Found problems: 85335
2017 CMIMC Algebra, 10
Let $c$ denote the largest possible real number such that there exists a nonconstant polynomial $P$ with \[P(z^2)=P(z-c)P(z+c)\] for all $z$. Compute the sum of all values of $P(\tfrac13)$ over all nonconstant polynomials $P$ satisfying the above constraint for this $c$.
2003 Vietnam National Olympiad, 3
Let $\mathcal{F}$ be the set of all functions $f : (0,\infty)\to (0,\infty)$ such that $f(3x) \geq f( f(2x) )+x$ for all $x$. Find the largest $A$ such that $f(x) \geq A x$ for all $f\in\mathcal{F}$ and all $x$.
1997 Moscow Mathematical Olympiad, 4
Along a circular railroad, $n$ trains circulate in the same direction at equal distances between them. Stations $A, B$ and $C$ on this railroad (denoted as the trains pass them) form an equilateral triangle. Ira enters station $A$ at the same time as Alex enters station $B$ in order to take the nearest train. It is knows that if they enter the stations at the same time as the driver Roma passes a forest, then Ira takes her train earlier than Alex; otherwise Alex takes the train earlier than or simultaneously with Ira. What part of the railroad goes through the forest (between which stations)?
2002 Irish Math Olympiad, 3
Find all triples of positive integers $ (p,q,n)$, with $ p$ and $ q$ primes, satisfying:
$ p(p\plus{}3)\plus{}q(q\plus{}3)\equal{}n(n\plus{}3)$.
2021 CMIMC, 1.5
Suppose $f$ is a degree 42 polynomial such that for all integers $0\le i\le 42$,
$$f(i)+f(43+i)+f(2\cdot43+i)+\cdots+f(46\cdot43+i)=(-2)^i$$
Find $f(2021)-f(0)$.
[i]Proposed by Adam Bertelli[/i]
1953 AMC 12/AHSME, 28
In triangle $ ABC$, sides $ a,b$ and $ c$ are opposite angles $ A,B$ and $ C$ respectively. $ AD$ bisects angle $ A$ and meets $ BC$ at $ D$. Then if $ x \equal{} \overline{CD}$ and $ y \equal{} \overline{BD}$ the correct proportion is:
$ \textbf{(A)}\ \frac {x}{a} \equal{} \frac {a}{b \plus{} c} \qquad\textbf{(B)}\ \frac {x}{b} \equal{} \frac {a}{a \plus{} c} \qquad\textbf{(C)}\ \frac {y}{c} \equal{} \frac {c}{b \plus{} c} \\
\textbf{(D)}\ \frac {y}{c} \equal{} \frac {a}{b \plus{} c} \qquad\textbf{(E)}\ \frac {x}{y} \equal{} \frac {c}{b}$
2010 Saudi Arabia IMO TST, 2
Find all functions $f,g : N \to N$ such that for all $m ,n \in N$ the following relation holds: $$f(m ) - f(n) = (m - n)(g(m) + g(n))$$.
Note: $N = \{0,1,2,...\}$
2017 AIME Problems, 13
For each integer $n\ge 3$, let $f(n)$ be the number of 3-element subsets of the vertices of a regular $n$-gon that are the vertices of an isosceles triangle (including equilateral triangles). Find the sum of all values of $n$ such that $f(n+1)=f(n)+78$.
III Soros Olympiad 1996 - 97 (Russia), 11.6
On the coordinate plane, draw a set of points $M(x,y)$, the coordinates of which satisfy the inequality $$\log_{x+y} (x^2+y^2) \le 1.$$
Croatia MO (HMO) - geometry, 2013.3
Given a pointed triangle $ABC$ with orthocenter $H$. Let $D$ be the point such that the quadrilateral $AHCD$ is parallelogram. Let $p$ be the perpendicular to the direction $AB$ through the midpoint $A_1$ of the side $BC$. Denote the intersection of the lines $p$ and $AB$ with $E$, and the midpoint of the length $A_1E$ with $F$. The point where the parallel to the line $BD$ through point $A$ intersects $p$ denote by $G$. Prove that the quadrilateral $AFA_1C$ is cyclic if and only if the lines $BF$ passes through the midpoint of the length $CG$.
1991 AMC 12/AHSME, 17
A positive integer $N$ is a [i]palindrome[/i] if the integer obtained by reversing the sequence of digits of $N$ is equal to $N$. The year 1991 is the only year in the current century with the following two properties:
(a) It is a palindrome
(b) It factors as a product of a 2-digit prime palindrome and a 3-digit prime palindrome.
How many years in the millennium between 1000 and 2000 (including the year 1991) have properties (a) and (b)?
$ \textbf{(A)}\ 1\qquad\textbf{(B)}\ 2\qquad\textbf{(C)}\ 3\qquad\textbf{(D)}\ 4\qquad\textbf{(E)}\ 5 $
2009 BAMO, 1
A square grid of $16$ dots (see the figure) contains the corners of nine $1\times1$ squares, four $2\times 2$ squares, and one $3\times3$ square, for a total of $14$ squares whose sides are parallel to the sides of the grid. What is the smallest possible number of dots you can remove so that, after removing those dots, each of the $14$ squares is missing at least one corner?
