Found problems: 85335
2011-2012 SDML (High School), 11
Eight points are equally spaced around a circle of radius $r$. If we draw a circle of radius $1$ centered at each of the eight points, then each of these circles will be tangent to two of the other eight circles that are next to it. IF $r^2=a+b\sqrt{2}$, where $a$ and $b$ are integers, then what is $a+b$?
$\text{(A) }3\qquad\text{(B) }4\qquad\text{(C) }5\qquad\text{(D) }6\qquad\text{(E) }7$
2007 Iran MO (3rd Round), 1
Let $ a,b$ be two complex numbers. Prove that roots of $ z^{4}\plus{}az^{2}\plus{}b$ form a rhombus with origin as center, if and only if $ \frac{a^{2}}{b}$ is a non-positive real number.
1976 Swedish Mathematical Competition, 3
If $a$, $b$, $c$ are rational, show that
\[
\frac{1}{(b-c)^2}+\frac{1}{(c-a)^2}+\frac{1}{(a-b)^2}
\]
is the square of a rational.
2014 Balkan MO Shortlist, A1
$\boxed{\text{A1}}$Let $a,b,c$ be positive reals numbers such that $a+b+c=1$.Prove that $2(a^2+b^2+c^2)\ge \frac{1}{9}+15abc$
2025 Ukraine National Mathematical Olympiad, 11.4
A pair of positive integer numbers \((a, b)\) is given. It turns out that for every positive integer number \(n\), for which the numbers \((n - a)(n + b)\) and \(n^2 - ab\) are positive, they have the same number of divisors. Is it necessarily true that \(a = b\)?
[i]Proposed by Oleksii Masalitin[/i]
2014 Singapore Senior Math Olympiad, 24
Find the number of integers $x$ which satisfy the equation $(x^2-5x+5)^{x+5}=1$.
2002 AIME Problems, 6
Find the integer that is closest to $ 1000 \sum_{n=3}^{10000}\frac{1}{n^{2}-4}.$
2016 Canada National Olympiad, 3
Find all polynomials $P(x)$ with integer coefficients such that $P(P(n) + n)$ is a prime number for infinitely many integers $n$.
2011 Indonesia MO, 8
Given a triangle $ABC$. Its incircle is tangent to $BC, CA, AB$ at $D, E, F$ respectively. Let $K, L$ be points on $CA, AB$ respectively such that $K \neq A \neq L, \angle EDK = \angle ADE, \angle FDL = \angle ADF$. Prove that the circumcircle of $AKL$ is tangent to the incircle of $ABC$.
2021 Brazil EGMO TST, 8
Let $n$ be a positive integer, such that $125n+22$ is a power of $3$. Prove that $125n+29$ has a prime factor greater than $100$.
1980 IMO Longlists, 18
Given a sequence $\{a_n\}$ of real numbers such that $|a_{k+m} - a_k - a_m| \leq 1$ for all positive integers $k$ and $m$, prove that, for all positive integers $p$ and $q$, \[|\frac{a_p}{p} - \frac{a_q}{q}| < \frac{1}{p} + \frac{1}{q}.\]
MOAA Accuracy Rounds, 2021.6
Let $\triangle ABC$ be a triangle in a plane such that $AB=13$, $BC=14$, and $CA=15$. Let $D$ be a point in three-dimensional space such that $\angle{BDC}=\angle{CDA}=\angle{ADB}=90^\circ$. Let $d$ be the distance from $D$ to the plane containing $\triangle ABC$. The value $d^2$ can be expressed as $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Compute $m+n$.
[i]Proposed by William Yue[/i]
2003 National High School Mathematics League, 3
Line passes the focal point $F$ of parabola $y^2=8(x+2)$ with bank angle of $60^{\circ}$ intersects the parabola at $A,B$. Perpendicular bisector of $AB$ intersects $x$-axis at $P$, then the length of $PF$ is
$\text{(A)}\frac{16}{3}\qquad\text{(B)}\frac{8}{3}\qquad\text{(C)}\frac{16}{3}\sqrt3\qquad\text{(D)}8\sqrt3$
2017 India PRMO, 12
In a class, the total numbers of boys and girls are in the ratio $4 : 3$. On one day it was found that $8$ boys and $14$ girls were absent from the class, and that the number of boys was the square of the number of girls. What is the total number of students in the class?
