Found problems: 85335
In the 75th Annual Putnam Games, participants compete at mathematical games. Patniss and Keeta play a game in which they take turns choosing an element from the group of invertible $n\times n$ matrices with entries in the field $\mathbb{Z}/p\mathbb{Z}$ of integers modulo $p,$ where $n$ is a fixed positive integer and $p$ is a fixed prime number. The rules of the game are:
(1) A player cannot choose an element that has been chosen by either player on any previous turn.
(2) A player can only choose an element that commutes with all previously chosen elements.
(3) A player who cannot choose an element on his/her turn loses the game.
Patniss takes the first turn. Which player has a winning strategy?
Compute the sum of the series
$\sum_{k=0}^{\infty} \frac{1}{(4k+1)(4k+2)(4k+3)(4k+4)} = \frac{1}{1\cdot2\cdot3\cdot4} + \frac{1}{5\cdot6\cdot7\cdot8} + ...$
Show that there exists a set of infinite positive integers such that the sum of an arbitrary finite subset of these is never a perfect square. What happens if we change the condition from not being a perfect square to not being a perfect power?
Let $A',B',C'$ be the midpoints of the sides $BC, CA, AB$, respectively, of an acute non-isosceles triangle $ABC$, and let $D,E,F$ be the feet of the altitudes through the vertices $A,B,C$ on these sides respectively. Consider the arc $DA'$ of the nine point circle of triangle $ABC$ lying outside the triangle. Let the point of trisection of this arc closer to $A'$ be $A''$. Define analogously the points $B''$ (on arc $EB'$) and $C''$(on arc $FC'$). Show that triangle $A''B''C''$ is equilateral.
A positive integer $n$ is defined as a $\textit{stepstool number}$ if $n$ has one less positive divisor than $n + 1$. For example, $3$ is a stepstool number, as $3$ has $2$ divisors and $4$ has $2 + 1 = 3$ divisors. Find the sum of all stepstool numbers less than $300$.
[i]Proposed by [b]Th3Numb3rThr33[/b][/i]
Let $\alpha$ be a real number, $1<\alpha<2$.
(a) Show that $\alpha$ can uniquely be represented as the infinte product \[ \alpha = \left(1+\dfrac1{n_1}\right)\left(1+\dfrac1{n_2}\right)\cdots \] with $n_i$ positive integers satisfying $n_i^2\le n_{i+1}$.
(b) Show that $\alpha\in\mathbb{Q}$ iff from some $k$ onwards we have $n_{k+1}=n_k^2$.
Given natural numbers $m, n$ and $X$ such that $X \ge m$ and $X \ge n$. Prove that one can find two integers $u$ and $v$ such that $|u| + |v| > 0$, $|u| \le \sqrt{X}$, $|v| \le \sqrt{X}$ and
$$0 \le mu + nv \le 2 \sqrt{X}.$$
Let $\triangle{ABC}$ be an acute, scalene triangle with orthocenter $H$, and let $AH$ meet the circumcircle of $\triangle{ABC}$ at a point $D \neq A$. Points $E$ and $F$ are chosen on $AC$ and $AB$ such that $DE \perp AC$ and $DF \perp AB$. Show that $BE$, $CF$, and the line through $H$ parallel to $EF$ concur.
Does there exist a triangle with length $a,b,c$ such that:
[b]a)[/b] $\begin{cases} a+b+c=6 \\ a^2+b^2+c^2=13 \\ a^3+b^3+c^3=28 \end{cases}$
[b]b)[/b] $\begin{cases} a+b+c=6 \\ a^2+b^2+c^2=13 \\ a^3+b^3+c^3=30 \end{cases}$
Let $a,b$ be positive real numbers and $n\geq 2$ a positive integer. Prove that if $x^n \leq ax+b$ holds for a positive real number $x$, then it also satisfies the inequality $x < \sqrt[n-1]{2a} + \sqrt[n]{2b}.$
Let $P(x)\in \mathbb{Z}[x]$ be a monic polynomial with even degree. Prove that, if for infinitely many integers $x$, the number $P(x)$ is a square of a positive integer, then there exists a polynomial $Q(x)\in\mathbb{Z}[x]$ such that $P(x)=Q(x)^2$.
Let $x$ be the positive integer satisfying $5^2 +28^2 +39^2 = 24^2 +35^2 + x^2$. Find $x$.
Ryan Murphy is playing poker. He is dealt a hand of $5$ cards. Given that the probability that he has a straight hand (the ranks are all consecutive; e.g. $3,4,5,6,7$ or $9,10,J,Q,K$) or $3$ of a kind (at least $3$ cards of the same rank; e.g. $5, 5, 5, 7, 7$ or $5, 5, 5, 7,K$) is $m/n$ , where $m$ and $n$ are relatively prime positive integers, find $m +n$.
