Found problems: 85335
2017 Israel Oral Olympiad, 4
What is the shortest possible side length of a four-dimensional hypercube that contains a regular octahedron with side 1?
2014 Contests, 3
Let $a$, $b$ and $c$ be rational numbers for which $a+bc$, $b+ac$ and $a+b$ are all non-zero and for which we have
\[\frac{1}{a+bc}+\frac{1}{b+ac}=\frac{1}{a+b}.\]
Prove that $\sqrt{(c-3)(c+1)}$ is rational.
2011 Abels Math Contest (Norwegian MO), 2b
The diagonals $AD, BE$, and $CF$ of a convex hexagon $ABCDEF$ intersect in a common point.
Show that $a(ABE) a(CDA) a(EFC) = a(BCE) a(DEA) a(FAC)$,
where $a(KLM)$ is the area of the triangle $KLM$.
[img]https://cdn.artofproblemsolving.com/attachments/0/a/bcbbddedde159150fe3c26b1f0a2bfc322aa1a.png[/img]
1984 IMO Longlists, 60
Determine all pairs $(a, b)$ of positive real numbers with $a \neq 1$ such that
\[\log_a b < \log_{a+1} (b + 1).\]
1984 Polish MO Finals, 4
A coin is tossed $n$ times, and the outcome is written in the form ($a_1,a_2,...,a_n$), where $a_i = 1$ or $2$ depending on whether the result of the $i$-th toss is the head or the tail, respectively. Set $b_j = a_1 +a_2 +...+a_j$ for $j = 1,2,...,n$, and let $p(n)$ be the probability that the sequence $b_1,b_2,...,b_n$ contains the number $n$. Express $p(n)$ in terms of $p(n-1)$ and $p(n-2)$.
2015 BMT Spring, Tie 1
Compute the surface area of a rectangular prism with side lengths $2, 3, 4$.
2015 EGMO, 3
Let $n, m$ be integers greater than $1$, and let $a_1, a_2, \dots, a_m$ be positive integers not greater than $n^m$. Prove that there exist positive integers $b_1, b_2, \dots, b_m$ not greater than $n$, such that \[ \gcd(a_1 + b_1, a_2 + b_2, \dots, a_m + b_m) < n, \] where $\gcd(x_1, x_2, \dots, x_m)$ denotes the greatest common divisor of $x_1, x_2, \dots, x_m$.
2017 Dutch BxMO TST, 4
A quadruple $(a; b; c; d)$ of positive integers with $a \leq b \leq c \leq d$ is called good if we can colour each integer red, blue, green or purple, in such a way that
$i$ of each $a$ consecutive integers at least one is coloured red;
$ii$ of each $b$ consecutive integers at least one is coloured blue;
$iii$ of each $c$ consecutive integers at least one is coloured green;
$iiii$ of each $d$ consecutive integers at least one is coloured purple.
Determine all good quadruples with $a = 2.$
2020 USMCA, 29
Let $ABC$ be a triangle with circumcircle $\Gamma$ and let $D$ be the midpoint of minor arc $BC$. Let $E, F$ be on $\Gamma$ such that $DE \bot AC$ and $DF \bot AB$. Lines $BE$ and $DF$ meet at $G$, and lines $CF$ and $DE$ meet at $H$. Given that $AB = 8, AC = 10$, and $\angle BAC = 60^\circ$, find the area of $BCHG$.
[i] Note: this is a modified version of Premier #2 [/i]
1988 Irish Math Olympiad, 8
Let $x_1,x_2,x_3,\dots$ be sequence of nonzero real numbers satisfying $$x_n=\frac{x_{n-2}x_{n-1}}{2x_{n-2}-x_{n-1}}, \quad \quad n=3,4,5,\dots$$ Establish necessary and sufficient conditions on $x_1,x_2$ for $x_n$ to be an integer for infinitely many values of $n$.
1986 Traian Lălescu, 2.3
Discuss $ \lim_{x\to 0}\frac{\lambda +\sin\frac{1}{x} \pm\cos\frac{1}{x}}{x} . $
2002 Tournament Of Towns, 4
Point $P$ is chosen in the plane of triangle $ABC$ such that $\angle{ABP}$ is congruent to $\angle{ACP}$ and $\angle{CBP}$ is congruent to $\angle{CAP}$. Show $P$ is the orthocentre.
2020 Princeton University Math Competition, A1/B2
Joey is playing with a $2$-by-$2$-by-$2$ Rubik’s cube made up of $ 8$ $1$-by-$1$-by-$1$ cubes (with two of these smaller cubes along each of the sides of the bigger cubes). Each face of the Rubik’s cube is distinct color. However, one day, Joey accidentally breaks the cube! He decides to put the cube back together into its solved state, placing each of the pieces one by one. However, due to the nature of the cube, he is only able to put in a cube if it is adjacent to a cube he already placed. If different orderings of the ways he chooses the cubes are considered distinct, determine the number of ways he can reassemble the cube.
