Found problems: 85335
Write $t(G)$ for the number of complete quadrilaterals in the graph $G$ and $e_G(S)$ for the number of edges spanned by a subset $S$ of vertices of $G$. Let $G_1, G_2$ be two (simple) graphs on a common underlying set $V$ of vertices, $|V|-n$, and assume that $|e_{G_1}(S)-e_{G_2}(S)|<\frac{n^2}{1000}$ holds for any subset $S\subseteq V$. Prove that $|t(G_1)-t(G_2)|\le \frac{n^4}{1000}$.
There are two paper figures: an equilateral triangle and a rectangle. The height of rectangle is equal to the height of the triangle and the base of the rectangle is equal to the base of the triangle. Divide the triangle into three parts and the rectangle into two, using straight cuts, so that with the five pieces can be assembled, without gaps or overlays, a equilateral triangle. To assemble the figure, each part can be rotated and / or turned around.
Source: 2018 Canadian Open Math Challenge Part A Problem 2
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Let $v$, $w$, $x$, $y$, and $z$ be five distinct integers such that $45 = v\times w\times x\times y\times z.$ What is the sum of the integers?
Let $|a|<\frac{\pi}{2}$.
Evaluate
\[\int_0^{\frac{\pi}{2}} \frac{dx}{\{\sin (a+x)+\cos x\}^2}\]
The hexagon $ABLCDK$ is inscribed and the line $LK$ intersects the segments $AD, BC, AC$ and $BD$ in points $M, N, P$ and $Q$, respectively. Prove that $NL \cdot KP \cdot MQ = KM \cdot PN \cdot LQ$.
A test consists of $30$ true or false questions. After the test (answering all $30$ questions), Victor gets his score: the number of correct answers. Victor is allowed to take the test (the same questions ) several times. Can Victor work out a strategy that insure him to get a perfect score after
[b](a) [/b] $30$th attempt?
[b](b)[/b] $25$th attempt?
(Initially, Victor does not know any answer)
On the sides $AB , BC$ of the convex quadrilateral $ABCD$ lie points $M$ and $N$ such that $AN$ and $CM$ each divide the quadrilateral $ABCD$ into two equal area parts. Prove that the line $MN$ and $AC$ are parallel.
Find the index of the singular point $0$ of the vector field with components
\[(x^4+y^4+z^4,x^3y-xy^3,xyz^2)\]
Show that if $a+b+c=0$ then $(\frac{a}{b-c}+\frac{b}{c-a}+\frac{c}{a-b})(\frac{b-c}{a}+\frac{c-a}{b}+\frac{a-b}{c})=9$.
a) Let $(a_n)$ be the sequence defined by $a_n=\ln (2n^2+1)-\ln (n^2+n+1)\,\,\forall n\geq 1.$ Prove that the set $\{n\in\mathbb{N}|\,\{a_n\}<\dfrac{1}{2}\}$ is a finite set;
b) Let $(b_n)$ be the sequence defined by $a_n=\ln (2n^2+1)+\ln (n^2+n+1)\,\,\forall n\geq 1$. Prove that the set $\{n\in\mathbb{N}|\,\{b_n\}<\dfrac{1}{2016}\}$ is an infinite set.
Consider a $m\times n$ rectangular grid with $m$ rows and $n$ columns. There are water fountains on some of the squares. A water fountain can spray water onto any of it's adjacent squares, or a square in the same column such that there is exactly one square between them. Find the minimum number of fountains such that each square can be sprayed in the case that
a) $m=4$;
b) $m=3$.
Let $A$ and $B$ be fixed points in the Euclidean plane with $AB = 6$. Let $R$ be the region of points in the plane such that, for each $P \in R$, there exists a point $C$ such that $AC = 3$ and $P$ does not lie outside $\vartriangle ABC$. Compute the greatest integer less than or equal to the area of $R$.
Let $ A_1A_2A_3$ be a non-isosceles triangle with incenter $ I.$ Let $ C_i,$ $ i \equal{} 1, 2, 3,$ be the smaller circle through $ I$ tangent to $ A_iA_{i\plus{}1}$ and $ A_iA_{i\plus{}2}$ (the addition of indices being mod 3). Let $ B_i, i \equal{} 1, 2, 3,$ be the second point of intersection of $ C_{i\plus{}1}$ and $ C_{i\plus{}2}.$ Prove that the circumcentres of the triangles $ A_1 B_1I,A_2B_2I,A_3B_3I$ are collinear.
