Found problems: 85335
2007 China Second Round Olympiad, 2
In a $7\times 8$ chessboard, $56$ stones are placed in the squares. Now we have to remove some of the stones such that after the operation, there are no five adjacent stones horizontally, vertically or diagonally. Find the minimal number of stones that have to be removed.
2024 Chile National Olympiad., 2
On a table, there are many coins and a container with two coins. Vale and Diego play the following game, where Vale starts and then Diego plays, alternating turns. If at the beginning of a turn the container contains \( n \) coins, the player can add a number \( d \) of coins, where \( d \) divides exactly into \( n \) and \( d < n \). The first player to complete at least 2024 coins in the container wins. Prove that there exists a strategy for Vale to win, no matter the decisions made by Diego.
2023 MOAA, 20
Big Bad Brandon is assigning groups of his Big Bad Burglars to attack 7 different towers. Each Burglar can only belong to one attack group and Brandon takes over a tower if the number of Burglars attacking the tower strictly exceeds the number of knights guarding it. He knows there the total number of knights guarding the towers is 99 but does not know the exact number of knights guarding each tower. What is the minimum number of Burglars that Brandon needs to guarantee he can take over at least 4 of the 7 towers?
[i]Proposed by Eric Wang[/i]
2015 Romania Team Selection Test, 4
Let $k$ be a positive integer congruent to $1$ modulo $4$ which is not a perfect square and let $a=\frac{1+\sqrt{k}}{2}$.
Show that $\{\left \lfloor{a^2n}\right \rfloor-\left \lfloor{a\left \lfloor{an}\right \rfloor}\right \rfloor : n \in \mathbb{N}_{>0}\}=\{1 , 2 , \ldots ,\left \lfloor{a}\right \rfloor\}$.
2008 ITest, 24
In order to earn her vacation spending money, Alexis helped her mother remove weeds from the garden. When she was done, she came into the house to put away her gardening gloves and change into clean clothes.
On her way to her room she notices Joshua with his face to the floor in the family room, looking pretty silly. "Josh, did you know you lose IQ points for sniffing the carpet?"
"Shut up. I'm $\textit{not}$ sniffing the carpet. I'm $\textit{doing something}$."
"Sure, if $\textit{sniffing the carpet}$ counts as $\textit{doing something}.$" At this point Alexis stands over her twin brother grinning, trying to see how silly she can make him feel.
Joshua climbs to his feet and stands on his toes to make himself a half inch taller than his sister, who is ordinarily a half inch taller than Joshua. "I'm measuring something. I'm $\textit{designing}$ something."
Alexis stands on her toes too, reminding her brother that she is still taller than he. "When you're done, can you design me a dress?"
"Very funny." Joshua walks to the table and points to some drawings. "I'm designing the sand castle I want to build at the beach. Everything needs to be measured out so that I can build something awesome."
"And this requires sniffing carpet?" inquires Alexis, who is just a little intrigued by her brother's project.
"I was imagining where to put the base of a spiral staircase. Everything needs to be measured out correctly. See, the castle walls will be in the shape of a rectangle, like this room. The center of the staircase will be $9$ inches from one of the corners, $15$ inches from another, $16$ inches from another, and some whole number of inches from the furthest corner." Joshua shoots Alexis a wry smile. The twins liked to challenge each other, and Alexis knew she had to find the distance from the center of the staircase to the fourth corner of the castle on her own, or face Joshua's pestering, which might last for hours or days.
Find the distance from the center of the staircase to the furthest corner of the rectangular castle, assuming all four of the distances to the corners are described as distances on the same plane (the ground).
2013 Indonesia MO, 1
In a $4 \times 6$ grid, all edges and diagonals are drawn (see attachment). Determine the number of parallelograms in the grid that uses only the line segments drawn and none of its four angles are right.
2009 Canada National Olympiad, 4
Find all ordered pairs of integers $(a,b)$ such that $3^a + 7^b$ is a perfect square.
Ukraine Correspondence MO - geometry, 2016.7
The circle $\omega$ inscribed in an isosceles triangle $ABC$ ($AC = BC$) touches the side $BC$ at point $D$ .On the extensions of the segment $AB$ beyond points $A$ and $B$, respectively mark the points $K$ and $L$ so that $AK = BL$, The lines $KD$ and $LD$ intersect the circle $\omega$ for second time at points $G$ and $H$, respectively. Prove that point $A$ belongs to the line $GH$.
2018 Rio de Janeiro Mathematical Olympiad, 1
Let $ABC$ be a triangle and $k < 1$ a positive real number. Let $A_1$, $B_1$, $C_1$ be points on the sides $BC$, $AC$, $AB$ such that $$\frac{A_1B}{BC} = \frac{B_1C}{AC} = \frac{C_1A}{AB} = k.$$
[b](a)[/b] Compute, in terms of $k$, the ratio between the areas of the triangles $A_1B_1C_1$ and $ABC$.
[b](b)[/b] Generally, for each $n \ge 1$, the triangle $A_{n+1}B_{n+1}C_{n+1}$ is built such that $A_{n+1}$, $B_{n+1}$, $C_{n+1}$ are points on the sides $B_nC_n$, $A_nC_n$ e $A_nB_n$ satisfying $$\frac{A_{n+1}B_n}{B_nC_n} = \frac{B_{n+1}C_n}{A_nC_n} = \frac{C_{n+1}A_n}{A_nB_n} = k.$$
Compute the values of $k$ such that the sum of the areas of every triangle $A_nB_nC_n$, for $n = 1, 2, 3, \dots$ is equal to $\dfrac{1}{3}$ of the area of $ABC$.
