Found problems: 85335
2009 Postal Coaching, 2
Solve for prime numbers $p, q, r$ : $$\frac{p}{q} - \frac{4}{r + 1}= 1$$
LMT Accuracy Rounds, 2022 S1
Kevin colors a ninja star on a piece of graph paper where each small square has area $1$ square inch. Find the area of the region colored, in square inches.
[img]https://cdn.artofproblemsolving.com/attachments/3/3/86f0ae7465e99d3e4bd3a816201383b98dc429.png[/img]
2025 SEEMOUS, P2
Calculate $$\lim_{n\rightarrow\infty}n\int_0^{\infty} e^{-x}\sqrt[n]{e^x - 1 -\frac{x}{1!} - \frac{x^2}{2!} - \dots -\frac{x^n}{n!}}\,dx.$$
2018 China Team Selection Test, 3
Prove that there exists a constant $C>0$ such that
$$H(a_1)+H(a_2)+\cdots+H(a_m)\leq C\sqrt{\sum_{i=1}^{m}i a_i}$$
holds for arbitrary positive integer $m$ and any $m$ positive integer $a_1,a_2,\cdots,a_m$, where $$H(n)=\sum_{k=1}^{n}\frac{1}{k}.$$
2014 BMT Spring, 1
A [i]festive [/i] number is a four-digit integer containing one of each of the digits $0, 1, 2$, and $4$ in its decimal representation. How many festive numbers are there?
1989 IMO, 6
A permutation $ \{x_1, x_2, \ldots, x_{2n}\}$ of the set $ \{1,2, \ldots, 2n\}$ where $ n$ is a positive integer, is said to have property $ T$ if $ |x_i \minus{} x_{i \plus{} 1}| \equal{} n$ for at least one $ i$ in $ \{1,2, \ldots, 2n \minus{} 1\}.$ Show that, for each $ n$, there are more permutations with property $ T$ than without.
2016 Iran Team Selection Test, 4
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]
1987 Traian Lălescu, 1.3
Let $ ABCD $ be a tetahedron and $ M,N $ the middlepoints of $ AB, $ respectively, $ CD. $ Show that any plane that contains $ M $ and $ N $ cuts the tetrahedron in two polihedra that have same volume.
PEN N Problems, 13
One member of an infinite arithmetic sequence in the set of natural numbers is a perfect square. Show that there are infinitely many members of this sequence having this property.
1999 IMC, 1
a) Show that $\forall n \in \mathbb{N}_0, \exists A \in \mathbb{R}^{n\times n}: A^3=A+I$.
b) Show that $\det(A)>0, \forall A$ fulfilling the above condition.
2003 Romania National Olympiad, 4
Let $ P$ be a plane. Prove that there exists no function $ f: P\rightarrow P$ such that for every convex quadrilateral $ ABCD$, the points $ f(A),f(B),f(C),f(D)$ are the vertices of a concave quadrilateral.
[i]Dinu Şerbănescu[/i]
2013 Bosnia And Herzegovina - Regional Olympiad, 2
If $x$ and $y$ are real numbers, prove that $\frac{4x^2+1}{y^2+2}$ is not integer
1989 China Team Selection Test, 3
$1989$ equal circles are arbitrarily placed on the table without overlap. What is the least number of colors are needed such that all the circles can be painted with any two tangential circles colored differently.
May Olympiad L1 - geometry, 2021.1
In a forest there are $5$ trees $A, B, C, D, E$ that are in that order on a straight line. At the midpoint of $AB$ there is a daisy, at the midpoint of $BC$ there is a rose bush, at the midpoint of $CD$ there is a jasmine, and at the midpoint of $DE$ there is a carnation. The distance between $A$ and $E$ is $28$ m; the distance between the daisy and the carnation is $20$ m. Calculate the distance between the rose bush and the jasmine.
