Found problems: 85335
For a ring $ A, $ and an element $ a $ of it, define $ s_a,d_a:A\longrightarrow A, s_a(x)=ax,d_a=xa.$
[b]a)[/b] Prove that if $ A $ is finite, then $ s_a $ is injective if and only if $ d_a $ is injective.
[b]b)[/b] Give example of a ring which has an element $ b $ for which $ s_b $ is injective and $ d_b $ is not, or, conversely, $ s_b $ is not injective, but $ d_b $ is.
Let be three positive real numbers $ a,b,c. $ Show that the function $ f:\mathbb{R}\longrightarrow\mathbb{R} , $
$$ f(x)=\frac{a^x}{b^x+c^x} +\frac{b^x}{a^x+c^x} +\frac{c^x}{a^x+b^x} , $$
is nondecresing on the interval $ \left[ 0,\infty \right) $ and nonincreasing on the interval $ \left( -\infty ,0 \right] . $
Consider the sequence $2, 3, 5, 6, 7, 8, 10, ...$ of all positive integers that are not perfect squares.
Determine the $2011^{th}$ term of the sequence.
How many solutions does the equation $a ! = b ! c !$ have where $a$, $b$, $c$ are integers greater than $1$?
$
\textbf{(A)}\ 1
\qquad\textbf{(B)}\ 2
\qquad\textbf{(C)}\ 6
\qquad\textbf{(D)}\ 8
\qquad\textbf{(E)}\ \text{Infinitely many}
$
It is known that for $0\le x \le 1$ the square trinomial $f (x)$ satisfies the condition $|f(x) | \le 1$. Show that $| f '(0) | \le 8.$
Evaluate $\int_0^1 (x+3)\sqrt{xe^x}\ dx.$
Let $I$ be the incenter of the $ABC$ triangle. The circumference that passes through $I$ and has center
in $A$ intersects the circumscribed circumference of the $ABC$ triangle at points $M$ and
$N$. Prove that the line $MN$ is tangent to the inscribed circle of the $ABC$ triangle.
Point $D$ lies on side $BC$ of $\triangle ABC$ so that $\overline{AD}$ bisects $\angle BAC$. The perpendicular bisector of $\overline{AD}$ intersects the bisectors of $\angle ABC$ and $\angle ACB$ in points $E$ and $F$, respectively. Given that $AB=4$, $BC=5$, $CA=6$, the area of $\triangle AEF$ can be written as $\tfrac{m\sqrt n}p$, where $m$ and $p$ are relatively prime positive integers, and $n$ is a positive integer not divisible by the square of any prime. Find $m+n+p$.
Let $P_0P_5Q_5Q_0$ be a rectangular chocolate bar, one half dark chocolate and one half white chocolate, as shown in the diagram below. We randomly select $4$ points on the segment $P_0P_5$, and immediately after selecting those points, we label those $4$ selected points $P_1, P_2, P_3, P_4$ from left to right. Similarly, we randomly select $4$ points on the segment $Q_0Q_5$, and immediately after selecting those points, we label those $4$ points $Q_1, Q_2, Q_3, Q_4$ from left to right. The segments $P_1Q_1, P_2Q_2, P_3Q_3, P_4Q_4$ divide the rectangular chocolate bar into $5$ smaller trapezoidal pieces of chocolate. The probability that exactly $3$ pieces of chocolate contain both dark and white chocolate can be written as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.
[Diagram in the individuals file for this exam on the Chmmc website]
Find all positive integers $a$ for which the equation $7an -3n! = 2020$ has a positive integer solution $n$.
(Richard Henner)
There are integers $a_1, a_2, a_3,...,a_{240}$ such that $x(x + 1)(x + 2)(x + 3) ... (x + 239) =\sum_{n=1}^{240}a_nx^n$. Find the number of integers $k$ with $1\le k \le 240$ such that ak is a multiple of $3$.
Let $\vartriangle ABC$ have sidelengths $BC = 7$, $CA = 8$, and, $AB = 9$, and let $\Omega$ denote the circumcircle of $\vartriangle ABC$. Let circles $\omega_A$, $\omega_B$, $\omega_C$ be internally tangent to the minor arcs $BC$, $CA$, $AB$ of $\Omega$, respectively, and tangent to the segments $BC$, $CA$, $AB$ at points $X$, $Y$, and, $Z$, respectively. Suppose that $\frac{BX}{XC} = \frac{CY}{Y A} = \frac{AZ}{ZB} = \frac12$ . Let $t_{AB}$ be the length of the common external tangent of $\omega_A$ and $\omega_B$, let $t_{BC}$ be the length of the common external tangent of $\omega_B$ and $\omega_C$, and let $t_{CA}$ be the length of the common external tangent of $\omega_C$ and $\omega_A$. If $t_{AB} + t_{BC} + t_{CA}$ can be expressed as $\frac{m}{n}$ for relatively prime positive integers $m, n$, find $m + n$.
