Found problems: 85335
PEN J Problems, 9
Show that the set of all numbers $\frac{\phi(n+1)}{\phi(n)}$ is dense in the set of all positive reals.
1936 Eotvos Mathematical Competition, 1
Prove that for all positive integers $n$,
$$\frac{1}{1 \cdot 2}+\frac{1}{3 \cdot 4}+ ...+ \frac{1}{(2n - 1)2n}=\frac{1}{n + 1}\frac{1}{n + 2}+ ... +\frac{1}{2n}$$
2020 AMC 12/AHSME, 1
Carlos took $70\%$ of a whole pie. Maria took one third of the remainder. What portion of the whole pie was left?
$\textbf{(A)}\ 10\%\qquad\textbf{(B)}\ 15\%\qquad\textbf{(C)}\ 20\%\qquad\textbf{(D)}\ 30\%\qquad\textbf{(E)}\ 35\%$
2017 Moldova Team Selection Test, 2
Let $$f(X)=a_{n}X^{n}+a_{n-1}X^{n-1}+\cdots +a_{1}X+a_{0}$$be a polynomial with real coefficients which satisfies $$a_{n}\geq a_{n-1}\geq \cdots \geq a_{1}\geq a_{0}>0.$$Prove that for every complex root $z$ of this polynomial, we have $|z|\leq 1$.
2007 Today's Calculation Of Integral, 198
Compare the values of the following definite integrals.
\[\int_{0}^{\infty}\ln \left(x+\frac{1}{x}\right)\frac{dx}{1+x^{2}},\ \ \int_{0}^{\frac{\pi}{2}}\left(\frac{\theta}{\sin \theta}\right)^{2}d\theta\]
2022 Taiwan TST Round 1, G
Let $ABCD$ be a parallelogram with $AC=BC.$ A point $P$ is chosen on the extension of ray $AB$ past $B.$ The circumcircle of $ACD$ meets the segment $PD$ again at $Q.$ The circumcircle of triangle $APQ$ meets the segment $PC$ at $R.$ Prove that lines $CD,AQ,BR$ are concurrent.
2013 Harvard-MIT Mathematics Tournament, 4
Spencer is making burritos, each of which consists of one wrap and one filling. He has enough filling for up to four beef burritos and three chicken burritos. However, he only has five wraps for the burritos; in how many orders can he make exactly five burritos?
2000 IMO Shortlist, 6
A nonempty set $ A$ of real numbers is called a $ B_3$-set if the conditions $ a_1, a_2, a_3, a_4, a_5, a_6 \in A$ and $ a_1 \plus{} a_2 \plus{} a_3 \equal{} a_4 \plus{} a_5 \plus{} a_6$ imply that the sequences $ (a_1, a_2, a_3)$ and $ (a_4, a_5, a_6)$ are identical up to a permutation. Let $A = \{a_0 = 0 < a_1 < a_2 < \cdots \}$, $B = \{b_0 = 0 < b_1 < b_2 < \cdots \}$ be infinite sequences of real numbers with $ D(A) \equal{} D(B),$ where, for a set $ X$ of real numbers, $ D(X)$ denotes the difference set $ \{|x\minus{}y|\mid x, y \in X \}.$ Prove that if $ A$ is a $ B_3$-set, then $ A \equal{} B.$
1985 Bundeswettbewerb Mathematik, 2
The insphere of any tetrahedron has radius $r$. The four tangential planes parallel to the side faces of the tetrahedron cut from the tetrahedron four smaller tetrahedrons whose in-sphere radii are $r_1, r_2, r_3$ and $r_4$. Prove that $$r_1 + r_2 + r_3 + r_4 = 2r$$
2010 Thailand Mathematical Olympiad, 9
Let $a, b, c$ be real numbers so that all roots of the equation $2x^5 + 5x^4 + 5x^3 + ax^2 + bx + c = 0$ are real. Find the smallest real root of the equation above.
2016 India IMO Training Camp, 2
Find all functions $f:\mathbb{R}\to\mathbb{R}$ such that $$f\left(x^3+f(y)\right)=x^2f(x)+y,$$for all $x,y\in\mathbb{R}.$ (Here $\mathbb{R}$ denotes the set of all real numbers.)
2016 Romania Team Selection Tests, 2
Let $ABC$ be a triangle with $CA \neq CB$. Let $D$, $F$, and $G$ be the midpoints of the sides $AB$, $AC$, and $BC$ respectively. A circle $\Gamma$ passing through $C$ and tangent to $AB$ at $D$ meets the segments $AF$ and $BG$ at $H$ and $I$, respectively. The points $H'$ and $I'$ are symmetric to $H$ and $I$ about $F$ and $G$, respectively. The line $H'I'$ meets $CD$ and $FG$ at $Q$ and $M$, respectively. The line $CM$ meets $\Gamma$ again at $P$. Prove that $CQ = QP$.
