Found problems: 85335
We say that an integer $x\in\{1,\dots,102\}$ is $\textit{square-ish}$ if there exists some integer $n$ such that $x\equiv n^2+n\pmod{103}$. Compute the product of all $\textit{square-ish}$ integers modulo $103$.
Determine whether there exists positive integers $a_{1}<a_{2}< \cdot \cdot \cdot <a_{k}$ such that all sums $ a_{i}+a_{j}$, where 1 $\leq i < j \leq k$, are unique, and among those sums, there are $1000$ consecutive integers.
Let $K$ be the number of sequences $A_1$, $A_2$, $\dots$, $A_n$ such that $n$ is a positive integer less than or equal to $10$, each $A_i$ is a subset of $\{1, 2, 3, \dots, 10\}$, and $A_{i-1}$ is a subset of $A_i$ for each $i$ between $2$ and $n$, inclusive. For example, $\{\}$, $\{5, 7\}$, $\{2, 5, 7\}$, $\{2, 5, 7\}$, $\{2, 5, 6, 7, 9\}$ is one such sequence, with $n = 5$. What is the remainder when $K$ is divided by $10$?
$\textbf{(A) } 1 \qquad \textbf{(B) } 3 \qquad \textbf{(C) } 5 \qquad \textbf{(D) } 7 \qquad \textbf{(E) } 9$
It is given function $f : A \rightarrow \mathbb{R}$, $(A\subseteq \mathbb{R})$ such that $$f(x+y)=f(x)\cdot f(y)-f(xy)+1; (\forall x,y \in A)$$ If $f : A \rightarrow \mathbb{R}$, $(\mathbb{N} \subseteq A\subseteq \mathbb{R})$ is solution of given functional equation, prove that: $$f(n)=\begin{cases}
\frac{c^{n+1}-1}{c-1} \text{, } \forall n \in \mathbb{N}, c \neq 1 \\
n+1 \text{, } \forall n \in \mathbb{N}, c = 1
\end{cases}$$
where $c=f(1)-1$
$a)$ Solve given functional equation for $A=\mathbb{N}$
$b)$ With $A=\mathbb{Q}$, find all functions $f$ which are solutions of the given functional equation and also $f(1997) \neq f(1998)$
Find the remainder of
$$\prod_{n = 2}^{99} (1 - n^2 + n^4)(1 - 2n^2 + n^4)$$
when divided by $101$.
Let $P(x)$ be given a polynomial of degree 4, having 4 positive roots. Prove that the equation
\[(1-4 \cdot x) \cdot \frac{P(x)}{x^2} + (x^2 + 4 \cdot x - 1) \cdot \frac{P'(x)}{x^2} - P''(x) = 0\]
has also 4 positive roots.
A kangaroo jumps from lattice poin to lattice point in the coordinate plane. It can make only two kinds of jumps: $ (A)$ $ 1$ to right and $ 3$ up, and $ (B)$ $ 2$ to the left and $ 4$ down.
$ (a)$ The start position of the kangaroo is $ (0,0)$. Show that it can jump to the point $ (19,95)$ and determine the number of jumps needed.
$ (b)$ Show that if the start position is $ (1,0)$, then it cannot reach $ (19,95)$.
$ (c)$ If the start position is $ (0,0)$, find all points $ (m,n)$ with $ m,n \ge 0$ which the kangaroo can reach.
Let $\mathbb N$ denote the set of positive integers. Let $f: \mathbb N \to \mathbb N$ be a function such that the following conditions hold:
(a) For any $n\in \mathbb N$, we have $f(n) | n^{2016}$.
(b) For any $a,b,c\in \mathbb N$ satisfying $a^2+b^2=c^2$, we have $f(a)f(b)=f(c)$.
Over all possible functions $f$, determine the number of distinct values that can be achieved by $f(2014)+f(2)-f(2016)$.
[i]Proposed by Vincent Huang[/i]
In triangle $ABC$, angle $C$ is right and the two catheti are both length $1$. For one given the choice of the point $P$ on the cathetus $BC$, the point $Q$ on the hypotenuse and the point $R$ are plotted on the second cathetus so that $PQ$ is parallel to $AC$ and $QR$ is parallel to $BC$. Thereby the triangle is divided into three parts. Determine the locations of point $P$ for which the rectangular part has a larger area than each of the other two parts.
Find all binary operations $\diamondsuit: \mathbb R_{>0}\times \mathbb R_{>0}\to \mathbb R_{>0}$ (meaning $\diamondsuit$ takes pairs of positive real numbers to positive real numbers) such that for any real numbers $a, b, c > 0$,
[list]
[*] the equation $a\,\diamondsuit\, (b\,\diamondsuit \,c) = (a\,\diamondsuit \,b)\cdot c$ holds; and
[*] if $a\ge 1$ then $a\,\diamondsuit\, a\ge 1$.
