Found problems: 85335
2021 Indonesia MO, 5
Let $P(x) = x^2 + rx + s$ be a polynomial with real coefficients. Suppose $P(x)$ has two distinct real roots, both of which are less than $-1$ and the difference between the two is less than $2$. Prove that $P(P(x)) > 0$ for all real $x$.
1996 IMO Shortlist, 7
Let $ABC$ be an acute triangle with circumcenter $O$ and circumradius $R$. $AO$ meets the circumcircle of $BOC$ at $A'$, $BO$ meets the circumcircle of $COA$ at $B'$ and $CO$ meets the circumcircle of $AOB$ at $C'$. Prove that \[OA'\cdot OB'\cdot OC'\geq 8R^{3}.\] Sorry if this has been posted before since this is a very classical problem, but I failed to find it with the search-function.
2015 Purple Comet Problems, 17
How many subsets of {1,2,3,4,5,6,7,8,9,10,11,12} have the property that no two of its elements differ by more than 5? For example, count the sets {3}, {2,5,7}, and {5,6,7,8,9} but not the set {1,3,5,7}.
1940 Moscow Mathematical Olympiad, 061
Given two lines on a plane, find the locus of all points with the difference between the distance to one line and the distance to the other equal to the length of a given segment.
2018 India IMO Training Camp, 2
Let $A,B,C$ be three points in that order on a line $\ell$ in the plane, and suppose $AB>BC$. Draw semicircles $\Gamma_1$ and $\Gamma_2$ respectively with $AB$ and $BC$ as diameters, both on the same side of $\ell$. Let the common tangent to $\Gamma_1$ and $\Gamma_2$ touch them respectively at $P$ and $Q$, $P\ne Q$. Let $D$ and $E$ be points on the segment $PQ$ such that the semicircle $\Gamma_3$ with $DE$ as diameter touches $\Gamma_2$ in $S$ and $\Gamma_1$ in $T$.
[list=1][*]Prove that $A,C,S,T$ are concyclic.
[*]Prove that $A,C,D,E$ are concyclic.[/list]
Durer Math Competition CD 1st Round - geometry, 2012.C5
In a triangle, the line between the center of the inscribed circle and the center of gravity is parallel to one of the sides. Prove that the sidelengths form an arithmetic sequence.
1985 Tournament Of Towns, (086) 2
The integer part $I (A)$ of a number $A$ is the greatest integer which is not greater than $A$ , while the fractional part $F(A)$ is defined as $A - I(A)$ .
(a) Give an example of a positive number $A$ such that $F(A) + F( 1/A) = 1$ .
(b) Can such an $A$ be a rational number?
(I. Varge, Romania)
1984 Swedish Mathematical Competition, 3
Prove that if $a,b$ are positive numbers, then
$$\left( \frac{a+1}{b+1}\right)^{b+1} \ge \left( \frac{a}{b}\right)^{b}$$
2013 IMO Shortlist, G4
Let $ABC$ be a triangle with $\angle B > \angle C$. Let $P$ and $Q$ be two different points on line $AC$ such that $\angle PBA = \angle QBA = \angle ACB $ and $A$ is located between $P$ and $C$. Suppose that there exists an interior point $D$ of segment $BQ$ for which $PD=PB$. Let the ray $AD$ intersect the circle $ABC$ at $R \neq A$. Prove that $QB = QR$.
2021 Korea Winter Program Practice Test, 2
Find all functions $f:R^+\rightarrow R^+$ such that for all positive reals $x$ and $y$
$$4f(x+yf(x))=f(x)f(2y)$$
1980 AMC 12/AHSME, 11
If the sum of the first 10 terms and the sum of the first 100 terms of a given arithmetic progression are 100 and 10, respectively, then the sum of first 110 terms is:
$\text{(A)} \ 90 \qquad \text{(B)} \ -90 \qquad \text{(C)} \ 110 \qquad \text{(D)} \ -110 \qquad \text{(E)} \ -100$
JOM 2014, 3.
There is a complete graph $G$ with $4027$ vertices drawn on the whiteboard. Ivan paints all the edges by red or blue colour. Find all ordered pairs $(r, b)$ such that Ivan can paint the edges so that every vertex is connected to exactly $r$ red edges and $b$ blue edges.
2003 May Olympiad, 1
Four digits $a, b, c, d$, different from each other and different from zero, are chosen and the list of all the four-digit numbers that are obtained by exchanging the digits $a, b, c, d$ is written. What digits must be chosen so that the list has the greatest possible number of four-digit numbers that are multiples of $36$?
