Found problems: 85335
Consider the equation \[\frac{xy-C}{x+y}= k ,\] where all symbols used are positive integers.
1. Show that, for any (fixed) values $C, k$ this equation has at least a solution $x, y$;
2. Show that, for any (fixed) values $C, k$ this equation has at most a finite number of solutions $x, y$;
3. Show that, for any $C, n$ there exists $k = k(C,n)$ such that the equation has more than $n$ solutions $x, y$.
How many integers $n$ are there such that $(n+1!)(n+2!)(n+3!)\cdots(n+2013!)$ is divisible by $210$ and $1 \le n \le 210$?
[i]Proposed by Lewis Chen[/i]
Let $F=A_0A_1...A_n$ be a convex polygon in the plane. Define for all $1 \leq k \leq n-1$ the operation $f_k$ which replaces $F$ with a new polygon $f_k(F)=A_0A_1..A_{k-1}A_k^\prime A_{k+1}...A_n$ where $A_k^\prime$ is the symmetric of $A_k$ with respect to the perpendicular bisector of $A_{k-1}A_{k+1}$. Prove that $(f_1\circ f_2 \circ f_3 \circ...\circ f_{n-1})^n(F)=F$.
Consider the $9\times 9$ grid of lattice points $\{(x,y) | 0 \le x, y \le 8\}$. How many rectangles with nonzero area and sides parallel to the $x, y$ axes are there such that each corner is one of the lattice points and the point $(4, 4)$ is not contained within the interior of the rectangle? ($(4,4)$ is allowed to lie on the boundary of the rectangle).
Let $ABCD$ be a parallelogram. Let $X$ and $Y$ be arbitrary points on sides $BC$ and $CD$, respectively. Segments $BY$ and $DX$ intersect at $P$.
Prove that the line going through the midpoints of segments $BD$ and $XY$ is either parallel to or coincides with line $AP$.
Let $\, a$, and $b \,$ be odd positive integers. Define the sequence $\{f_n\}_{n\ge 1}$ by putting $\, f_1 = a,$ $f_2 = b, \,$ and by letting $\, f_n \,$ for $\, n \geq 3 \,$ be the greatest odd divisor of $\, f_{n-1} + f_{n-2}$. Show that $\, f_n \,$ is constant for sufficiently large $\, n \,$ and determine the eventual value as a function of $\, a \,$ and $\, b$.
Let regular tetrahedron $ABCD$ have center $O$. Find $\tan^2(\angle AOB)$.
[i]2022 CCA Math Bonanza Individual Round #6[/i]
Jacob uses the following procedure to write down a sequence of numbers. First he chooses the first term to be $ 6$. To generate each succeeding term, he flips a fair coin. If it comes up heads, he doubles the previous term and subtracts $ 1$. If it comes up tails, he takes half of the previous term and subtracts $ 1$. What is the probability that the fourth term in Jacob's sequence is an integer?
$ \textbf{(A)}\ \frac{1}{6} \qquad
\textbf{(B)}\ \frac{1}{3} \qquad
\textbf{(C)}\ \frac{1}{2} \qquad
\textbf{(D)}\ \frac{5}{8} \qquad
\textbf{(E)}\ \frac{3}{4}$
On a $5\times 5$ chessboard, two players play the following game: The first player places a knight on some square. Then the players alternately move the knight according to the rules of chess, starting with the second player. It is not allowed to move the knight to a square that was visited previously. The player who cannot move loses. Which of the two players has a winning strategy?
A set of five-digit numbers $\{N_1,... ,N_k\}$ is such that any five-digit a number whose digits are all in ascending order is the same in at least one digit with at least one of the numbers $N_1$,$...$ ,$N_k$. Find the smallest possible value of $k$.
Let the integers $a, b, c, d$ be relatively prime to $$m = ad - bc.$$
Prove that the pairs of integers $(x,y)$ for which $ax+by$ is a multiple of $m$ are identical with those for which $cx + dy$ is a multiple of $m$.
For any function $f:[0,1]\to [0,1]$, let $P_n (f)$ denote the number of fixed points of the function $\underbrace{f(f(\dotsc f}_{n} (x)\dotsc )$, i.e., the number of points $x\in [0,1]$ satisfying $\underbrace{f(f(\dotsc f}_{n} (x)\dotsc )=x$. Construct a piecewise linear, continuous, surjective function $f:[0,1] \to [0,1]$ such that for a suitable $2<A<3$, the sequence $\frac{P_n(f)}{A^n}$ converges.
[i]Based on the 8th problem of the Miklós Schweitzer competition, 2018[/i]
Solve in $N$:
$$\begin{cases} a^3=b^3+c^3+12a \\ a^2=5(b+c) \end{cases}$$
Take any point $P_1$ on the side $BC$ of a triangle $ABC$ and draw the following chain of lines: $P_1P_2$ parallel to $AC$; $P_2P_3$ parallel to $BC$; $P_3P_4$ parallel to $AB$ ; $P_4P_5$ parallel to $CA$; and $P_5P_6$ parallel to $BC$, Here, $P_2,P_5$ lie on $AB$; $P_3,P_6$ lie on $CA$ and $P_4$ on $BC$> Show that $P_6P_1$ is parallel to $AB$.
