Found problems: 85335
2015 Hanoi Open Mathematics Competitions, 11
Given a convex quadrilateral $ABCD$. Let $O$ be the intersection point of diagonals $AC$ and $BD$ and let $I , K , H$ be feet of perpendiculars from $B , O , C$ to $AD$, respectively. Prove that $AD \times BI \times CH \le AC \times BD \times OK$.
2017 Iran MO (3rd round), 2
Let $ABCD$ be a trapezoid ($AB<CD,AB\parallel CD$) and $P\equiv AD\cap BC$. Suppose that $Q$ be a point inside $ABCD$ such that $\angle QAB=\angle QDC=90-\angle BQC$. Prove that $\angle PQA=2\angle QCD$.
2004 Belarusian National Olympiad, 5
Suppose that $A$ and $B$ are sets of real numbers such that
$$A\subset B+\alpha \mathbb{Z}\quad \text{and}\quad B\subset A+\alpha\mathbb{Z}\quad \text{for all}\quad \alpha>0$$
(where $X+\alpha\mathbb=\{x+\alpha n|x\in\mathbb{X}, n\in\mathbb{Z}\}$
(a) Does it follow that $A=B$
(b) The same question, with the assumption that $B$ is bounded
2009 Hanoi Open Mathematics Competitions, 1
Let $a,b, c$ be $3$ distinct numbers from $\{1, 2,3, 4, 5, 6\}$
Show that $7$ divides $abc + (7 - a)(7 - b)(7 - c)$
2010 Malaysia National Olympiad, 1
In the diagram, congruent rectangles $ABCD$ and $DEFG$ have a common vertex $D$. Sides $BC$ and $EF$ meet at $H$. Given that $DA = DE = 8$, $AB = EF = 12$, and $BH = 7$. Find the area of $ABHED$.
[img]https://cdn.artofproblemsolving.com/attachments/f/b/7225fa89097e7b20ea246b3aa920d2464080a5.png[/img]
VMEO II 2005, 6
For a given cyclic quadrilateral $ABCD$, let $I$ be a variable point on the diagonal $AC$ such that $I$ and $A$ are on the same side of the diagonal $BD$. Assume $E,F$ lie on the diagonal $BD$ such that $IE\parallel AB$ and $IF\parallel AD$. Show that $\angle BIE =\angle DCF $
2023 Romanian Master of Mathematics, 3
Let $n\geq 2$ be an integer and let $f$ be a $4n$-variable polynomial with real coefficients. Assume that, for any $2n$ points $(x_1,y_1),\dots,(x_{2n},y_{2n})$ in the Cartesian plane, $f(x_1,y_1,\dots,x_{2n},y_{2n})=0$ if and only if the points form the vertices of a regular $2n$-gon in some order, or are all equal.
Determine the smallest possible degree of $f$.
(Note, for example, that the degree of the polynomial $$g(x,y)=4x^3y^4+yx+x-2$$ is $7$ because $7=3+4$.)
[i]Ankan Bhattacharya[/i]
2007 Today's Calculation Of Integral, 223
Evaluate $ \int_{0}^{\pi}\sqrt{(\cos x\plus{}\cos 2x\plus{}\cos 3x)^{2}\plus{}(\sin x\plus{}\sin 2x\plus{}\sin 3x)^{2}}\ dx$.
1987 IMO Shortlist, 20
Let $n\ge2$ be an integer. Prove that if $k^2+k+n$ is prime for all integers $k$ such that $0\le k\le\sqrt{n\over3}$, then $k^2+k+n$ is prime for all integers $k$ such that $0\le k\le n-2$.[i](IMO Problem 6)[/i]
[b][i]Original Formulation[/i][/b]
Let $f(x) = x^2 + x + p$, $p \in \mathbb N.$ Prove that if the numbers $f(0), f(1), \cdots , f( \sqrt{p\over 3} )$ are primes, then all the numbers $f(0), f(1), \cdots , f(p - 2)$ are primes.
[i]Proposed by Soviet Union. [/i]
2023 Pan-American Girls’ Mathematical Olympiad, 4
In an acute-angled triangle $ABC$, let $D$ be a point on the segment $BC$. Let $R$ and $S$ be the feet of the perpendiculars from $D$ to $AC$ and $AB$, respectively. The line $DR$ intersects the circumcircle of $BDS$ at $X$, with $X \neq D$. Similarly, the line $DS$ intersects the circumcircle of $CDR$ at $Y$, with $Y \neq D$. Prove that if $XY$ is parallel to $RS$, then $D$ is the midpoint of $BC$.
2008 AMC 10, 22
Three red beads, two white beads, and one blue bead are placed in a line in random order. What is the probability that no two neighboring beads are the same color?
