Found problems: 85335
2013 National Olympiad First Round, 17
Let $ABC$ be an equilateral triangle with side length $10$ and $P$ be a point inside the triangle such that $|PA|^2+ |PB|^2 + |PC|^2 = 128$. What is the area of a triangle with side lengths $|PA|,|PB|,|PC|$?
$
\textbf{(A)}\ 6\sqrt 3
\qquad\textbf{(B)}\ 7 \sqrt 3
\qquad\textbf{(C)}\ 8 \sqrt 3
\qquad\textbf{(D)}\ 9 \sqrt 3
\qquad\textbf{(E)}\ 10 \sqrt 3
$
1991 Arnold's Trivium, 33
Find the linking coefficient of the phase trajectories of the equation of small oscillations $\ddot{x}=-4x$, $\ddot{y}=-9y$ on a level surface of the total energy.
2005 France Team Selection Test, 3
In an international meeting of $n \geq 3$ participants, 14 languages are spoken. We know that:
- Any 3 participants speak a common language.
- No language is spoken more that by the half of the participants.
What is the least value of $n$?
1965 AMC 12/AHSME, 5
When the repeating decimal $ 0.363636\ldots$ is written in simplest fractional form, the sum of the numerator and denominator is:
$ \textbf{(A)}\ 15 \qquad \textbf{(B)}\ 45 \qquad \textbf{(C)}\ 114 \qquad \textbf{(D)}\ 135 \qquad \textbf{(E)}\ 150$
2021 Iran MO (3rd Round), 1
Positive real numbers $a, b, c$ and $d$ are given such that $a+b+c+d = 4$ prove that
$$\frac{ab}{a^2-\frac{4}{3}a+\frac{4}{3}} + \frac{bc}{b^2-\frac{4}{3}b+ \frac{4}{3}} + \frac{cd}{c^2-\frac{4}{3}c+ \frac{4}{3}} + \frac{da}{d^2-\frac{4}{3}d+ \frac{4}{3}}\leq 4.$$
2023 Harvard-MIT Mathematics Tournament, 4
Philena and Nathan are playing a game. First, Nathan secretly chooses an ordered pair $(x, y)$ of positive integers such that $x \leq 20$ and $y \leq 23$. (Philena knows that Nathan’s pair must satisfy $x \leq 20$ and $y \leq 23$.) The game then proceeds in rounds; in every round, Philena chooses an ordered pair $(a, b)$ of positive integers and tells it to Nathan; Nathan says YES if $x \leq a$ and $y \leq b$, and NO otherwise. Find, with proof, the smallest positive integer $N$ for which Philena has a strategy that guarantees she can be certain of Nathan’s pair after at most $N$ rounds.
2022 BMT, 2
The equation
$$4^x -5 \cdot 2^{x+1} +16 = 0$$
has two integer solutions for $x.$ Find their sum.
2022 DIME, 3
An up-right path from lattice points $P$ and $Q$ on the $xy$-plane is a path in which every move is either one unit right or one unit up. The probability that a randomly chosen up-right path from $(0,0)$ to $(10,3)$ does not intersect the graph of $y=x^2+0.5$ can be written as $\tfrac mn$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.
[i]Proposed by [b]HrishiP[/b][/i]
2025 Harvard-MIT Mathematics Tournament, 32
In the coordinate plane, a closed lattice loop of length $2n$ is a sequence of lattice points $P_0, P_1, P_2, \ldots, \ldots, P_{2n}$ such that $P_0$ and $P_{2n}$ are both the origin and $P_{i}P_{i+1}=1$ for each $i.$ A closed lattice loop of length $2026$ is chosen uniformly at random from all such loops. Let $k$ be the maximum integer such that the line $\ell$ with equation $x+y=k$ passes through at least one point of the loop. Compute the expected number of indices $i$ such that $0 \le i \le 2025$ and $P_i$ lies on $\ell.$
(A lattice point is a point with integer coordinates.)
2002 Tournament Of Towns, 1
There are many $a\times b$ rectangular cardboard pieces ($a,b\in\mathbb{N}$ such that $a<b$). It is given that by putting such pieces together without overlapping one can make $49\times 51$ rectangle, and $99\times 101$ rectangle. Can one uniquely determine $a,b$ from this?
2018 Azerbaijan Junior NMO, 5
For a positive integer $n$, define $f(n)=n+P(n)$ and $g(n)=n\cdot S(n)$, where $P(n)$ and $S(n)$ denote the product and sum of the digits of $n$, respectively. Find all solutions to $f(n)=g(n)$
2014 May Olympiad, 4
Let $ABC$ be a right triangle and isosceles, with $\angle C = 90^o$. Let $M$ be the midpoint of $AB$ and $N$ the midpoint of $AC$. Let $ P$ be such that $MNP$ is an equilateral triangle with $ P$ inside the quadrilateral $MBCN$. Calculate the measure of $\angle CAP$
2014 Indonesia MO, 4
Determine all polynomials with integral coefficients $P(x)$ such that if $a,b,c$ are the sides of a right-angled triangle, then $P(a), P(b), P(c)$ are also the sides of a right-angled triangle. (Sides of a triangle are necessarily positive. Note that it's not necessary for the order of sides to be preserved; if $c$ is the hypotenuse of the first triangle, it's not necessary that $P(c)$ is the hypotenuse of the second triangle, and similar with the others.)
