Found problems: 85335
Let $a, b, c$ positive reals such that $a+b+c=1$. Prove that
$$\min\{a(1-b),b(1-c),c(1-a)\}\leq \frac{1}{4}$$
$$\max\{a(1-b),b(1-c),c(1-a)\}\geq \frac{2}{9}$$
Let $ABC$ be an acute-angled triangle with circumcenter $O$. Let $I$ be a circle with center on the altitude from $A$ in $ABC$, passing through vertex $A$ and points $P$ and $Q$ on sides $AB$ and $AC$. Assume that
\[BP\cdot CQ = AP\cdot AQ.\] Prove that $I$ is tangent to the circumcircle of triangle $BOC$.
A natural number $n$ has exactly $1995$ units in its binary representation. Show that $n!$ is divisible by $2^{n-1995}$.
What is the hundreds digit of $(20!-15!)?$
$\textbf{(A) } 0 \qquad\textbf{(B) } 1 \qquad\textbf{(C) } 2 \qquad\textbf{(D) } 4 \qquad\textbf{(E) } 5$
Does there exist an increasing arithmetic progression of
(a) $11$
(b) $10000$
(c) infinitely many
positive integers such that the sums of their digits in base $10$ also form an increasing arithmetic progression?
(A Shapovalov)
Find the number of integer solutions of the equation
$x^{2016} + (2016! + 1!) x^{2015} + (2015! + 2!) x^{2014} + ... + (1! + 2016!) = 0$
If $ r$ and $ s$ are the roots of $ x^2\minus{}px\plus{}q\equal{}0$, then $ r^2\plus{}s^2$ equals:
$ \textbf{(A)}\ p^2\plus{}2q \qquad
\textbf{(B)}\ p^2\minus{}2q \qquad
\textbf{(C)}\ p^2\plus{}q^2 \qquad
\textbf{(D)}\ p^2\minus{}q^2 \qquad
\textbf{(E)}\ p^2$
Let $f$ be a function on a set $X$. Prove that $$f(X-f(X))=f(X)-f(f(X)),$$ where for a set $S$, the notation $f(S)$ means $\{f(a) | a \in S\}$.
Prove that for any integer $n\geq 2$ it holds that
$\dbinom {2n}{n}>\frac {4^n}{2n}$.
Determine all continuous, strictly monotone functions $\phi : \mathbb{R}^+\to\mathbb{R}$ such that $$F(x,y)=\phi^{-1} \left(\frac{x\phi(x)+y\phi(y)}{x+y}\right) + \phi^{-1} \left(\frac{y\phi(x)+x\phi(y)}{x+y}\right) $$ is homogeneous of degree 1, ie $F(tx,ty)=tF(x,y) , \forall x,y,t\in\mathbb{R}^+$
[hide=Note]F(x,y)=F(y,x) and F(x,x)=2x[/hide]
Find all prime number pairs $(p, q)$ such that \[p^q+q^p+p+q-5pq\] is a perfect square.
[i]Proposed by chengbilly[/i]
Prove that if a person a has infinitely many descendants (children, their children, etc.), then a has an infinite sequence $a_0, a_1, \ldots$ of descendants (i.e., $a = a_0$ and for all $n \geq 1, a_{n+1}$ is always a child of $a_n$). It is assumed that no-one can have infinitely many children.
[i]Variant 1[/i]. Prove that if $a$ has infinitely many ancestors, then $a$ has an infinite descending sequence of ancestors (i.e., $a_0, a_1, \ldots$ where $a = a_0$ and $a_n$ is always a child of $a_{n+1}$).
[i]Variant 2.[/i] Prove that if someone has infinitely many ancestors, then all people cannot descend from $A(dam)$ and $E(ve)$.
Consider a $3 \times 3$ grid of squares. A circle is inscribed in the lower left corner, the middle square of the top row, and the rightmost square of the middle row, and a circle $O$ with radius $r$ is drawn such that $O$ is externally tangent to each of the three inscribed circles. If the side length of each square is 1, compute $r$.
At the beginning of a game, I write the numbers $1$, $2$, ..., $2004$ onto a desk. A move consists of
- selecting some numbers standing on the desk;
- calculating the rest of the sum of these numbers under division by $11$ and writing this rest onto the desk;
- deleting the selected numbers.
In such a game, after a number of moves, only two numbers remained on the desk. One of them was $1000$. What was the other one?
