Found problems: 85335
2020 Brazil Team Selection Test, 5
Let $n \geq 3$ be a fixed integer. The number $1$ is written $n$ times on a blackboard. Below the blackboard, there are two buckets that are initially empty. A move consists of erasing two of the numbers $a$ and $b$, replacing them with the numbers $1$ and $a+b$, then adding one stone to the first bucket and $\gcd(a, b)$ stones to the second bucket. After some finite number of moves, there are $s$ stones in the first bucket and $t$ stones in the second bucket, where $s$ and $t$ are positive integers. Find all possible values of the ratio $\frac{t}{s}$.
2019 AMC 12/AHSME, 16
There are lily pads in a row numbered 0 to 11, in that order. There are predators on lily pads 3 and 6, and a morsel of food on lily pad 10. Fiona the frog starts on pad 0, and from any given lily pad, has a $\tfrac{1}{2}$ chance to hop to the next pad, and an equal chance to jump 2 pads. What is the probability that Fiona reaches pad 10 without landing on either pad 3 or pad 6?
$\textbf{(A) } \frac{15}{256} \qquad \textbf{(B) } \frac{1}{16} \qquad \textbf{(C) } \frac{15}{128}\qquad \textbf{(D) } \frac{1}{8} \qquad \textbf{(E) } \frac14$
2014 ELMO Shortlist, 3
Let $t$ and $n$ be fixed integers each at least $2$. Find the largest positive integer $m$ for which there exists a polynomial $P$, of degree $n$ and with rational coefficients, such that the following property holds: exactly one of \[ \frac{P(k)}{t^k} \text{ and } \frac{P(k)}{t^{k+1}} \] is an integer for each $k = 0,1, ..., m$.
[i]Proposed by Michael Kural[/i]
1995 All-Russian Olympiad Regional Round, 9.5
Find all prime numbers $p$ for which number $p^2 + 11$ has exactly six different divisors (counting $1$ and itself).
1974 IMO Longlists, 22
The variables $a,b,c,d,$ traverse, independently from each other, the set of positive real values. What are the values which the expression \[ S= \frac{a}{a+b+d} + \frac{b}{a+b+c} + \frac{c}{b+c+d} + \frac{d}{a+c+d} \] takes?
2007 Romania Team Selection Test, 1
Let $\mathcal{F}$ be the set of all the functions $f : \mathcal{P}(S) \longrightarrow \mathbb{R}$ such that for all $X, Y \subseteq S$, we have $f(X \cap Y) = \min (f(X), f(Y))$, where $S$ is a finite set (and $\mathcal{P}(S)$ is the set of its subsets). Find
\[\max_{f \in \mathcal{F}}| \textrm{Im}(f) |. \]
2011 Morocco TST, 1
Find all pairs $(m,n)$ of nonnegative integers for which \[m^2 + 2 \cdot 3^n = m\left(2^{n+1} - 1\right).\]
[i]Proposed by Angelo Di Pasquale, Australia[/i]
1985 Vietnam National Olympiad, 3
A parallelepiped with the side lengths $ a$, $ b$, $ c$ is cut by a plane through its intersection of diagonals which is perpendicular to one of these diagonals. Calculate the area of the intersection of the plane and the parallelepiped.
2001 Manhattan Mathematical Olympiad, 4
How many digits has the number $2^{100}$?
1997 German National Olympiad, 5
We are given $n$ discs in a plane, possibly overlapping, whose union has the area $1$. Prove that we can choose some of them which are mutually disjoint and have the total area greater than $1/9$.
1987 AIME Problems, 9
Triangle $ABC$ has right angle at $B$, and contains a point $P$ for which $PA = 10$, $PB = 6$, and $\angle APB = \angle BPC = \angle CPA$. Find $PC$.
[asy]
pair A=(0,5), B=origin, C=(12,0), D=rotate(-60)*C, F=rotate(60)*A, P=intersectionpoint(A--D, C--F);
draw(A--P--B--A--C--B^^C--P);
dot(A^^B^^C^^P);
pair point=P;
label("$A$", A, dir(point--A));
label("$B$", B, dir(point--B));
label("$C$", C, dir(point--C));
label("$P$", P, NE);[/asy]
2015 India Regional MathematicaI Olympiad, 3
Let $P(x)$ be a polynomial whose coefficients are positive integers. If $P(n)$ divides $P(P(n)-2015)$ for every natural number $n$, prove that $P(-2015)=0$.
