Found problems: 85335
The roots of the polynomial $x^3 - \frac32 x^2 - \frac14 x + \frac38 = 0$ are in arithmetic progression. What are they?
Let $d_1,d_2,\dots,d_{12}$ be real numbers in the open interval $(1,12).$ Show that there exist distinct indices $i,j,k$ such that $d_i,d_j,d_k$ are the side lengths of an acute triangle.
Let $F_n{}$ denote the $n{}$-th Fibonacci number. Prove that $3^{2023}$ divides \[3^2\cdot F_4+3^3\cdot F_6+3^4\cdot F_8+\dots+3^{2023}F_{4046}.\][i]Proposed by Dylan Toh[/i]
There are ${n}$ tokens in a pack. Some of them (at least one, but not all) are white and the rest are black. All tokens are extracted randomly from the pack, one by one, without putting them back. Let ${X_i}$ be the ratio of white tokens in the pack before the ${i^{\text{th}}}$ extraction and let
\[ \displaystyle T =\max \{ |X_i-X_j| : 1 \leq i \leq j \leq n\}.\]
Prove that ${\Bbb{E}(T) \leq H(\Bbb{E}(X_1))},$ where ${H(x)=-x\ln x -(1-x)\ln(1-x)}.$
[i]Proposed by Tamás Móri[/i]
Let $a$, $b$, $c$ be positive reals such that $a+b+c=3$. Show that $$\sqrt{\frac{a}{b+c}} + \sqrt{\frac{b}{c+a}} + \sqrt{\frac{c}{a+b}} \leq \frac{6}{\sqrt(a+b)(b+c)(c+a)}$$
Prove that there exists a sequence of unbounded positive integers $a_1\leq a_2\leq a_3\leq\cdots$, such that there exists a positive integer $M$ with the following property: for any integer $n\geq M$, if $n+1$ is not prime, then any prime divisor of $n!+1$ is greater than $n+a_n$.
The set $S$ is a subset of $\{1, 2, \dots, 2025\}$ such that no two elements of $S$ differ by $2$ or by $7$. What is the largest number of elements that $S$ can have?
Initially on the blackboard all the integers from $1$ to $2002$ inclusive are written in one line and in some order, without repetitions. In each step, the first and second numbers of the line are deleted and the absolute value of the subtraction of the two numbers that have just been deleted is written at the beginning of the line; the other numbers are not modified in that step, and there is a new line that has one less number than the previous step. After completing $2001$ steps, only one number remains on the board.
Determine all possible values of the number left on the board by varying the order of the $2002$ numbers on the initial line (and performing the $2001$ steps).
Let $P$ be a point inside a triangle $ABC$ such that
\[\angle P AB = \angle P BC = \angle P CA\]
Suppose $AP, BP, CP$ meet the circumcircles of triangles $P BC, P CA, P AB$ at $X, Y, Z$ respectively $(\neq P)$ . Prove that
\[[XBC] + [Y CA] + [ZAB] \ge 3[ABC]\]
Let $d$ and $n$ be positive integers such that $d$ divides $n$, $n > 1000$, and $n$ is not a perfect square. The minimum possible value of $\left\lvert d - \sqrt{n} \right\rvert$ can be written in the form $a\sqrt{b} + c$, where $b$ is a positive integer not divisible by the square of any prime, and $a$ and $c$ are nonzero integers (not necessarily positive). Compute $a+b+c$.
[i]Proposed by Matthew Lerner-Brecher[/i]
A regular $n$-gon is inscribed in a circle radius $1$. Let $X$ be the set of all arcs $PQ$, where $P, Q$ are distinct vertices of the $n$-gon. $5$ elements $L_1, L_2, ... , L_5$ of $X$ are chosen at random (so two or more of the $L_i$ can be the same). Show that the expected length of $L_1 \cap L_2 \cap L_3 \cap L_4 \cap L_5$ is independent of $n$.
The trapezoid is composed of three conguent right isosceles triangles as shown in the figure. It is necessary to cut it into $4$ equal parts. How to do it?
[img]https://cdn.artofproblemsolving.com/attachments/f/e/87b07ae823190f26b70bfa22824679a829e649.png[/img]
Four boxes with ball capacity $3, 5, 7,$ and $8$ are given. In how many ways can $19$ same balls be put into these boxes?
$\textbf{(A)}\ 34 \qquad\textbf{(B)}\ 35 \qquad\textbf{(C)}\ 36 \qquad\textbf{(D)}\ 40 \qquad\textbf{(E)}\ \text{None}$
Let $m$ be a positive integer. Find all bounded sequences of integers $a_1, a_2, a_3,... $for which $a_n + a_{n+1} + a_{n+m }= 0$ for all $n \in N$.
