Found problems: 85335
Bangladesh Mathematical Olympiad 2020 Final, #11
A prime number$ q $is called[b][i] 'Kowai' [/i][/b]number if $ q = p^2 + 10$ where $q$, $p$, $p^2-2$, $p^2-8$, $p^3+6$ are prime numbers. WE know that, at least one [b][i]'Kowai'[/i][/b] number can be found. Find the summation of all [b][i]'Kowai'[/i][/b] numbers.
2007 Denmark MO - Mohr Contest, 1
Triangle $ABC$ lies in a regular decagon as shown in the figure.
What is the ratio of the area of the triangle to the area of the entire decagon?
Write the answer as a fraction of integers.
[img]https://1.bp.blogspot.com/-Ld_-4u-VQ5o/Xzb-KxPX0wI/AAAAAAAAMWg/-qPtaI_04CQ3vvVc1wDTj3SoonocpAzBQCLcBGAsYHQ/s0/2007%2BMohr%2Bp1.png[/img]
2020 Harvard-MIT Mathematics Tournament, 2
Let $ABC$ be a triangle with $AB=5$, $AC=8$, and $\angle BAC=60^\circ$. Let $UVWXYZ$ be a regular hexagon that is inscribed inside $ABC$ such that $U$ and $V$ lie on side $BA$, $W$ and $X$ lie on side $AC$, and $Z$ lies on side $CB$. What is the side length of hexagon $UVWXYZ$?
[i]Proposed by Ryan Kim.[/i]
2016 PUMaC Geometry B, 1
A circle of radius 1 has four circles $\omega_1, \omega_2, \omega_3$, and $\omega_4$ of equal radius internally tangent to it, so that $\omega_1$ is tangent to $\omega_2$, which is tangent to $\omega_3$, which is tangent to $\omega_4$, which is tangent to $\omega_1$, as shown. The radius of the circle externally tangent to $\omega_1, \omega_2, \omega_3$, and $\omega_4$ has radius r. If $r = a -\sqrt{b}$ for positive integers $a$ and $b$, compute $a + b$.
[img]https://cdn.artofproblemsolving.com/attachments/e/3/c23f66333c0b4c0bf31b704cec665e50816149.png[/img]
2019 Online Math Open Problems, 3
Let $k$ be a positive real number. Suppose that the set of real numbers $x$ such that $x^2+k|x| \leq 2019$ is an interval of length $6$. Compute $k$.
[i]Proposed by Luke Robitaille[/i]
2019 MIG, 5
$3$ builders are scheduled to build a house in $60$ days. However, they suffer from a bout of procrastination and thus do nothing for the first $50$ days. Panicked, they realize in order to build the house on time, they must hire more workers [i]and[/i] work twice as fast as they would have originally. If the new workers they hire also will work at the doubled rate, how many new workers will they need to hire? Assume each builder works at the same rate as the others and they do not get in each other's way.
2013 India Regional Mathematical Olympiad, 2
In a triangle $ABC$, $AD$ is the altitude from $A$, and $H$ is the orthocentre. Let $K$ be the centre of the circle passing through $D$ and tangent to $BH$ at $H$. Prove that the line $DK$ bisects $AC$.
2021 Israel TST, 2
Find all unbounded functions $f:\mathbb Z \rightarrow \mathbb Z$ , such that $f(f(x)-y)|x-f(y)$ holds for any integers $x,y$.
1996 Tournament Of Towns, (491) 4
A rook stands at a corner of an $m \times n$ squared board. Two players move the rook in turn (vertically or horizontally through any numbers of squares). As the rook moves, it paints the squares that it visits (stopping or passing through). The rook is not allowed to pass through or stop at the painted squares. The player who cannot move, loses. Who has a guaranteed win: the first player (who starts the game) or the other, and how should he/she play?
(B Begun)
2024-25 IOQM India, 1
The smallest positive integer that does not divide $1 \times 2 \times 3 \times 4 \times 5 \times 6 \times 7 \times 8 \times 9$ is:
2014 Abels Math Contest (Norwegian MO) Final, 2
The points $P$ and $Q$ lie on the sides $BC$ and $CD$ of the parallelogram $ABCD$ so that $BP = QD$. Show that the intersection point between the lines $BQ$ and $DP$ lies on the line bisecting $\angle BAD$.
2012 Bundeswettbewerb Mathematik, 4
A rectangle with the side lengths $a$ and $b$ with $a <b$ should be placed in a right-angled coordinate system so that there is no point with integer coordinates in its interior or on its edge.
Under what necessary and at the same time sufficient conditions for $a$ and $b$ is this possible?
2019 Istmo Centroamericano MO, 5
Gabriel plays to draw triangles using the vertices of a regular polygon with $2019$ sides, following these rules:
(i) The vertices used by each triangle must not have been previously used.
(ii) The sides of the triangle to be drawn must not intersect with the sides of the triangles previously drawn.
If Gabriel continues to draw triangles until it is no longer possible, determine the minimum number of triangles that he drew.
2023 Denmark MO - Mohr Contest, 3
In a field, $2023$ friends are standing in such a way that all distances between them are distinct. Each of them fires a water pistol at the friend that stands closest. Prove that at least one person does not get wet.