[hide]One additional condition must be given that $P$ is non-constant, which even though is understood.[/hide]
2014 China Team Selection Test, 1
Let the circumcenter of triangle $ABC$ be $O$. $H_A$ is the projection of $A$ onto $BC$. The extension of $AO$ intersects the circumcircle of $BOC$ at $A'$. The projections of $A'$ onto $AB, AC$ are $D,E$, and $O_A$ is the circumcentre of triangle $DH_AE$. Define $H_B, O_B, H_C, O_C$ similarly.
Prove: $H_AO_A, H_BO_B, H_CO_C$ are concurrent
1989 Chile National Olympiad, 6
The function $f$, with domain on the set of non-negative integers, is defined by the following :
$\bullet$ $f (0) = 2$
$\bullet$ $(f (n + 1) -1)^2 + (f (n)-1) ^2 = 2f (n) f (n + 1) + 4$, taking $f (n)$ the largest possible value.
Determine $f (n)$.
2004 AMC 8, 14
What is the area enclosed by the geoboard quadrilateral below?
[asy]
int i,j;
for(i=0; i<11; i=i+1) {
for(j=0; j<11; j=j+1) {
dot((i,j));
}
}
draw((0,5)--(4,0)--(10,10)--(3,4)--cycle, linewidth(0.7));
[/asy]
$\textbf{(A)} 15\qquad
\textbf{(B)} 18\tfrac12\qquad
\textbf{(C)} 22\tfrac12\qquad
\textbf{(D)} 27\qquad
\textbf{(E)} 41\qquad$
1962 All Russian Mathematical Olympiad, 025
Given $a_0, a_1, ... , a_n$. It is known that $$a_0=a_n=0, a_{k-1}-2a_k+a_{k+1}\ge 0$$ for all $k = 1, 2, ... , k-1$.Prove that all the numbers are nonnegative.
2019 Moroccan TST, 4
Let $p$ be a prime number. Find all the positive integers $n$ such that $p+n$ divides $pn$
1991 Arnold's Trivium, 51
Calculate the integral
\[\int_{-\infty}^{+\infty}e^{ikx}\frac{1-e^x}{1+e^x}dx\]
2017 OMMock - Mexico National Olympiad Mock Exam, 5
Let $k$ be a positive real number. Determine all functions $f:[-k, k]\rightarrow[0, k]$ satisfying the equation
$$f(x)^2+f(y)^2-2xy=k^2+f(x+y)^2$$
for any $x, y\in[-k, k]$ such that $x+y\in[-k, k]$.
[i]Proposed by Maximiliano Sánchez[/i]
2007 IMO Shortlist, 7
Let $ \alpha < \frac {3 \minus{} \sqrt {5}}{2}$ be a positive real number. Prove that there exist positive integers $ n$ and $ p > \alpha \cdot 2^n$ for which one can select $ 2 \cdot p$ pairwise distinct subsets $ S_1, \ldots, S_p, T_1, \ldots, T_p$ of the set $ \{1,2, \ldots, n\}$ such that $ S_i \cap T_j \neq \emptyset$ for all $ 1 \leq i,j \leq p$
[i]Author: Gerhard Wöginger, Austria[/i]
2018 India PRMO, 4
The equation $166\times 56 = 8590$ is valid in some base $b \ge 10$ (that is, $1, 6, 5, 8, 9, 0$ are digits in base $b$ in the above equation). Find the sum of all possible values of $b \ge 10$ satisfying the equation.
1998 Estonia National Olympiad, 1
Let $d_1$ and $d_2$ be divisors of a positive integer $n$. Suppose that the greatest common divisor of $d_1$ and $n/d_2$ and the greatest common divisor of $d_2$ and $n/d_1$ are equal. Show that $d_1 = d_2$.
2021 Israel TST, 1
Let $ABCDEFGHIJ$ be a regular $10$-gon. Let $T$ be a point inside the $10$-gon, such that the $DTE$ is isosceles: $DT = ET$ , and its angle at the apex is $72^\circ$. Prove that there exists a point $S$ such that $FTS$ and $HIS$ are both isosceles, and for both of them the angle at the apex is $72^\circ$.
2018 PUMaC Live Round, 1.2
Define a function given the following $2$ rules:
$\qquad$ 1) for prime $p$, $f(p)=p+1$.
$\qquad$ 2) for positive integers $a$ and $b$, $f(ab)=f(a)\cdot f(b)$.
For how many positive integers $n\leq 100$ is $f(n)$ divisible by $3$?
1967 IMO Longlists, 24
In a sports meeting a total of $m$ medals were awarded over $n$ days. On the first day one medal and $\frac{1}{7}$ of the remaining medals were awarded. On the second day two medals and $\frac{1}{7}$ of the remaining medals were awarded, and so on. On the last day, the remaining $n$ medals were awarded. How many medals did the meeting last, and what was the total number of medals ?