2006 Flanders Math Olympiad, 3
Elfs and trolls are seated at a round table, 60 creatures in total. Trolls always lie, and all elfs always speak the truth, except when they make a little mistake.
Everybody claims to sit between an elf and a troll, but exactly two elfs made a mistake! How many trolls are there at this table?
2021 Math Prize for Girls Problems, 17
In the coordinate plane, let $A = (-8, 0)$, $B = (8, 0)$, and $C = (t, 6)$. What is the maximum value of $\sin m\angle CAB \cdot \sin m\angle CBA$, over all real numbers $t$?
1973 Canada National Olympiad, 2
Find all real numbers that satisfy the equation $|x+3|-|x-1|=x+1$. (Note: $|a| = a$ if $a\ge 0$; $|a|=-a$ if $a<0$.)
2022 Purple Comet Problems, 24
Find the number of permutations of the letters $AAABBBCCC$ where no letter appears in a position that originally contained that letter. For example, count the permutations $BBBCCCAAA$ and $CBCAACBBA$ but not the permutation $CABCACBAB$.
2016 Brazil Undergrad MO, 3
Let it \(k \geq 1\) be an integer. Define the sequence \((a_n)_{n \geq 1}\) by \(a_0=0,a_1=1\) and
\[ a_{n+2} = ka_{n+1}+a_n \]
for \(n \geq 0\).
Let it \(p\) an odd prime number.
Denote \(m(p)\) as the smallest positive integer \(m\) such that \(p | a_m\).
Denote \(T(p)\) as the smallest positive integer \(T\) such that for every natural \(j\) we gave \(p | (a_{T+j}-a_j)\).
[list='i']
[*] Show that \(T(p) \leq (p-1) \cdot m(p)\).
[*] Show that if \(T(p) = (p-1) \cdot m(p)\) then
\[ \prod_{1 \leq j \leq T(p)-1}^{j \not \equiv 0 \pmod{m(p)}}{a_j} \equiv (-1)^{m(p)-1} \pmod{p} \]
[/list]
2016 IMC, 1
Let $f : \left[ a, b\right]\rightarrow\mathbb{R}$ be continuous on $\left[ a, b\right]$ and differentiable on $\left( a, b\right)$. Suppose that $f$ has infinitely many zeros, but there is no $x\in \left( a, b\right)$ with $f(x)=f'(x)=0$.
(a) Prove that $f(a)f(b)=0$.
(b) Give an example of such a function on $\left[ 0, 1\right]$.
(Proposed by Alexandr Bolbot, Novosibirsk State University)
2012 ELMO Shortlist, 6
In $\triangle ABC$, $H$ is the orthocenter, and $AD,BE$ are arbitrary cevians. Let $\omega_1, \omega_2$ denote the circles with diameters $AD$ and $BE$, respectively. $HD,HE$ meet $\omega_1,\omega_2$ again at $F,G$. $DE$ meets $\omega_1,\omega_2$ again at $P_1,P_2$ respectively. $FG$ meets $\omega_1,\omega_2$ again $Q_1,Q_2$ respectively. $P_1H,Q_1H$ meet $\omega_1$ at $R_1,S_1$ respectively. $P_2H,Q_2H$ meet $\omega_2$ at $R_2,S_2$ respectively. Let $P_1Q_1\cap P_2Q_2 = X$, and $R_1S_1\cap R_2S_2=Y$. Prove that $X,Y,H$ are collinear.
[i]Ray Li.[/i]
2006 Tuymaada Olympiad, 1
Seven different odd primes are given. Is it possible that for any two of them, the difference of their eight powers to be divisible by all the remaining ones ?
[i]Proposed by F. Petrov, K. Sukhov[/i]
2009 Indonesia TST, 4
Sixteen people for groups of four people such that each two groups have at most two members in common. Prove that there exists a set of six people in which every group is not properly contained in it.
2016 Hong Kong TST, 3
Let $a,b,c$ be positive real numbers satisfying $abc=1$. Determine the smallest possible value of
$$\frac{a^3+8}{a^3(b+c)}+\frac{b^3+8}{b^3(a+c)}+\frac{c^3+8}{c^3(b+a)}$$
1989 IMO Longlists, 88
Prove that the sequence $ (a_n)_{n \geq 0,}, a_n \equal{} [n \cdot \sqrt{2}],$ contains an infinite number of perfect squares.
2024 Auckland Mathematical Olympiad, 6
There are $50$ coins in a row; each coin has a value. Two people are playing a game alternating moves. In one move a player can take either the leftmost or the rightmost coin. Who can always accumulate coins whose total value is at least the value of the coins of the opponent?