Found problems: 85335
2004 Romania National Olympiad, 4
Let $\mathcal K$ be a field of characteristic $p$, $p \equiv 1 \left( \bmod 4 \right)$.
(a) Prove that $-1$ is the square of an element from $\mathcal K.$
(b) Prove that any element $\neq 0$ from $\mathcal K$ can be written as the sum of three squares, each $\neq 0$, of elements from $\mathcal K$.
(c) Can $0$ be written in the same way?
[i]Marian Andronache[/i]
1996 Iran MO (3rd Round), 5
Let $O$ be the circumcenter and $H$ the orthocenter of an acute-angled triangle $ABC$ such that $BC>CA$. Let $F$ be the foot of the altitude $CH$ of triangle $ABC$. The perpendicular to the line $OF$ at the point $F$ intersects the line $AC$ at $P$. Prove that $\measuredangle FHP=\measuredangle BAC$.
2012 NIMO Problems, 2
A permutation $(a_1, a_2, a_3, \dots, a_{100})$ of $(1, 2, 3, \dots, 100)$ is chosen at random. Denote by $p$ the probability that $a_{2i} > a_{2i - 1}$ for all $i \in \{1, 2, 3, \dots, 50\}$. Compute the number of ordered pairs of positive integers $(a, b)$ satisfying $\textstyle\frac{1}{a^b} = p$.
[i]Proposed by Aaron Lin[/i]
2014 AMC 8, 12
A magazine printed photos of three celebrities along with three photos of the celebrities as babies. The baby pictures did not identify the celebrities. Readers were asked to match each celebrity with the correct baby pictures. What is the probability that a reader guessing at random will match all three correctly?
$\textbf{(A) }\frac{1}{9}\qquad\textbf{(B) }\frac{1}{6}\qquad\textbf{(C) }\frac{1}{4}\qquad\textbf{(D) }\frac{1}{3}\qquad \textbf{(E) }\frac{1}{2}$
2018 CMIMC Team, 4-1/4-2
Define an integer $n \ge 0$ to be \textit{two-far} if there exist integers $a$ and $b$ such that $a$, $b$, and $n + a + b$ are all powers of two. If $N$ is the number of two-far integers less than 2048, find the remainder when $N$ is divided by 100.
Let $T = TNYWR$. Let $CMU$ be a triangle with $CM=13$, $MU=14$, and $UC=15$. Rectangle $WEAN$ is inscribed in $\triangle CMU$ with points $W$ and $E$ on $\overline{MU}$, point $A$ on $\overline{CU}$, and point $N$ on $\overline{CM}$. If the area of $WEAN$ is $T$, what is its perimeter?
1978 All Soviet Union Mathematical Olympiad, 268
Consider a sequence $$x_n=(1+\sqrt2+\sqrt3)^n$$ Each member can be represented as $$x_n=q_n+r_n\sqrt2+s_n\sqrt3+t_n\sqrt6$$ where $q_n, r_n, s_n, t_n$ are integers. Find the limits of the fractions $r_n/q_n, s_n/q_n, t_n/q_n$.
2009 Purple Comet Problems, 6
Wiles county contains eight townships as shown on the map. If there are four colors available, in how many ways can the the map be colored so that each township is colored with one color and no two townships that share a border are colored with the same color?
[asy]
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(0,0)--(13,0)--(13,11)--(0,11)--cycle,
(5,0)--(13,6)--(13,0)--cycle,
(13,7)--(13,11)--(7,11)--cycle,
(0,0)--(7,0)--(7,11)--(0,11)--cycle,
(0,5)--(0,11)--(11,11)--cycle,
circle((4,7),2.5),
(0,0)--(5,0)--(2,11)--(0,11)--cycle,
(0,5)--(0,11)--(5,11)--cycle,
};
for(int k=0;k<P.length;++k)
{
unfill(P[k]);
draw(P[k]);
}[/asy]
2020 Australian Maths Olympiad, 6
Let $ABCD$ be a square. For a point $P$ inside $ABCD$, a $\emph{windmill}$ centred at $P$ consists of two perpendicular lines $l_1$ and $l_2$ passing through $P$, such that
$\quad\bullet$ $l_1$ intersects the sides $AB$ and $CD$ at $W$ and $Y$, respectively, and
$\quad\bullet$ $l_2$ intersects the sides $BC$ and $DA$ at $X$ and $Z$, respectively.