A token is placed at each vertex of a regular $2n$-gon. A [i]move[/i] consists in choosing an edge of the $2n$-gon and swapping the two tokens placed at the endpoints of that edge. After a finite number of moves have been performed, it turns out that every two tokens have been swapped exactly once. Prove that some edge has never been chosen.
The King decided to reduce his Council consisting of thousand wizards. He placed them in a line and placed hats with numbers from $1$ to $1001$ on their heads not necessarily in this order (one hat was hidden). Each wizard can see the numbers on the hats of all those before him but not on himself or on anyone who stayed behind him. By King's command, starting from the end of the line each wizard calls one integer from $1$ to $1001$ so that every wizard in the line can hear it. No number can be repeated twice.
In the end each wizard who fails to call the number on his hat is removed from the Council. The wizards knew the conditions of testing and could work out their strategy prior to it.
(a) Can the wizards work out a strategy which guarantees that more than $500$ of them remain in the Council?
(b) Can the wizards work out a strategy which guarantees that at least $999$ of them remain in the Council?
Let $M=\{1,2,\dots,49\}$ be the set of the first $49$ positive integers. Determine the maximum integer $k$ such that the set $M$ has a subset of $k$ elements such that there is no $6$ consecutive integers in such subset. For this value of $k$, find the number of subsets of $M$ with $k$ elements with the given property.
For a positive integer $n$ let $f(n)$ denote the number of ways to put $n$ objects into pairs if the only thing that matters is which object each object gets paired with. Find the sum of all $f(f(2k))$, where $k$ ranges from 1 to 2023.
[i]Lightning 2.2[/i]
Let $$x_0 = 5\,\, ,\, \,\,x_{n+1} = x_n +\frac{1}{x_n}.$$
Prove that $45 < x_{1000} < 45.1$.
The difference between the product and the sum of two different integers is equal to the sum of their GCD (greatest common divisor) and LCM (least common multiple). Findall these pairs of numbers. Justify your answer.
D is the a point on BC，I1 is the heart of a triangle ABD, I2 is the heart of a triangle ACD，O1 is the Circumcenter of triangle AI1D, O2 is the Circumcenter of the triangle AI2D，P is the intersection point of O1I2 and O2I1，Prove: PD is perpendicular to BC.
(a) $K_1,K_2,..., K_n$ are the feet of the perpendiculars from an arbitrary point $M$ inside a given regular $n$-gon to its sides (or sides produced). Prove that the sum $\overrightarrow{MK_1} + \overrightarrow{MK_2} + ... + \overrightarrow{MK_n}$ equals $\frac{n}{2}\overrightarrow{MO}$, where $O$ is the centre of the $n$-gon.
(b) Prove that the sum of the vectors whose origin is an arbitrary point $M$ inside a given regular tetrahedron and whose endpoints are the feet of the perpendiculars from $M$ to the faces of the tetrahedron equals $\frac43 \overrightarrow{MO}$, where $O$ is the centre of the tetrahedron.
(VV Prasolov, Moscow)
Let $n\ge 2$ be a positive integer. There are $n$ real coefficient polynomials $P_1(x),P_2(x),\cdots ,P_n(x)$ which is not all the same, and their leading coefficients are positive. Prove that
$$\deg(P_1^n+P_2^n+\cdots +P_n^n-nP_1P_2\cdots P_n)\ge (n-2)\max_{1\le i\le n}(\deg P_i)$$
and find when the equality holds.
We know $\mathbb Z_{210} \cong \mathbb Z_2 \times \mathbb Z_3 \times \mathbb Z_5 \times \mathbb Z_7$.
Moreover,\begin{align*}
53 & \equiv 1 \pmod{2} \\
53 & \equiv 2 \pmod{3} \\
53 & \equiv 3 \pmod{5} \\
53 & \equiv 4 \pmod{7}.
\end{align*}
Let
\[ M = \left(
\begin{array}{ccc}
53 & 158 & 53 \\
23 & 93 & 53 \\
50 & 170 & 53
\end{array}
\right). \]
Based on the above, find $\overline{(M \mod{2})(M \mod{3})(M \mod{5})(M \mod{7})}$.
Csenge and Eszter ate a whole basket of cherries. Csenge ate a quarter of all cherries while Eszter ate four-sevenths of all cherries and forty more. How many cherries were in the basket in total?
For real numbers $a, b$, and $c$ the polynomial $p(x) = 3x^7 - 291x^6 + ax^5 + bx^4 + cx^2 + 134x - 2$ has $7$ real roots whose sum is $97$. Find the sum of the reciprocals of those $7$ roots.