Found problems: 85335
2020 Jozsef Wildt International Math Competition, W17
Let $(K,+,\cdot)$ be a field with the property $-x=x^{-1},\forall x\in K,x\ne0$. Prove that:
$$(K,+,\cdot)\simeq(\mathbb Z_2,+,\cdot)$$
[i]Proposed by Ovidiu Pop[/i]
2003 JHMMC 8, 9
Compute the product of the integers from $-5$ to $5$, inclusive.
1974 Bundeswettbewerb Mathematik, 2
Seven polygons of area $1$ lie in the interior of a square with side length $2$. Show that there are two of these polygons whose intersection has an area of at least $1\slash 7.$
2020 LMT Fall, A19
Euhan and Minjune are playing a game. They choose a number $N$ so that they can only say integers up to $N$. Euhan starts by saying the $1$, and each player takes turns saying either $n+1$ or $4n$ (if possible), where $n$ is the last number said. The player who says $N$ wins. What is the smallest number larger than $2019$ for which Minjune has a winning strategy?
[i]Proposed by Janabel Xia[/i]
2000 Stanford Mathematics Tournament, 3
A twelve foot tree casts a five foot shadow. How long is Henry's shadow (at the same time of day) if he is five and a half feet tall?
2022 Brazil National Olympiad, 5
Let $n$ be a positive integer number. Define $S(n)$ to be the least positive integer such that $S(n) \equiv n \pmod{2}$, $S(n) \geq n$, and such that there are [b]not[/b] positive integers numbers $k,x_1,x_2,...,x_k$ such that $n=x_1+x_2+...+x_k$ and $S(n)=x_1^2+x_2^2+...+x_k^2$. Prove that there exists a real constant $c>0$ and a positive integer $n_0$ such that, for all $n \geq n_0$, $S(n) \geq cn^{\frac{3}{2}}$.
LMT Theme Rounds, 5
Pixar Prison, for Pixar villains, is shaped like a 600 foot by 1000 foot rectangle with a 300 foot by 500 foot rectangle removed from it, as shown below. The warden separates the prison into three congruent polygonal sections for villains from The Incredibles, Finding Nemo, and Cars. What is the perimeter of each of these sections?
[asy]
draw((0,0)--(0,6)--(10,6)--(10,0)--(8,0)--(8,3)--(3,3)--(3,0)--(0,0));
label("600", (1,3.5));
label("1000", (5.5,6.5));
label("300", (4,1.5));
label("500", (5.5,3.5));
label("300", (1.5,-0.5));
[/asy]
[i]Proposed by Peter Rowley
1991 Vietnam National Olympiad, 3
Prove that:
$ \frac {x^{2}y}{z} \plus{} \frac {y^{2}z}{x} \plus{} \frac {z^{2}x}{y}\geq x^{2} \plus{} y^{2} \plus{} z^{2}$
where $ x;y;z$ are real numbers saisfying $ x \geq y \geq z \geq 0$
1998 German National Olympiad, 2
Two pupils $A$ and $B$ play the following game. They begin with a pile of $1998$ matches and $A$ plays first. A player who is on turn must take a nonzero square number of matches from the pile. The winner is the one who makes the last move. Decide who has the winning strategy and give one such strategy.
1990 IMO Shortlist, 21
Let $ n$ be a composite natural number and $ p$ a proper divisor of $ n.$ Find the binary representation of the smallest natural number $ N$ such that
\[ \frac{(1 \plus{} 2^p \plus{} 2^{n\minus{}p})N \minus{} 1}{2^n}\]
is an integer.
2022 MIG, 17
Jane and Jena sit at non-adjacent chairs of a four-chair circular table. In a turn, one person can move to an adjacent chair without a person. Jane moves in the first turn, and alternates with Jena afterwards. In how many ways can Jena be adjacent to Jane after nine moves?
$\textbf{(A) }16\qquad\textbf{(B) }18\qquad\textbf{(C) }32\qquad\textbf{(D) }162\qquad\textbf{(E) }512$
1999 Hungary-Israel Binational, 3
Find all functions $ f:\mathbb{Q}\to\mathbb{R}$ that satisfy $ f(x\plus{}y)\equal{}f(x)f(y)\minus{}f(xy)\plus{}1$ for every $x,y\in\mathbb{Q}$.
1988 IMO Longlists, 39
[b]i.)[/b] Let $g(x) = x^5 + x^4 + x^3 + x^2 + x + 1.$ What is the remainder when the polynomial $g(x^{12}$ is divided by the polynomial $g(x)$?
