Found problems: 85335
Let $\mathbb N$ be the set of positive integers. Find all functions $f: \mathbb N \to \mathbb N$ that satisfy the equation
\[ f^{abc-a}(abc) + f^{abc-b}(abc) + f^{abc-c}(abc) = a + b + c \]
for all $a,b,c \ge 2$.
(Here $f^1(n) = f(n)$ and $f^k(n) = f(f^{k-1}(n))$ for every integer $k$ greater than $1$.)
In Skinien there 2009 towns where each of them is connected with exactly 1004 other town by a highway. Prove that starting in an arbitrary town one can make a round trip along the highways such that each town is passed exactly once and finally one returns to its starting point.
In triangle $ABC$ with $AB = 26$, $BC = 28$, and $C A = 30$, let $M$ be the midpoint of $AB$ and let $N$ be the midpoint of $C A$. The circumcircle of triangle $BCM$ intersects $AC$ at $X\ne C$, and the circumcircle of triangle $BCN $intersects $AB$ at $Y\ne B$. Lines $MX$ and $NY$ intersect $BC$ at $P$ and $Q$, respectively. The area of quadrilateral $PQY X$ can be expressed as $\frac{p}{q}$ for positive integers $p$ and $q$ such that gcd$(p,q) = 1$. Find $q$.
For each integer $n\ge 1,$ compute the smallest possible value of \[\sum_{k=1}^{n}\left\lfloor\frac{a_k}{k}\right\rfloor\] over all permutations $(a_1,\dots,a_n)$ of $\{1,\dots,n\}.$
[i]Proposed by Shahjalal Shohag, Bangladesh[/i]
Let $n \ge 3$ be an integer. On a circle, there are $n$ points. Each of them is labelled with a real number at most $1$ such that each number is the absolute value of the difference of the two numbers immediately preceding it in clockwise order. Determine the maximal possible value of the sum of all numbers as a function of $n$.
(Walther Janous)
Determine all functions $f: \mathbb{Q^+} \rightarrow \mathbb{Q^+}$ that satisfy:
\[f\left(x+\frac{y}{x}\right) = f(x)+f\left(\frac{y}{x}\right)+2y \:\text{for all}\: x, y \in \mathbb{Q^+}\]
Let $k$ be the incircle of the triangle $ABC$ with the center of the incircle $I$. The circle $k$ touches the sides $BC, CA$ and $AB$ in points $D, E$ and $F$. Let $G$ be the intersection of the straight line $AI$ and the circle $k$, which lies between $A$ and $I$. Assume $BE$ and $FG$ are parallel. Show that $BD = EF$.
For $ a\in\mathbb{R}$, let $ M(a)$ be the maximum value of the function $ f(x)\equal{}\int_{0}^{\pi}\sin (x\minus{}t)\sin (2t\minus{}a)\ dt$.
Evaluate $ \int_{0}^{\frac{\pi}{2}}M(a)\sin (2a)\ da$.
Let $A=\{1,2,3,\cdots ,17\}$. A mapping $f:A\rightarrow A$ is defined as follows: $f^{[1]}(x)=f(x), f^{[k+1]}(x)=f(f^{[k]}(x))$ for $k\in\mathbb{N}$. Suppose that $f$ is bijective and that there exists a natural number $M$ such that:
i) when $m<M$ and $1\le i\le 16$, we have $f^{[m]}(i+1)- f^{[m]}(i) \not=\pm 1\pmod{17}$ and $f^{[m]}(1)- f^{[m]}(17) \not=\pm 1\pmod{17}$;
ii) when $1\le i\le 16$, we have $f^{[M]}(i+1)- f^{[M]}(i)=\pm 1 \pmod{17}$ and $f^{[M]}(1)- f^{[M]}(17)=\pm 1\pmod{17}$.
Find the maximal value of $M$.
Let $X$ be a non-empty and finite set, $A_1,...,A_k$ $k$ subsets of $X$, satisying:
(1) $|A_i|\leq 3,i=1,2,...,k$
(2) Any element of $X$ is an element of at least $4$ sets among $A_1,....,A_k$.
Show that one can select $[\frac{3k}{7}] $ sets from $A_1,...,A_k$ such that their union is $X$.
Find all functions $f:\mathbb{R}\to\mathbb{R}$ such that
\[f(x^3)+f(y^3)=(x+y)f(x^2)+f(y^2)- f(xy)\]
for all $x\in\mathbb{R}$.
Given a triangle $ ABC$ with angle $ C \geq 60^{\circ}$. Prove that:
$ \left(a \plus{} b\right) \cdot \left(\frac {1}{a} \plus{} \frac {1}{b} \plus{} \frac {1}{c} \right) \geq 4 \plus{} \frac {1}{\sin\left(\frac {C}{2}\right)}.$
To be continuous at $ x \equal{} \minus{} 1$, the value of $ \frac {x^3 \plus{} 1}{x^2 \minus{} 1}$ is taken to be:
$ \textbf{(A)}\ \minus{} 2 \qquad\textbf{(B)}\ 0 \qquad\textbf{(C)}\ \frac {3}{2} \qquad\textbf{(D)}\ \infty \qquad\textbf{(E)}\ \minus{} \frac {3}{2}$
Stekel and Prick play a game on an $ m \times n$ board, where $m$ and $n$ are positive are integers. They alternate turns, with Stekel starting. Spine bets on his turn, he always takes a pawn on a square where there is no pawn yet. Prick does his turn the same, but his pawn must always come into a square adjacent to the square that Spike just placed a pawn in on his previous turn. Prick wins like the whole board is full of pawns. Spike wins if Prik can no longer move a pawn on his turn, while there is still at least one empty square on the board. Determine for all pairs $(m, n)$ who has a winning strategy.