[i]Proposed by Aditya Rao[/i]
Anna took a number \(N\), which is written in base 10 and has fewer than 9 digits, and duplicated it by writing another \(N\) to its left, creating a new number with twice as many digits. Bob computed the sum of all integers from 1 to \(N\). It turned out that Anna's new number is 7 times as large as the sum computed by Bob. Determine \(N\).
[i]Proposed by Bachana Kutsia, Georgia [/i]
Let $ABC$ be a triangle such that it's circumcircle radius is equal to the radius of outer inscribed circle with respect to $A$.
Suppose that the outer inscribed circle with respect to $A$ touches $BC,AC,AB$ at $M,N,L$.
Prove that $O$ (Center of circumcircle) is the orthocenter of $MNL$.
Find all positive integers $n$ for which the equation $$(x^2 + y^2)^n = (xy)^{2016}$$ has positive integer solutions.
Given a circle of radius $\sqrt{13}$, let $A$ be a point at a distance $4 + \sqrt{13}$ from the center $O$ of the circle. Let $B$ be the point on the circle nearest to point $A$. A line passing through the point $A$ intersects the circle at points $K$ and $L$. The maximum possible area for $\triangle BKL$ can be written in the form $\tfrac{a-b\sqrt{c}}{d}$, where $a$, $b$, $c$, and $d$ are positive integers, $a$ and $d$ are relatively prime, and $c$ is not divisible by the square of any prime. Find $a+b+c+d$.
Suppose that $P(x)$ is a polynomial with the property that there exists another polynomial $Q(x)$ to satisfy $P(x)Q(x)=P(x^2)$. $P(x)$ and $Q(x)$ may have complex coefficients. If $P(x)$ is a quintic with distinct complex roots $r_1,\dots,r_5$, find all possible values of $|r_1|+\dots+|r_5|$.
In Eldorado a year has $20$ months, and each month has $20$ days. One day Brigi asked Adél who lives in Eldorado what day her birthday is. Adél answered that she is only going to tell her the product of the month and the day in her birthday. (For example, if she was born on the $19$th day of the $4$th month, she would say $4 \cdot 19 = 76$.) From this, Brigi was able to tell Adél’s birthday. Based on this information, how many days of the year can be Adél’s birthday?
A cuboctahedron is the convex hull of (smallest convex set containing) the $12$ points $(\pm 1, \pm 1, 0), (\pm 1, 0, \pm 1), (0, \pm 1, \pm 1)$. Find the cosine of the solid angle of one of the triangular faces, as viewed from the origin. (Take a figure and consider the set of points on the unit sphere centered on the origin such that the ray from the origin through the point intersects the fi gure. The area of that set is the solid angle of the fi gure as viewed from the origin.)
Mike has $1000$ unit cubes. Each has $2$ opposite red faces, $2$ opposite blue faces and $2$ opposite white faces. Mike assembles them into a $10 \times 10 \times 10$ cube. Whenever two unit cubes meet face to face, these two faces have the same colour. Prove that an entire face of the $10 \times 10 \times 10$ cube has the same colour.
[i](6 points)[/i]
Lenth of a right angle triangle sides are posive integer. Prove that double area of the triangle divides 12.
Let $n\ge 1$ be an integer. In town $X$ there are $n$ girls and $n$ boys, and each girl knows each boy. In town $Y$ there are $n$ girls, $g_1,g_2,\ldots ,g_n$, and $2n-1$ boys, $b_1,b_2,\ldots ,b_{2n-1}$. For $i=1,2,\ldots ,n$, girl $g_i$ knows boys $b_1,b_2,\ldots ,b_{2i-1}$ and no other boys. Let $r$ be an integer with $1\le r\le n$. In each of the towns a party will be held where $r$ girls from that town and $r$ boys from the same town are supposed to dance with each other in $r$ dancing pairs. However, every girl only wants to dance with a boy she knows. Denote by $X(r)$ the number of ways in which we can choose $r$ dancing pairs from town $X$, and by $Y(r)$ the number of ways in which we can choose $r$ dancing pairs from town $Y$. Prove that $X(r)=Y(r)$ for $r=1,2,\ldots ,n$.
Let the roots of $ax^2+bx+c=0$ be $r$ and $s$. The equation with roots $ar+b$ and $as+b$ is:
$ \textbf{(A)}\ x^2-bx-ac=0$
$\qquad\textbf{(B)}\ x^2-bx+ac=0$
$\qquad\textbf{(C)}\ x^2+3bx+ca+2b^2=0$
${\qquad\textbf{(D)}\ x^2+3bx-ca+2b^2=0 }$
${\qquad\textbf{(E)}\ x^2+bx(2-a)+a^2c+b^2(a+1)=0} $
Let $AD, BE$ be the altitudes and $CF$ be the angle bissector of acute non-isosceles triangle $ABC$ and $AE+BD=AB.$ Denote by $I_A, I_B, I_C$ the incentres of triangles $AEF,$ $BDF,$ $CDE$ respectively. Prove that points $D, E, F, I_A, I_B$ and $I_C$ lie on the same circle.