Found problems: 85335
2011 USA Team Selection Test, 5
Let $c_n$ be a sequence which is defined recursively as follows: $c_0 = 1$, $c_{2n+1} = c_n$ for $n \geq 0$, and $c_{2n} = c_n + c_{n-2^e}$ for $n > 0$ where $e$ is the maximal nonnegative integer such that $2^e$ divides $n$. Prove that
\[\sum_{i=0}^{2^n-1} c_i = \frac{1}{n+2} {2n+2 \choose n+1}.\]
2006 Finnish National High School Mathematics Competition, 3
The numbers $p, 4p^2 + 1,$ and $6p^2 + 1$ are primes. Determine $p.$
1986 Tournament Of Towns, (114) 1
For which natural number $k$ does $\frac{k^2}{1.001^k}$ attain its maximum value?
2024 Mathematical Talent Reward Programme, 1
The Integration Premier League has $n$ teams competing. The tournament follows a round-robin system, that is, where every pair of teams play each other exactly once. So every team plays exactly $n-1$ matches. The top $m \leq n$ temas at the end of the tournament qualify for the playoffs. Assume there are no tied matches.
Let $A(m,n)$ be the minimum number of matches a team has to win to gurantee selection for the playoffs, regardless of what their run rate is. For example, $A(n,n) = 0$ (everyone qualifies anyway so no need to win!) and $A(1,n) = n-1$ (even if you lose to just one other team, they might defeat everyone and qualify instead of you). Answer the following:
$(A)$ FInd the value of $A(2,4),A(2,6)$ and $A(4,10)$ with proof (explain why a smaller value can still lead to the team not qualifying, and show that the respective values themselves are enough).
$(B)$ Show that $A(n-1,n) = \frac{n}{2}$ when $n$ even and $ = \frac{n+1}{2}$ when $n$ odd.
$(C)$ For bonus marks, try to find a general pattern for $A(m,n)$.
2020 Peru EGMO TST, 5
Let $AD$ be the diameter of a circle $\omega$ and $BC$ is a chord of $\omega$ which is perpendicular to $AD$. Let $M,N,P$ be points on the segments $AB,AC,BC$ respectively, such that $MP\parallel AC$ and $PN\parallel AB$. The line $MN$ cuts the line $PD$ in the point $Q$ and the angle bisector of $\angle MPN$ in the point $R$. Prove that the points $B,R,Q,C$ are concyclic.
VII Soros Olympiad 2000 - 01, 8.8
Is there a quadrilateral, any vertex of which can be moved to another location so that the new quadrilateral is congruent to the original one?
2010 Romanian Master of Mathematics, 6
Given a polynomial $f(x)$ with rational coefficients, of degree $d \ge 2$, we define the sequence of sets $f^0(\mathbb{Q}), f^1(\mathbb{Q}), \ldots$ as $f^0(\mathbb{Q})=\mathbb{Q}$, $f^{n+1}(\mathbb{Q})=f(f^{n}(\mathbb{Q}))$ for $n\ge 0$. (Given a set $S$, we write $f(S)$ for the set $\{f(x)\mid x\in S\})$.
Let $f^{\omega}(\mathbb{Q})=\bigcap_{n=0}^{\infty} f^n(\mathbb{Q})$ be the set of numbers that are in all of the sets $f^n(\mathbb{Q})$, $n\geq 0$. Prove that $f^{\omega}(\mathbb{Q})$ is a finite set.
[i]Dan Schwarz, Romania[/i]
2005 ISI B.Stat Entrance Exam, 6
Let $f$ be a function defined on $(0, \infty )$ as follows:
\[f(x)=x+\frac1x\]
Let $h$ be a function defined for all $x \in (0,1)$ as
\[h(x)=\frac{x^4}{(1-x)^6}\]
Suppose that $g(x)=f(h(x))$ for all $x \in (0,1)$.
(a) Show that $h$ is a strictly increasing function.
(b) Show that there exists a real number $x_0 \in (0,1)$ such that $g$ is strictly decreasing in the interval $(0,x_0]$ and strictly increasing in the interval $[x_0,1)$.
2024 Israel TST, P3
Let $ABCD$ be a parallelogram. Let $\omega_1$ be the circle passing through $D$ tangent to $AB$ at $A$. Let $\omega_2$ be the circle passing through $A$ tangent to $CD$ at $D$. The tangents from $B$ to $\omega_1$ touch it at $A$ and $P$. The tangents from $C$ to $\omega_2$ touch it at $D$ and $Q$. Lines $AP$ and $DQ$ intersect at $X$. The perpendicular bisector of $BC$ intersects $AD$ at $R$.
Show that the circumcircles of triangles $\triangle PQX$, $\triangle BCR$ are concentric.
1998 Canada National Olympiad, 5
Let $m$ be a positive integer. Define the sequence $a_0, a_1, a_2, \cdots$ by $a_0 = 0,\; a_1 = m,$ and $a_{n+1} = m^2a_n - a_{n-1}$ for $n = 1,2,3,\cdots$.
Prove that an ordered pair $(a,b)$ of non-negative integers, with $a \leq b$, gives a solution to the equation
\[ {\displaystyle \frac{a^2 + b^2}{ab + 1} = m^2} \]
if and only if $(a,b)$ is of the form $(a_n,a_{n+1})$ for some $n \geq 0$.
