Found problems: 85335
For $p_0,\dots,p_d\in\mathbb{R}^d$, let
\[
S(p_0,\dots,p_d)=\left\{ \alpha_0p_0+\dots+\alpha_dp_d : \alpha_i\le 1, \sum_{i=0}^d \alpha_i =1 \right\}.
\]
Let $\pi$ be an arbitrary probability distribution on $\mathbb{R}^d$, and choose $p_0,\dots,p_d$ independently with distribution $\pi$. Prove that the expectation of $\pi(S(p_0,\dots,p_d))$ is at least $1/(d+2)$.
Can number $2012^n-3^n$ be perfect square, while $n$ is positive integer
Suppose that f is a function $f : R_{\ge 0} \to R$ so that for all $x, y \in R_{\ge 0}$ (nonnegative reals) we have that $$f(x)+f(y) = f(x+y+xy)+f(x)f(y).$$ Given that $f\left(\frac{3}{5} \right) = \frac12$ and$ f(1) = 3$, determine
$$\lfloor \log_2 (-f(10^{2021} - 1)) \rfloor.$$
The sequence $\left(a_n \right)$ is defined by $a_1=1, \ a_2=2$ and
$$a_{n+2} = 2a_{n+1}-pa_n, \ \forall n \ge 1,$$ for some prime $p.$ Find all $p$ for which there exists $m$ such that $a_m=-3.$
Find all prime numbers $p, q, r$ such that
\[15p+7pq+qr=pqr.\]
Show that there are infinitely many triples $(x, y, z)$ of real numbers such that $$\displaystyle{x^2+y = y^2+z= z^2 + x}$$ and $x\ne y\ne z \ne x.$
Compute the number of ordered pairs of complex numbers $(u, v)$ such that $uv = 10$ and such that the real and imaginary parts of $u$ and $v$ are integers.
Given points $P_1, P_2,\cdots,P_7$ on a straight line, in the order stated (not necessarily evenly spaced). Let $P$ be an arbitrarily selected point on the line and let $s$ be the sum of the undirected lengths
\[PP_1, PP_2, \cdots , PP_7.\] Then $s$ is smallest if and only if the point $P$ is:
$\textbf{(A)}\ \text{midway between }P_1\text{ and }P_7\qquad
\textbf{(B)}\ \text{midway between }P_2\text{ and }P_6\qquad
\textbf{(C)}\ \text{midway between }P_3\text{ and }P_5 \qquad$
$
\textbf{(D)}\ \text{at }P_4 \qquad
\textbf{(E)}\ \text{at }P_1$
Let $P_0(x) = x^3 + 313x^2 - 77x - 8$. For integers $n \ge 1$, define $P_n(x) = P_{n - 1}(x - n)$. What is the coefficient of $x$ in $P_{20}(x)$?
Can $77$ blocks each $3 \times 3 \times1$ be assembled to form a $7 \times 9 \times 11$ block?
Brennan chooses a set $A = \{a, b,c, d, e \}$ of five real numbers with $a \leq b \leq c \leq d \leq e.$ Delaney determines the subsets of $A$ containing three numbers and adds up the numbers in these subsets. She obtains the sums $0, 3; 4, 8; 9, 10, 11, 12, 14, 19.$ What are the five numbers in Brennan's set?
Two perpendicular planes intersect a sphere in two circles. These circles intersect in two points, $A$ and $B$, such that $AB=42$. If the radii of the two circles are $54$ and $66$, find $R^2$, where $R$ is the radius of the sphere.
El Chapulín observed that the number $2014$ has an unusual property. By placing its eight positive divisors in increasing order, the fifth divisor is equal to three times the third minus $4$. A number of eight divisors with this unusual property is called the [i]red[/i] number . How many [i]red[/i] numbers smaller than $2014$ exist?
$P$ is an monic integer coefficient polynomial which has no integer roots. deg$P=n$ and define
$A$ $:=${$v_2(P(m))|m\in Z, v_2(P(m)) \ge 1$}. If $|A|=n$, show that all of the elements of $A$ is smaller than $\frac{3}{2}n^2$.
