Found problems: 85335
1993 Turkey Team Selection Test, 6
Determine all functions $f: \mathbb{Q^+} \rightarrow \mathbb{Q^+}$ that satisfy:
\[f\left(x+\frac{y}{x}\right) = f(x)+f\left(\frac{y}{x}\right)+2y \:\text{for all}\: x, y \in \mathbb{Q^+}\]
2014 Iran Team Selection Test, 3
let $m,n\in \mathbb{N}$ and $p(x),q(x),h(x)$ are polynomials with real Coefficients such that $p(x)$ is Descending.
and for all $x\in \mathbb{R}$
$p(q(nx+m)+h(x))=n(q(p(x))+h(x))+m$ .
prove that dont exist function $f:\mathbb{R}\rightarrow \mathbb{R}$ such that for all $x\in \mathbb{R}$
$f(q(p(x))+h(x))=f(x)^{2}+1$
1988 Spain Mathematical Olympiad, 6
For all integral values of parameter $t$, find all integral solutions $(x,y)$ of the equation
$$ y^2 = x^4-22x^3+43x^2+858x+t^2+10452(t+39)$$ .
Kyiv City MO Juniors Round2 2010+ geometry, 2019.9.3
The equilateral triangle $ABC$ is inscribed in the circle $w$. Points $F$ and $E$ on the sides $AB$ and $AC$, respectively, are chosen such that $\angle ABE+ \angle ACF = 60^o$. The circumscribed circle of $\vartriangle AFE$ intersects the circle $w$ at the point $D$ for the second time. The rays $DE$ and $DF$ intersect the line $BC$ at the points $X$ and $Y$, respectively. Prove that the center of the inscribed circle of $\vartriangle DXY$ does not depend on the choice of points $F$ and $E$.
(Hilko Danilo)
1960 Polish MO Finals, 6
On the perimeter of a rectangle, point $ M $ is chosen. Find the shortest path whose beginning and end are point $ M $ and which has a point in common with each side of the rectangle.
2005 Miklós Schweitzer, 8
Determine all continuous, strictly monotone functions $\phi : \mathbb{R}^+\to\mathbb{R}$ such that $$F(x,y)=\phi^{-1} \left(\frac{x\phi(x)+y\phi(y)}{x+y}\right) + \phi^{-1} \left(\frac{y\phi(x)+x\phi(y)}{x+y}\right) $$ is homogeneous of degree 1, ie $F(tx,ty)=tF(x,y) , \forall x,y,t\in\mathbb{R}^+$
[hide=Note]F(x,y)=F(y,x) and F(x,x)=2x[/hide]
2004 Harvard-MIT Mathematics Tournament, 9
Find the positive constant $c_0$ such that the series \[ \displaystyle\sum_{n = 0}^{\infty} \dfrac {n!}{(cn)^n} \] converges for $c>c_0$ and diverges for $0<c<c_0$.
2006 AIME Problems, 6
Square $ABCD$ has sides of length 1. Points $E$ and $F$ are on $\overline{BC}$ and $\overline{CD}$, respectively, so that $\triangle AEF$ is equilateral. A square with vertex $B$ has sides that are parallel to those of $ABCD$ and a vertex on $\overline{AE}$. The length of a side of this smaller square is $\displaystyle \frac{a-\sqrt{b}}{c}$, where $a$, $b$, and $c$ are positive integers and $b$ is not divisible by the square of any prime. Find $a+b+c$.
1978 AMC 12/AHSME, 12
In $\triangle ADE$, $\measuredangle ADE=140^\circ$, points $B$ and $C$ lie on sides $AD$ and $AE$, respectively, and points $A,~B,~C,~D,~E$ are distinct.* If lengths $AB,~BC,~CD,$ and $DE$ are all equal, then the measure of $\measuredangle EAD$ is
$\textbf{(A) }5^\circ\qquad\textbf{(B) }6^\circ\qquad\textbf{(C) }7.5^\circ\qquad\textbf{(D) }8^\circ\qquad \textbf{(E) }10^\circ$
[size=50]* The specification that points $A,B,C,D,E$ be distinct was not included in the original statement of the problem. If $B=D$, then $C=E$ and $\measuredangle EAD=20^\circ$.[/size]
2007 Princeton University Math Competition, 1
Suppose that $A$ is a set of integers. Denote the number of elements in $A$ by $|A|$. Define $A+A = \{a_1+a_2: a_1, a_2 \in A\}$ and $A-A = \{a_1-a_2:a_1, a_2 \in A\}$. Prove or disprove: for any set $A$, we have the inequality $|A-A| \ge |A+A|$.
1969 All Soviet Union Mathematical Olympiad, 123
Every city in the certain state is connected by airlines with no more than with three other ones, but one can get from every city to every other city changing a plane once only or directly. What is the maximal possible number of the cities?
2014 PUMaC Geometry A, 1
Let $x=\frac pq$ for $p$, $q$ coprime. Find $p+q$.
