Found problems: 85335
2021 LMT Spring, A17
Given that the value of \[\sum_{k=1}^{2021} \frac{1}{1^2+2^2+3^2+\cdots+k^2}+\sum_{k=1}^{1010} \frac{6}{2k^2-k}+\sum_{k=1011}^{2021} \frac{24}{2k+1}\] can be expressed as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers, find $m+n$.
[i]Proposed by Aidan Duncan[/i]
1979 IMO Longlists, 2
For a finite set $E$ of cardinality $n \geq 3$, let $f(n)$ denote the maximum number of $3$-element subsets of $E$, any two of them having exactly one common element. Calculate $f(n)$.
2012 Online Math Open Problems, 48
Suppose that \[\sum_{i=1}^{982} 7^{i^2}\] can be expressed in the form $983q + r$, where $q$ and $r$ are integers and $0 \leq r \leq 492$. Find $r$.
[i]Author: Alex Zhu[/i]
2017 Princeton University Math Competition, A1/B3
Let $a \diamond b = ab-4(a+b)+20$. Evaluate
\[1\diamond(2\diamond(3\diamond(\cdots(99\diamond100)\cdots))).\]
2019 Balkan MO Shortlist, C4
A town-planner has built an isolated city whose road network consists of $2N$ roundabouts, each connecting exactly three roads. A series of tunnels and bridges ensure that all roads in the town meet only at roundabouts. All roads are two-way, and each roundabout is oriented clockwise.
Vlad has recently passed his driving test, and is nervous about roundabouts. He starts driving from his house, and always takes the first edit at each roundabout he encounters. It turns out his journey incluldes every road in the town in both directions before he arrives back at the starting point in the starting direction. For what values of $N$ is this possible?
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.