Found problems: 85335
2008 HMNT, 3
How many diagonals does a regular undecagon ($11$-sided polygon) have?
2008 Oral Moscow Geometry Olympiad, 4
A circle can be circumscribed around the quadrilateral $ABCD$. Point $P$ is the foot of the perpendicular drawn from point $A$ on line $BC$, and respectively $Q$ from $A$ on $DC$, $R$ from $D$ on $AB$ and $T$ from $D$ on $BC$ . Prove that points $P,Q,R$ and $T$ lie on the same circle.
(A. Myakishev)
2023 Austrian MO Beginners' Competition, 1
Let $x, y, z$ be nonzero real numbers with $$\frac{x + y}{z}=\frac{y + z}{x}=\frac{z + x}{y}.$$
Determine all possible values of $$\frac{(x + y)(y + z)(z + x)}{xyz}.$$
[i](Walther Janous)[/i]
2021 Balkan MO Shortlist, C3
In an exotic country, the National Bank issues coins that can take any value in the interval $[0, 1]$. Find the smallest constant $c > 0$ such that the following holds, no matter the situation in that country:
[i]Any citizen of the exotic country that has a finite number of coins, with a total value of no more than $1000$, can split those coins into $100$ boxes, such that the total value inside each box is at most $c$.[/i]
1990 Balkan MO, 1
The sequence $ (a_{n})_{n\geq 1}$ is defined by $ a_{1} \equal{} 1, a_{2} \equal{} 3$, and $ a_{n \plus{} 2} \equal{} (n \plus{} 3)a_{n \plus{} 1} \minus{} (n \plus{} 2)a_{n}, \forall n \in \mathbb{N}$. Find all values of $ n$ for which $ a_{n}$ is divisible by $ 11$.
2012 NIMO Problems, 2
A permutation $(a_1, a_2, a_3, \dots, a_{100})$ of $(1, 2, 3, \dots, 100)$ is chosen at random. Denote by $p$ the probability that $a_{2i} > a_{2i - 1}$ for all $i \in \{1, 2, 3, \dots, 50\}$. Compute the number of ordered pairs of positive integers $(a, b)$ satisfying $\textstyle\frac{1}{a^b} = p$.
[i]Proposed by Aaron Lin[/i]
2021 Princeton University Math Competition, 8
The new PUMaC tournament hosts $2020$ students, numbered by the following set of labels $1, 2, . . . , 2020$. The students are initially divided up into $20$ groups of $101$, with each division into groups equally likely. In each of the groups, the contestant with the lowest label wins, and the winners advance to the second round. Out of these $20$ students, we chose the champion uniformly at random. If the expected value of champion’s number can be written as $\frac{a}{b}$, where $a, b$ are relatively prime integers, determine $a + b$.
1987 Putnam, A4
Let $P$ be a polynomial, with real coefficients, in three variables and $F$ be a function of two variables such that
\[
P(ux, uy, uz) = u^2 F(y-x,z-x) \quad \mbox{for all real $x,y,z,u$},
\]
and such that $P(1,0,0)=4$, $P(0,1,0)=5$, and $P(0,0,1)=6$. Also let $A,B,C$ be complex numbers with $P(A,B,C)=0$ and $|B-A|=10$. Find $|C-A|$.
2009 Jozsef Wildt International Math Competition, W. 23
If $x_k \in \mathbb{R}$ ($k=1, 2, \cdots , n$) and $m \in \mathbb{N}$ then
[list=1]
[*] $\sum \limits_{cyc} \left (x_1^2 -x_1x_2+x_2^2 \right )^m \leq 3^m \sum \limits_{k=1}^n x_k^{2m}$
[*] $\prod \limits_{cyc} \left (x_1^2 -x_1x_2+x_2^2 \right )^m \leq \left (\frac{3^m}{n}\right )^m \left (\sum \limits_{k=1}^n x_k^{2m}\right )^n$
[/list]
2020 CMIMC Algebra & Number Theory, 10
We call a polynomial $P$ [i]square-friendly[/i] if it is monic, has integer coefficients, and there is a polynomial $Q$ for which $P(n^2)=P(n)Q(n)$ for all integers $n$. We say $P$ is [i]minimally square-friendly[/i] if it is square-friendly and cannot be written as the product of nonconstant, square-friendly polynomials. Determine the number of nonconstant, minimally square-friendly polynomials of degree at most $12$.
2001 Tuymaada Olympiad, 2
Solve the equation
\[(a^{2},b^{2})+(a,bc)+(b,ac)+(c,ab)=199.\]
in positive integers.
(Here $(x,y)$ denotes the greatest common divisor of $x$ and $y$.)
[i]Proposed by S. Berlov[/i]
1995 Taiwan National Olympiad, 5
Let $P$ be a point on the circumcircle of a triangle $A_{1}A_{2}A_{3}$, and let $H$ be the orthocenter of the triangle. The feet $B_{1},B_{2},B_{3}$ of the perpendiculars from $P$ to $A_{2}A_{3},A_{3}A_{1},A_{1}A_{2}$ lie on a line. Prove that this line bisects the segment $PH$.
