Found problems: 85335
2012 Balkan MO Shortlist, G1
Let $A$, $B$ and $C$ be points lying on a circle $\Gamma$ with centre $O$. Assume that $\angle ABC > 90$. Let $D$ be the point of intersection of the line $AB$ with the line perpendicular to $AC$ at $C$. Let $l$ be the line through $D$ which is perpendicular to $AO$. Let $E$ be the point of intersection of $l$ with the line $AC$, and let $F$ be the point of intersection of $\Gamma$ with $l$ that lies between $D$ and $E$.
Prove that the circumcircles of triangles $BFE$ and $CFD$ are tangent at $F$.
1972 IMO Longlists, 19
Let $S$ be a subset of the real numbers with the following
properties:
$(i)$ If $x \in S$ and $y \in S$, then $x - y \in S$;
$(ii)$ If $x \in S$ and $y \in S$, then $xy \in S$;
$(iii)$ $S$ contains an exceptional number $x'$ such that there is no number $y$ in $S$ satisfying $x'y + x' + y = 0$;
$(iv)$ If $x \in S$ and $x \neq x'$ , there is a number $y$ in $S$ such that $xy+x+y = 0$.
Show that
$(a)$ $S$ has more than one number in it;
$(b)$ $x' \neq -1$ leads to a contradiction;
$(c)$ $x \in S$ and $x \neq 0$ implies $1/x \in S$.
2023 Pan-American Girls’ Mathematical Olympiad, 1
An integer \(n \geq 2\) is said to be [i]tuanis[/i] if, when you add the smallest prime divisor of \(n\) and the largest prime divisor of \(n\) (these divisors can be the same), you obtain an odd result. Calculate the sum of all [i]tuanis[/i] numbers that are less or equal to \(2023\).
2001 Moldova National Olympiad, Problem 4
Let $P(x)=x^n+a_1x^{n-1}+\ldots+a_n$ ($n\ge2$) be a polynomial with integer coefficients having $n$ real roots $b_1,\ldots,b_n$. Prove that for $x_0\ge\max\{b_1,\ldots,b_n\}$,
$$P(x_0+1)\left(\frac1{x_0-b_1}+\ldots+\frac1{x_0-b_n}\right)\ge2n^2.$$
2022 Putnam, B1
Suppose that $P(x)=a_1x+a_2x^2+\ldots+a_nx^n$ is a polynomial with integer coefficients, with $a_1$ odd. Suppose that $e^{P(x)}=b_0+b_1x+b_2x^2+\ldots$ for all $x.$ Prove that $b_k$ is nonzero for all $k \geq 0.$
2017 IMAR Test, 1
Let $P$ be a point in the interior $\triangle ABC$, and $AD,BE,CF$ 3 concurrent cevians through $P$, with $D,E,F$ on $BC,CA,AB$. The circle with the diameter $BC$ intersects the circle with the diameter $AD$ in $D_1,D_2$. Analogously we define $E_1,E_2$ and $F_1,F_2$. Prove that $D_1,D_2,E_1,E_2,F_1,F_2$ are concylic.
1978 Bundeswettbewerb Mathematik, 4
A prime number has the property that however its decimal digits are permuted, the obtained number is also prime. Prove that this number has at most three different digits. Also prove a stronger statement.
2022 HMNT, 3
A polygon $\mathcal{P}$ is drawn on the $2\text{D}$ coordinate plane. Each side of $\mathcal{P}$ is either parallel to the $x$ axis or the $y$ axis (the vertices of $\mathcal{P}$ do not have to be lattice points). Given that the interior of $\mathcal{P}$ includes the interior of the circle $x^2+y^2=2022,$ find the minimum possible perimeter of $\mathcal{P}.$
Estonia Open Senior - geometry, 2014.2.3
The angles of a triangle are $22.5^o, 45^o$ and $112.5^o$. Prove that inside this triangle there exists a point that is located on the median through one vertex, the angle bisector through another vertex and the altitude through the third vertex.
Today's calculation of integrals, 864
Let $m,\ n$ be positive integer such that $2\leq m<n$.
(1) Prove the inequality as follows.
\[\frac{n+1-m}{m(n+1)}<\frac{1}{m^2}+\frac{1}{(m+1)^2}+\cdots +\frac{1}{(n-1)^2}+\frac{1}{n^2}<\frac{n+1-m}{n(m-1)}\]
(2) Prove the inequality as follows.
\[\frac 32\leq \lim_{n\to\infty} \left(1+\frac{1}{2^2}+\cdots+\frac{1}{n^2}\right)\leq 2\]
(3) Prove the inequality which is made precisely in comparison with the inequality in (2) as follows.
\[\frac {29}{18}\leq \lim_{n\to\infty} \left(1+\frac{1}{2^2}+\cdots+\frac{1}{n^2}\right)\leq \frac{61}{36}\]
2006 Junior Tuymaada Olympiad, 4
The sum of non-negative numbers $ x $, $ y $ and $ z $ is $3$. Prove the inequality
$$ {1 \over x ^ 2 + y + z} + {1 \over x + y ^ 2 + z} + {1 \over x + y + z ^ 2} \leq 1. $$
2011 USAMTS Problems, 3
Find all integers $b$ such that there exists a positive real number $x$ with \[ \dfrac {1}{b} = \dfrac {1}{\lfloor 2x \rfloor} + \dfrac {1}{\lfloor 5x \rfloor} \] Here, $\lfloor y \rfloor$ denotes the greatest integer that is less than or equal to $y$.
