Found problems: 85335
1964 IMO Shortlist, 3
A circle is inscribed in a triangle $ABC$ with sides $a,b,c$. Tangents to the circle parallel to the sides of the triangle are contructe. Each of these tangents cuts off a triagnle from $\triangle ABC$. In each of these triangles, a circle is inscribed. Find the sum of the areas of all four inscribed circles (in terms of $a,b,c$).
2016 China Western Mathematical Olympiad, 1
Let $a,b,c,d$ be real numbers such that $abcd>0$. Prove that:There exists a permutation $x,y,z,w$ of $a,b,c,d$ such that $$2(xy+zw)^2>(x^2+y^2)(z^2+w^2)$$.
2021 Junior Balkan Team Selection Tests - Romania, P1
Let $a,b,c>0$ be real numbers with the property that $a+b+c=1$. Prove that \[\frac{1}{a+bc}+\frac{1}{b+ca}+\frac{1}{c+ab}\geq\frac{7}{1+abc}.\]
1990 National High School Mathematics League, 11
$\frac{1}{2^{1990}}(1-3\text{C}_{1990}^2+3^2\text{C}_{1990}^4-3^3\text{C}_{1990}^6+\cdots+3^{994}\text{C}_{1990}^{1988}-3^{995}\text{C}_{1990}^{1990})=$________.
JOM 2015 Shortlist, A6
Let $(a_{n})_{n\ge 0}$ and $(b_{n})_{n\ge 0}$ be two sequences with arbitrary real values $a_0, a_1, b_0, b_1$. For $n\ge 1$, let $a_{n+1}, b_{n+1}$ be defined in this way:
$$a_{n+1}=\dfrac{b_{n-1}+b_{n}}{2}, b_{n+1}=\dfrac{a_{n-1}+a_{n}}{2}$$
Prove that for any constant $c>0$ there exists a positive integer $N$ s.t. for all $n>N$, $|a_{n}-b_{n}|<c$.
2009 BAMO, 2
The Fibonacci sequence is the list of numbers that begins $1, 2, 3, 5, 8, 13$ and continues with each subsequent number being the sum of the previous two.
Prove that for every positive integer $n$ when the first $n$ elements of the Fibonacci sequence are alternately added and subtracted, the result is an element of the sequence or the negative of an element of the sequence.
For example, when $n = 4$ we have $1-2+3-5 = -3$ and $3$ is an element of the Fibonacci sequence.
2011 All-Russian Olympiad, 4
Perimeter of triangle $ABC$ is $4$. Point $X$ is marked at ray $AB$ and point $Y$ is marked at ray $AC$ such that $AX=AY=1$. Line segments $BC$ and $XY$ intersectat point $M$. Prove that perimeter of one of triangles $ABM$ or $ACM$ is $2$.
(V. Shmarov).
2021 OlimphÃada, 5
Let $p$ be an odd prime. The numbers $1, 2, \ldots, d$ are written on a blackboard, where $d \geq p-1$ is a positive integer. A valid operation is to delete two numbers $x$ and $y$ and write $x + y - c \cdot xy$ in their place, where $c$ is a positive integer. One moment there is only one number $A$ left on the board. Show that if there is an order of operations such that $p$ divides $A$, then $p | d$ or $p | d + 1$.
2000 China National Olympiad, 2
Find all positive integers $n$ such that there exists integers $n_1,\ldots,n_k\ge 3$, for some integer $k$, satisfying
\[n=n_1n_2\cdots n_k=2^{\frac{1}{2^k}(n_1-1)\cdots (n_k-1)}-1.\]
2011 Middle European Mathematical Olympiad, 2
Let $a, b, c$ be positive real numbers such that
\[\frac{a}{1+a}+\frac{b}{1+b}+\frac{c}{1+c}=2.\]
Prove that
\[\frac{\sqrt a + \sqrt b+\sqrt c}{2} \geq \frac{1}{\sqrt a}+\frac{1}{\sqrt b}+\frac{1}{\sqrt c}.\]
1953 AMC 12/AHSME, 6
Charles has $ 5q \plus{} 1$ quarters and Richard has $ q \plus{} 5$ quarters. The difference in their money in dimes is:
$ \textbf{(A)}\ 10(q \minus{} 1) \qquad\textbf{(B)}\ \frac {2}{5}(4q \minus{} 4) \qquad\textbf{(C)}\ \frac {2}{5}(q \minus{} 1) \\
\textbf{(D)}\ \frac {5}{2}(q \minus{} 1) \qquad\textbf{(E)}\ \text{none of these}$
2004 Thailand Mathematical Olympiad, 1
Given that $\cos 4A =\frac13$ and $-\frac{\pi}{4} \le A \le \frac{\pi}{4}$ , find the value of $\cos^8 A - \sin^8 A$.
