Found problems: 85335
Let $P,Q$ be two monic polynomials with complex coefficients such that $P(P(x))=Q(Q(x))$ for all $x$. Prove that $P=Q$.
[i]Marius Cavachi[/i]
If $ a\plus{}1\equal{}b\plus{}2\equal{}c\plus{}3\equal{}d\plus{}4\equal{}a\plus{}b\plus{}c\plus{}d\plus{}5$, then $ a\plus{}b\plus{}c\plus{}d$ is
$ \text{(A)}\ \minus{}5 \qquad
\text{(B)}\ \minus{}10/3 \qquad
\text{(C)}\ \minus{}7/3 \qquad
\text{(D)}\ 5/3 \qquad
\text{(E)}\ 5$
Let $\vartriangle ABC$ be an acute triangle with $AB > AC$. Let $P$ be the foot of the altitude from $C$ to $AB$ and let $Q$ be the foot of the altitude from $B$ to $AC$. Let $X$ be the intersection of $PQ$ and $BC$. Let the intersection of the circumcircles of triangle $\vartriangle AXC$ and triangle $\vartriangle PQC$ be distinct points: $C$ and $Y$ . Prove that $PY$ bisects $AX$.
Claudia has 12 coins, each of which is a 5-cent coin or a 10-cent coin. There are exactly 17 different values that can be obtained as combinations of one or more of her coins. How many 10-cent coins does Claudia have?
$ \textbf{(A) }3\qquad\textbf{(B) }4\qquad\textbf{(C) }5\qquad\textbf{(D) }6\qquad\textbf{(E) }7 $
Two mirror walls are placed to form an angle of measure $\alpha$. There is a candle inside the angle. How many reflections of the candle can an observer see?
Consider two functions $f , \,g \,:\mathbb{R} \to \mathbb{R}$ such that from some $a>0$ holds $g(x)=f(x+a)$ for any $x \in \mathbb{R}$. If $f$ is even and $g$ is odd, prove that both functions are periodic.
Compute
$$\int_0^13x^2dx.$$
In every unit square of a$ n \times n$ table ($n \ge 11$) a real number is written, such that the sum of the numbers in any $10 \times 10$ square is positive and the sum of the numbers in any $11\times 11$ square is negative. Determine all possible values for $n$
Find all functions $f : R \to R$ satisfying the condition $f(x- f(y)) = 1+x-y$ for all $x,y \in R$.
In a cube, let $M$ be the midpoint of one of the segments. Choose two vertices of the cube, $A$ and $B$. What is the number of distinct possible triangles $\triangle AMB$ up to congruency?
[i]Proposed by Harry Kim[/i]
Are there infinite increasing sequence of natural numbers, such that sum of every 2 different numbers are relatively prime with sum of every 3 different numbers?
The sequence $\{u_n\}$ is defined by $u_1 = 1, u_2 = 1, u_n = u_{n-1} + 2u_{n-2} for n \geq 3$. Prove that for any positive integers $n, p \ (p > 1), u_{n+p} = u_{n+1}u_{p} + 2u_nu_{p-1}$. Also find the greatest common divisor of $u_n$ and $u_{n+3}.$
For postive integers $k,n$, let
$$f_k(n)=\sum_{m\mid n,m>0}m^k$$
Find all pairs of positive integer $(a,b)$ such that $f_a(n)\mid f_b(n)$ for every positive integer $n$.
Positive integers $(p,a,b,c)$ called [i]good quadruple[/i] if
a) $p $ is odd prime,
b) $a,b,c $ are distinct ,
c) $ab+1,bc+1$ and $ca+1$ are divisible by $p $.
Prove that for all good quadruple $p+2\le \frac {a+b+c}{3} $, and show the equality case.
Without lifting pen from paper, we draw a polygon in such away that from every two adjacent sides one of them is vertical.
In addition, while drawing the polygon all vertical sides have been drawn from up to down. Prove that this polygon has cut itself.
Prove that all solutions of the equation $0.001x^3 + x^2 - 1 = 0$ are irrational numbers. (A number $x$ is said to be [i]irrational[/i], if one cannot write $x = m/n$, with $m$ and $n$ integer numbers.)
Given triangle $ABC$. The midperpendicular of side $AB$ meets one of the remaining sides at point $C'$. Points $A'$ and $B'$ are defined similarly. Find all triangles $ABC$ such that triangle $A'B'C'$ is regular.
There are five guys named Alan, Bob, Casey, Dan, and Eric. Each one either always tells the truth or always lies. You overhear the following discussion between them:
Alan: [i]"All of us are truth-tellers."[/i]
Bob: [i]"No, only Alan and I are truth-tellers."[/i]
Casey: [i]"You are both liars."[/i]
Dan:[i] "If Casey is a truth-teller, then Eric is too."[/i]
Eric: [i]"An odd number of us are liars."[/i]
Who are the liars?
The (Fibonacci) sequence $f_n$ is defined by $f_1=f_2=1$ and $f_{n+2}=f_{n+1}+f_n$ for
$n\ge1$. Prove that the area of the triangle with the sides $\sqrt{f_{2n+1}},\sqrt{f_{2n+2}},$ and $\sqrt{f_{2n+3}}$ is equal to $\frac12$.
Let $S = \{1, 2, \ldots, 1990\}$. A $31$-element subset of $S$ is called "good" if the sum of its elements is divisible by $5$. Find the number of good subsets of $S.$
Consider five points in the plane, with no three of them collinear. Every pair of points among them is joined by a line. In how many ways can we color these lines by red or blue, so taht no three of the points form a triangle with lines of the same colour.
A right regular hexagonal prism has bases $ABCDEF$, $A'B'C'D'E'F'$ and edges $AA'$, $BB'$, $CC'$, $DD'$, $EE'$, $FF'$, each of which is perpendicular to both hexagons. The height of the prism is $5$ and the side length of the hexagons is $6$. The plane $P$ passes through points $A$, $C'$, and $E$. The area of the portion of $P$ contained in the prism can be expressed as $m\sqrt{n}$, where $n$ is not divisible by the square of any prime. Find $m+n$.
If the product of $(\sqrt2 +\sqrt3+\sqrt5) (\sqrt2 +\sqrt3-\sqrt5) (\sqrt2 -\sqrt3+\sqrt5) (-\sqrt2 +\sqrt3+\sqrt5)$ is $12\sqrt6+ 6\sqrt{x}$ , find $x$.
([i]0 points[/i] - [b]THROWN OUT[/b])
We call a string of characters [i]neat [/i] when it has an even length and its first half is identical to the other half (eg. [i]abab[/i]). We call a string [i]nice [/i] if it can be split on several neat strings (e.g. [i]abcabcdedef [/i]to [i]abcabc[/i], [i]dede[/i], and [i]ff[/i]). By string [i]reduction[/i] we call an operation in which we wipe two identical adjacent characters from the string (e.g. the string [i]abbac[/i] can be reduced to [i]aac[/i] and further to [i]c[/i]). Prove any string containing each of its characters in even numbers can be obtained by a series of reductions from a suitable nice string.
(Martin Melicher)
How many integers $n$ with $10 \le n \le 500$ have the property that the hundreds digit of $17n$ and $17n+17$ are different?
[i]Proposed by Evan Chen[/i]