2019 CMI B.Sc. Entrance Exam, 3
Evaluate $\int_{ 0 }^{ \infty } ( 1 + x^2 )^{-( m + 1 )} \mathrm{d}x$ where $m \in \mathbb{N} $
2015 Postal Coaching, Problem 2
Let $ n$ be a positive integer. Find the number of odd coefficients of the polynomial
\[ u_n(x) \equal{} (x^2 \plus{} x \plus{} 1)^n.
\]
MOAA Team Rounds, 2022.8
Raina the frog is playing a game in a circular pond with six lilypads around its perimeter numbered clockwise from $1$ to $6$ (so that pad $1$ is adjacent to pad $6$). She starts at pad $1$, and when she is on pad i, she may jump to one of its two adjacent pads, or any pad labeled with $j$ for which $j - i$ is even. How many jump sequences enable Raina to hop to each pad exactly once?
2000 Moldova National Olympiad, Problem 2
For $n\in\mathbb N$, define
$$a_n=\frac1{\binom n1}+\frac1{\binom n2}+\ldots+\frac1{\binom nn}.$$
(a) Prove that the sequence $b_n=a_n^n$ is convergent and determine the limit.
(b) Show that $\lim_{n\to\infty}b_n>\left(\frac32\right)^{\sqrt3+\sqrt2}$.
2023 Harvard-MIT Mathematics Tournament, 10
Let $x_0 = x_{101} = 0$. The numbers $x_1, x_2,...,x_{100}$ are chosen at random from the interval $[0, 1]$ uniformly and independently. Compute the probability that $2x_i \ge x_{i-1} + x_{i+1}$ for all $i = 1, 2,..., 100.$
1997 AMC 8, 10
What fraction of this square region is shaded? Stripes are equal in width, and the figure is drawn to scale.
[asy]
unitsize(8);
fill((0,0)--(6,0)--(6,6)--(0,6)--cycle,black);
fill((0,0)--(5,0)--(5,5)--(0,5)--cycle,white);
fill((0,0)--(4,0)--(4,4)--(0,4)--cycle,black);
fill((0,0)--(3,0)--(3,3)--(0,3)--cycle,white);
fill((0,0)--(2,0)--(2,2)--(0,2)--cycle,black);
fill((0,0)--(1,0)--(1,1)--(0,1)--cycle,white);
draw((0,6)--(0,0)--(6,0));
[/asy]
$\textbf{(A)}\ \dfrac{5}{12} \qquad \textbf{(B)}\ \dfrac{1}{2} \qquad \textbf{(C)}\ \dfrac{7}{12} \qquad \textbf{(D)}\ \dfrac{2}{3} \qquad \textbf{(E)}\ \dfrac{5}{6}$
2011 Saudi Arabia Pre-TST, 4
Points $A ,B ,C ,D$ lie on a line in this order. Draw parallel lines $a$ and $b$ through $A$ and $B$, respectively, and parallel lines $c$ and $d$ through $C$ and $D$, respectively, such that their points of intersection are vertices of a square. Prove that the side length of this square does not depend on the length of segment $BC$.
1997 Tournament Of Towns, (528) 5
$E$ is the midpoint of the side $AD$ of a parallelogram $ABCD$. $F$ is the foot of the perpendicular from the vertex $B$ to the line $CE$. Prove that $ABF$ is an isosceles triangle.
(MA Bolchkevich)
2016 Azerbaijan IMO TST First Round, 3
Find the solution of the equation $8x(2x^2-1)(8x^4-8x^2+1)=1$ in the interval $[0,1]$?
2021 CMIMC Integration Bee, 1
$$\int_0^5 \max(2x,x^2)\,dx$$
[i]Proposed by Connor Gordon[/i]
2022 Novosibirsk Oral Olympiad in Geometry, 2
A ball was launched on a rectangular billiard table at an angle of $45^o$ to one of the sides. Reflected from all sides (the angle of incidence is equal to the angle of reflection), he returned to his original position . It is known that one of the sides of the table has a length of one meter. Find the length of the second side.
[img]https://cdn.artofproblemsolving.com/attachments/3/d/e0310ea910c7e3272396cd034421d1f3e88228.png[/img]
2006 All-Russian Olympiad Regional Round, 9.2
Each cell of the infinite checkered plane contains one from the numbers $1, 2, 3, 4$ so that each number appears at least once. Let's call a cell [i]correct [/i] if the number of distinct numbers written in four adjacent (side) cells to it, equal to the number written in this cell. Can all the cells of the plane be [i]correct[/i]?
2020 March Advanced Contest, 4
Let \(\mathbb{Z}^2\) denote the set of points in the Euclidean plane with integer coordinates. Find all functions \(f : \mathbb{Z}^2 \to [0,1]\) such that for any point \(P\), the value assigned to \(P\) is the average of all the values assigned to points in \(\mathbb{Z}^2\) whose Euclidean distance from \(P\) is exactly 2020.