Let $n \ge 3$ be an integer. Zagi the squirrel sits at a vertex of a regular $n$-gon. Zagi plans to make a journey of $n-1$ jumps such that in the $i$-th jump, it jumps by $i$ edges clockwise, for $i \in \{1, \ldots,n-1 \}$. Prove that if after $\lceil \tfrac{n}{2} \rceil$ jumps Zagi has visited $\lceil \tfrac{n}{2} \rceil+1$ distinct vertices, then after $n-1$ jumps Zagi will have visited all of the vertices.
([i]Remark.[/i] For a real number $x$, we denote by $\lceil x \rceil$ the smallest integer larger or equal to $x$.)
Which of the following is equal to $\sqrt{9-6\sqrt{2}}+\sqrt{9+6\sqrt{2}}$?
$\textbf{(A)}\,3\sqrt2 \qquad\textbf{(B)}\,2\sqrt6 \qquad\textbf{(C)}\,\frac{7\sqrt2}{2} \qquad\textbf{(D)}\,3\sqrt3 \qquad\textbf{(E)}\,6$
Determine all composite positive integers $n$ for which it is possible to arrange all divisors of $n$ that are greater than 1 in a circle so that no two adjacent divisors are relatively prime.
Determine whether there exist non-constant polynomials $P(x)$ and $Q(x)$ with real coefficients satisfying
$$P(x)^{10}+P(x)^9 = Q(x)^{21}+Q(x)^{20}.$$
Given a convex pentagon $ABCDE$ with $\angle ABC= \angle ADE$ and $\angle AEC= \angle ADB$ . Prove that $\angle BAC = \angle DAE$ .
Two quadratic polynomials $A(x)$ and $B(x)$ have a leading term of $x^2$. For some real numbers $a$ and $b$, the roots of $A(x)$ are $1$ and $a$, and the roots of $B(x)$ are $6$ and $b$. If the roots of $A(x) + B(x)$ are $a + 3$ and $b + \frac1 2$ , then compute $a^2 + b^2$.
Suppose $S$ cans of soda can be purchased from a vending machine for $Q$ quarters. Which of the following expressions describes the number of cans of soda that can be purchased for $D$ dollars, where $1$ dollar is worth $4$ quarters?
$\textbf{(A) }\frac{4DQ}S\qquad\textbf{(B) }\frac{4DS}Q\qquad\textbf{(C) }\frac{4Q}{DS}\qquad\textbf{(D) }\frac{DQ}{4S}\qquad\textbf{(E) }\frac{DS}{4Q}$
A $n \times n$ matrix is filled with non-negative real numbers such that on each line and column the sum of the elements is $1$. Prove that one can choose n positive entries from the matrix, such that each of them lies on a different line and different column.
A box contains 36 pink, 18 blue, 9 green, 6 red, and 3 purple cubes that are identical in size. If a cube is selected at random, what is the probability that it is green?
$\textbf{(A)}\ \dfrac{1}{9} \qquad \textbf{(B)}\ \dfrac{1}{8} \qquad \textbf{(C)}\ \dfrac{1}{5} \qquad \textbf{(D)}\ \dfrac{1}{4} \qquad \textbf{(E)}\ \dfrac{9}{70}$
Given an odd $n$, prove that there exist $2n$ integers $a_1,a_2,\cdots ,a_n$; $b_1,b_2,\cdots ,b_n$, such that for any integer $k$ ($0<k<n$), the following $3n$ integers:
$a_i+a_{i+1}, a_i+b_i, b_i+b_{i+k}$ ($i=1,2,\cdots ,n; a_{n+1}=a_1, b_{n+j}=b_j, 0<j<n$) are of different remainders on division by $3n$.
Let $O$ be the circumcenter and $I$ be the incenter of an acute triangle $ABC$ with $m(\widehat{B}) \neq m(\widehat{C})$. Let $D$, $E$, $F$ be the midpoints of the sides $[BC]$, $[CA]$, $[AB]$, respectively. Let $T$ be the foot of perpendicular from $I$ to $[AB]$. Let $P$ be the circumcenter of the triangle $DEF$ and $Q$ be the midpoint of $[OI]$. If $A$, $P$, $Q$ are collinear, prove that \[\dfrac{|AO|}{|OD|}-\dfrac{|BC|}{|AT|}=4.\]
Suppose $x,y,,a,b,c,d,e,f$ are real numbers satifying
i)$\max{(a,0)}+\max{(b,0)}<x+ay+bz<1+\min{(a,0)}+\min{(b,0)}$, and
ii)$\max{(c,0)}+\max{(d,0)}<cx+y+dz<1+\min{(c,0)}+\min{(d,0)}$, and
iii)$\max{(e,0)}+\max{(f,0)}<ex+fy+z<1+\min{(e,0)}+\min{(f,0)}$.
Prove that $0<x,y,z<1$.
In a three-digit positive integer $M$, the number of hundreds is less than the number of tenths and the number of tenths is less than the number of ones. The arithmetic mean of the integer three-digit numbers obtained by arranging the number $M$ and its numbers ends with the number $5$. Find all such three-digit numbers $M$.