Found problems: 85335
Find all functions $ f:\mathbb{R}\longrightarrow\mathbb{R} $ that verify the relation
$$ f\left( x^3+y^3 \right) =xf\left( y^2 \right) + yf\left( x^2 \right) , $$
for all real numbers $ x,y. $
Evaluate $\int_1^e \frac{\{1-(x-1)e^{x}\}\ln x}{(1+e^x)^2}dx.$
Show that
\[ \frac{1 - s^a}{1 - s} \leq (1 + s)^{a-1}\]
holds for every $1 \neq s > 0$ real and $0 < a \leq 1$ rational.
Let $ABC$ be a triangle and $D, E, F$ the midpoints of the sides $BC, CA, AB$ respectively. Show that $\angle DAC = \angle ABE$ if and only if $\angle AFC = \angle BDA$
Fix an odd prime number $p$. Find the largest positive integer $n$ such that there exist points $A_1,A_2,\cdots,A_n$ in the plane with integral coordinates, no three points are collinear. Moreover, for any $1\le i<j<k\le n$, $p \nmid 2S_{\Delta A_iA_j A_k}.$
We are given two point sets $A$ and $B$ which are both composed of finite disjoint arcs on the unit circle. Moreover, the length of each arc in $B$ is equal to $\dfrac{\pi}{m}$ ($m \in \mathbb{N}$). We denote by $A^j$ the set obtained by a counterclockwise rotation of $A$ about the center of the unit circle for $\dfrac{j\pi}{m}$ ($j=1,2,3,\dots$). Show that there exists a natural number $k$ such that $l(A^k\cap B)\ge \dfrac{1}{2\pi}l(A)l(B)$.(Here $l(X)$ denotes the sum of lengths of all disjoint arcs in the point set $X$)
Find all positive integers $n$ such that \[3^n+4^n+\cdots+(n+2)^n=(n+3)^n.\]
Consider any complex units $Z,W$ with $\text{Im}\ Z\ge0,\text{Re}\,W\ge 0.$ Determine and draw the locus of all possible sums $S=Z+W$ in the complex plane.
A sequence of positive real numbers $a_1, a_2, a_3, ... $ satisfies $a_n = a_{n-1} + a_{n-2}$ for all $n \ge 3$. A sequence $b_1, b_2, b_3, ...$ is defined by equations
$b_1 = a_1$ ,
$b_n = a_n + (b_1 + b_3 + ...+ b_{n-1})$ for even $n > 1$ ,
$b_n = a_n + (b_2 + b_4 + ... +b_{n-1})$ for odd $n > 1$.
Prove that if $n\ge 3$, then $\frac13 < \frac{b_n}{n \cdot a_n} < 1$
Let $ABC$ be a triangle with $AB=AC$, and let $M$ be the midpoint of $BC$. Let $P$ be a point such that $PB<PC$ and $PA$ is parallel to $BC$. Let $X$ and $Y$ be points on the lines $PB$ and $PC$, respectively, so that $B$ lies on the segment $PX$, $C$ lies on the segment $PY$, and $\angle PXM=\angle PYM$. Prove that the quadrilateral $APXY$ is cyclic.
Find all triples $(x,y, z)$ of integers such that $$\begin{cases} x^2y + y^2z + z^2x= 2010^2 \\ xy^2 + yz^2 + zx^2= -2010 \end{cases}$$
In regular pentagon $ABCDE$, let $O \in CE$ be the center of circle $\Gamma$ tangent to $DA$ and $DE$. $\Gamma$ meets $DE$ at $X$ and $DA$ at $Y$ . Let the altitude from $B$ meet $CD$ at $P$. If $CP = 1$, the area of $\vartriangle COY$ can be written in the form $\frac{a}{b} \frac{\sin c^o}{\cos^2 c^o}$ , where $a$ and $b$ are relatively prime positive integers and $c$ is an integer in the range $(0, 90)$. Find $a + b + c$.
Let $r$, $s$ be real numbers, find maximum $t$ so that if $a_1, a_2, \ldots$ is a sequence of positive real numbers satisfying
\[ a_1^r + a_2^r + \cdots + a_n^r \le 2023 \cdot n^t \]
for all $n \ge 2023$ then the sum
\[ b_n = \frac 1{a_1^s} + \cdots + \frac 1{a_n^s} \]
is unbounded, i.e for all positive reals $M$ there is an $n$ such that $b_n > M$.
For every positive integer $n$, let $p(n)$ denote the number of ways to express $n$ as a sum of positive integers. For instance, $p(4)=5$ because
\[4=3+1=2+2=2+1+1=1+1+1.\]
Also define $p(0)=1$.
Prove that $p(n)-p(n-1)$ is the number of ways to express $n$ as a sum of integers each of which is strictly greater than 1.