2024 Azerbaijan Senior NMO, 4
Let $P(x)$ be a polynomial with the coefficients being $0$ or $1$ and degree $2023$. If $P(0)=1$, then prove that every real root of this polynomial is less than $\frac{1-\sqrt{5}}{2}$.
2019 India PRMO, 6
Let $ABC$ be a triangle such that $AB=AC$. Suppose the tangent to the circumcircle of ABC at B is perpendicular to AC. Find angle ABC measured in degrees
Novosibirsk Oral Geo Oly VIII, 2021.6
Inside the equilateral triangle $ABC$, points $P$ and $Q$ are chosen so that the quadrilateral $APQC$ is convex, $AP = PQ = QC$ and $\angle PBQ = 30^o$. Prove that $AQ = BP$.
Kvant 2023, M2758
The numbers $2,4,\ldots,2^{100}$ are written on a board. At a move, one may erase the numbers $a,b$ from the board and replace them with $ab/(a+b).$ Prove that the last numer on the board will be greater than 1.
[i]From the folklore[/i]
1988 Tournament Of Towns, (191) 4
(a) Two identical cogwheels with $14$ teeth each are given . One is laid horizontally on top of the other in such a way that their teeth coincide (thus the projections of the teeth on the horizontal plane are identical ) . Four pairs of coinciding teeth are cut off. Is it always possible to rotate the two cogwheels with respect to each other so that their common projection looks like that of an entire cogwheel?
(The cogwheels may be rotated about their common axis, but not turned over.)
(b) Answer the same question , but with two $13$-tooth cogwheels and four pairs of cut-off teeth.
2024 JHMT HS, 16
Let $N_{15}$ be the answer to problem 15.
For a positive integer $x$ expressed in base ten, let $x'$ be the result of swapping its first and last digits (for example, if $x = 2024$, then $x' = 4022$). Let $C$ be the number of $N_{15}$-digit positive integers $x$ with a nonzero leading digit that satisfy the property that both $x$ and $x'$ are divisible by $11$ (note: $x'$ is allowed to have a leading digit of zero). Compute the sum of the digits of $C$ when $C$ is expressed in base ten.
2013 NIMO Problems, 4
Let $\mathcal F$ be the set of all $2013 \times 2013$ arrays whose entries are $0$ and $1$. A transformation $K : \mathcal F \to \mathcal F$ is defined as follows: for each entry $a_{ij}$ in an array $A \in \mathcal F$, let $S_{ij}$ denote the sum of all the entries of $A$ sharing either a row or column (or both) with $a_{ij}$. Then $a_{ij}$ is replaced by the remainder when $S_{ij}$ is divided by two.
Prove that for any $A \in \mathcal F$, $K(A) = K(K(A))$.
[i]Proposed by Aaron Lin[/i]
2024 Mongolian Mathematical Olympiad, 3
A set $X$ consisting of $n$ positive integers is called $\textit{good}$ if the following condition holds:
For any two different subsets of $X$, say $A$ and $B$, the number $s(A) - s(B)$ is not divisible by $2^n$.
(Here, for a set $A$, $s(A)$ denotes the sum of the elements of $A$)
Given $n$, find the number of good sets of size $n$, all of whose elements is strictly less than $2^n$.
2010 All-Russian Olympiad, 2
There are $100$ random, distinct real numbers corresponding to $100$ points on a circle. Prove that you can always choose $4$ consecutive points in such a way that the sum of the two numbers corresponding to the points on the outside is always greater than the sum of the two numbers corresponding to the two points on the inside.
2016 Saudi Arabia BMO TST, 3
Find all integers $n$ such that there exists a polynomial $P(x)$ with integer coefficients satisfying
$$P(\sqrt[3]{n^2} + \sqrt[3]{ n}) = 2016n + 20\sqrt[3]{n^2} + 16\sqrt[3]{n}$$
2021 AMC 12/AHSME Spring, 14
What is the value of $$\left(\sum_{k=1}^{20} \log_{5^k} 3^{k^2}\right)\cdot\left(\sum_{k=1}^{100} \log_{9^k} 25^k\right)?$$
$\textbf{(A) }21 \qquad \textbf{(B) }100\log_5 3 \qquad \textbf{(C) }200\log_3 5 \qquad \textbf{(D) }2,200\qquad \textbf{(E) }21,000$
2018 APMO, 5
Find all polynomials $P(x)$ with integer coefficients such that for all real numbers $s$ and $t$, if $P(s)$ and $P(t)$ are both integers, then $P(st)$ is also an integer.
1982 All Soviet Union Mathematical Olympiad, 348
The $KLMN$ tetrahedron (triangle pyramid) vertices are situated inside or on the faces or on the edges of the $ABCD$ tetrahedron. Prove that perimeter of $KLMN$ is less than $4/3$ perimeter of $ABCD$.
2012 Centers of Excellency of Suceava, 1
Let be a natural number $ n $ and a $ n\times n $ nilpotent real matrix $ A. $
Prove that $ 0=\det\left( A+\text{adj} A \right) . $
[i]Neculai Moraru[/i]
2023 HMNT, 10
Compute the number of ways a non-self-intersecting concave quadrilateral can be drawn in the plane such that two of its vertices are $(0, 0)$ and $(1, 0)$, and the other two vertices are two distinct lattice points $(a, b)$, $(c, d)$ with $0 \le a$, $c \le 59$ and $1 \le b$, $d \le 5.$
(A concave quadrilateral is a quadrilateral with an angle strictly larger than $180^o$. A lattice point is a point with both coordinates integers.)