Found problems: 85335
2013 Hanoi Open Mathematics Competitions, 9
Solve the following system in positive numbers $\begin{cases} x+y\le 1 \\
\frac{2}{xy} +\frac{1}{x^2+y^2}=10\end{cases}$
1977 Miklós Schweitzer, 6
Let $ f$ be a real function defined on the positive half-axis for which $ f(xy)\equal{}xf(y)\plus{}yf(x)$ and $ f(x\plus{}1) \leq f(x)$ hold for every positive $ x$ and $ y$. Show that if $ f(1/2)\equal{}1/2$, then \[ f(x)\plus{}f(1\minus{}x) \geq \minus{}x \log_2 x \minus{}(1\minus{}x) \log_2 (1\minus{}x)\] for every $ x\in (0,1)$.
[i]Z. Daroczy, Gy. Maksa[/i]
2019 IMO Shortlist, G7
Let $I$ be the incentre of acute triangle $ABC$ with $AB\neq AC$. The incircle $\omega$ of $ABC$ is tangent to sides $BC, CA$, and $AB$ at $D, E,$ and $F$, respectively. The line through $D$ perpendicular to $EF$ meets $\omega$ at $R$. Line $AR$ meets $\omega$ again at $P$. The circumcircles of triangle $PCE$ and $PBF$ meet again at $Q$.
Prove that lines $DI$ and $PQ$ meet on the line through $A$ perpendicular to $AI$.
[i]Proposed by Anant Mudgal, India[/i]
2014 Irish Math Olympiad, 9
Let $n$ be a positive integer and $a_1,...,a_n$ be positive real numbers.
Let $g(x)$ denote the product $(x + a_1)\cdot ... \cdot (x + a_n)$ .
Let $a_0$ be a real number and let
$f(x) = (x - a_0)g(x)= x^{n+1} + b_1x^n + b_2x^{n-1}+...+ b_nx + b_{n+1}$ .
Prove that all the coeffcients $b_1,b_2,..., b_{n+1}$ of the polynomial $f(x)$ are negative if and only if $a_0 > a_1 + a_2 +...+ a_n$.
2018 Putnam, A6
Suppose that $A$, $B$, $C$, and $D$ are distinct points, no three of which lie on a line, in the Euclidean plane. Show that if the squares of the lengths of the line segments $AB$, $AC$, $AD$, $BC$, $BD$, and $CD$ are rational numbers, then the quotient
\[\frac{\mathrm{area}(\triangle ABC)}{\mathrm{area}(\triangle ABD)}\]
is a rational number.
2022 China Second Round, 2
Integer $n$ has $k$ different prime factors. Prove that $\sigma (n) \mid (2n-k)!$
2007 ITest, 9
Suppose that $m$ and $n$ are positive integers such that $m<n$, the geometric mean of $m$ and $n$ is greater than $2007$, and the arithmetic mean of $m$ and $n$ is less than $2007$. How many pairs $(m,n)$ satisfy these conditions?
$\textbf{(A) }0\hspace{14em}\textbf{(B) }1\hspace{14em}\textbf{(C) }2$
$\textbf{(D) }3\hspace{14em}\textbf{(E) }4\hspace{14em}\textbf{(F) }5$
$\textbf{(G) }6\hspace{14em}\textbf{(H) }7\hspace{14em}\textbf{(I) }2007$
1999 Flanders Math Olympiad, 1
Determine all 6-digit numbers $(abcdef)$ so that $(abcdef) = (def)^2$ where $\left( x_1x_2...x_n \right)$ is no multiplication but an n-digit number.
2012 Sharygin Geometry Olympiad, 1
In triangle $ABC$ point $M$ is the midpoint of side $AB$, and point $D$ is the foot of altitude $CD$. Prove that $\angle A = 2\angle B$ if and only if $AC = 2 MD$.
2010 Contests, 3
Two rectangles of unit area overlap to form a convex octagon. Show that the area of the octagon is at least $\dfrac {1} {2}$.
[i]Kvant Magazine [/i]
2012 Nordic, 4
The number $1$ is written on the blackboard. After that a sequence of numbers is created as follows: at each step each number $a$ on the blackboard is replaced by the numbers $a - 1$ and $a + 1$; if the number $0$ occurs, it is erased immediately; if a number occurs more than once, all its occurrences are left on the blackboard. Thus the blackboard will show $1$ after $0$ steps; $2$ after $1$ step; $1, 3$ after $2$ steps; $2, 2, 4$ after $3$ steps, and so on. How many numbers will there be on the blackboard after $n$ steps?