2009 Miklós Schweitzer, 3
Prove that there exist positive constants $ c$ and $ n_0$ with the following property. If $ A$ is a finite set of integers, $ |A| \equal{} n > n_0$, then
\[ |A \minus{} A| \minus{} |A \plus{} A| \leq n^2 \minus{} c n^{8/5}.\]
2017 BMT Spring, 6
Consider the function $f(x, y, z) = (x-y +z,y -z +x, z-x+y)$ and denote by $f^{(n)}(x, y,z)$ the function $f$ applied $n$ times to the tuple $(x,y,z)$. Let $r_1$, $r_2$, $r_3$ be the three roots of the equation $x^3- 4x^2 + 12 = 0$ and let $g(x) = x^3 + a_2x^2 + a_1x + a_0$ be the cubic polynomial with the tuple $f^{(3)}(r_1, r_2, r_3)$ as roots. Find the value of $a_1$.
2013 Miklós Schweitzer, 5
A subalgebra $\mathfrak{h}$ of a Lie algebra $\mathfrak g$ is said to have the $\gamma$ property with respect to a scalar product ${\langle \cdot,\cdot \rangle}$ given on ${\mathfrak g}$ if ${X \in \mathfrak{h}}$ implies ${\langle [X,Y],X\rangle =0}$ for all ${Y \in \mathfrak g}$. Prove that the maximum dimension of ${\gamma}$-property subalgebras of a given ${2}$ step nilpotent Lie algebra with respect to a scalar product is independent of the selection of the scalar product.
[i]Proposed by Péter Nagy Tibor[/i]
LMT Team Rounds 2021+, A18
Points $X$ and $Y$ are on a parabola of the form $y=\frac{x^2}{a^2}$ and $A$ is the point $(x, y) = (0, a)$. Assume $XY$ passes through $A$ and hits the line $y=-a$ at a point $B$. Let $\omega$ be the circle passing through $(0, -a)$, $A$, and $B$. A point $P$ is chosen on $\omega$ such that $PA = 8$. Given that $X$ is between $A$ and $B$, $AX=2$, and $XB=10$, find $PX \cdot PY$.
[i]Proposed by Kevin Zhao[/i]
2014 Hanoi Open Mathematics Competitions, 1
Let $a$ and $b$ satisfy the conditions $\begin{cases}
a^3 - 6a^2 + 15a = 9 \\
b^3 - 3b^2 + 6b = -1 \end{cases}$ .
The value of $(a - b)^{2014}$ is:
(A): $1$, (B): $2$, (C): $3$, (D): $4$, (E) None of the above.
1984 Swedish Mathematical Competition, 4
Find all positive integers $p$ and $q$ such that all the roots of the polynomial $(x^2 - px+q)(x^2 -qx+ p)$ are positive integers.
2020 Final Mathematical Cup, 1
Find all such functions $f:\mathbb{R} \to \mathbb{R}$ that for any real $x,y$ the following equation is true.
$$f(f(x)+y)+1=f(x^2+y)+2f(x)+2y$$
2024 VJIMC, 2
Let $n$ be a positive integer and let $A$, $B$ be two complex nonsingular $n \times n$ matrices such that
\[A^2B-2ABA+BA^2=0.\]
Prove that the matrix $AB^{-1}A^{-1}B-I_n$ is nilpotent.
2020 Ecuador NMO (OMEC), 2
Find all pairs $(n, q)$ such that $n$ is a positive integer, $q$ is a not integer rational and
$$n^q-q$$
is an integer.
2018 Romanian Masters in Mathematics, 6
Fix a circle $\Gamma$, a line $\ell$ to tangent $\Gamma$, and another circle $\Omega$ disjoint from $\ell$ such that $\Gamma$ and $\Omega$ lie on opposite sides of $\ell$. The tangents to $\Gamma$ from a variable point $X$ on $\Omega$ meet $\ell$ at $Y$ and $Z$. Prove that, as $X$ varies over $\Omega$, the circumcircle of $XYZ$ is tangent to two fixed circles.
2019 AMC 12/AHSME, 24
For how many integers $n$ between $1$ and $50$, inclusive, is
\[
\frac{(n^2-1)!}{(n!)^n}
\]an integer? (Recall that $0! = 1$.)
$\textbf{(A) } 31 \qquad \textbf{(B) } 32 \qquad \textbf{(C) } 33 \qquad \textbf{(D) } 34 \qquad \textbf{(E) } 35$