Found problems: 85335
Alice and the Mad Hatter are playing a game. At the start of the game, three $2024$’s are written on the blackboard. Then, Alice and the Mad Hatter alternate turns, with the Mad Hatter starting. On the Mad Hatter’s turn, he must pick one of the numbers on the blackboard and increase it by $1$. On Alice’s turn, she must:
- pick one of the numbers on the blackboard and decrease it by 1, and then
- replace the two numbers $a$ and $b$ on the blackboard which were not chosen by the Mad Hatter on the previous turn with $\sqrt{ab}$.
Alice wins if, on the start of her turn, any of the three numbers are less than $1$. Can the Mad Hatter prevent Alice from winning?
Let $ABC$ be a triangle with $AB<AC$ and circuncircle $\omega$. Let $M$ and $N$ be the midpoints of $AC$ and $AB$ respectively and $G$ is the centroid of $ABC$. Let $P$ be the foot of perpendicular of $A$ to the line $BC$, and the point $Q$ is the intersection of $GP$ and $\omega$($Q,P,G$ are collinears in this order). The line $QM$ cuts $\omega$ in $M_1$ and the line $QN$ cuts $\omega$ in $N_1$. If $K$ is the intersection of $BM_1$ and $CN_1$ prove that $P$, $G$ and $K$ are collinears.
In right triangle $ABC$ with right angle $C$, $CA=30$ and $CB=16$. Its legs $\overline{CA}$ and $\overline{CB}$ are extended beyond $A$ and $B$. Points $O_{1}$ and $O_{2}$ lie in the exterior of the triangle and are the centers of two circles with equal radii. The circle with center $O_{1}$ is tangent to the hypotenuse and to the extension of leg CA, the circle with center $O_{2}$ is tangent to the hypotenuse and to the extension of leg CB, and the circles are externally tangent to each other. The length of the radius of either circle can be expressed as $p/q$, where $p$ and $q$ are relatively prime positive integers. Find $p+q$.
In a circle of center $O$ and radius $r$, a triangle $ABC$ of orthocenter $H$ is inscribed. It is considered a triangle $A'B'C'$ whose sides have by length the measurements of the segments $AB, CH$ and $2r$. Determine the triangle $ABC$ so that the area of the triangle $A'B'C'$ is maximum.
A man travels $ m$ feet due north at $ 2$ minutes per mile. He returns due south to his starting point at $ 2$ miles per minute. The average rate in miles per hour for the entire trip is:
$ \textbf{(A)}\ 75\qquad
\textbf{(B)}\ 48\qquad
\textbf{(C)}\ 45\qquad
\textbf{(D)}\ 24\qquad\\
\textbf{(E)}\ \text{impossible to determine without knowing the value of }{m}$
Let the complex function $F(z)$ be regular on the punctuated disk $\{ 0<|z| < R\}$. By a [i]level curve[/i] we mean a component of the level set of $\mathrm{Re}F(z)$, that is, a maximal connected set on which $\mathrm{Re}F(z)$ is constant. Denote by $A(r)$ the union of those level curves that are entirely contained in the punctuated disk $\{ 0<|z|<r\}$. Prove that if the number of components of $A(r)$ has an upper bound independent of $r$ then $F(z)$ can only have a pole type singularity at $0$.
Prove that there exists infinitely many positive integers $n$ for which the equation$$x^2+y^{11}-z^{2022!}=n$$has no solution $(x,y,z)$ over the integers.
For a quadrilateral $ABCD$ is given that $\angle CBD=2\angle ADB$, $\angle ABD=2\angle CDB$, and
$AB=CB$. Prove that $AD=CD$.
Let it \(k \geq 1\) be an integer. Define the sequence \((a_n)_{n \geq 1}\) by \(a_0=0,a_1=1\) and
\[ a_{n+2} = ka_{n+1}+a_n \]
for \(n \geq 0\).
Let it \(p\) an odd prime number.
Denote \(m(p)\) as the smallest positive integer \(m\) such that \(p | a_m\).
Denote \(T(p)\) as the smallest positive integer \(T\) such that for every natural \(j\) we gave \(p | (a_{T+j}-a_j)\).
[list='i']
[*] Show that \(T(p) \leq (p-1) \cdot m(p)\).
[*] Show that if \(T(p) = (p-1) \cdot m(p)\) then
\[ \prod_{1 \leq j \leq T(p)-1}^{j \not \equiv 0 \pmod{m(p)}}{a_j} \equiv (-1)^{m(p)-1} \pmod{p} \]
[/list]
A permutation $ (a_1,a_2,a_3,a_4,a_5)$ of $ (1,2,3,4,5)$ is heavy-tailed if $ a_1 \plus{} a_2 < a_4 \plus{} a_5$. What is the number of heavy-tailed permutations?