[b]ii.)[/b] If $k$ is a positive number and $f$ is a function such that, for every positive number $x, f(x^2 + 1 )^{\sqrt{x}} = k.$ Find the value of
\[ f( \frac{9 +y^2}{y^2})^{\sqrt{ \frac{12}{y} }} \] for every positive number $y.$
[b]iii.)[/b] The function $f$ satisfies the functional equation $f(x) + f(y) = f(x+y) - x \cdot y - 1$ for every pair $x,y$ of real numbers. If $f(1) = 1,$ then find the numbers of integers $n,$ for which $f(n) = n.$
2011 Saudi Arabia IMO TST, 3
Find all functions $f : R \to R$ such that $$2f(x) =f(x+y)+f(x+2y)$$, for all $x \in R$ and for all $y \ge 0$.
2017 Math Prize for Girls Problems, 11
Let $S(N)$ be the number of 1's in the binary representation of an integer $N$, and let $D(N) = S(N + 1) - S(N)$. Compute the sum of $D(N)$ over all $N$ such that $1 \le N \le 2017$ and $D(N) < 0$.
1969 IMO Shortlist, 1
$(BEL 1)$ A parabola $P_1$ with equation $x^2 - 2py = 0$ and parabola $P_2$ with equation $x^2 + 2py = 0, p > 0$, are given. A line $t$ is tangent to $P_2.$ Find the locus of pole $M$ of the line $t$ with respect to $P_1.$
VI Soros Olympiad 1999 - 2000 (Russia), 10.4
Can we say that two triangles are congruent if the radii of the inscribed circles, the radii of the circumscribed circles, and the areas of these triangles are equal?
2004 Estonia Team Selection Test, 5
Find all natural numbers $n$ for which the number of all positive divisors of the number lcm $(1,2,..., n)$ is equal to $2^k$ for some non-negative integer $k$.
2009 China Team Selection Test, 1
Let $ n$ be a composite. Prove that there exists positive integer $ m$ satisfying $ m|n, m\le\sqrt {n},$ and $ d(n)\le d^3(m).$ Where $ d(k)$ denotes the number of positive divisors of positive integer $ k.$
Kvant 2022, M2719
For an odd positive integer $n>1$ define $S_n$ to be the set of the residues of the powers of two, modulo $n{}$. Do there exist distinct $n{}$ and $m{}$ whose corresponding sets $S_n$ and $S_m$ coincide?
[i]Proposed by D. Kuznetsov[/i]
2009 IMO Shortlist, 6
Suppose that $ s_1,s_2,s_3, \ldots$ is a strictly increasing sequence of positive integers such that the sub-sequences \[s_{s_1},\, s_{s_2},\, s_{s_3},\, \ldots\qquad\text{and}\qquad s_{s_1+1},\, s_{s_2+1},\, s_{s_3+1},\, \ldots\] are both arithmetic progressions. Prove that the sequence $ s_1, s_2, s_3, \ldots$ is itself an arithmetic progression.
[i]Proposed by Gabriel Carroll, USA[/i]
2016 Estonia Team Selection Test, 12
The circles $k_1$ and $k_2$ intersect at points $M$ and $N$. The line $\ell$ intersects with the circle $k_1$ at points $A$ and $C$ and with circle $k_2$ at points $B$ and $D$, so that points $A, B, C$ and $D$ are on the line $\ell$ in that order. Let $X$ be a point on line $MN$ such that the point $M$ is between points $X$ and $N$. Lines $AX$ and $BM$ intersect at point $P$ and lines $DX$ and $CM$ intersect at point $Q$. Prove that $PQ \parallel \ell $.
1965 AMC 12/AHSME, 11
Consider the statements: I: $ (\sqrt { \minus{} 4})(\sqrt { \minus{} 16}) \equal{} \sqrt {( \minus{} 4)( \minus{} 16)}$, II: $ \sqrt {( \minus{} 4)( \minus{} 16)} \equal{} \sqrt {64}$, and $ \sqrt {64} \equal{} 8$. Of these the following are [u]incorrect[/u].
$ \textbf{(A)}\ \text{none} \qquad \textbf{(B)}\ \text{I only} \qquad \textbf{(C)}\ \text{II only} \qquad \textbf{(D)}\ \text{III only} \qquad \textbf{(E)}\ \text{I and III only}$
2017 Czech-Polish-Slovak Match, 2
Let ${\omega}$ be the circumcircle of an acute-angled triangle ${ABC}$. Point ${D}$ lies on the arc ${BC}$ of ${\omega}$ not containing point ${A}$. Point ${E}$ lies in the interior of the triangle ${ABC}$, does not lie on the line ${AD}$, and satis fies ${\angle DBE =\angle ACB}$ and ${\angle DCE = \angle ABC}$. Let ${F}$ be a point on the line ${AD}$ such that lines ${EF}$ and ${BC}$ are parallel, and let ${G}$ be a point on ${\omega}$ different from ${A}$ such that ${AF = FG}$. Prove that points ${D,E, F,G}$ lie on one circle.
(Slovakia)
2008 Baltic Way, 1
Determine all polynomials $p(x)$ with real coefficients such that $p((x+1)^3)=(p(x)+1)^3$ and $p(0)=0$.