Found problems: 85335
2003 JBMO Shortlist, 1
Is there is a convex quadrilateral which the diagonals divide into four triangles with areas of distinct primes?
1996 AMC 12/AHSME, 15
Two opposite sides of a rectangle are each divided into $n$ congruent segments, and the endpoints of one segment are joined to the center to form triangle $A$. The other sides are each divided into $m$ congruent segments, and the endpoints of one of these segments are joined to the center to form triangle $B$. [See figure for $n = 5, m = 7$.] What is the ratio of the area of triangle $A$ to the area of triangle $B$?
[asy]
int i;
for(i=0; i<8; i=i+1) {
dot((i,0)^^(i,5));
}
for(i=1; i<5; i=i+1) {
dot((0,i)^^(7,i));
}
draw(origin--(7,0)--(7,5)--(0,5)--cycle, linewidth(0.8));
pair P=(3.5, 2.5);
draw((0,4)--P--(0,3)^^(2,0)--P--(3,0));
label("$B$", (2.3,0), NE);
label("$A$", (0,3.7), SE);[/asy]
$\text{(A)} \ 1 \qquad \text{(B)} \ m/n \qquad \text{(C)} \ n/m \qquad \text{(D)} \ 2m/n \qquad \text{(E)} \ 2n/m$
2021 Balkan MO Shortlist, N7
A [i]super-integer[/i] triangle is defined to be a triangle whose lengths of all sides and at least
one height are positive integers. We will deem certain positive integer numbers to be [i]good[/i] with
the condition that if the lengths of two sides of a super-integer triangle are two (not necessarily
different) good numbers, then the length of the remaining side is also a good number. Let $5$ be
a good number. Prove that all integers larger than $2$ are good numbers.
1991 Mexico National Olympiad, 2
A company of $n$ soldiers is such that
(i) $n$ is a palindrome number (read equally in both directions);
(ii) if the soldiers arrange in rows of $3, 4$ or $5$ soldiers, then the last row contains $2, 3$ and $5$ soldiers, respectively.
Find the smallest $n$ satisfying these conditions and prove that there are infinitely many such numbers $n$.
1978 Polish MO Finals, 5
For a given real number $a$, define the sequence $(a_n)$ by $a_1 = a$ and
$$a_{n+1} =\begin{cases}
\dfrac12 \left(a_n -\dfrac{1}{a_n}\right) \,\,\, if \,\,\, a_n \ne 0, \\
0 \,\,\, if \,\,\, a_n = 0 \end{cases}$$
Prove that the sequence $(a_n)$ contains infinitely many nonpositive terms.
2008 Mediterranean Mathematics Olympiad, 4
The sequence of polynomials $(a_n)$ is defined by $a_0=0$, $ a_1=x+2$ and $a_n=a_{n-1}+3a_{n-1}a_{n-2} +a_{n-2}$ for $n>1$.
(a) Show for all positive integers $k,m$: if $k$ divides $m$ then $a_k$ divides $a_m$.
(b) Find all positive integers $n$ such that the sum of the roots of polynomial $a_n$ is an integer.
2011 Postal Coaching, 1
Let $X$ be the set of all positive real numbers. Find all functions $f : X \longrightarrow X$ such that
\[f (x + y) \ge f (x) + yf (f (x))\]
for all $x$ and $y$ in $X$.
2023 Silk Road, 1
Let $ABCD$ be a trapezoid with $AD\parallel BC$. A point $M $ is chosen inside the trapezoid, and a point $N$ is chosen inside the triangle $BMC$ such that $AM\parallel CN$, $BM\parallel DN$. Prove that triangles $ABN$ and $CDM$ have equal areas.
Durer Math Competition CD Finals - geometry, 2020.D2
Let $ABC$ be an acute triangle where $AC > BC$. Let $T$ denote the foot of the altitude from vertex $C$, denote the circumcentre of the triangle by $O$. Show that quadrilaterals $ATOC$ and $BTOC$ have equal area.
1987 IMO Longlists, 50
Let $P,Q,R$ be polynomials with real coefficients, satisfying $P^4+Q^4 = R^2$. Prove that there exist real numbers $p, q, r$ and a polynomial $S$ such that $P = pS, Q = qS$ and $R = rS^2$.
[hide="Variants"]Variants. (1) $P^4 + Q^4 = R^4$; (2) $\gcd(P,Q) = 1$ ; (3) $\pm P^4 + Q^4 = R^2$ or $R^4.$[/hide]
2016 BMT Spring, 3
How many five-card hands from a standard deck of $52$ cards are full houses? A full house consists of $3$ cards of one rank and $2$ cards of another rank.
1998 French Mathematical Olympiad, Problem 4
Let there be given two lines $D_1$ and $D_2$ which intersect at point $O$, and a point $M$ not on any of these lines. Consider two variable points $A\in D_1$ and $b\in D_2$ such that $M$ belongs to the segment $AB$.
(a) Prove that there exists a position of $A$ and $B$ for which the area of triangle $OAB$ is minimal. Construct such points $A$ and $B$.
(b) Prove that there exists a position of $A$ and $B$ for which the area of triangle $OAB$ is minimal. Show that for such $A$ and $B$, the perimeters of $\triangle OAM$ and $\triangle OBM$ are equal, and that $\frac{AM}{\tan\frac12\angle OAM}=\frac{BM}{\tan\frac12\angle OBM}$. Construct such points $A$ and $B$.