May Olympiad L2 - geometry, 2015.3
Let $ABCDEFGHI$ be a regular polygon of $9$ sides. The segments $AE$ and $DF$ intersect at $P$. Prove that $PG$ and $AF$ are perpendicular.
2022 Kazakhstan National Olympiad, 1
Given a triangle $ABC$ draw the altitudes $AD$, $BE$, $CF$. Take points $P$ and $Q$ on $AB$ and $AC$, respectively such that $PQ \parallel BC$. Draw the circles with diameters $BQ$ and $CP$ and let them intersect at points $R$
and $T$ where $R$ is closer to $A$ than $T$. Draw the altitudes $BN$ and $CM$ in the triangle $BCR$. Prove that $FM$, $EN$ and $AD$ are concurrent.\\
2023 Bangladesh Mathematical Olympiad, P5
Consider an integrable function $f:\mathbb{R} \rightarrow \mathbb{R}$ such that $af(a)+bf(b)=0$ when $ab=1$. Find the value of the following integration:
$$ \int_{0}^{\infty} f(x) \,dx $$
2019 Iran Team Selection Test, 4
Let $1<t<2$ be a real number. Prove that for all sufficiently large positive integers like $d$, there is a monic polynomial $P(x)$ of degree $d$, such that all of its coefficients are either $+1$ or $-1$ and
$$\left|P(t)-2019\right| <1.$$
[i]Proposed by Navid Safaei[/i]
2021 Harvard-MIT Mathematics Tournament., 1
Let $a$ and $b$ be positive integers with $a>b$. Suppose that
$$\sqrt{\sqrt{a}+\sqrt{b}}+\sqrt{\sqrt{a}-\sqrt{b}}$$
is an integer.
(a) Must $\sqrt{a}$ be an integer?
(b) Must $\sqrt{b}$ be an integer?
1949-56 Chisinau City MO, 32
Determine the locus of points that are the midpoints of segments of equal length, the ends of which lie on the sides of a given right angle.
2007 Federal Competition For Advanced Students, Part 2, 1
Let $ M$ be the set of all polynomials $ P(x)$ with pairwise distinct integer roots, integer coefficients and all absolut values of the coefficients less than $ 2007$. Which is the highest degree among all the polynomials of the set $ M$?
1990 IMO Longlists, 67
Let $a + bi$ and $c + di$ be two roots of the equation $x^n = 1990$, where $n \geq 3$ is an integer and $a,b,c,d \in \mathbb R$. Under the linear transformation $f =\left(\begin{array}{cc}a&c\\b &d\end{array}\right)$, we have $(2, 1) \to (1, 2)$. Denote $r$ to be the distance from the image of $(2, 2)$ to the origin. Find the range of $r.$
2011 Finnish National High School Mathematics Competition, 1
An equilateral triangle has been drawn inside the circle. Split the triangle to two parts with equal area by a line segment parallel to the triangle side. Draw an inscribed circle inside this smaller triangle. What is the ratio of the area of this circle compared to the area of original circle.
1951 Miklós Schweitzer, 11
Prove that, for every pair $ n$, $r$ of positive integers, there can be found a polynomial $ f(x)$ of degree $ n$ with integer coefficients, so that every polynomial $ g(x)$ of degree at most $ n$, for which the coefficients of the polynomial $ f(x)\minus{}g(x)$ are integers with absolute value not greater than $ r$, is irreducible over the field of rational numbers.
2018 Brazil National Olympiad, 5
Consider the sequence in which $a_1 = 1$ and $a_n$ is obtained by juxtaposing the decimal representation of $n$ at the end of the decimal representation of $a_{n-1}$. That is, $a_1 = 1$, $a_2 = 12$, $a_3 = 123$, $\dots$ , $a_9 = 123456789$, $a_{10} = 12345678910$ and so on. Prove that infinitely many numbers of this sequence are multiples of $7$.
2022 Kyiv City MO Round 1, Problem 1
Represent $\frac{1}{2021}$ as a difference of two irreducible fractions with smaller denominators.
[i](Proposed by Bogdan Rublov)[/i]
2014 India IMO Training Camp, 1
Prove that in any set of $2000$ distinct real numbers there exist two pairs $a>b$ and $c>d$ with $a \neq c$ or $b \neq d $, such that \[ \left| \frac{a-b}{c-d} - 1 \right|< \frac{1}{100000}. \]
2009 Danube Mathematical Competition, 1
Let be $\triangle ABC$ .Let $A'$, $B'$, $C'$ be the foot of perpendiculars from $A$, $B$ and $C$ respectively.
The points $E$ and $F$ are on the sides $CB'$ and $BC'$ respectively, such that $B'E\cdot C'F = BF\cdot CE$.
Show that $AEA'F$ is cyclic.
2003 Tournament Of Towns, 1
Johnny writes down quadratic equation
\[ax^2 + bx + c = 0.\]
with positive integer coefficients $a, b, c$. Then Pete changes one, two, or none “$+$” signs to “$-$”. Johnny wins, if both roots of the (changed) equation are integers. Otherwise (if there are no real roots or at least one of them is not an integer), Pete wins. Can Johnny choose the coefficients in such a way that he will always win?
2014 May Olympiad, 3
There are nine boxes. In the first there is $1$ stone, in the second there are $2$ stones, in the third there are $3$ stones, and thus continuing, in the eighth there are $8$ stones and in the ninth there are $9$ stones. The allowed operation is to remove the same number of stones from two different boxes and place them in a third box. The goal is that all stones are in a single box. Describe how to do it with the minimum number of operations allowed.
Explain why it is impossible to achieve it with fewer operations.