$ \textbf{(A)}\ 36 \qquad
\textbf{(B)}\ 40 \qquad
\textbf{(C)}\ 44 \qquad
\textbf{(D)}\ 48 \qquad
\textbf{(E)}\ 52$
$ AL$, $ BM$, and $ CN$ are the medians of $ \triangle ABC$. $ K$ is the intersection of medians. If $ C,K,L,M$ are concyclic and $ AB \equal{} \sqrt 3$, then the median $ CN$ = ?
$\textbf{(A)}\ 1 \qquad\textbf{(B)}\ \sqrt 3 \qquad\textbf{(C)}\ \frac {3\sqrt3}{2} \qquad\textbf{(D)}\ 3 \qquad\textbf{(E)}\ \text{None}$
Find all the triples $(x,y,z)$ of positive integers such that $xy+yz+zx-xyz=2$.
Let $\triangle AB\varGamma$ be an acute-angled triangle with $AB < A\varGamma$, and let $O$ be the center of the circumcircle of the triangle. On the sides $AB$ and $A \varGamma$ we select points $T$ and $P$ respectively such that $OT=OP$. Let $M,K$ and $\varLambda$ be the midpoints of $PT,PB$ and $\varGamma T$ respectively. Prove that $\angle TMK = \angle M\varLambda K$.
Find all pairs of natural numbers $x$ and $y$ for which $x^2+3y$ and $y^2+3x$ are simultaneously squares of natural numbers.
Prove the following statement:
If a polynomial $p(x) = x^3 + Ax^2 + Bx +C$ has three real positve roots at least two of which are distinct, then $A^2 +B^2 +18C > 0$.
The sum of all but one of the interior angles of a convex polygon equals $2570^\circ$. The remaining angle is
$\textbf{(A)} \ 90^\circ \qquad \textbf{(B)} \ 105^\circ \qquad \textbf{(C)} \ 120^\circ \qquad \textbf{(D)} \ 130^\circ \qquad \textbf{(E)} \ 144^\circ$
Let $A$ be the largest subset of $\{1,\dots,n\}$ such that for each $x\in A$, $x$ divides at most one other element in $A$. Prove that \[\frac{2n}3\leq |A|\leq \left\lceil \frac{3n}4\right\rceil. \]
Let $A$ be a finite ring and $a,b \in A,$ such that $(ab-1)b=0.$ Prove that $b(ab-1)=0.$
The sum of three numbers is $96$. The first number is $6$ times the third number, and the third number is $40$ less than the second number. What is the absolute value of the difference between the first and second numbers?
$\textbf{(A) } 1 \qquad \textbf{(B) } 2 \qquad \textbf{(C) } 3 \qquad \textbf{(D) } 4 \qquad \textbf{(E) } 5$
Find $k$ if $P,Q,R,$ and $S$ are points on the sides of quadrilateral $ABCD$ so that \[ \frac{AP}{PB} = \frac{BQ}{QC} = \frac{CR}{RD} = \frac{DS}{SA} = k, \] and the area of the quadrilateral $PQRS$ is exactly 52% of the area of the quadrilateral $ABCD$.
For picture, go [url=http://www.cms.math.ca/Competitions/IMTS/imts3.html]here[/url].
Determine all functions $f: \mathbb R \to \mathbb R$ which satisfy the inequality
$$f(a^2) - f(b^2) \leq \left( f(a) + b\right)\left( a - f(b)\right),$$
for all $a,b \in \mathbb R$.
Let $f(x)$ be a polynomial with integer coefficients. We assume that there exists a positive integer $k$ and $k$ consecutive integers $n, n+1, ... , n+k -1$ so that none of the numbers $f(n), f(n+ 1),... , f(n + k - 1)$ is divisible by $k$.
Show that the zeroes of $f(x)$ are not integers.
Do there exist $ 14$ positive integers, upon increasing each of them by $ 1$,their product increases exactly $ 2008$ times?
Find the smallest prime which is not the difference (in some order) of a power of $2$ and a power of $3$.
Let the sequence $ \{a_n\}_{n\geq 1}$ be defined by $ a_1 \equal{} 20$, $ a_2 \equal{} 30$ and $ a_{n \plus{} 2} \equal{} 3a_{n \plus{} 1} \minus{} a_n$ for all $ n\geq 1$. Find all positive integers $ n$ such that $ 1 \plus{} 5a_n a_{n \plus{} 1}$ is a perfect square.