[asy]
import olympiad;
size(200);
pen qq=font("phvb");
defaultpen(linewidth(0.6)+fontsize(10pt));
pair A=(-2.25,7),B=(-5,0),C=(5,0),D=waypoint(A--B,3/7), E=waypoint(A--C,1/2),F=intersectionpoint(C--D, B--E);
draw(A--B--C--cycle^^B--E^^C--D);
label("$A$",A,NW);
label("$B$",B,SW);
label("$C$",C,SE);
label("$D$",D,NW);
label("$E$",E,NE);
label("$F$",F,N);
label(scale(2.5)*"X",centroid(A,D,E),qq);
label(scale(2.5)*"3",centroid(B,D,F),0.5*N,qq);
label(scale(2.5)*"6",centroid(B,F,C),0.25*dir(180),qq);
label(scale(2.5)*"2",centroid(C,E,F),dir(140),qq);
[/asy]
2012 Harvard-MIT Mathematics Tournament, 7
Let $S$ be the set of the points $(x_1, x_2, . . . , x_{2012})$ in $2012$-dimensional space such that $|x_1|+|x_2|+...+|x_{2012}| \le 1$. Let $T$ be the set of points in $2012$-dimensional space such that $\max^{2012}_{i=1}|x_i| = 2$. Let $p$ be a randomly chosen point on $T$. What is the probability that the closest point in $S$ to $p$ is a vertex of $S$?
IV Soros Olympiad 1997 - 98 (Russia), 11.10
Let $a_n = \frac{\pi}{2n}$, where $n$ is a natural number. Prove that for any $k = 1$,$2$,$...$, $n$ holds the equality $$\frac{\sin ka_n}{1-\cos a_n}+\frac{\sin 5ka_n}{1-\cos 5a_n}+\frac{\sin 9ka_n}{1-\cos 9a_n}+...+\frac{\sin (4n-3)a_n}{1-\cos (4n-3)a_n}=kn$$
2009 ISI B.Math Entrance Exam, 2
Let $c$ be a fixed real number. Show that a root of the equation
\[x(x+1)(x+2)\cdots(x+2009)=c\]
can have multiplicity at most $2$. Determine the number of values of $c$ for which the equation has a root of multiplicity $2$.
2022 Bulgarian Spring Math Competition, Problem 9.2
Let $\triangle ABC$ have median $CM$ ($M\in AB$) and circumcenter $O$. The circumcircle of $\triangle AMO$ bisects $CM$. Determine the least possible perimeter of $\triangle ABC$ if it has integer side lengths.
2006 JBMO ShortLists, 3
Let $ n\ge 3$ be a natural number. A set of real numbers $ \{x_1,x_2,\ldots,x_n\}$ is called [i]summable[/i] if $ \sum_{i\equal{}1}^n \frac{1}{x_i}\equal{}1$. Prove that for every $ n\ge 3$ there always exists a [i]summable[/i] set which consists of $ n$ elements such that the biggest element is:
a) bigger than $ 2^{2n\minus{}2}$
b) smaller than $ n^2$
2010 AMC 10, 13
What is the sum of all the solutions of $ x \equal{} |2x \minus{} |60\minus{}2x\parallel{}$?
$ \textbf{(A)}\ 32\qquad\textbf{(B)}\ 60\qquad\textbf{(C)}\ 92\qquad\textbf{(D)}\ 120\qquad\textbf{(E)}\ 124$
2005 Estonia National Olympiad, 2
Let $a, b$ and $c$ be real numbers such that $\frac{a}{b + c}+\frac{b}{c + a}+\frac{c}{a + b}= 1$.
Prove that $\frac{a^2}{b + c}+\frac{b^2}{c + a}+\frac{c^2}{a + b}= 0$.
2011 Albania National Olympiad, 1
[b](a) [/b] Find the minimal distance between the points of the graph of the function $y=\ln x$ from the line $y=x$.
[b](b)[/b] Find the minimal distance between two points, one of the point is in the graph of the function $y=e^x$ and the other point in the graph of the function $y=ln x$.
1999 Ukraine Team Selection Test, 3
Let $m,n$ be positive integers with $m \le n$, and let $F$ be a family of $m$-element subsets of $\{1,2,...,n\}$ satisfying $A \cap B \ne \varnothing$ for all $A,B \in F$. Determine the maximum possible number of elements in $F$.
2010 Harvard-MIT Mathematics Tournament, 2
A rectangular piece of paper is folded along its diagonal (as depicted below) to form a non-convex pentagon that has an area of $\tfrac{7}{10}$ of the area of the original rectangle. Find the ratio of the longer side of the rectangle to the shorter side of the rectangle.
[asy]
size(150);
pair A = (-5,0);
pair B = (5,0);
pair C = (-3,4);
pair D = (3,4);
pair E = intersectionpoint(B--C,A--D);
draw(A--B--D--cycle);
draw(A--C);
draw(C--E);
draw(E--B,dashed);
markscalefactor=0.06;
draw(rightanglemark(A,C,B));
[/asy]
1986 Spain Mathematical Olympiad, 3
Find all natural numbers $n$ such that $5^n+3$ is a power of $2$
2016 Romania National Olympiad, 4
Let $K$ be a finite field with $q$ elements, $q \ge 3.$ We denote by $M$ the set of polynomials in $K[X]$ of degree $q-2$ whose coefficients are nonzero and pairwise distinct. Find the number of polynomials in $M$ that have $q-2$ distinct roots in $K.$
[i]Marian Andronache[/i]
2009 Thailand Mathematical Olympiad, 4
In triangle $\vartriangle ABC$, $D$ is the midpoint of $BC$. Points $E$ and $F$ are chosen on side $AC$ so that $AF = F E = EC$. Let $AD$ intersect $BE$ and $BF$ and $G$ and $H$, respectively. Find the ratio of the areas of $\vartriangle BGH$ and $\vartriangle ABC$.