2024 Junior Balkan Team Selection Tests - Romania, P2
Let $ABC$ be a scalene triangle, with circumcircle $\omega$ and incentre $I.{}$ The tangent line at $C$ to $\omega$ intersects the line $AB$ at $D.{}$ The angle bisector of $BDC$ meets $BI$ at $P{}$ and $AI{}$ at $Q{}.$ Let $M{}$ be the midpoint of the segment $PQ.$ Prove that the line $IM$ passes through the middle of the arc $ACB$ of $\omega.$
[i]Dana Heuberger[/i]
2023 District Olympiad, P4
Consider the functions $f,g,h:\mathbb{R}_{\geqslant 0}\to\mathbb{R}_{\geqslant 0}$ and the binary operation $*:\mathbb{R}_{\geqslant 0}\times \mathbb{R}_{\geqslant 0}\to \mathbb{R}_{\geqslant 0}$ defined as \[x*y=f(x)+g(y)+h(x)\cdot|x-y|,\]for all $x,y\in\mathbb{R}_{\geqslant 0}$. Suppose that $(\mathbb{R}_{\geqslant 0},*)$ is a commutative monoid. Determine the functions $f,g,h$.
1989 AMC 12/AHSME, 22
A child has a set of $96$ distinct blocks. Each block is one of $2$ materials ([i]plastic, wood[/i]), $3$ sizes ([i]small, medium, large[/i]), $4$ colors ([i]blue, green, red, yellow[/i]), and $4$ shapes ([i]circle, hexagon, square, triangle[/i]). How many blocks in the set are different from the "[i]plastic medium red circle[/i]" in exactly two ways? (The "[i]wood medium red square[/i]" is such a block.)
$ \textbf{(A)}\ 29 \qquad\textbf{(B)}\ 39 \qquad\textbf{(C)}\ 48 \qquad\textbf{(D)}\ 56 \qquad\textbf{(E)}\ 62 $
1996 Portugal MO, 6
In a regular polygon with $134$ sides, $67$ diagonals are drawn so that exactly one diagonal emerges from each vertex. We call the [i]length[/i] of a diagonal the number of sides of the polygon included between the vertices of the diagonal and which is less than or equal to $67$. If we order the [i]lengths [/i] of the diagonals in ascending order, we obtain a succession of $67$ numbers $(d_1,d_2,...,d_{67})$. It will be possible to draw diagonals such that
a) $(d_1,d_2,...,d_{67})=\underbrace{2 ... 2}_{6},\underbrace{3 ... 3}_{61}$ ?
b) $(d_1,d_2,...,d_{67}) =\underbrace{3 ... 3}_{8},\underbrace{6 ... 6}_{55}.\underbrace{8 ... 8}_{4} $ ?
2007 AIME Problems, 4
Three planets revolve about a star in coplanar circular orbits with the star at the center. All planets revolve in the same direction, each at a constant speed, and the periods of their orbits are 60, 84, and 140 years. The positions of the star and all three planets are currently collinear. They will next be collinear after $n$ years. Find $n$.
2006 Moldova National Olympiad, 12.4
Let $P(x)= x^n+a_{1}x^{n-1}+...+a_{n-1}x+(-1)^{n}$ , $a_{i} \in C$ , $n\geq 2$ with
all roots having same modulo. Prove that $P(-1) \in R$
2001 China Team Selection Test, 2
If the sum of all positive divisors (including itself) of a positive integer $n$ is $2n$, then $n$ is called a perfect number. For example, the sum of the positive divisors of 6 is $1 + 2 + 3 + 6 = 2 \times 6$, hence 6 is a perfect number.
Prove: There does not exist a perfect number of the form $p^a q^b r^c$, where $a, b, c$ are positive integers, and $p, q, r$ are odd primes.
1989 IMO Shortlist, 28
Consider in a plane $ P$ the points $ O,A_1,A_2,A_3,A_4$ such that \[ \sigma(OA_iA_j) \geq 1 \quad \forall i, j \equal{} 1, 2, 3, 4, i \neq j.\] where $ \sigma(OA_iA_j)$ is the area of triangle $ OA_iA_j.$ Prove that there exists at least one pair $ i_0, j_0 \in \{1, 2, 3, 4\}$ such that \[ \sigma(OA_iA_j) \geq \sqrt{2}.\]
LMT Team Rounds 2021+, 10
In a country with $5$ distinct cities, there may or may not be a road between each pair of cities. It’s possible to get from any city to any other city through a series of roads, but there is no set of three cities $\{A,B,C\}$ such that there are roads between $A$ and $B$, $B$ and $C$, and $C$ and $A$. How many road systems between the five cities are possible?
1980 IMO Longlists, 11
Ten gamblers started playing with the same amount of money. Each turn they cast (threw) five dice. At each stage the gambler who had thrown paid to each of his 9 opponents $\frac{1}{n}$ times the amount which that opponent owned at that moment. They threw and paid one after the other. At the 10th round (i.e. when each gambler has cast the five dice once), the dice showed a total of 12, and after payment it turned out that every player had exactly the same sum as he had at the beginning. Is it possible to determine the total shown by the dice at the nine former rounds ?
1999 Greece Junior Math Olympiad, 2
Let $n$ be a fixed positive integer and let $x, y$ be positive integers such that $xy = nx+ny$.
Determine the minimum and the maximum of $x$ in terms of $n$.