Found problems: 85335
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?
MBMT Team Rounds, 2020.42
$\vartriangle ABC$ has side lengths $AB = 4$ and $AC = 9$. Angle bisector $AD$ bisects angle $A$ and intersects $BC$ at $D$. Let $k$ be the ratio $\frac{BD}{AB}$ . Given that the length $AD$ is an integer, find the sum of all possible $k^2$
.
2011 Morocco National Olympiad, 4
Let $ABC$ be a triangle with area $1$ and $P$ the middle of the side $[BC]$. $M$ and $N$ are two points of $[AB]-\left \{ A,B \right \} $ and $[AC]-\left \{ A,C \right \}$ respectively such that $AM=2MB$ and$CN=2AN$. The two lines $(AP)$ and $(MN)$ intersect in a point $D$. Find the area of the triangle $ADN$.
1961 AMC 12/AHSME, 36
In triangle $ABC$ the median from $A$ is given perpendicular to the median from $B$. If $BC=7$ and $AC=6$, find the length of $AB$.
${{ \textbf{(A)}\ 4\qquad\textbf{(B)}\ \sqrt{17} \qquad\textbf{(C)}\ 4.25\qquad\textbf{(D)}\ 2\sqrt{5} }\qquad\textbf{(E)}\ 4.5} $
2022 May Olympiad, 5
Vero had an isosceles triangle made of paper. Using scissors, he divided it into three smaller triangles and painted them blue, red and green. Having done so, he observed that:
$\bullet$ with the blue triangle and the red triangle an isosceles triangle can be formed,
$\bullet$ with the blue triangle and the green triangle an isosceles triangle can be formed,
$\bullet$ with the red triangle and the green triangle an isosceles triangle can be formed.
Show what Vero's triangle looked like and how he might have made the cuts to make this situation be possible.
2012 Putnam, 5
Prove that, for any two bounded functions $g_1,g_2 : \mathbb{R}\to[1,\infty),$ there exist functions $h_1,h_2 : \mathbb{R}\to\mathbb{R}$ such that for every $x\in\mathbb{R},$\[\sup_{s\in\mathbb{R}}\left(g_1(s)^xg_2(s)\right)=\max_{t\in\mathbb{R}}\left(xh_1(t)+h_2(t)\right).\]
2022 Romania EGMO TST, P1
Determine all functions $f:\mathbb{R}\to\mathbb{R}$ such that all real numbers $x$ and $y$ satisfy \[f(f(x)+y)=f(x^2-y)+4f(x)y.\]
2008 National Chemistry Olympiad, 1
Which element is a liquid at $25^\circ\text{C}$ and $1.0 \text{ atm}$?
$\textbf{(A)}\hspace{.05in}\text{bromine} \qquad\textbf{(B)}\hspace{.05in}\text{krypton} \qquad\textbf{(C)}\hspace{.05in}\text{phosphorus} \qquad\textbf{(D)}\hspace{.05in}\text{xenon} \qquad$
1993 Turkey MO (2nd round), 6
$n_{1},\ldots ,n_{k}, a$ are integers that satisfies the above conditions A)For every $i\neq j$, $(n_{i}, n_{j})=1$ B)For every $i, a^{n_{i}}\equiv 1 (mod n_{i})$ C)For every $i, X^{a-1}\equiv 0(mod n_{i})$.
Prove that $a^{x}\equiv 1(mod x)$ congruence has at least $2^{k+1}-2$ solutions. ($x>1$)
2016 Moldova Team Selection Test, 9
Let $\alpha \in \left( 0, \dfrac{\pi}{2}\right)$.Find the minimum value of the expression
$$ P = (1+\cos\alpha)\left(1+\frac{1}{\sin \alpha} \right)+(1+\sin \alpha)\left(1+\frac{1}{\cos \alpha} \right) .$$
2008 Purple Comet Problems, 6
Three friends who live in the same apartment building leave the building at the same time to go rock climbing on a cliff just outside of town. Susan walks to the cliff and climbs to the top of the cliff. Fred runs to the cliff twice as fast as Susan walks and climbs the top of the cliff at a rate that is only two-thirds as fast as Susan climbs. Ralph bikes to the cliff at a speed twice as fast as Fred runs and takes two hours longer to climb to the top of the cliff than Susan does. If all three friends reach the top of the cliff at the same time, how many minutes after they left home is that?
2018 Purple Comet Problems, 22
Positive integers $a$ and $b$ satisfy $a^3 + 32b + 2c = 2018$ and $b^3 + 32a + 2c = 1115$. Find $a^2 + b^2 + c^2$.
2015 Tuymaada Olympiad, 5
There is some natural number $n>1$ on the board. Operation is adding to number on the board it maximal non-trivial divisor. Prove, that after some some operations we get number, that is divisible by $3^{2000}$
[i]A. Golovanov[/i]
2013 Sharygin Geometry Olympiad, 6
Dear Mathlinkers,
1. A, B the end points of an arch circle
2. (O) a circle tangent to AB intersecting the arch in question
3. T the point of contact of (O) and AB
4. C, D the points of intersection of (O) with the arch in the order A, D, C, B
5. E, F the points of intersection of AC and DT, BD and CT.
Prove : EF is parallel to AB.
Sincerely
Jean-Louis