Found problems: 85335
1977 IMO Longlists, 13
Describe all closed bounded figures $\Phi$ in the plane any two points of which are connectable by a semicircle lying in $\Phi$.
KoMaL A Problems 2017/2018, A. 713
We say that a sequence $a_1,a_2,\cdots$ is [i]expansive[/i] if for all positive integers $j,\; i<j$ implies $|a_i-a_j|\ge \tfrac 1j$. Find all positive real numbers $C$ for which one can find an expansive sequence in the interval $[0,C]$.
1957 AMC 12/AHSME, 28
If $ a$ and $ b$ are positive and $ a\not\equal{} 1,\,b\not\equal{} 1$, then the value of $ b^{\log_b{a}}$ is:
$ \textbf{(A)}\ \text{dependent upon }{b} \qquad
\textbf{(B)}\ \text{dependent upon }{a}\qquad
\textbf{(C)}\ \text{dependent upon }{a}\text{ and }{b}\qquad
\textbf{(D)}\ \text{zero}\qquad
\textbf{(E)}\ \text{one}$
2025 239 Open Mathematical Olympiad, 4
Positive numbers $a$, $b$ and $c$ are such that $a^2+b^2+c^2+abc=4$. Prove that \[\sqrt{2-a}+\sqrt{2-b}+\sqrt{2-c}\geqslant 2+\sqrt{(2-a)(2-b)(2-c)}.\]
2023 Israel National Olympiad, P5
Let $ABC$ be an equilateral triangle whose sides have length $1$. The midpoints of $AB,BC$ are $M,N$ respectively. Points $K,L$ were chosen on $AC$ so that $KLMN$ is a rectangle. Inside this rectangle are three semi-circles with the same radius, as in the picture (the endpoints are on the edges of the rectangle, and the arcs are tangent).
Find the minimum possible value of the radii of the semi-circles.
2006 Junior Balkan Team Selection Tests - Romania, 1
Let $ABCD$ be a cyclic quadrilateral of area 8. If there exists a point $O$ in the plane of the quadrilateral such that $OA+OB+OC+OD = 8$, prove that $ABCD$ is an isosceles trapezoid.
2016 ASDAN Math Tournament, 8
There are $n$ integers $a$ such that $0\leq a<91$ and $a$ is a solution to $x^3+8x^2-x+83\equiv0\pmod{91}$. What is $n$?
2013 Romania Team Selection Test, 3
Let $S$ be the set of all rational numbers expressible in the form \[\frac{(a_1^2+a_1-1)(a_2^2+a_2-1)\ldots (a_n^2+a_n-1)}{(b_1^2+b_1-1)(b_2^2+b_2-1)\ldots (b_n^2+b_n-1)}\] for some positive integers $n, a_1, a_2 ,\ldots, a_n, b_1, b_2, \ldots, b_n$. Prove that there is an infinite number of primes in $S$.
2025 AIME, 6
An isosceles trapezoid has an inscribed circle tangent to each of its four sides. The radius of the circle is $3$, and the area of the trapezoid is $72$. Let the parallel sides of the trapezoid have lengths $r$ and $s$, with $r \neq s$. Find $r^2+s^2$
2006 Iran Team Selection Test, 2
Let $n$ be a fixed natural number.
[b]a)[/b] Find all solutions to the following equation :
\[ \sum_{k=1}^n [\frac x{2^k}]=x-1 \]
[b]b)[/b] Find the number of solutions to the following equation ($m$ is a fixed natural) :
\[ \sum_{k=1}^n [\frac x{2^k}]=x-m \]
2024 USEMO, 4
Find all sequences $a_1$, $a_2$, $\dots$ of nonnegative integers such that for all positive integers $n$, the polynomial \[1+x^{a_1}+x^{a_2}+\dots+x^{a_n}\] has at least one integer root. (Here $x^0=1$.)
[i]Kornpholkrit Weraarchakul[/i]
PEN H Problems, 11
Find all $(x,y,n) \in {\mathbb{N}}^3$ such that $\gcd(x, n+1)=1$ and $x^{n}+1=y^{n+1}$.
1977 IMO Longlists, 44
Let $E$ be a finite set of points in space such that $E$ is not contained in a plane and no three points of $E$ are collinear. Show that $E$ contains the vertices of a tetrahedron $T = ABCD$ such that $T \cap E = \{A,B,C,D\}$ (including interior points of $T$ ) and such that the projection of $A$ onto the plane $BCD$ is inside a triangle that is similar to the triangle $BCD$ and whose sides have midpoints $B,C,D.$
1976 AMC 12/AHSME, 26
[asy]
size(150);
dotfactor=4;
draw(circle((0,0),4));
draw(circle((10,-6),3));
pair O,A,P,Q;
O = (0,0);
A = (10,-6);
P = (-.55, -4.12);
Q = (10.7, -2.86);
dot("$O$", O, NE);
dot("$O'$", A, SW);
dot("$P$", P, SW);
dot("$Q$", Q, NE);
draw((2*sqrt(2),2*sqrt(2))--(10 + 3*sqrt(2)/2, -6 + 3*sqrt(2)/2)--cycle);
draw((-1.68*sqrt(2),-2.302*sqrt(2))--(10 - 2.6*sqrt(2)/2, -6 - 3.4*sqrt(2)/2)--cycle);
draw(P--Q--cycle);
//Credit to happiface for the diagram[/asy]
In the adjoining figure, every point of circle $\mathit{O'}$ is exterior to circle $\mathit{O}$. Let $\mathit{P}$ and $\mathit{Q}$ be the points of intersection of an internal common tangent with the two external common tangents. Then the length of $PQ$ is
$\textbf{(A) }\text{the average of the lengths of the internal and external common tangents}\qquad$
$\textbf{(B) }\text{equal to the length of an external common tangent if and only if circles }\mathit{O}\text{ and }\mathit{O'}\text{ have equal radii}\qquad$
$\textbf{(C) }\text{always equal to the length of an external common tangent}\qquad$
$\textbf{(D) }\text{greater than the length of an external common tangent}\qquad$
$\textbf{(E) }\text{the geometric mean of the lengths of the internal and external common tangents}$
2023 pOMA, 4
Let $x_1,x_2,\ldots,x_n$ be positive real numbers such that
\[
x_1+\frac{1}{x_2} = x_2+\frac{1}{x_3} = x_3+\frac{1}{x_4} = \dots = x_{n-1}+\frac{1}{x_n} = x_n+\frac{1}{x_1}.