2008 Bulgarian Autumn Math Competition, Problem 8.3
Prove that there exists a prime number $p$, such that the sum of digits of $p$ is a composite odd integer. Find the smallest such $p$.
1994 India Regional Mathematical Olympiad, 1
A leaf is torn from a paperback novel. The sum of the numbers on the remaining pages is $15000$. What are the page numbers on the torn leaf?
2019 Iranian Geometry Olympiad, 3
Three circles $\omega_1$, $\omega_2$ and $\omega_3$ pass through one common point, say $P$. The tangent line to $\omega_1$ at $P$ intersects $\omega_2$ and $\omega_3$ for the second time at points $P_{1,2}$ and $P_{1,3}$, respectively. Points $P_{2,1}$, $P_{2,3}$, $P_{3,1}$ and $P_{3,2}$ are similarly defined. Prove that the perpendicular bisector of segments $P_{1,2}P_{1,3}$, $P_{2,1}P_{2,3}$ and $P_{3,1}P_{3,2}$ are concurrent.
[i]Proposed by Mahdi Etesamifard[/i]
2007 Romania National Olympiad, 1
In a triangle $ ABC$, where $ a \equal{} BC$, $ b \equal{} CA$ and $ c \equal{} AB$, it is known that: $ a \plus{} b \minus{} c \equal{} 2$ and $ 2ab \minus{} c^2 \equal{} 4$. Prove that $ ABC$ is an equilateral triangle.
2024 Belarusian National Olympiad, 11.5
On the chord $AB$ of the circle $\omega$ points $C$ and $D$ are chosen such that $AC=BD$ and point $C$ lies between $A$ and $D$. On $\omega$ point $E$ and $F$ are marked, they lie on different sides with respect to line $AB$ and lines $EC$ and $FD$ are perpendicular to $AB$. The angle bisector of $AEB$ intersects line $DF$ at $R$
Prove that the circle with center $F$ and radius $FR$ is tangent to $\omega$
[i]V. Kamenetskii, D. Bariev[/i]
2023 Taiwan TST Round 3, 4
Find all positive integers $a$, $b$ and $c$ such that $ab$ is a square, and
\[a+b+c-3\sqrt[3]{abc}=1.\]
[i]Proposed by usjl[/i]
2011 Uzbekistan National Olympiad, 3
Given an acute triangle $ABC$ with altituties AD and BE. O circumcinter of $ABC$.If o lies on the segment DE then find the value of $sinAsinBcosC$
1994 AMC 12/AHSME, 11
Three cubes of volume $1, 8$ and $27$ are glued together at their faces. The smallest possible surface area of the resulting configuration is
$ \textbf{(A)}\ 36 \qquad\textbf{(B)}\ 56 \qquad\textbf{(C)}\ 70 \qquad\textbf{(D)}\ 72 \qquad\textbf{(E)}\ 74 $
2007 Stanford Mathematics Tournament, 5
Two disks of radius 1 are drawn so that each disk's circumference passes through the center of the other disk. What is the circumference of the region in which they overlap?
1996 USAMO, 6
Determine (with proof) whether there is a subset $X$ of the integers with the following property: for any integer $n$ there is exactly one solution of $a + 2b = n$ with $a,b \in X$.
1986 Iran MO (2nd round), 2
[b](a)[/b] Sketch the diagram of the function $f$ if
\[f(x)=4x(1-|x|) , \quad |x| \leq 1.\]
[b](b)[/b] Does there exist derivative of $f$ in the point $x=0 \ ?$
[b](c)[/b] Let $g$ be a function such that
\[g(x)=\left\{\begin{array}{cc}\frac{f(x)}{x} \quad : x \neq 0\\ \text{ } \\ 4 \ \ \ \ \quad : x=0\end{array}\right.\]
Is the function $g$ continuous in the point $x=0 \ ?$
[b](d)[/b] Sketch the diagram of $g.$