A windmill is called $\emph{round}$ if the quadrilateral $WXYZ$ is cyclic.
Determine all points $P$ inside $ABCD$ such that every windmill centred at $P$ is round.
1984 Bundeswettbewerb Mathematik, 3
The sequences $a_1, a_2, a_3,...$ and $b_1, b_2, b_3,... $suffices for all positive integers $n$ of the following recursion:
$a_{n+1} = a_n - b_n$ and $b_{n+1} = 2b_n$, if $a_n \ge b_n$,
$a_{n+1} = 2a_n$ and $b_{n+1} = b_n - a_n$, if $a_n < b_n$.
For which pairs $(a_1, b_1)$ of positive real initial terms is there an index $k$ with $a_k = 0$?
2015 Putnam, A5
Let $q$ be an odd positive integer, and let $N_q$ denote the number of integers $a$ such that $0<a<q/4$ and $\gcd(a,q)=1.$ Show that $N_q$ is odd if and only if $q$ is of the form $p^k$ with $k$ a positive integer and $p$ a prime congruent to $5$ or $7$ modulo $8.$
2018 Romania National Olympiad, 4
Let $n \in \mathbb{N}_{\geq 2}.$ For any real numbers $a_1,a_2,...,a_n$ denote $S_0=1$ and for $1 \leq k \leq n$ denote
$$S_k=\sum_{1 \leq i_1 < i_2 < ... <i_k \leq n}a_{i_1}a_{i_2}...a_{i_k}$$
Find the number of $n-$tuples $(a_1,a_2,...a_n)$ such that $$(S_n-S_{n-2}+S_{n-4}-...)^2+(S_{n-1}-S_{n-3}+S_{n-5}-...)^2=2^nS_n.$$
1978 All Soviet Union Mathematical Olympiad, 265
Given a simple number $p>3$. Consider the set $M$ of the pairs $(x,y)$ with the integer coordinates in the plane such that $0 \le x < p, 0 \le y < p$. Prove that it is possible to mark $p$ points of $M$ such that not a triple of marked points will belong to one line and there will be no parallelogram with the vertices in the marked points.
2006 AMC 10, 4
A digital watch displays hours and minutes with $ \text c{AM}$ and $ \text c{PM}$. What is the largest possible sum of the digits in the display?
$ \textbf{(A) } 17\qquad \textbf{(B) } 19\qquad \textbf{(C) } 21\qquad \textbf{(D) } 22\qquad \textbf{(E) } 23$
2001 Junior Balkan Team Selection Tests - Romania, 3
In the interior of a circle centred at $O$ consider the $1200$ points $A_1,A_2,\ldots ,A_{1200}$, where for every $i,j$ with $1\le i\le j\le 1200$, the points $O,A_i$ and $A_j$ are not collinear. Prove that there exist the points $M$ and $N$ on the circle, with $\angle MON=30^{\circ}$, such that in the interior of the angle $\angle MON$ lie exactly $100$ points.