2002 AMC 12/AHSME, 20
Suppose that $ a$ and $ b$ are digits, not both nine and not both zero, and the repeating decimal $ 0.\overline{ab}$ is expressed as a fraction in lowest terms. How many different denominators are possible?
$ \textbf{(A)}\ 3 \qquad \textbf{(B)}\ 4 \qquad \textbf{(C)}\ 5 \qquad \textbf{(D)}\ 8 \qquad \textbf{(E)}\ 9$
2009 AMC 12/AHSME, 19
Andrea inscribed a circle inside a regular pentagon, circumscribed a circle around the pentagon, and calculated the area of the region between the two circles. Bethany did the same with a regular heptagon (7 sides). The areas of the two regions were $ A$ and $ B$, respectively. Each polygon had a side length of $ 2$. Which of the following is true?
$ \textbf{(A)}\ A\equal{}\frac{25}{49}B\qquad \textbf{(B)}\ A\equal{}\frac{5}{7}B\qquad \textbf{(C)}\ A\equal{}B\qquad \textbf{(D)}\ A\equal{}\frac{7}{5}B\qquad \textbf{(E)}\ A\equal{}\frac{49}{25}B$
2024 OMpD, 2
Let \( ABCDE \) be a convex pentagon whose vertices lie on a circle \( \Gamma \). The tangents to \( \Gamma \) at \( C \) and \( E \) intersect at \( X \), and the segments \( CE \) and \( AD \) intersect at \( Y \). Given that \( CE \) is perpendicular to \( BD \), that \( XY \) is parallel to \( BD \), that \( AY = BD \), and that \( \angle BAD = 30^\circ \), what is the measure of the angle \( \angle BDA \)?
Proposed by João Gilberti Alves Tavares
2001 Italy TST, 3
Find all pairs $ (p, q)$ of prime numbers such that $ p$ divides $ 5^q \plus{} 1$ and $ q$ divides $ 5^p \plus{} 1$.
2011 All-Russian Olympiad, 4
Do there exist any three relatively prime natural numbers so that the square of each of them is divisible by the sum of the two remaining numbers?
2020 Tournament Of Towns, 4
We say that a nonconstant polynomial $p(x)$ with real coefficients is split into two squares if it is represented as $a(x) +b(x)$ where $a(x)$ and $b(x)$ are squares of polynomials with real coefficients. Is there such a polynomial $p(x)$ that it may be split into two squares:
a) in exactly one way;
b) in exactly two ways?
Note: two splittings that differ only in the order of summands are considered to be the same.
Sergey Markelov
2011 Polish MO Finals, 2
The incircle of triangle $ABC$ is tangent to $BC,CA,AB$ at $D,E,F$ respectively. Consider the triangle formed by the line joining the midpoints of $AE,AF$, the line joining the midpoints of $BF,BD$, and the line joining the midpoints of $CD,CE$. Prove that the circumcenter of this triangle coincides with the circumcenter of triangle $ABC$.
2003 Baltic Way, 4
Let $a,b,c$ be positive real numbers. Prove that
\[ \frac{2a}{a^{2}+bc}+\frac{2b}{b^{2}+ca}+\frac{2c}{c^{2}+ab}\leq\frac{a}{bc}+\frac{b}{ca}+\frac{c}{ab} \]
2004 IMO Shortlist, 1
Let $\tau(n)$ denote the number of positive divisors of the positive integer $n$. Prove that there exist infinitely many positive integers $a$ such that the equation $ \tau(an)=n $ does not have a positive integer solution $n$.
2020 Serbia National Math Olympiad, 1
Find all monic polynomials $P(x)$ such that the polynomial $P(x)^2-1$ is divisible by the polynomial $P(x+1)$.
1998 China Team Selection Test, 3
For a fixed $\theta \in \lbrack 0, \frac{\pi}{2} \rbrack$, find the smallest $a \in \mathbb{R}^{+}$ which satisfies the following conditions:
[b]I. [/b] $\frac{\sqrt a}{\cos \theta} + \frac{\sqrt a}{\sin \theta} >
1$.
[b]II.[/b] There exists $x \in \lbrack 1 - \frac{\sqrt a}{\sin \theta},
\frac{\sqrt a}{\cos \theta} \rbrack$ such that $\lbrack (1 -
x)\sin \theta - \sqrt{a - x^2 \cos^{2} \theta} \rbrack^{2} +
\lbrack x\cos \theta - \sqrt{a - (1 - x)^2 \sin^{2} \theta}
\rbrack^{2} \leq a$.
2017 China Team Selection Test, 4
Given a circle with radius 1 and 2 points C, D given on it. Given a constant l with $0<l\le 2$. Moving chord of the circle AB=l and ABCD is a non-degenerated convex quadrilateral. AC and BD intersects at P. Find the loci of the circumcenters of triangles ABP and BCP.
2022 USAMTS Problems, 5
A lattice point is a point on the coordinate plane with integer coefficients. Prove or disprove : there exists a finite set $S$ of lattice points such that for every line $l$ in the plane with slope $0,1,-1$, or undefined, either $l$ and $S$ intersect at exactly $2022$ points, or they do not intersect.