Found problems: 85335
A box contains red, blue, and white balls, $100$ balls in total. It is known that among any $26$ of them there are always $10$ balls of the same color. Find the minimal number $N$ such that among any $N$ balls there are always $30$ balls of the same color.
Compute the largest integer $N \le 2012$ with four distinct digits.
[i]Proposed by Evan Chen[/i]
A binary string is a string consisting of only 0’s and 1’s (for instance, 001010, 101, etc.). What is the probability that a randomly chosen binary string of length 10 has 2 consecutive 0’s? Express your answer as a fraction.
Let $p > 5$ be a prime. Suppose that $$\frac{1}{2^2} + \frac{1}{4^2}+ \frac{1}{6^2}+ ...+ \frac{1}{(p -1)^2} =\frac{a}{b}$$ where $a/b$ is a fraction in lowest terms. Show that $p | a$.
In triangle $ABC$, $F$ is on segment $AB$ such that $CF$ bisects $\angle ACB$. Points $D$ and $E$ are on line $CF$ such that lines $AD,BE$ are perpendicular to $CF$. $M$ is the midpoint of $AB$. If $ME=13$, $AD=15$, and $BE=25$, find $AC+CB$.
[i]Ray Li[/i]
A sequence $x_1,x_2,x_3,...$ has the following properties:
(a) $1 = x_1 < x_2 < x_3 < ...$
(b) $x_{n+1} \le 2n$ for all $n \in N$.
Prove that for each positive integer $k$ there exist indices $i$ and $j$ such that $k =x_i -x_j$.
Compute the largest integer $N$ such that one can select $N$ different positive integers, none of which is larger than $17$, and no two of which share a common divisor greater than $1$.
Let $n\in\mathbb{N}^*$. Solve the equation $\sum_{k=0}^n C_n^k\cos2kx=\cos nx$ in $\mathbb{R}$.
The points $A$ and $B$ lie on a horizontal line over a vertical plane. We consider the semicircumference passing through $A$ and $B$ that lies under the horizontal line. A segment of length $a$, with the same diameter that the semicircumference, moves in a way that always contains the point $A$ and one of its extremes lies always on the semicircumference. Determine the value of the cosine of the angle between this segment and the horizontal line that makes the medium point of the segment to be as down as possible.
Show that the product of every $k$ consecutive members of the Fibonacci sequence is divisible by $f_1f_2\ldots f_k$ (where $f_0=0$ and $f_1=1$).
Let $A$ and $B$ be positive self-adjoint operators on a complex Hilbert space $H$. Prove that
\[\limsup_{n \to \infty} \|A^n x\|^{1/n} \le
\limsup_{n \to \infty} \|B^n x\|^{1/n}\]
holds for every $x \in H$ if and only if $A^n \le B^n$ for each positive integer $n$.
The [i]runcible[/i] positive integers are defined recursively as follows:
[list]
[*]$1$ and $2$ are runcible
[*]If $a$ and $b$ are runcible (where $a$ and $b$ are not necessarily distinct) then $2a + 3b$ is runcible.
[/list]
Is $2024$ runcible?
Consider the complex numbers $x,y,z$ such that
$|x|=|y|=|z|=1$. Define the number
$$a=\left (1+\frac xy\right )\left (1+\frac yz\right )\left (1+\frac zx\right ).$$
$\textbf{(a)}$ Prove that $a$ is a real number.
$\textbf{(b)}$ Find the minimal and maximal value $a$ can achieve, when $x,y,z$ vary subject to $|x|=|y|=|z|=1$.
[i] (Stefan Bălăucă & Vlad Robu)[/i]
In trapezoid $ABCD$, $AD \parallel BC$ and $\angle ABC + \angle CDA = 270^{\circ}$. Compute $AB^2$ given that $AB \cdot \tan(\angle BCD) = 20$ and $CD = 13$.