2014 Turkey MO (2nd round), 4
Let $P$ and $Q$ be the midpoints of non-parallel chords $k_1$ and $k_2$ of a circle $\omega$, respectively. Let the tangent lines of $\omega$ passing through the endpoints of $k_1$ intersect at $A$ and the tangent lines passing through the endpoints of $k_2$ intersect at $B$. Let the symmetric point of the orthocenter of triangle $ABP$ with respect to the line $AB$ be $R$ and let the feet of the perpendiculars from $R$ to the lines $AP, BP, AQ, BQ$ be $R_1, R_2, R_3, R_4$, respectively. Prove that
\[ \frac{AR_1}{PR_1} \cdot \frac{PR_2}{BR_2} = \frac{AR_3}{QR_3} \cdot \frac{QR_4}{BR_4} \]
2019 Iran MO (3rd Round), 3
We are given a natural number $d$. Find all open intervals of maximum length $I \subseteq R$ such that for all real numbers $a_0,a_1,...,a_{2d-1}$ inside interval $I$, we have that the polynomial $P(x)=x^{2d}+a_{2d-1}x^{2d-1}+...+a_1x+a_0$ has no real roots.
2004 BAMO, 5
Find (with proof) all monic polynomials $f(x)$ with integer coefficients that satisfy the following two conditions.
1. $f (0) = 2004$.
2. If $x$ is irrational, then $f (x)$ is also irrational.
(Notes: Apolynomial is monic if its highest degree term has coefficient $1$. Thus, $f (x) = x^4-5x^3-4x+7$ is an example of a monic polynomial with integer coefficients.
A number $x$ is rational if it can be written as a fraction of two integers. A number $x$ is irrational if it is a real number which cannot be written as a fraction of two integers. For example, $2/5$ and $-9$ are rational, while $\sqrt2$ and $\pi$ are well known to be irrational.)
1996 Cono Sur Olympiad, 2
Consider a sequence of real numbers defined by:
$a_{n + 1} = a_n + \frac{1}{a_n}$ for $n = 0, 1, 2, ...$
Prove that, for any positive real number $a_0$, is true that $a_{1996}$ is greater than $63$.
1963 AMC 12/AHSME, 21
The expression $x^2-y^2-z^2+2yz+x+y-z$ has:
$\textbf{(A)}\ \text{no linear factor with integer coeficients and integer exponents} \qquad$
$
\textbf{(B)}\ \text{the factor }-x+y+z \qquad$
$
\textbf{(C)}\ \text{the factor }x-y-z+1 \qquad$
$
\textbf{(D)}\ \text{the factor }x+y-z+1 \qquad$
$
\textbf{(E)}\ \text{the factor }x-y+z+1$
2020 Bundeswettbewerb Mathematik, 1
Show that there are infinitely many perfect squares of the form $50^m-50^n$, but no perfect square of the form $2020^m+2020^n$, where $m$ and $n$ are positive integers.
2013 Costa Rica - Final Round, 1
Determine and justify all solutions $(x,y, z)$ of the system of equations:
$x^2 = y + z$
$y^2 = x + z$
$z^2 = x + y$
2010 Contests, 3
In an $m\times n$ rectangular chessboard,there is a stone in the lower leftmost square. Two persons A,B move the stone alternately. In each step one can move the stone upward or rightward any number of squares. The one who moves it into the upper rightmost square wins. Find all $(m,n)$ such that the first person has a winning strategy.
1994 Baltic Way, 12
The inscribed circle of the triangle $A_1A_2A_3$ touches the sides $A_2A_3,A_3A_1,A_1A_2$ at points $S_1,S_2,S_3$, respectively. Let $O_1,O_2,O_3$ be the centres of the inscribed circles of triangles $A_1S_2S_3, A_2S_3S_1,A_3S_1S_2$, respectively. Prove that the straight lines $O_1S_1,O_2S_2,O_3S_3$ intersect at one point.
2016 Korea National Olympiad, 7
Let $N=2^a p_1^{b_1} p_2^{b_2} \ldots p_k^{b_k}$. Prove that there are $(b_1+1)(b_2+1)\ldots(b_k+1)$ number of $n$s which satisfies these two conditions.
$\frac{n(n+1)}{2}\le N$, $N-\frac{n(n+1)}{2}$ is divided by $n$.
2003 South africa National Olympiad, 5
Prove that the sum of the squares of two consecutive positive integers cannot be equal to a sum of the fourth powers of two consecutive positive integers.