Determine whether there exists a function $ f: \mathbb{N}\longrightarrow \mathbb{N}$ such that
$ f(n)\equal{}f(f(n\minus{}1))\plus{}f(f(n\plus{}1))$ for all natural numbers $ n\ge 2$.
Alice, Bob, and Charlie are visiting Princeton and decide to go to the Princeton U-Store to buy some tiger plushies. They each buy at least one plushie at price $p$. A day later, the U-Store decides to give a discount on plushies and sell them at $p'$ with $0 < p' < p$. Alice, Bob, and Charlie go back to the U-Store and buy some more plushies with each buying at least one again. At the end of that day, Alice has $12$ plushies, Bob has $40$, and Charlie has $52$ but they all spent the same amount of money: $\$42$. How many plushies did Alice buy on the first day?
What is the smallest $n$ for which there is a solution to $$\begin{cases} \sin x_1 + \sin x_2 + ... + \sin x_n = 0 \\ \sin x_1 + 2 \sin x_2 + ... + n \sin x_n = 100 \end{cases}$$ ?
Tommy takes a 25-question true-false test. He answers each question correctly with independent probability $\frac{1}{2}$. Tommy earns bonus points for correct streaks: the first question in a streak is worth 1 point, the second question is worth 2 points, and so on. For instance, the sequence TFFTTTFT is worth 1 + 1 + 2 + 3 + 1 = 8 points. Compute the expected value of Tommy’s score.
Let $ P(z)$ be a polynomial of degree $ n$ with complex coefficients, \[ P(0)\equal{}1, \;\textrm{and}\ \;|P(z)|\leq M\ \;\textrm{for}\ \;|z| \leq 1\ .\] Prove that every root of $ P(z)$ in the closed unit disc has multiplicity at most $ c\sqrt{n}$, where $ c\equal{}c(M) >0$ is a constant depending only on $ M$.
[i]G. Halasz[/i]
Let $a, b, c, d$ be positive reals such that $abcd = 1$. Prove that
$$\frac{1}{a(b + 1)} +\frac{1}{b(c + 1)} +\frac{1}{c(d + 1)} +\frac{1}{d(a + 1)} \ge 2.$$
On a checkered square $10 \times 10$ the cells of the upper left $5 \times 5$ square are black and all the other cells are white. What is the maximal $n$ such that the original square can be dissected (along the borders of the cells) into $n$ polygons such that in each of them the number of black cells is three times less than the number of white cells? (The polygons need not be congruent or even equal in area.)
Let $ABC$ be a triangle. Take $n$ point lying on the side $AB$ (different from $A$ and $B$) and connect all of them by straight lines to the vertex $C$. Similarly, take $n$ points on the side $AC$ and connect them to $B$. Into how many regions is the triangle $ABC$ partitioned by these lines?
Further, take $n$ points on the side $BC$ also and join them with $A$. Assume that no three straight lines meet at a point other than $A,B$ and $C$. Into how many regions is the triangle $ABC$ partitioned now?
Let $\triangle BC$ be a triangle with side lengths $AB = 9, BC = 10, CA = 11$. Let $O$ be the circumcenter of $\triangle ABC$. Denote $D = AO \cap BC, E = BO \cap CA, F = CO \cap AB$. If $\frac{1}{AD} + \frac{1}{BE} + \frac{1}{FC}$ can be written in simplest form as $\frac{a \sqrt{b}}{c}$, find $a + b + c$.
On the sides $AB$ and $BC$ arbitrarily mark points $M$ and $N$, respectively. Let $P$ be the point of intersection of segments $AN$ and $BM$. In addition, we note the points $Q$ and $R$ such that quadrilaterals $MCNQ$ and $ACBR$ are parallelograms. Prove that the points $P,Q$ and $R$ lie on one line.
The shapes in the image consist of six unit cubes. Which of the following 3D objects can be filled up with the aforementioned shapes:
a) a cube with side length $3$, from which one edge has been removed (i.e. three layers of the shape [img]https://i.imgur.com/vUqgHS2.png[/img] )?
b) a rectangular prism of size $5 \times 4 \times 3$, from which two edges of length $3$ have been removed from one of the $5 \times 3$ sides (i.e. three layers of the shape [img]https://imgur.com/W4pfEfz.png[/img] )?
We can use each of shapes at most once, no two shapes can overlap, nor protrude from the 3D object and every unit cube of the 3D object must be covered by a unit cube of one of the constituent shapes.
[center][img]https://imgur.com/evAmuep.png[/img][/center]
[i]Proposed by Ilija Jovčeski[/i]