Found problems: 85335
2003 Brazil National Olympiad, 2
Let $f(x)$ be a real-valued function defined on the positive reals such that
(1) if $x < y$, then $f(x) < f(y)$,
(2) $f\left(2xy\over x+y\right) \geq {f(x) + f(y)\over2}$ for all $x$.
Show that $f(x) < 0$ for some value of $x$.
PEN A Problems, 5
Let $x$ and $y$ be positive integers such that $xy$ divides $x^{2}+y^{2}+1$. Show that \[\frac{x^{2}+y^{2}+1}{xy}=3.\]
2023 Harvard-MIT Mathematics Tournament, 1
For any positive integer $a$, let $\tau(a)$ be the number of positive divisors of $a$. Find, with proof, the largest possible value of $4\tau(n)-n$ over all positive integers $n$.
2007 Germany Team Selection Test, 1
For a multiple of $ kb$ of $ b$ let $ a \% kb$ be the greatest number such that $ a \% kb \equal{} a \bmod b$ which is smaller than $ kb$ and not greater than $ a$ itself. Let $ n \in \mathbb{Z}^ \plus{} .$ Determine all integer pairs $ (a,b)$ with:
\[ a\%b \plus{} a\%2b \plus{} a\%3b \plus{} \ldots \plus{} a\%nb \equal{} a \plus{} b
\]
2004 239 Open Mathematical Olympiad, 7
$200n$ diagonals are drawn in a convex $n$-gon. Prove that one of them intersects at least 10000 others.
[b]proposed by D. Karpov, S. Berlov[/b]
2023 MMATHS, 10
Consider the recurrence relation $x_{n+2}=2x_{n+1}+x_n,$ with $x_0=0, x_1=1.$ What is the greatest common divisor of $x_{2023}$ and $x_{721}$?
1983 IMO Longlists, 49
Given positive integers $k,m, n$ with $km \leq n$ and non-negative real numbers $x_1, \ldots , x_k$, prove that
\[n \left( \prod_{i=1}^k x_i^m -1 \right) \leq m \sum_{i=1}^k (x_i^n-1).\]
2023 CCA Math Bonanza, T6
$ABC$ is an equilateral triangle and $l$ is a line such that the distances from $A, B,$ and $C$ to $l$ are $39, 35,$ and $13$, respectively. Find the largest possible value of $AB$.
[i]Team #6[/i]
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]