For any real number $a$, find all real numbers $x$ that satisfy the following equation.
$$(2x + 1)^4 + ax(x + 1) - \frac{x}{2}= 0$$
Convex pentagon $ABCDE$ has $\overline{BC}=17$, $\overline{AB}=2\overline{CD}$, and $\angle E=90^\circ$. Additionally, $\overline{BD}-\overline{CD}=\overline{AC}$, and $\overline{BD}+\overline{CD}=25$. Let $M$ and $N$ be the midpoints of $BC$ and $AD$ respectively. Ray $EA$ is extended out to point $P$, and a line parallel to $AD$ is drawn through $P$, intersecting line $EM$ at $Q$. Let $G$ be the midpoint of $AQ$. Given that $N$ and $G$ lie on $EM$ and $PM$ respectively, and the perimeter of $\triangle QBC$ is $42$, find the length of $\overline{EM}$.
[i]Proposed by Adam Bertelli[/i]
There are two points $A$ and $B$ in the plane.
a) Determine the set $M$ of all points $C$ in the plane for which $|AC|^2 +|BC|^2 = 2\cdot|AB|^2.$
b) Decide whether there is a point $C\in M$ such that $\angle ACB$ is maximal and if so, determine this angle.
Let $(a_n)_{n=0}^{\infty}$ be a sequence of real numbers defined as follows:
[list]
[*] $a_0 = 3$, $a_1 = 2$, and $a_2 = 12$; and
[*] $2a_{n + 3} - a_{n + 2} - 8a_{n + 1} + 4a_n = 0$ for $n \geq 0$.
[/list]
Show that $a_n$ is always a strictly positive integer.
How many positive integers which are less or equal with $2013$ such that $3$ or $5$ divide the number.
A circle is divided by the chord $AB$ into two segments and one of them is rotated about the point $A$ by a certain angle, the point $B$ being taken to $B'$. Prove that the line segments joining the midpoints of the two arcs (i.e. the arc $AB$ which had not been rotated and the rotated arc $AB'$) with the midpoint of $BB'$ are perpendicular.
(F. Nazyrov, 11th form student, Obninsk)
Let
\[f(x)=\frac{(x-18)(x-72)(x-98)(x-k)}{x}.\]
There exist exactly three positive real values of $k$ such that $f$ has a minimum at exactly two real values of $x$. Find the sum of these three values of $k$.
Let $XY Z$ be a triangle. The convex hexagon $ABCDEF$ is such that $AB; CD$ and $EF$ are parallel and equal to $XY; Y Z$ and $ZX$, respectively. Prove that area of triangle with vertices at the midpoints of $BC; DE$ and $FA$ is no less than area of triangle $XY Z.$
[i](8 points)[/i]
At summer camp, there are $20$ campers in each of the swimming class, the archery class, and the rock climbing class. Each camper is in at least one of these classes. If $4$ campers are in all three classes, and $24$ campers are in exactly one of the classes, how many campers are in exactly two classes?
$\text{(A) }10\qquad\text{(B) }11\qquad\text{(C) }12\qquad\text{(D) }13\qquad\text{(E) }14$
Let $ u_1, u_2, \ldots, u_m$ be $ m$ vectors in the plane, each of length $ \leq 1,$ with zero sum. Show that one can arrange $ u_1, u_2, \ldots, u_m$ as a sequence $ v_1, v_2, \ldots, v_m$ such that each partial sum $ v_1, v_1 \plus{} v_2, v_1 \plus{} v_2 \plus{} v_3, \ldots, v_1, v_2, \ldots, v_m$ has length less than or equal to $ \sqrt {5}.$
Let $x$ and $y$ be positive real numbers such that $x + y = 1$. Show that $$\left( \frac{x+1}{x} \right)^2 + \left( \frac{y+1}{y} \right)^2 \geq 18.$$