Justify your answer by showing both that the number of dots you claim is sufficient and by explaining why no smaller number of dots will work.
[img]https://cdn.artofproblemsolving.com/attachments/0/9/bf091a769dbec40eceb655f5588f843d4941d6.png[/img]
2012 Today's Calculation Of Integral, 855
Let $f(x)$ be a function which is differentiable twice and $f''(x)>0$ on $[0,\ 1]$.
For a positive integer $n$, find $\lim_{n\to\infty} n\left\{\int_0^1 f(x)\ dx-\frac{1}{n}\sum_{k=0}^{n-1} f\left(\frac{k}{n}\right)\right\}.$
MBMT Guts Rounds, 2015.20
How many lattice points are exactly twice as close to $(0,0)$ as they are to $(15,0)$? (A lattice point is a point $(a,b)$ such that both $a$ and $b$ are integers.)
2007 Germany Team Selection Test, 3
For all positive integers $n$, show that there exists a positive integer $m$ such that $n$ divides $2^{m} + m$.
[i]Proposed by Juhan Aru, Estonia[/i]
2015 Irish Math Olympiad, 10
Prove that, for all pairs of nonnegative integers, $j,n$, $$\sum_{K=0}^{n}k^j\binom n k \ge 2^{n-j} n^j$$
1999 Irish Math Olympiad, 2
Show that there is a positive number in the Fibonacci sequence which is divisible by $ 1000$.
1991 IMO Shortlist, 26
Let $ n \geq 2, n \in \mathbb{N}$ and let $ p, a_1, a_2, \ldots, a_n, b_1, b_2, \ldots, b_n \in \mathbb{R}$ satisfying $ \frac{1}{2} \leq p \leq 1,$ $ 0 \leq a_i,$ $ 0 \leq b_i \leq p,$ $ i \equal{} 1, \ldots, n,$ and \[ \sum^n_{i\equal{}1} a_i \equal{} \sum^n_{i\equal{}1} b_i.\] Prove the inequality: \[ \sum^n_{i\equal{}1} b_i \prod^n_{j \equal{} 1, j \neq i} a_j \leq \frac{p}{(n\minus{}1)^{n\minus{}1}}.\]
2000 AMC 12/AHSME, 15
Let $ f$ be a function for which $ f(x/3) \equal{} x^2 \plus{} x \plus{} 1$. Find the sum of all values of $ z$ for which $ f(3z) \equal{} 7$.
$ \textbf{(A)}\ \minus{} 1/3 \qquad \textbf{(B)}\ \minus{} 1/9 \qquad \textbf{(C)}\ 0 \qquad \textbf{(D)}\ 5/9 \qquad \textbf{(E)}\ 5/3$
2007 Purple Comet Problems, 22
Let $a=3^{1/223}+1$ and for all $n \ge 3$ let \[f(n)= \dbinom{n}{0} a^{n-1} - \dbinom{n}{1} a^{n-2} + \dbinom{n}{2} a^{n-3}- ... +(-1)^{n-1} \dbinom{n}{n-1} a^0.\] Find $f(2007)+f(2008).$
2003 Romania Team Selection Test, 4
Prove that among the elements of the sequence $\left\{ \left\lfloor n\sqrt{2003} \right\rfloor \right\}_{n\geq 1}$ one can find a geometric progression having any number of terms, and having the ratio bigger than $k$, where $k$ can be any positive integer.
[i]Radu Gologan[/i]
2005 Lithuania Team Selection Test, 3
The sequence $a_1, a_2,..., a_{2000}$ of real numbers satisfies the condition
\[a_1^3+a_2^3+...+a_n^3=(a_1+a_2+...+a_n)^2\]
for all $n$, $1\leq n \leq 2000$. Prove that every element of the sequence is an integer.
2017 IFYM, Sozopol, 3
$ABC$ is a triangle with a circumscribed circle $k$, center $I$ of its inscribed circle $\omega$, and center $I_a$ of its excircle $\omega _a$, opposite to $A$. $\omega$ and $\omega _a$ are tangent to $BC$ in points $P$ and $Q$, respectively, and $S$ is the middle point of the arc $\widehat{BC}$ that doesn’t contain $A$. Consider a circle that is tangent to $BC$ in point $P$ and to $k$ in point $R$. Let $RI$ intersect $k$ for a second time in point $L$. Prove that, $LI_a$ and $SQ$ intersect in a point that lies on $k$.
2018 CCA Math Bonanza, I12
For how many integers $n\neq1$ does $\left(n-1\right)^3$ divide $n^{2018\left(n-1\right)}-1$?
[i]2018 CCA Math Bonanza Individual Round #12[/i]
2020 IMC, 1
Let $n$ be a positive integer. Compute the number of words $w$ that satisfy the following three properties.
1. $w$ consists of $n$ letters from the alphabet $\{a,b,c,d\}.$
2. $w$ contains an even number of $a$'s
3. $w$ contains an even number of $b$'s.
For example, for $n=2$ there are $6$ such words: $aa, bb, cc, dd, cd, dc.$