[i]Proposed by El Salvador[/i]
2018 South Africa National Olympiad, 4
Let $ABC$ be a triangle with circumradius $R$, and let $\ell_A, \ell_B, \ell_C$ be the altitudes through $A, B, C$ respectively. The altitudes meet at $H$. Let $P$ be an arbitrary point in the same plane as $ABC$. The feet of the perpendicular lines through $P$ onto $\ell_A, \ell_B, \ell_C$ are $D, E, F$ respectively. Prove that the areas of $DEF$ and $ABC$ satisfy the following equation:
$$
\operatorname{area}(DEF) = \frac{{PH}^2}{4R^2} \cdot \operatorname{area}(ABC).
$$
2014 Romania National Olympiad, 2
Let be a function $ f:\mathbb{N}\longrightarrow\mathbb{N} $ satisfying
$ \text{(i)} f(1)=1 $
$ \text{(ii)} f(p)=1+f(p-1), $ for any prime $ p $
$ \text{(iii)} f(p_1p_2\cdots p_u)=f(p_1)+f(p_2)+\cdots f(p_u), $ for any natural number $ u $ and any primes $ p_1,p_2,\ldots ,p_u. $
Show that $ 2^{f(n)}\le n^3\le 3^{f(n)}, $ for any natural $ n\ge 2. $
1966 Miklós Schweitzer, 7
Does there exist a function $ f(x,y)$ of two real variables that takes natural numbers as its values and for which $ f(x,y)\equal{}f(y,z)$ implies $ x\equal{}y\equal{}z?$
[i]A. Hajnal[/i]
2013 Argentina Cono Sur TST, 6
Let $m\geq 4$ and $n\geq 4$. An integer is written on each cell of a $m \times n$ board. If each cell has a number equal to the arithmetic mean of some pair of numbers written on its neighbouring cells, determine the maximum amount of distinct numbers that the board may have.
Note: two neighbouring cells share a common side.
2003 Brazil National Olympiad, 3
$ABCD$ is a rhombus. Take points $E$, $F$, $G$, $H$ on sides $AB$, $BC$, $CD$, $DA$ respectively so that $EF$ and $GH$ are tangent to the incircle of $ABCD$. Show that $EH$ and $FG$ are parallel.
2005 MOP Homework, 7
Let $n$ be a positive integer with $n>1$, and let $a_1$, $a_2$, ..., $a_n$ be positive integers such that $a_1<a_2<...<a_n$ and $\frac{1}{a_1}+\frac{1}{a_2}+...+\frac{1}{a_n} \le 1$.
Prove that
$(\frac{1}{a_1^2+x^2}+\frac{1}{a_2^2+x^2}+...+\frac{1}{a_n^2+x^2})^2 \le \frac{1}{2} \cdot \frac{1}{a_1(a_1-1)+x^2}$
for all real numbers $x$.
2024 ELMO Shortlist, G5
Let $ABC$ be a triangle with circumcenter $O$ and circumcircle $\omega$. Let $D$ be the foot of the altitude from $A$ to $\overline{BC}$. Let $P$ and $Q$ be points on the circumcircles of triangles $AOB$ and $AOC$, respectively, such that $A$, $P$, and $Q$ are collinear. Prove that if the circumcircle of triangle $OPQ$ is tangent to $\omega$ at $T$, then $\angle BTD=\angle CAP$.
[i]Tiger Zhang[/i]
1960 Polish MO Finals, 3
On the circle 6 distinct points $ A $, $ B $, $ C $, $ D $, $ E $, $ F $ are chosen in such a way that $ AB $ is parallel to $ DE $, and $ DC $ is parallel to $ AF $. Prove that $ BC $ is parallel to $ EF $
2025 PErA, P4
Let \( ABC \) be an acute-angled scalene triangle. Let \( B_1 \) and \( B_2 \) be points on the rays \( BC \) and \( BA \), respectively, such that \( BB_1 = BB_2 = AC \). Similarly, let \( C_1 \) and \( C_2 \) be points on the rays \( CB \) and \( CA \), respectively, such that \( CC_1 = CC_2 = AB \). Prove that if \( B_1B_2 \) and \( C_1C_2 \) intersect at \( K \), then \( AK \) is parallel to \( BC \).
2022 JHMT HS, 7
Two rays emanate from the origin $O$ and form a $45^\circ$ angle in the first quadrant of the Cartesian coordinate plane. For some positive numbers $X$, $Y$, and $S$, the ray with the larger slope passes through point $A = (X, S)$, and the ray with the smaller slope passes through point $B = (S, Y)$. If $6X + 6Y + 5S = 600$, then determine the maximum possible area of $\triangle OAB$.
2000 Brazil Team Selection Test, Problem 2
For a positive integer $n$, let $A_n$ be the set of all positive numbers greater than $1$ and less than $n$ which are coprime to $n$. Find all $n$ such that all the elements of $A_n$ are prime numbers.
2018 CMIMC Algebra, 3
Let $P(x)=x^2+4x+1$. What is the product of all real solutions to the equation $P(P(x))=0$?
2009 Tournament Of Towns, 5
A country has two capitals and several towns. Some of them are connected by roads. Some of the roads are toll roads where a fee is charged for driving along them. It is known that any route from the south capital to the north capital contains at least ten toll roads. Prove that all toll roads can be distributed among ten companies so that anybody driving from the south capital to the north capital must pay each of these companies.
[i](5 points)[/i]