2020 Purple Comet Problems, 18
Wendy randomly chooses a positive integer less than or equal to $2020$. The probability that the digits in Wendy's number add up to $10$ is $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Find $m + n$.
2017 Balkan MO Shortlist, A6
Find all functions $f : \mathbb R\to\mathbb R $ such that \[f(x+yf(x^2))=f(x)+xf(xy)\] for all real numbers $x$ and $y$.
2020-2021 OMMC, 9
There is a $4 \times 4$ array of integers $A$, all initially equal to $0$. An operation may be performed on the array for any row or column such that every number in that row or column has $1$ added to it, and then is replaced with its remainder modulo $3$. Given a random $4 \times 4$ array of integers between $0$ and $2$ not identical to $A$, the probability that it can be reached through a series of operations on $A$ is $\frac{p}{q},$ where $p,q$ are relatively prime positive integers. Find $p$.
2015 Junior Balkan Team Selection Tests - Moldova, 8
Determine the number of all ordered triplets of positive integers $(a, b, c)$, which satisfy the equalities:
$$[a, b] =1000, [b, c] = 2000, [c, a] =2000.$$
([x, y]represents the least common multiple of positive integers x,y)
1991 AMC 12/AHSME, 9
From time $t = 0$ to time $t = 1$ a population increased by $i\%$, and from time $t = 1$ to time $t = 2$ the population increased by $j\%$. Therefore, from time $t = 0$ to time $t = 2$ the population increased by
$ \textbf{(A)}\ (i + j)\%\qquad\textbf{(B)}\ ij\%\qquad\textbf{(C)}\ (i+ij)\%\qquad\textbf{(D)}\ \left(i + j + \frac{ij}{100}\right)\%\qquad\textbf{(E)}\left( i + j + \frac{i + j}{100}\right)\% $
2014 BMT Spring, 19
A number $k$ is [i]nice [/i] in base $b$ if there exists a $k$-digit number $n$ such that $n, 2n, . . . kn$ are each some cyclic shifts of the digits of $n$ in base $b$ (for example, $2$ is [i]nice [/i] in base $5$ because $2\cdot 135 = 315$). Determine all nice numbers in base $18$.
2000 Moldova National Olympiad, Problem 7
In a trapezoid $ABCD$ with $AB\parallel CD$, the diagonals $AC$ and $BD$ meet at $O$. Let $M$ and $N$ be the centers of the regular hexagons constructed on the sides $AB$ and $CD$ in the exterior of the trapezoid. Prove that $M,O$ and $N$ are collinear.
2001 All-Russian Olympiad, 1
The polynomial $ P(x)\equal{}x^3\plus{}ax^2\plus{}bx\plus{}d$ has three distinct real roots. The polynomial $ P(Q(x))$, where $ Q(x)\equal{}x^2\plus{}x\plus{}2001$, has no real roots. Prove that $ P(2001)>\frac{1}{64}$.
2018 CCA Math Bonanza, I11
Square $ABCD$ has side length $1$; circle $\Gamma$ is centered at $A$ with radius $1$. Let $M$ be the midpoint of $BC$, and let $N$ be the point on segment $CD$ such that $MN$ is tangent to $\Gamma$. Compute $MN$.
[i]2018 CCA Math Bonanza Individual Round #11[/i]
2018 Sharygin Geometry Olympiad, 2
A rectangle $ABCD$ and its circumcircle are given. Let $E$ be an arbitrary point on the minor arc $BC$. The tangent to the circle at $B$ meets $CE$ at point $G$. The segments $AE$ and $BD$ meet at point $K$. Prove that $GK$ and $AD$ are perpendicular.
1996 Nordic, 4
The real-valued function $f$ is defined for positive integers, and the positive integer $a$ satisfies
$f(a) = f(1995), f(a+1) = f(1996), f(a+2) = f(1997), f(n + a) = \frac{f(n) - 1}{f(n) + 1}$ for all positive integers $n$.
(i) Show that $f(n+ 4a) = f(n)$ for all positive integers $n$.
(ii) Determine the smallest possible $a$.
1982 AMC 12/AHSME, 17
How many real numbers $x$ satisfy the equation $3^{2x+2}-3^{x+3}-3^{x}+3=0$?
$\textbf {(A) } 0 \qquad \textbf {(B) } 1 \qquad \textbf {(C) } 2 \qquad \textbf {(D) } 3 \qquad \textbf {(E) } 4$
2023 Oral Moscow Geometry Olympiad, 4
Given isosceles tetrahedron $PABC$ (faces are equal triangles). Let $A_0$, $B_0$ and $C_0$ be the touchpoints of the circle inscribed in the triangle $ABC$ with sides $BC$, $AC$ and $AB$ respectively, $A_1$, $B_1$ and $C_1$ are the touchpoints of the excircles of triangles $PCA$, $PAB$ and $PBC$ with extensions of sides $PA$, $PB$ and $PC$, respectively (beyond points $A$, $B$, $C$). Prove that the lines $A_0A_1$, $B_0B_1$ and $C_0C_1$ intersect at one point.