[i]Proposed by Fedor Duzhin, Nanyang Technological University.[/i]
Fix positive integers $k$ and $n$.Derive a simple expression involving Fibonacci numbers for the number of sequences $(T_1,T_2,\ldots,T_k)$ of subsets $T_i$ of $[n]$ such that $T_1\subseteq T_2\supseteq T_3\subseteq T_4\supseteq\ldots$.
[color=#008000]Moderator says: and the original source for this one is Richard Stanley, [i]Enumerative Combinatorics[/i] vol. 1 (1st edition), exercise 1.15.[/color]
To fold a paper airplane, Austin starts with a square paper $F OLD$ with side length $2$. First, he folds corners $L$ and $D$ to the square’s center. Then, he folds corner $F$ to corner $O$. What is the longest distance between two corners of the resulting figure?
Given a triangle $A, B, C, X$ is on side $AB$, $Y$ is on side $AC$, and $P$ and $Q$ are on side $BC$ such that $AX = AY , BX = BP$ and $CY = CQ$. Let $XP$ and $YQ$ intersect at $T$. Prove that $AT$ passes through the midpoint of $PQ$.
Let $p$ be a prime and $k$ a positive integer such that $k \le p$. We know that $f(x)$ is a polynomial in $\mathbb Z[x]$ such that for all $x \in \mathbb{Z}$ we have $p^k | f(x)$.
[b](a)[/b] Prove that there exist polynomials $A_0(x),\ldots,A_k(x)$ all in $\mathbb Z[x]$ such that
\[ f(x)=\sum_{i=0}^{k} (x^p-x)^ip^{k-i}A_i(x),\]
[b](b)[/b] Find a counter example for each prime $p$ and each $k > p$.
For which integers $n\ge 2$ can the numbers $1$ to $16$ be written each in one square of a squared $4\times 4$ paper such that the $8$ sums of the numbers in rows and columns are all different and divisible by $n$?
Let $\Omega$ be the circle circumscribed to the triangle $ABC$. Tangents taken to the circle $\Omega$ at points $A$ and $B$ intersects at the point $P$ , and the perpendicular bisector of $ (BC)$ cuts line $AC$ at point $Q$. Prove that lines $BC$ and $PQ$ are parallel.
Given vector $\mathbf{u}=\left(\frac{1}{3}, \frac{1}{3}, \frac{1}{3} \right)\in\mathbb{R}^3$ and recursively defined sequence of vectors $\{\mathbf{v}_n\}_{n\geq 0}$
$$\mathbf{v}_0 = (1,2,3),\quad \mathbf{v}_n = \mathbf{u}\times\mathbf{v}_{n-1}$$
Evaluate the value of infinite series $\sum_{n=1}^\infty (3,2,1)\cdot \mathbf{v}_{2n}$.
Let be a point $ P $ on the diagonal $ BD $ (excluding its endpoints) of a quadrilateral $ ABCD, $ and $ Q $ be a point in the interior of $ ABD. $ The projections of $ P $ on $ AB,AD $ are $ P_1,P_2, $ respectively, and the projections of $ Q $ on $ AB,AD $ are $ Q_1,Q_2, $ respectively, and verify the equations $ AQ_1=\frac{1}{4}AB $ and $ AQ_2=\frac{1}{4}AD. $ Show that the point $ Q $ is not in the interior of $ AP_1P_2. $
Let $\Omega$ be the circumcircle of a triangle $ABC$ with $\angle BAC > 90^{\circ}$ and $AB > AC$. The tangents
of $\Omega$ at $B$ and $C$ cross at $D$ and the tangent of $\Omega$ at $A$ crosses the line $BC$ at $E$. The line
through $D$, parallel to $AE$, crosses the line $BC$ at $F$. The circle with diameter $EF$ meets the
line $AB$ at $P$ and $Q$ and the line $AC$ at $X$ and $Y$. Prove that one of the angles $\angle AEB$, $\angle PEQ$, $\angle XEY$ is equal to the sum of the other two.
Let $ABC$ be an acute angle triangle.
Suppose that $D,E,F$ are the feet of perpendicluar lines from $A,B,C$ to $BC,CA,AB$.
Let $P,Q,R$ be the feet of perpendicular lines from $A,B,C$ to $EF,FD,DE$.
Prove that
\[ 2(PQ+QR+RP)\geq DE+EF+FD \]
Consider a grid of points where each point is coloured either white or black, such that no two rows have the same sequence of colours and no two columns have the same sequence of colours. Let a [i]table[/i] denote four points on the grid that form the vertices of a rectangle with sides parallel to those of the grid. A table is called [i]balanced[/i] if one diagonal pair of points are coloured white and the other diagonal pair black.
Determine all possible values of \(k \geq 2\) for which there exists a colouring of a \(k\times 2019\) grid with no balanced tables.
[i]proposed by the ICMC Problem Committee[/i]