1994 Vietnam Team Selection Test, 1
Given a parallelogram $ABCD$. Let $E$ be a point on the side $BC$ and $F$ be a point on the side $CD$ such that the triangles $ABE$ and $BCF$ have the same area. The diaogonal $BD$ intersects $AE$ at $M$ and intersects $AF$ at $N$. Prove that:
[b]I. [/b] There exists a triangle, three sides of which are equal to $BM, MN, ND$.
[b]II.[/b] When $E, F$ vary such that the length of $MN$ decreases, the radius of the circumcircle of the above mentioned triangle also decreases.
1992 India Regional Mathematical Olympiad, 7
Solve the system \begin{eqnarray*} \\ (x+y)(x+y+z) &=& 18 \\ (y+z)(x+y+z) &=& 30 \\ (x+z)(x+y+z) &=& 2A \end{eqnarray*} in terms of the parameter $A$.
1972 Kurschak Competition, 1
A triangle has side lengths $a, b, c$. Prove that
$$a(b -c)^2 + b(c - a)^2 + c(a - b)^2 + 4abc > a^3 + b^3 + c^3$$
2023 HMNT, 1
The formula to convert Celsius to Fahrenheit is $$F^\circ = 1.8 \cdot C^\circ + 32.$$ In Celcius, it is $10^\circ$ warmer in New York right now than in Boston. In Fahrenheit, how much warmer is it in New York than in Boston?
PEN Q Problems, 8
Show that a polynomial of odd degree $2m+1$ over $\mathbb{Z}$, \[f(x)=c_{2m+1}x^{2m+1}+\cdots+c_{1}x+c_{0},\] is irreducible if there exists a prime $p$ such that \[p \not\vert c_{2m+1}, p \vert c_{m+1}, c_{m+2}, \cdots, c_{2m}, p^{2}\vert c_{0}, c_{1}, \cdots, c_{m}, \; \text{and}\; p^{3}\not\vert c_{0}.\]
1989 APMO, 2
Prove that the equation \[ 6(6a^2 + 3b^2 + c^2) = 5n^2 \] has no solutions in integers except $a = b = c = n = 0$.
2016 Bosnia and Herzegovina Team Selection Test, 3
For an infinite sequence $a_1<a_2<a_3<...$ of positive integers we say that it is [i]nice[/i] if for every positive integer $n$ holds $a_{2n}=2a_n$. Prove the following statements:
$a)$ If there is given a [i]nice[/i] sequence and prime number $p>a_1$, there exist some term of the sequence which is divisible by $p$.
$b)$ For every prime number $p>2$, there exist a [i]nice[/i] sequence such that no terms of the sequence are divisible by $p$.
2006 Hanoi Open Mathematics Competitions, 8
In $\vartriangle ABC, PQ // BC$ where $P$ and $Q$ are points on $AB$ and $AC$ respectively. The lines $PC$ and $QB$ intersect at $G$. It is also given $EF//BC$, where $G \in EF, E \in AB$ and $F\in AC$ with $PQ = a$ and $EF = b$. Find value of $BC$.
2015 Singapore MO Open, 5
Let n > 3 be a given integer. Find the largest integer d (in terms of n) such that for
any set S of n integers, there are four distinct (but not necessarily disjoint) nonempty
subsets, the sum of the elements of each of which is divisible by d.
2025 China Team Selection Test, 7
Let $k$, $a$, and $b$, be fixed integers such that $0 \le a < k$, $0 \le b < k+1$, and $a$, $b$ are not both zero.
The sequence $\{T_n\}_{n \ge k}$ satisfies $T_n = T_{n-1}+T_{n-2} \pmod{n}$, $0 \le T_n < n$, $T_k = a$, and $T_{k+1} = b$. Let the decimal expression of $T_n$ form a sequence $x=\overline{0.T_kT_{k+1} \dots}$. For instance, when $k = 66, a = 5, b = 20$, we get $T_{66}=5$, $T_{67}=20$, $T_{68}=25$, $T_{69}=45$, $T_{70}=0$, $T_{71}=45, \dots$, and thus $x=0.522545045 \dots$.
Prove that $x$ is irrational.
2016 NIMO Problems, 8
For a complex number $z \neq 3$,$4$, let $F(z)$ denote the real part of $\tfrac{1}{(3-z)(4-z)}$. If \[
\int_0^1 F \left( \frac{\cos 2 \pi t + i \sin 2 \pi t}{5} \right) \; dt = \frac mn
\] for relatively prime positive integers $m$ and $n$, find $100m+n$.
[i]Proposed by Evan Chen[/i]
2018 Sharygin Geometry Olympiad, 7
Let $B_1,C_1$ be the midpoints of sides $AC,AB$ of a triangle $ABC$ respectively. The tangents to the circumcircle at $B$ and $C$ meet the rays $CC_1,BB_1$ at points $K$ and $L$ respectively. Prove that $\angle BAK = \angle CAL$.