2021 Greece Junior Math Olympiad, 1
If positive reals $x,y$ are such that $2(x+y)=1+xy$, find the minimum value of expression $$A=x+\frac{1}{x}+y+\frac{1}{y}$$
2009 Math Prize For Girls Problems, 20
Let $ y_0$ be chosen randomly from $ \{0, 50\}$, let $ y_1$ be chosen randomly from $ \{40, 60, 80\}$, let $ y_2$ be chosen randomly from $ \{10, 40, 70, 80\}$, and let $ y_3$ be chosen randomly from $ \{10, 30, 40, 70, 90\}$. (In each choice, the possible outcomes are equally likely to occur.) Let $ P$ be the unique polynomial of degree less than or equal to $ 3$ such that $ P(0) \equal{} y_0$, $ P(1) \equal{} y_1$, $ P(2) \equal{} y_2$, and $ P(3) \equal{} y_3$. What is the expected value of $ P(4)$?
2013 Today's Calculation Of Integral, 882
Find $\lim_{n\to\infty} \sum_{k=1}^n \frac{1}{n+k}(\ln (n+k)-\ln\ n)$.
May Olympiad L2 - geometry, 1999.2
In a unit circle where $O$ is your circumcenter, let $A$ and $B$ points in the circle with $\angle BOA = 90$. In the arc $AB$(minor arc) we have the points $P$ and $Q$ such that $PQ$ is parallel to $AB$. Let $X$ and $Y$ be the points of intersections of the line $PQ$ with $OA$ and $OB$ respectively. Find the value of $PX^2 + PY^2$
2019 Saudi Arabia JBMO TST, 4
Let $AD$ be the perpendicular to the hypotenuse $BC$ of the right triangle $ABC$. Let $DE$ be the height of the triangle $ADB$ and $DZ$ be the height of the triangle $ADC$. On the line $AB$ is chosen the point $N$ so that $CN$ is parallel to $EZ$. Let $A'$ be symmetrical of $A$ to $EZ$ and $I, K$ projections of $A'$ on $AB$, respectively, on $AC$. Prove that $<$ $NA'T$ $=$ $<$ $ADT$, where $T$ is the point of intersection of $IK$ and $DE$.
2016 LMT, 2
Mike rides a bike for $30$ minutes, traveling $8$ miles. He started riding at $20$ miles per hour, but by the end of his journey he was only traveling at $10$ miles per hour. What was his average speed, in miles per hour?
[i]Proposed by Nathan Ramesh
1985 Miklós Schweitzer, 10
Show that any two intervals $A, B\subseteq \mathbb R$ of positive lengths can be countably disected into each other, that is, they can be written as countable unions $A=A_1\cup A_2\cup\ldots\,$ and $B=B_1\cup B_2\cup\ldots\,$ of pairwise disjoint sets, where $A_i$ and $B_i$ are congruent for every $i\in \mathbb N$ [Gy. Szabo]
2016 PUMaC Number Theory A, 2
For positive integers $i$ and $j$, define $d(i,j)$ as follows: $d(1,j) = 1, d(i,1) = 1$ for all $i$ and $j$, and for $i, j > 1$, $d(i,j) = d(i-1,j) + d(i,j-1) + d(i-1,j-1)$. Compute the remainder when $d(3,2016)$ is divided by $1000$.
2024 CMIMC Geometry, 3
Circles $C_1$, $C_2$, and $C_3$ are inside a rectangle $WXYZ$ such that $C_1$ is tangent to $\overline{WX}$, $\overline{ZW}$, and $\overline{YZ}$; $C_2$ is tangent to $\overline{WX}$ and $\overline{XY}$; and $C_3$ is tangent to $\overline{YZ}$, $C_1$, and $C_2$. If the radii of $C_1$, $C_2$, and $C_3$ are $1$, $\tfrac 12$, and $\tfrac 23$ respectively, compute the area of the triangle formed by the centers of $C_1$, $C_2$, and $C_3$.
[i]Proposed by Connor Gordon[/i]
2006 Purple Comet Problems, 4
A rogue spaceship escapes. $54$ minutes later the police leave in a spaceship in hot pursuit. If the police spaceship travels $12\%$ faster than the rogue spaceship along the same route, how many minutes will it take for the police to catch up with the rogues?
2009 Sharygin Geometry Olympiad, 13
In triangle $ ABC$, one has marked the incenter, the foot of altitude from vertex $ C$ and the center of the excircle tangent to side $ AB$. After this, the triangle was erased. Restore it.
2010 Contests, 3
Let $(x_n)_{n \in \mathbb{N}}$ be the sequence defined as $x_n = \sin(2 \pi n! e)$ for all $n \in \mathbb{N}$. Compute $\lim_{n \to \infty} x_n$.
2023 Taiwan TST Round 3, A
Show that there exists a positive constant $C$ such that, for all positive reals $a$ and $b$ with $a + b$ being an integer, we have
$$\left\{a^3\right\} + \left\{b^3\right\} + \frac{C}{(a+b)^6} \le 2. $$
Here $\{x\} = x - \lfloor x\rfloor$ is the fractional part of $x$.
[i]Proposed by Li4 and Untro368.[/i]
2002 Turkey Junior National Olympiad, 3
Find all ordered positive integer pairs of $(m,n)$ such that $2^n-1$ divides $2^m+1$.