Found problems: 85335
2012 Today's Calculation Of Integral, 821
Prove that : $\ln \frac{11}{27}<\int_{\frac 14}^{\frac 34} \frac{1}{\ln (1-x)}\ dx<\ln \frac{7}{15}.$
2008 Romania Team Selection Test, 3
Let $ \mathcal{P}$ be a square and let $ n$ be a nonzero positive integer for which we denote by $ f(n)$ the maximum number of elements of a partition of $ \mathcal{P}$ into rectangles such that each line which is parallel to some side of $ \mathcal{P}$ intersects at most $ n$ interiors (of rectangles). Prove that
\[ 3 \cdot 2^{n\minus{}1} \minus{} 2 \le f(n) \le 3^n \minus{} 2.\]
2014 Contests, 3
Isabella had a week to read a book for a school assignment. She read an average of $36$ pages per day for the first three days and an average of $44$ pages per day for the next three days. She then finished the book by reading $10$ pages on the last day. How many pages were in the book?
$\textbf{(A) }240\qquad\textbf{(B) }250\qquad\textbf{(C) }260\qquad\textbf{(D) }270\qquad \textbf{(E) }280$
2004 Putnam, B3
Determine all real numbers $a>0$ for which there exists a nonnegative continuous function $f(x)$ defined on $[0,a]$ with the property that the region
$R=\{(x,y): 0\le x\le a, 0\le y\le f(x)\}$
has perimeter $k$ units and area $k$ square units for some real number $k$.
2019 India PRMO, 30
For any real number $x$, let $\lfloor x \rfloor$ denote the integer part of $x$; $\{ x \}$ be the fractional part of $x$ ($\{x\}$ $=$ $x-$ $\lfloor x \rfloor$). Let $A$ denote the set of all real numbers $x$ satisfying
$$\{x\} =\frac{x+\lfloor x \rfloor +\lfloor x + (1/2) \rfloor }{20}$$
If $S$ is the sume of all numbers in $A$, find $\lfloor S \rfloor$
2015 May Olympiad, 3
Let $ABCDEFGHI$ be a regular polygon of $9$ sides. The segments $AE$ and $DF$ intersect at $P$. Prove that $PG$ and $AF$ are perpendicular.
2012 China National Olympiad, 2
Consider a square-free even integer $n$ and a prime $p$, such that
1) $(n,p)=1$;
2) $p\le 2\sqrt{n}$;
3) There exists an integer $k$ such that $p|n+k^2$.
Prove that there exists pairwise distinct positive integers $a,b,c$ such that $n=ab+bc+ca$.
[i]Proposed by Hongbing Yu[/i]
2017 India PRMO, 17
Suppose the altitudes of a triangle are $10, 12$ and $15$. What is its semi-perimeter?
2006 Princeton University Math Competition, 3
Find the fifth root of $14348907$.
PEN D Problems, 4
Let $n$ be a positive integer. Prove that $n$ is prime if and only if \[{{n-1}\choose k}\equiv (-1)^{k}\pmod{n}\] for all $k \in \{ 0, 1, \cdots, n-1 \}$.
1989 Bundeswettbewerb Mathematik, 1
For a given positive integer $n$, let $f(x) =x^{n}$. Is it possible for the decimal number
$$0.f(1)f(2)f(3)\ldots$$
to be rational? (Example: for $n=2$, we are considering $0.1491625\ldots$)
1984 Canada National Olympiad, 5
Given any $7$ real numbers, prove that there are two of them $x,y$ such that $0\le\frac{x-y}{1+xy}\le\frac{1}{\sqrt{3}}$.
2014 AIME Problems, 9
Let $x_1<x_2<x_3$ be three real roots of equation $\sqrt{2014}x^3-4029x^2+2=0$. Find $x_2(x_1+x_3)$.
2013 CHMMC (Fall), 3
Let $p_n$ be the product of the $n$th roots of $1$. For integral $x > 4$, let $f(x) = p_1 - p_2 + p_3 - p_4 + ... + (-1)^{x+1}p_x$. What is $f(2010)$?