[/list]
[i]Evan Chen[/i]
For a positive integer $n$ and a subset $S$ of $\{1, 2, \dots, n\}$, let $S$ be "$n$-good" if and only if for any $x$, $y\in S$ (allowed to be same), if $x+y\leq n$, then $x+y\in S$. Let $r_n$ be the smallest real number such that for any positive integer $m\leq n$, there is always a $m$-element "$n$-good" set, so that the sum of its elements is not more than $m\cdot r_n$. Prove that there exists a real number $\alpha$ such that for any positive integer $n$, $|r_n-\alpha n|\leq 2024.$
Using a compass and a ruler, construct a triangle $ABC$ given the sides $b, c$ and the segment $AI$, where$ I$ is the center of the inscribed circle of this triangle.
Let n be an integer such that if d | n then d + 1 | n + 1. Show that n is a prime number.
For how many values of $ x$ in $ [0,\pi]$ is $ \sin^{\minus{}1}(\sin 6x)\equal{}\cos^{\minus{}1}(\cos x)$?
Note: The functions $ \sin^{\minus{}1}\equal{}\arcsin$ and $ \cos^{\minus{}1}\equal{}\arccos$ denote inverse trigonometric functions.
$ \textbf{(A)}\ 3\qquad
\textbf{(B)}\ 4\qquad
\textbf{(C)}\ 5\qquad
\textbf{(D)}\ 6\qquad
\textbf{(E)}\ 7$
In a dance class there are $10$ boys and $10$ girls. It is known that for each $1\le k\le 10$ and for each group of $k$ boys, the number of girls who are friends with at least one boy in the group is not less than $k$. Prove that it is possible to pair up the boys and the girls for a dance so that each pair consists of a boy and a girl who are friends.
Let $P$, $A$, $B$, $C$, $D$ be points on a plane such that $PA = 9$, $PB = 19$, $PC = 9$, $PD = 5$, $\angle APB = 120^\circ$, $\angle BPC = 45^\circ$, $\angle CPD = 60^\circ$, and $\angle DPA = 135^\circ$. Let $G_1$, $G_2$, $G_3$, and $G_4$ be the centroids of triangles $PAB$, $PBC$, $PCD$, $PDA$. $[G_1G_2G_3G_4]$ can be expressed as $a\sqrt{b} + c\sqrt{d}$. Find $a+b+c+d$.
[i]2022 CCA Math Bonanza Individual Round #15[/i]
Evaluate $\textstyle\int_{0}^{\infty} t^{-1 / 2} e^{-1985\left(t+t^{-1}\right)} d t.$ You may assume that $\textstyle\int_{-\infty}^{\infty} e^{-x^{2}} d x=\sqrt{\pi}.$
To two circles $r_1$ and $r_2$, intersecting at points $A$ and $B$, their common tangent $CD$ is drawn ($C$ and $D$ are tangency points, respectively, point $B$ is closer to line $CB$ than $A$). Line passing through $A$ , intersects $r_1$ and $r_2$ for second time at points $K$ and $L$, respectively ($A$ lies between $K$ and $L$). Lines $KC$ and $LD$ intersect at point $P$. Prove that $PB$ is the symmedian of triangle $KPL$.
(Yu. Blinkov)
For any natural number $n$, expressed in base $10$, let $S(n)$ denote the sum of all digits of $n$. Find all natural numbers $n$ such that $n=2S(n)^2$.
A triangle whose all sides have lengths greater than $1$ is contained in a unit square. Show that the center of the square lies inside the triangle.
Let $ a, b, c, d,m, n \in \mathbb{Z}^\plus{}$ such that \[ a^2\plus{}b^2\plus{}c^2\plus{}d^2 \equal{} 1989,\]
\[ a\plus{}b\plus{}c\plus{}d \equal{} m^2,\] and the largest of $ a, b, c, d$ is $ n^2.$ Determine, with proof, the values of $m$ and $ n.$
Evaluate $\sum_{m=0}^{2009}\sum_{n=0}^{m}\binom{2009}{m}\binom{m}{n}$
A triangle $ ABC$ is given. Let $ B_1$ be the reflection of $ B$ across the line $ AC$, $ C_1$ the reflection of $ C$ across the line $ AB$, and $ O_1$ the reflection of the circumcentre of $ ABC$ across the line $ BC$. Prove that the circumcentre of $ AB_1C_1$ lies on the line $ AO_1$.
[i]Proposed by A. Akopyan[/i]
Suppose there are $n$ points in a plane no three of which are collinear with the property that if we label these points as $A_1,A_2,\ldots,A_n$ in any way whatsoever, the broken line $A_1A_2\ldots A_n$ does not intersect itself. Find the maximum value of $n$.
[i]Dinu Serbanescu, Romania[/i]
Several fleas sit on the squares of a $10\times 10$ chessboard (at most one fea per square). Every minute, all fleas simultaneously jump to adjacent squares. Each fea begins jumping in one of four directions (up, down, left, right), and keeps jumping in this direction while it is possible; otherwise, it reverses direction on the opposite. It happened that during one hour, no two fleas ever occupied the same square. Find the maximal possible number of fleas on the board.