Find all rational numbers $a$, $b$, $c$ and $d$ such that \[ 8a^2 - 3b^2 + 5c^2 + 16d^2 - 10ab + 42cd + 18a + 22b - 2c - 54d = 42, \] \[ 15a^2 - 3b^2 + 21c^2 - 5d^2 + 4ab +32cd - 28a + 14b - 54c - 52d = -22. \]
Find the first two values of $40!(\text{mod }1763)$
[hide=Answer]1311, 3074[\hide]
Prove that
$$(1 + x)(1 + x^2)(1 + x^4)(1 + x^8) ... (1 + x^{2^{k-1}} ) = 1 + x + x^2 + x^3 +... + x^{2^{k-1}}$$
Let $a_1$, $a_2, \dots, a_{2015}$ be a sequence of positive integers in $[1,100]$.
Call a nonempty contiguous subsequence of this sequence [i]good[/i] if the product of the integers in it leaves a remainder of $1$ when divided by $101$.
In other words, it is a pair of integers $(x, y)$ such that $1 \le x \le y \le 2015$ and \[a_xa_{x+1}\dots a_{y-1}a_y \equiv 1 \pmod{101}. \]Find the minimum possible number of good subsequences across all possible $(a_i)$.
[i]Proposed by Yang Liu[/i]
Prove that there exists a $2021$-digit positive integer $\overline{a_1a_2\ldots a_{2021}}$, with all its digits being non-zero, such that for every $1 \leq n \leq 2020$ the following equality holds
$$\overline{a_1a_2\ldots a_n} \cdot \overline{a_{n+1}a_{n+2}\ldots a_{2021}}=\overline{a_1a_2\ldots a_{2021-n}} \cdot \overline{a_{2022-n}a_{2023-n}\ldots a_{2021}}$$
and all four numbers in the equality are pairwise different.
Prove that there exist two natural numbers $a,b$ such that $|a-m|+|b-n|>1000$ for any relatively prime natural numbers $m,n$.
Given functions $f,g:N^*\rightarrow N^*$, $g$ is surjective and $2f(n)^2=n^2+g(n)^2$, $\forall n>0$. Prove that if $|f(n)-n|\le2005\sqrt n$, $\forall n>0$, then $f(n)=n$ for infinitely many $n$.
A simple pendulum of length $L$ is constructed from a point object of mass $m$ suspended by a massless string attached to a fixed pivot point. A small peg is placed a distance $2L/3$ directly below the fixed pivot point so that the pendulum would swing as shown in the figure below. The mass is displaced $5$ degrees from the vertical and released. How long does it take to return to its starting position?
[asy]
// Code by riben
size(275);
draw(circle((0,0),1),linewidth(2));
filldraw(circle((0,0),1),gray);
draw((0,0)--(0,-70.8));
draw(circle((0,-71.8),3));
filldraw(circle((0,-71.8),3),gray);
draw(circle((0,-45),1));
filldraw(circle((0,-45),1),gray);
filldraw(circle((15,-70),3),gray,linewidth(0.2));
filldraw(circle((-15,-67),3),gray,linewidth(0.2));
draw((0,0)--(14.5,-66.5),dashed);
draw((0,-45)--(-13,-65),dashed);
// Labels
label("Fixed Pivot Point",(0,0),4*E);
label("Small Peg",(0,-45),12*E);
label("Point Object of mass m",(0,-70),17*E);
draw((-40,1)--(-40,-76.8),EndArrow(size=5));
draw((-40,-76.8)--(-40,1),EndArrow(size=5));
label("L",(-40,-37.9),E*2);
[/asy]
(A) $\pi \sqrt{\frac{L}{g}} \left(1+\sqrt{\frac{2}{3}}\right)$
(B) $\pi \sqrt{\frac{L}{g}} \left(2+\frac{2}{\sqrt{3}}\right)$
(C) $\pi \sqrt{\frac{L}{g}} \left(1+\frac{1}{3}\right)$
(D) $\pi \sqrt{\frac{L}{g}} \left(1+\sqrt{3}\right)$
(E) $\pi \sqrt{\frac{L}{g}} \left(1+\frac{1}{\sqrt{3}}\right)$
We say that a triangle $ABC$ is great if the following holds: for any point $D$ on the side $BC$, if $P$ and $Q$ are the feet of the perpendiculars from $D$ to the lines $AB$ and $AC$, respectively, then the reflection of $D$ in the line $PQ$ lies on the circumcircle of the triangle $ABC$. Prove that triangle $ABC$ is great if and only if $\angle A = 90^{\circ}$ and $AB = AC$.
[i]Senior Problems Committee of the Australian Mathematical Olympiad Committee[/i]
Let $ a,b,c $ be three real numbers such that $ \cos a+\cos b+\cos c=\sin a+\sin b+\sin c=0. $ Prove that
[b]i)[/b] $ \cos 6a+\cos 6b+\cos 6c=3\cos (2a+2b+2c) $
[b]ii)[/b] $ \sin 6a+\sin 6b+\sin 6c=3\sin (2a+2b+2c) $
[i]Vasile Pop[/i]
In the acute triangle $ABC$, with $AB \ne BC$, let $T$ denote the midpoint of the side $[AC], A_1$ and $C_1$ denote the feet of the altitudes drawn from $A$ and $C$, respectively. Let $Z$ be the intersection point of the tangents in $A$ and $C $ to the circumcircle of triangle $ABC, X$ be the intersection point of lines $ZA$ and $A_1C_1$ and $Y$ be the intersection point of lines $ZC$ and $A_1C_1$.
a) Prove that $T$ is the incircle of triangle $XYZ$.
b) The circumcircles of triangles $ABC$ and $A_1BC_1$ meet again at $D$. Prove that the orthocenter $H$ of triangle $ABC$ is on the line $TD$.
c) Prove that the point $D$ lies on the circumcircle of triangle $XYZ$.