2013 Stanford Mathematics Tournament, 8
Find the sum of all real $x$ such that \[\frac{4x^2 + 15x + 17}{x^2 + 4x + 12}=\frac{5x^2 + 16x + 18}{2x^2 + 5x + 13}.\]
2013 China Western Mathematical Olympiad, 6
Let $PA, PB$ be tangents to a circle centered at $O$, and $C$ a point on the minor arc $AB$. The perpendicular from $C$ to $PC$ intersects internal angle bisectors of $AOC,BOC$ at $D,E$. Show that $CD=CE$
2022 Iran MO (2nd round), 6
we have an isogonal triangle $ABC$ such that $BC=AB$. take a random $P$ on the altitude from $B$ to $AC$.
The circle $(ABP)$ intersects $AC$ second time in $M$. Take $N$ such that it's on the segment $AC$ and $AM=NC$ and $M \neq N$.The second intersection of $NP$ and circle $(APB)$ is $X$ , ($X \neq P$) and the second intersection of $AB$ and circle $(APN)$ is $Y$ ,($Y \neq A$).The tangent from $A$ to the circle $(APN)$ intersects the altitude from $B$ at $Z$.
Prove that $CZ$ is tangent to circle $(PXY)$.
1999 Mediterranean Mathematics Olympiad, 2
A plane figure of area $A > n$ is given, where $n$ is a positive integer. Prove that
this figure can be placed onto a Cartesian plane so that it covers at least $n+1$
points with integer coordinates.
1986 IMO Longlists, 1
Let $k$ be one of the integers $2, 3,4$ and let $n = 2^k -1$. Prove the inequality
\[1+ b^k + b^{2k} + \cdots+ b^{nk} \geq (1 + b^n)^k\]
for all real $b \geq 0.$
2004 Irish Math Olympiad, 2
Each of the players in a tennis tournament played one match against each of
the others. If every player won at least one match, show that there is a group
A; B; C of three players for which A beat B, B beat C and C beat A.
2017 International Olympic Revenge, 1
Let $f(x)$ be the distance from $x$ to the nearest perfect square. For example, $f(\pi) = 4 - \pi$. Let $\alpha = \frac{3 + \sqrt{5}}{2}$ and let $m$ be an integer such that the sequence $a_n = f(m \; \alpha^n)$ is bounded. Prove that either $m=k^2$ or $m = 5k^2$ for some integer $k$.
[i]Proposed by Rodrigo Sanches Angelo (rsa365), Brazil[/i].
2017 Romanian Master of Mathematics Shortlist, G2
Let $ABC$ be a triangle. Consider the circle $\omega_B$ internally tangent to the sides $BC$ and $BA$, and to the circumcircle of the triangle $ABC$, let $P$ be the point of contact of the two circles. Similarly, consider the circle $\omega_C$ internally tangent to the sides $CB$ and $CA$, and to the circumcircle of the triangle $ABC$, let $Q$ be the point of contact of the two circles. Show that the incentre of the triangle $ABC$ lies on the segment $PQ$ if and only if $AB + AC = 3BC$.
proposed by Luis Eduardo Garcia Hernandez, Mexico
2015 Paraguayan Mathematical Olympiad, Problem 1
Alexa wrote the first $16$ numbers of a sequence:
\[1, 2, 2, 3, 4, 4, 5, 6, 6, 7, 8, 8, 9, 10, 10, 11, …\]
Then she continued following the same pattern, until she had $2015$ numbers in total.
What was the last number she wrote?
1987 IMO Longlists, 38
Let $S_1$ and $S_2$ be two spheres with distinct radii that touch externally. The spheres lie inside a cone $C$, and each sphere touches the cone in a full circle. Inside the cone there are $n$ additional solid spheres arranged in a ring in such a way that each solid sphere touches the cone $C$, both of the spheres $S_1$ and $S_2$ externally, as well as the two neighboring solid spheres. What are the possible values of $n$?
[i]Proposed by Iceland.[/i]
1996 Spain Mathematical Olympiad, 4
For each real value of $p$, find all real solutions of the equation $\sqrt{x^2 - p}+2\sqrt{x^2-1} = x$.