$2.$ The equations $x^2-4x+k=0$ and $x^2+kx-4=0,$ where $k$ is a real number, have exactly one common root. What is the value of $k ?$
In the triangle $ABC$, the lengths of the sides $BC, CA$ and $AB$ are $a,b$ and $c{}$ respectively. Several segments are drawn from the vertex $C{}$, which cut the triangle $ABC$ into several triangles. Find the smallest number $M{}$ for which, with each such cut, the sum of the radii of the circles inscribed in triangles does not exceed $M{}$.
[i]Porposed by O. Titov[/i]
Compute the number of subsets $S$ of $\{0,1,\dots,14\}$ with the property that for each $n=0,1,\dots,
6$, either $n$ is in $S$ or both of $2n+1$ and $2n+2$ are in $S$.
[i]Proposed by Evan Chen[/i]
Triangle $ABC$ is equilateral with side length $\sqrt{3}$ and circumcenter at $O$. Point $P$ is in the plane such that $(AP)(BP)(CP) = 7$. Compute the difference between the maximum and minimum possible values of $OP$.
[i]2015 CCA Math Bonanza Team Round #8[/i]
Let $a,b,c>0$ and $abc\ge1.$ Prove that $a^4+b^3+c^2\ge a^3+b^2+c.$
Let $(x_n)_{n\ge 0}$ a sequence of real numbers defined by $x_0>0$ and $x_{n+1}=x_n+\frac{1}{\sqrt{x_n}}$. Compute $\lim_{n\to \infty}x_n$ and $\lim_{n\to \infty} \frac{x_n^3}{n^2}$.
There are $n$ boys and $n$ girls in a school class, where $n$ is a positive integer. The heights of all the children in this class are distinct. Every girl determines the number of boys that are taller than her, subtracts the number of girls that are taller than her, and writes the result on a piece of paper. Every boy determines the number of girls that are shorter than him, subtracts the number of boys that are shorter than him, and writes the result on a piece of paper.
Prove that the numbers written down by the girls are the same as the numbers written down by the boys (up to a permutation).
[i]Proposed by Stephan Wagner, Austria[/i]
What is the remainder when
\[\sum_{k=0}^{100}10^k\]
is divided by $9$?
If $A$ and $B$ are vertices of a polyhedron, define the [i]distance[/i] $d(A, B)$ to be the minimum number of edges of the polyhedron one must traverse in order to connect $A$ and $B$. For example, if $\overline{AB}$ is an edge of the polyhedron, then $d(A, B) = 1$, but if $\overline{AC}$ and $\overline{CB}$ are edges and $\overline{AB}$ is not an edge, then $d(A, B) = 2$. Let $Q$, $R$, and $S$ be randomly chosen distinct vertices of a regular icosahedron (regular polyhedron made up of 20 equilateral triangles). What is the probability that $d(Q, R) > d(R, S)$?
$\textbf{(A)}~\frac{7}{22}\qquad\textbf{(B)}~\frac13\qquad\textbf{(C)}~\frac38\qquad\textbf{(D)}~\frac5{12}\qquad\textbf{(E)}~\frac12$
Evaluate $$2023 \cdot \frac{2023^6 +27}{(2023^2 +3)(2024^3 -1)}-2023^2.$$
[i]Proposed by Evin Liang[/i]
Right isosceles triangles are constructed on the sides of a 3-4-5 right triangle, as shown. A capital letter represents the area of each triangle. Which one of the following is true?
[asy]/* AMC8 2002 #16 Problem */
draw((0,0)--(4,0)--(4,3)--cycle);
draw((4,3)--(-4,4)--(0,0));
draw((-0.15,0.1)--(0,0.25)--(.15,0.1));
draw((0,0)--(4,-4)--(4,0));
draw((4,0.2)--(3.8,0.2)--(3.8,-0.2)--(4,-0.2));
draw((4,0)--(7,3)--(4,3));
draw((4,2.8)--(4.2,2.8)--(4.2,3));
label(scale(0.8)*"$Z$", (0, 3), S);
label(scale(0.8)*"$Y$", (3,-2));
label(scale(0.8)*"$X$", (5.5, 2.5));
label(scale(0.8)*"$W$", (2.6,1));
label(scale(0.65)*"5", (2,2));
label(scale(0.65)*"4", (2.3,-0.4));
label(scale(0.65)*"3", (4.3,1.5));[/asy]
$ \textbf{(A)}\ X\plus{}Z\equal{}W\plus{}Y \qquad \textbf{(B)}\ W\plus{}X\equal{}Z \qquad\textbf{(C)}\ 3X\plus{}4Y\equal{}5Z \qquad $
$\textbf{(D)}\ X\plus{}W\equal{}\frac{1}{2}(Y\plus{}Z) \qquad\textbf{(E)}\ X\plus{}Y\equal{}Z$