Let $P$ be an interior point of a circle and $A_1,A_2...,A_{10}$ be points on the circle such that $\angle A_1PA_2 = \angle A_2PA_3 = ... = \angle A_{10}PA_1 = 36^o$. Prove that $PA_1 + PA_3 + PA_5 + PA_7 +PA_9 = PA_2 + PA_4 + PA_6 + PA_8 + PA_{10}$.
The circle $C_{1}$ with radius $6$ and the circle $C_{2}$ with radius $8$ are externally tangent to each other at $A$. The circle $C_3$ which is externally tangent to $C_{1}$ and $C_{2}$ has a radius with length $21$. The common tangent of $C_{1}$ and $C_{2}$ which passes through $A$ meets $C_{3}$ at $B$ and $C$. What is $|BC|$?
$
\textbf{(A)}\ 24
\qquad\textbf{(B)}\ 25
\qquad\textbf{(C)}\ 14\sqrt{3}
\qquad\textbf{(D)}\ 24\sqrt{3}
\qquad\textbf{(E)}\ 25\sqrt{3}
$
Floyd the flea makes jumps on the positive integers. On the first day he can jump to any positive integer. From then on, every day he jumps to another number that is not more than twice his previous day's place.
[list=a]
[*]Show that Floyd can make infinitely many jumps in such a way that he never arrives at any number with the same sum of decimal digits as at a previous place.[/*]
[*]Can the flea jump this way if we consider the sum of binary digits instead of decimal digits?[/*]
[/list]
The sum of the length, width, and height of a rectangular parallelepiped will be called its size. Can it happen that one rectangular parallelepiped contains another one of greater size?
(A Shen)
Given several integers, the allowed operation is to replace two of them by their non-negative difference. The operation is repeated until only one number remains. If the initial numbers are $1, 2, … , 2010$, what can be the last remaining number?
Let $ABC$ be an acute-angled triangle with $AC > AB$, let $O$ be its circumcentre, and let $D$ be a point on the segment $BC$. The line through $D$ perpendicular to $BC$ intersects the lines $AO, AC,$ and $AB$ at $W, X,$ and $Y,$ respectively. The circumcircles of triangles $AXY$ and $ABC$ intersect again at $Z \ne A$.
Prove that if $W \ne D$ and $OW = OD,$ then $DZ$ is tangent to the circle $AXY.$
A permutation $\pi : \{1,2,\ldots,N\} \rightarrow \{1,2, \ldots,N\}$ is [i]very odd[/i] if the smallest positive integer $k$ such that $\pi^k(a) = a$ for all $1 \le a \le N$ is odd, where $\pi^k$ denotes $\pi$ composed with itself $k$ times. Let $X_0 = 1,$ and for $i \ge 1,$ let $X_i$ be the fraction of all permutations of $\{1,2,\ldots,i\}$ that are very odd. Let $S$ denote the set of all ordered $4$-tuples $(A,B,C,D)$ of nonnegative integers such that $A+B +C +D =2023.$ Find the last three digits of the integer $$2023\sum_{(A,B,C,D) \in S} X_AX_BX_CX_D.$$
a) Let $ x_1, x_2, ..., x_n $ ($ n> 2 $) be negative real numbers and $ x_1 + x_2 + ... + x_n = m. $
Determine the maximum value of the sum
$ S = x_1x_2 + x_1x_3 + \dots + x_1x_n + x_2x_3 + x_2x_4 + \dots + x_2x_n + \dots + x_ {n-1} x_n. $
b) Let $ x_1, x_2, ..., x_n $ ($ n> 2 $) be nonnegative natural numbers and $ x_1 + x_2 + ... + x_n = m. $
Determine the maximum value of the sum
$ S = x_1x_2 + x_1x_3 + \dots + x_1x_n + x_2x_3 + x_2x_4 + \dots + x_2x_n + \dots + x_ {n-1} x_n. $
Find all functions $f: R\to R$ such that:
\[ f(x^2)-f(y^2)=(x+y)(f(x)-f(y)), x,y \in R \]
If the rotational inertia of a sphere about an axis through the center of the sphere is $I$, what is the rotational inertia of another sphere that has the same density, but has twice the radius?
$ \textbf{(A)}\ 2I \qquad\textbf{(B)}\ 4I \qquad\textbf{(C)}\ 8I\qquad\textbf{(D)}\ 16I\qquad\textbf{(E)}\ 32I $
Bag A has three chips labeled 1, 3, and 5. Bag B has three chips labeled 2, 4, and 6. If one chip is drawn from each bag, how many different values are possible for the sum of the two numbers on the chips?
$ \textbf{(A)} 4 \qquad\textbf{(B)} 5 \qquad\textbf{(C)} 6 \qquad\textbf{(D)} 7 \qquad\textbf{(E)} 9 $