2023 OMpD, 2
Find all pairs $(a,b)$ of real numbers such that $\lfloor an + b \rfloor$ is a perfect square, for all positive integer $n$.
2016 Tournament Of Towns, 5
On a blackboard, several polynomials of degree $37$ are written, each of them has the leading coefficient equal to $1$. Initially all coefficients of each polynomial are non-negative. By one move it is allowed to erase any pair of polynomials $f, g$ and replace it by another pair of polynomials $f_1, g_1$ of degree $37$ with the leading coefficients equal to $1$ such that either $f_1+g_1 = f+g$ or $f_1g_1 = fg$. Prove that it is impossible that after some move each polynomial
on the blackboard has $37$ distinct positive roots. [i](8 points)[/i]
[i]Alexandr Kuznetsov[/i]
2020 CCA Math Bonanza, L2.2
A rectangular box with side lengths $1$, $2$, and $16$ is cut into two congruent smaller boxes with integer side lengths. Compute the square of the largest possible length of the space diagonal of one of the smaller boxes.
[i]2020 CCA Math Bonanza Lightning Round #2.2[/i]
2023 Middle European Mathematical Olympiad, 4
Let $c \geq 4$ be an even integer. In some football league, each team has a home uniform and anaway uniform. Every home uniform is coloured in two different colours, and every away uniformis coloured in one colour. A team’s away uniform cannot be coloured in one of the colours fromthe home uniform. There are at most $c$ distinct colours on all of the uniforms. If two teams havethe same two colours on their home uniforms, then they have different colours on their away uniforms. We say a pair of uniforms is clashing if some colour appears on both of them. Suppose that for every team $X$ in the league, there is no team $Y$ in the league such that the home uniform of $X$ is clashing with both uniforms of $Y$. Determine the maximum possible number of teams in the league.
2018 Junior Regional Olympiad - FBH, 4
Determine the last digit of number $18^1+18^2+...+18^{19}+18^{20}$
2015 Iran Geometry Olympiad, 5
a) Do there exist 5 circles in the plane such that every circle passes through centers
of exactly 3 circles?
b) Do there exist 6 circles in the plane such that every circle passes through centers
of exactly 3 circles?
1993 Putnam, A1
Let $O$ be the origin. $y = c$ intersects the curve $y = 2x - 3x^3$ at $P$ and $Q$ in the first quadrant and cuts the y-axis at $R$. Find $c$ so that the region $OPR$ bounded by the y-axis, the line $y = c$ and the curve has the same area as the region between $P$ and $Q$ under the curve and above the line $y = c$.
1975 IMO Shortlist, 10
Determine the polynomials P of two variables so that:
[b]a.)[/b] for any real numbers $t,x,y$ we have $P(tx,ty) = t^n P(x,y)$ where $n$ is a positive integer, the same for all $t,x,y;$
[b]b.)[/b] for any real numbers $a,b,c$ we have $P(a + b,c) + P(b + c,a) + P(c + a,b) = 0;$
[b]c.)[/b] $P(1,0) =1.$
2021 AMC 12/AHSME Fall, 11
Consider two concentric circles of radius $17$ and $19.$ The larger circle has a chord, half of which lies inside the smaller circle. What is the length of the chord in the larger circle?
$\textbf{(A)}\ 12\sqrt{2} \qquad\textbf{(B)}\ 10\sqrt{3} \qquad\textbf{(C)}\ \sqrt{17 \cdot 19} \qquad\textbf{(D)}\
18 \qquad\textbf{(E)}\ 8\sqrt{6}$
2009 Harvard-MIT Mathematics Tournament, 6
Let $x$ and $y$ be positive real numbers and $\theta$ an angle such that $\theta \neq \frac{\pi}{2}n$ for any integer $n$. Suppose
\[\frac{\sin\theta}{x}=\frac{\cos\theta}{y}\]
and
\[
\frac{\cos^4 \theta}{x^4}+\frac{\sin^4\theta}{y^4}=\frac{97\sin2\theta}{x^3y+y^3x}.
\]
Compute $\frac xy+\frac yx.$
2012 ISI Entrance Examination, 7
Let $\Gamma_1,\Gamma_2$ be two circles centred at the points $(a,0),(b,0);0<a<b$ and having radii $a,b$ respectively.Let $\Gamma$ be the circle touching $\Gamma_1$ externally and $\Gamma_2$ internally. Find the locus of the centre of of $\Gamma$
2023 Estonia Team Selection Test, 5
We say that distinct positive integers $n, m$ are $friends$ if $\vert n-m \vert$ is a divisor of both ${}n$ and $m$. Prove that, for any positive integer $k{}$, there exist $k{}$ distinct positive integers such that any two of these integers are friends.