\]
Prove that $x_1=x_2=x_3=\dots=x_n$.
2021 MMATHS, 7
Let $P_k(x) = (x-k)(x-(k+1))$. Kara picks four distinct polynomials from the set $\{P_1(x), P_2(x), P_3(x), \ldots ,$ $P_{12}(x)\}$ and discovers that when she computes the six sums of pairs of chosen polynomials, exactly two of the sums have two (not necessarily distinct) integer roots! How many possible combinations of four polynomials could Kara have picked?
[i]Proposed by Andrew Wu[/i]
2012 Sharygin Geometry Olympiad, 1
Determine all integer $n$ such that a surface of an $n \times n \times n$ grid cube can be pasted in one layer by paper $1 \times 2$ rectangles so that each rectangle has exactly five neighbors (by a line segment).
(A.Shapovalov)
2021 Sharygin Geometry Olympiad, 19
A point $P$ lies inside a convex quadrilateral $ABCD$. Common internal tangents to the incircles of triangles $PAB$ and $PCD$ meet at point $Q$, and common internal tangents to the incircles of $PBC,PAD$ meet at point $R$. Prove that $P,Q,R$ are collinear.
2001 Estonia National Olympiad, 2
Dividing a three-digit number by the number obtained from it by swapping its first and last digit we get $3$ as the quotient and the sum of digits of the original number as the remainder. Find all three-digit numbers with this property.
2006 Federal Competition For Advanced Students, Part 1, 2
Show that the sequence $ a_n \equal{} \frac {(n \plus{} 1)^nn^{2 \minus{} n}}{7n^2 \plus{} 1}$ is strictly monotonically increasing, where $ n \equal{} 0,1,2, \dots$.
2016 IMAR Test, 3
Fix an integer $n \ge 2$, let $Q_n$ be the graph consisting of all vertices and all edges of an $n$-cube, and let $T$ be a spanning tree in $Q_n$. Show that $Q_n$ has an edge whose adjunction to $T$ produces a simple cycle of length at least $2n$.
2005 IMO Shortlist, 4
Let $n\geq 3$ be a fixed integer. Each side and each diagonal of a regular $n$-gon is labelled with a number from the set $\left\{1;\;2;\;...;\;r\right\}$ in a way such that the following two conditions are fulfilled:
[b]1.[/b] Each number from the set $\left\{1;\;2;\;...;\;r\right\}$ occurs at least once as a label.
[b]2.[/b] In each triangle formed by three vertices of the $n$-gon, two of the sides are labelled with the same number, and this number is greater than the label of the third side.
[b](a)[/b] Find the maximal $r$ for which such a labelling is possible.
[b](b)[/b] [i]Harder version (IMO Shortlist 2005):[/i] For this maximal value of $r$, how many such labellings are there?
[hide="Easier version (5th German TST 2006) - contains answer to the harder version"]
[i]Easier version (5th German TST 2006):[/i] Show that, for this maximal value of $r$, there are exactly $\frac{n!\left(n-1\right)!}{2^{n-1}}$ possible labellings.[/hide]
[i]Proposed by Federico Ardila, Colombia[/i]
2015 ASDAN Math Tournament, 8
Lynnelle and Moor love toy cars, and together, they have $27$ red cars, $27$ purple cars, and $27$ green cars. The number of red cars Lynnelle has individually is the same as the number of green cars Moor has individually. In addition, Lynnelle has $17$ more cars of any color than Moor has of any color. How many purple cars does Lynnelle have?
The Golden Digits 2024, P2
Let $n\in\mathbb{Z}$, $n\geq 2$. Find all functions $f:\mathbb{R}_{>0}\rightarrow\mathbb{R}_{>0}$ such that $$f(x_1+\dots +x_n)^2=\sum_{i=1}^nf(x_i) ^2+ 2\sum_{i<j}f(x_ix_j),$$ for all $x_1,\dots ,x_n\in\mathbb{R}_{>0}$.
[i]Proposed by Andrei Vila[/i]
2015 Cuba MO, 1
Let $f$ be a function of the positive reals in the positive reals, such that
$$f(x) \cdot f(y) - f(xy) = \frac{x}{y} + \frac{y}{x} \ \ for \ \ all \ \ x, y > 0 .$$
(a) Find $f(1)$.
(b) Find $f(x)$.