$. Find the set of all points $M$, $N'$, $M'$, and $N$ when $A$ moves around the whole circle.
2024 Brazil Cono Sur TST, 4
In the cartesian plane, consider the subset of all the points with both integer coordinates. Prove that it is possible to choose a finite non-empty subset $S$ of these points in such a way that any line $l$ that forms an angle of $90^{\circ},0^{\circ},135^{\circ}$ or $45^{\circ}$ with the positive horizontal semi-axis intersects $S$ at exactly $2024$ points or at no points.
2007 Princeton University Math Competition, 8
$f(x) = x^3+3x^2 - 1$. Find the number of real solutions of $f(f(x)) = 0$.
2011 Purple Comet Problems, 16
Let $a$ and $b$ be nonzero real numbers such that $\tfrac{1}{3a}+\tfrac{1}{b}=2011$ and $\tfrac{1}{a}+\tfrac{1}{3b}=1$. What is the quotient when $a+b$ is divided by $ab$?
2011 IFYM, Sozopol, 7
We define the sequence
$x_1=n,y_1=1,x_{i+1}=[\frac{x_i+y_i}{2}],y_{i+1}=[\frac{n}{x_{i+1}} ]$.
Prove that $min\{ x_1, x_2, ..., x_n\}=[\sqrt{n}]$ .
2019 New Zealand MO, 1
How many positive integers less than $2019$ are divisible by either $18$ or $21$, but not both?
1988 IMO Longlists, 74
Let $ \{a_k\}^{\infty}_1$ be a sequence of non-negative real numbers such that:
\[ a_k \minus{} 2 a_{k \plus{} 1} \plus{} a_{k \plus{} 2} \geq 0
\]
and $ \sum^k_{j \equal{} 1} a_j \leq 1$ for all $ k \equal{} 1,2, \ldots$. Prove that:
\[ 0 \leq a_{k} \minus{} a_{k \plus{} 1} < \frac {2}{k^2}
\]
for all $ k \equal{} 1,2, \ldots$.
1961 All-Soviet Union Olympiad, 5
Consider a $2^k$-tuple of numbers $(a_1,a_2,\dots,a_{2^k})$ all equal to $1$ or $-1$. In one step, we transform it to $(a_1a_2,a_2a_3,\dots,a_{2^k}a_1)$. Prove that eventually, we will obtain a $2^k$-tuple consisting only of $1$'s.
1968 AMC 12/AHSME, 27
Let $S_n=1-2+3-4+\cdots +(-1)^{n-1}n,\ n=1, 2, \cdots$. Then $S_{17}+S_{33}+S_{50}$ equals:
$\textbf{(A)}\ 0 \qquad\textbf{(B)}\ 1 \qquad\textbf{(C)}\ 2 \qquad\textbf{(D)}\ -1 \qquad\textbf{(E)}\ -2$
2006 Alexandru Myller, 3
$ 5 $ points are situated in the plane so that any three of them form a triangle of area at most $ 1. $ Prove that there is a trapezoid of area at most $ 3 $ which contains all these points ('including' here means that the points can also be on the sides of the trapezoid).
2019 IMO Shortlist, N5
Let $a$ be a positive integer. We say that a positive integer $b$ is [i]$a$-good[/i] if $\tbinom{an}{b}-1$ is divisible by $an+1$ for all positive integers $n$ with $an \geq b$. Suppose $b$ is a positive integer such that $b$ is $a$-good, but $b+2$ is not $a$-good. Prove that $b+1$ is prime.
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 $\lfloor10S\rfloor$.
2002 AMC 10, 10
Let $a$ and $b$ be distinct real numbers for which \[\dfrac ab+\dfrac{a+10b}{b+10a}=2.\] Find $\dfrac ab$.
$\textbf{(A) }0.6\qquad\textbf{(B) }0.7\qquad\textbf{(C) }0.8\qquad\textbf{(D) }0.9\qquad\textbf{(E) }1$
2005 Austrian-Polish Competition, 2
Determine all polynomials $P$ with integer coefficients satisfying
\[P(P(P(P(P(x)))))=x^{28}\cdot P(P(x))\qquad \forall x\in\mathbb{R}\]
1992 Vietnam Team Selection Test, 1
In the plane let a finite family of circles be given satisfying the condition: every two circles, either are outside each other, either touch each other from outside and each circle touch at most 6 other circles. Suppose that every circle which does not touch 6 other circles be assigned a real number. Show that there exist at most one assignment to each remaining circle a real number equal to arithmetic mean of 6 numbers assigned to 6 circles which touch it.
1996 French Mathematical Olympiad, Problem 1
Consider a triangle $ABC$ and points $D,E,F,G,H,I$ in the plane such that $ABED$, $BCGF$ and $ACHI$ are squares exterior to the triangle. Prove that points $D,E,F,G,H,I$ are concyclic if and only if one of the following two statements hold:
(i) $ABC$ is an equilateral triangle.
(ii) $ABC$ is an isosceles right triangle.
2016 Azerbaijan BMO TST, 1
Find all $n$ natural numbers such that for each of them there exist $p , q$ primes such that these terms satisfy.
$1.$ $p+2=q$
$2.$ $2^n+p$ and $2^n+q$ are primes.
2016 BMT Spring, 5
Positive integers $x, y, z$ satisfy $(x + yi)^2 - 46i = z$. What is $x + y + z$?
2008 Dutch IMO TST, 4
Let $n$ be positive integer such that $\sqrt{1 + 12n^2}$ is an integer.
Prove that $2 + 2\sqrt{1 + 12n^2}$ is the square of an integer.
2013 Federal Competition For Advanced Students, Part 2, 2
Let $k$ be an integer. Determine all functions $f\colon \mathbb{R}\to\mathbb{R}$ with $f(0)=0$ and \[f(x^ky^k)=xyf(x)f(y)\qquad \mbox{for } x,y\neq 0.\]