[i]Proposed by Lewis Chen[/i]
For any set $S$, let $P(S)$ be its power set, the set of all of its subsets. Over all sets $A$ of $2015$ arbitrary finite sets, let $N$ be the maximum possible number of ordered pairs $(S,T)$ such that $S \in P(A), T \in P(P(A))$, $S \in T$, and $S \subseteq T$. (Note that by convention, a set may never contain itself.) Find the remainder when $N$ is divided by $1000.$
[i] Proposed by Ashwin Sah [/i]
Find the largest positive integer $n$ such that $\sigma(n) = 28$, where $\sigma(n)$ is the sum of the divisors of $n$, including $n$.
By number-theoretical functions, we will understand integer-valued functions defined on the set of all integers. Are there number-theoretical functions $ f_0(x),f_1(x),f_2(x),\dots$ such that every number theoretical function $ F(x)$ can be uniquely represented in the form
$ F(x)\equal{}\sum_{k\equal{}0}^{\infty}a_kf_k(x)$,
$ a_0,a_1,a_2,\dots$ being integers?
A student on vacation for $d$ days observed that $(1)$ it rained $7$ times, morning or afternoon $(2)$ when it rained in the afternoon, it was clear in the morning $(3)$ there were five clear afternoons $(4)$ there were six clear mornings. Then $d$ equals:
$ \textbf{(A)}\ 7\qquad\textbf{(B)}\ 9\qquad\textbf{(C)}\ 10\qquad\textbf{(D)}\ 11\qquad\textbf{(E)}\ 12 $
There is a unique quadruple of positive integers $(a,b,c,k)$ such that $c$ is not a perfect square and $a+\sqrt{b+\sqrt{c}}$ is a root of the polynomial $x^4-20x^3+108x^2-kx+9.$ Compute $c.$
Let $p$ be some odd prime number and let $k=\frac{p+1}{2}$. The natural numbers $a_1,a_2…a_k$ are such that $a_i\neq a_j$ and $a_i<p$ for $\forall i,j=1,2…k$. Prove that for each natural number $r<p$ there exist not necessarily different $a_i$ and $a_j$, for which $a_i a_j\equiv r\, (mod\, p)$.
You have a $4$-digit whole number that is a perfect square. Another number is built adding $ 1$ to the unit's digit, subtracting $ 1$ from the ten's digit, adding $ 1$ to the hundred's digit and subtracting $ 1$ from the ones digit of one thousand. If the number you get is also a perfect square, find the original number. It's unique?
Two operations are allowed:
(i) to write two copies of number $1$;
(ii) to replace any two identical numbers $n$ by $(n + 1)$ and $(n - 1)$.
Find the minimal number of operations that required to produce the number $2005$ (at the beginning there are no numbers).
[i](8 points)[/i]
Let $f : (0,+\infty) \to \mathbb R$ be a function having the property that $f(x) = f\left(\frac{1}{x}\right)$ for all $x > 0.$ Prove that there exists a function $u : [1,+\infty) \to \mathbb R$ satisfying $u\left(\frac{x+\frac 1x }{2} \right) = f(x)$ for all $x > 0.$
Let $\Gamma_i, i = 0, 1, 2, \dots$ , be a circle of radius $r_i$ inscribed in an angle of measure $2\alpha$ such that each $\Gamma_i$ is externally tangent to $\Gamma_{i+1}$ and $r_{i+1} < r_i$. Show that the sum of the areas of the circles $\Gamma_i$ is equal to the area of a circle of radius $r =\frac 12 r_0 (\sqrt{ \sin \alpha} + \sqrt{\text{csc} \alpha}).$
Let $n, m, k$ be natural numbers, with $m > n$. Which of the numbers is greater:
$$\sqrt{n+\sqrt{m+\sqrt{n+...}}}\,\,\, or \,\,\,\, \sqrt{m+\sqrt{n+\sqrt{m+...}}}\,\, ?$$
Note: Each of the expressions contains $k$ square root signs; $n, m$ alternate within each expression.
(N. Kurlandchik)