2010 District Olympiad, 4
Determine all the functions $ f: \mathbb{N}\rightarrow \mathbb{N}$ such that
\[ f(n)\plus{}f(n\plus{}1)\plus{}f(f(n))\equal{}3n\plus{}1, \quad \forall n\in \mathbb{N}.\]
2007 Greece Junior Math Olympiad, 2
If $n$ is is an integer such that $4n+3$ is divisible by $11,$ find the form of $n$ and the remainder of $n^{4}$ upon division by $11$.
1953 AMC 12/AHSME, 27
The radius of the first circle is $ 1$ inch, that of the second $ \frac{1}{2}$ inch, that of the third $ \frac{1}{4}$ inch and so on indefinitely. The sum of the areas of the circles is:
$ \textbf{(A)}\ \frac{3\pi}{4} \qquad\textbf{(B)}\ 1.3\pi \qquad\textbf{(C)}\ 2\pi \qquad\textbf{(D)}\ \frac{4\pi}{3} \qquad\textbf{(E)}\ \text{none of these}$
1957 AMC 12/AHSME, 21
Start with the theorem "If two angles of a triangle are equal, the triangle is isosceles," and the following four statements:
1. If two angles of a triangle are not equal, the triangle is not isosceles.
2. The base angles of an isosceles triangle are equal.
3. If a triangle is not isosceles, then two of its angles are not equal.
4. A necessary condition that two angles of a triangle be equal is that the triangle be isosceles.
Which combination of statements contains only those which are logically equivalent to the given theorem?
$ \textbf{(A)}\ 1,\,2,\,3,\,4 \qquad
\textbf{(B)}\ 1,\,2,\,3\qquad
\textbf{(C)}\ 2,\,3,\,4\qquad
\textbf{(D)}\ 1,\,2\qquad
\textbf{(E)}\ 3,\,4$
2024 India IMOTC, 14
Let $ABCD$ be a convex cyclic quadrilateral with circumcircle $\omega$. Let $BA$ produced beyond $A$ meet $CD$ produced beyond $D$, at $L$. Let $\ell$ be a line through $L$ meeting $AD$ and $BC$ at $M$ and $N$ respectively, so that $M,D$ (respectively $N,C$) are on opposite sides of $A$ (resp. $B$). Suppose $K$ and $J$ are points on the arc $AB$ of $\omega$ not containing $C,D$ so that $MK, NJ$ are tangent to $\omega$. Prove that $K,J,L$ are collinear.
[i]Proposed by Rijul Saini[/i]
2006 Finnish National High School Mathematics Competition, 2
Show that the inequality \[3(1 + a^2 + a^4)\geq (1 + a + a^2)^2\]
holds for all real numbers $a.$
1996 Iran MO (2nd round), 4
Let $n$ blue points $A_i$ and $n$ red points $B_i \ (i = 1, 2, \ldots , n)$ be situated on a line. Prove that
\[\sum_{i,j} A_i B_j \geq \sum_{i<j} A_iA_j + \sum_{i<j} B_iB_j.\]
2014 PUMaC Team, 1
The evilest number $666^{666}$ has $1881$ digits. Let $a$ be the sum of digits of $66^{666}$ and let $b$ be the sum of digits of $a$ and let $c$ be the sum of digits of $b$. Find $c$.
2008 China Team Selection Test, 3
Let $ z_{1},z_{2},z_{3}$ be three complex numbers of moduli less than or equal to $ 1$. $ w_{1},w_{2}$ are two roots of the equation $ (z \minus{} z_{1})(z \minus{} z_{2}) \plus{} (z \minus{} z_{2})(z \minus{} z_{3}) \plus{} (z \minus{} z_{3})(z \minus{} z_{1}) \equal{} 0$. Prove that, for $ j \equal{} 1,2,3$, $\min\{|z_{j} \minus{} w_{1}|,|z_{j} \minus{} w_{2}|\}\leq 1$ holds.
2005 Taiwan National Olympiad, 1
Let $a,b,c$ be three positive real numbers such that $abc=1$. Prove that: \[ 1+\frac{3}{a+b+c}\ge{\frac{6}{ab+bc+ca}} . \]
1966 IMO Shortlist, 27
Given a point $P$ lying on a line $g,$ and given a circle $K.$ Construct a circle passing through the point $P$ and touching the circle $K$ and the line $g.$
2016 NIMO Problems, 6
Let $ABC$ be a triangle with $AB=20$, $AC=34$, and $BC=42$. Let $\omega_1$ and $\omega_2$ be the semicircles with diameters $\overline{AB}$ and $\overline{AC}$ erected outwards of $\triangle ABC$ and denote by $\ell$ the common external tangent to $\omega_1$ and $\omega_2$. The line through $A$ perpendicular to $\overline{BC}$ intersects $\ell$ at $X$ and $BC$ at $Y$. The length of $\overline{XY}$ can be written in the form $m+\sqrt n$ where $m$ and $n$ are positive integers. Find $100m+n$.
[i]Proposed by David Altizio[/i]
2016 ASDAN Math Tournament, 3
Real numbers $x,y,z$ form an arithmetic sequence satisfying
\begin{align*}
x+y+z&=6\\
xy+yz+zx&=10.
\end{align*}
What is the absolute value of their common difference?