$ \textbf{(A)}\ \frac{1}{12} \qquad
\textbf{(B)}\ \frac{1}{10} \qquad
\textbf{(C)}\ \frac{1}{6} \qquad
\textbf{(D)}\ \frac{1}{3} \qquad
\textbf{(E)}\ \frac{1}{2}$
2000 Baltic Way, 6
Fredek runs a private hotel. He claims that whenever $ n \ge 3$ guests visit the hotel, it is possible to select two guests that have equally many acquaintances among the other guests, and that also have a common acquaintance or a common unknown among the guests. For which values of $ n$ is Fredek right? (Acquaintance is a symmetric relation.)
PEN G Problems, 10
Show that $\frac{1}{\pi} \arccos \left( \frac{1}{\sqrt{2003}} \right)$ is irrational.
2004 All-Russian Olympiad Regional Round, 10.1
The sum of positive numbers $a, b, c$ is equal to $\pi/2$. Prove that
$$\cos a + \cos b + \cos c > \sin a + \sin b + \sin c.$$
1989 AMC 12/AHSME, 27
Let $n$ be a positive integer. If the equation $2x+2y+z=n$ has $28$ solutions in positive integers $x, y$ and $z$, then $n$ must be either
$ \textbf{(A)}\ 14\ \text{or}\ 15 \qquad\textbf{(B)}\ 15\ \text{or}\ 16 \qquad\textbf{(C)}\ 16\ \text{or}\ 17 \qquad\textbf{(D)}\ 17\ \text{or}\ 18 \qquad\textbf{(E)}\ 18\ \text{or}\ 19 $
2024 CAPS Match, 2
For a positive integer $n$, an $n$-configuration is a family of sets $\left\langle A_{i,j}\right\rangle_{1\le i,j\le n}.$ An $n$-configuration is called [i]sweet[/i] if for every pair of indices $(i, j)$ with $1\le i\le n -1$ and $1\le j\le n$ we have $A_{i,j}\subseteq A_{i+1,j}$ and $A_{j,i}\subseteq A_{j,i+1}.$ Let $f(n, k)$ denote the number of sweet $n$-configurations such that $A_{n,n}\subseteq \{1, 2,\ldots , k\}$. Determine which number is larger: $f\left(2024, 2024^2\right)$ or $f\left(2024^2, 2024\right).$
1995 Rioplatense Mathematical Olympiad, Level 3, 6
A convex polygon with $2n$ sides is called [i]rhombic [/i] if its sides are equal and all pairs of opposite sides are parallel.
A rhombic polygon can be partitioned into rhombic quadrilaterals.
For what value of$ n$, a $2n$-sided rhombic polygon splits into $666$ rhombic quadrilaterals?
2010 All-Russian Olympiad Regional Round, 9.1
Three quadratic polynomials $f_1(x) = x^2+2a_1x+b_1$, $f_2(x) = x^2+2a_2x+b_2$,
$f_3(x) = x^2 + 2a_3x + b_3$ are such that $a_1a_2a_3 = b_1b_2b_3 > 1$. Prove that at
least one polynomial has two distinct roots.
2008 ISI B.Stat Entrance Exam, 9
Suppose $S$ is the set of all positive integers. For $a,b \in S$, define
\[a * b=\frac{\text{lcm}[a,b]}{\text{gcd}(a,b)}\]
For example $8*12=6$.
Show that [b]exactly two[/b] of the following three properties are satisfied:
(i) If $a,b \in S$, then $a*b \in S$.
(ii) $(a*b)*c=a*(b*c)$ for all $a,b,c \in S$.
(iii) There exists an element $i \in S$ such that $a *i =a$ for all $a \in S$.
2011 Tournament of Towns, 3
Three pairwise intersecting rays are given. At some point in time not on every ray from its beginning a point begins to move with speed. It is known that these three points form a triangle at any time, and the center of the circumscribed circle of this the triangle also moves uniformly and on a straight line. Is it true, that all these triangles are similar to each other?
1977 AMC 12/AHSME, 29
Find the smallest integer $n$ such that \[(x^2+y^2+z^2)^2\le n(x^4+y^4+z^4)\] for all real numbers $x,y,$ and $z$.
$\textbf{(A) }2\qquad\textbf{(B) }3\qquad\textbf{(C) }4\qquad\textbf{(D) }6\qquad \textbf{(E) }\text{There is no such integer }n.$
2022 Irish Math Olympiad, 1
1. For [i]n[/i] a positive integer, [i]n[/i]! = 1 $\cdot$ 2 $\cdot$ 3 $\dots$ ([i]n[/i] - 1) $\cdot$ [i]n[/i] is the product of the positive integers from 1 to [i]n[/i].
Determine, with proof, all positive integers [i]n[/i] for which [i]n[/i]! + 3 is a power of 3.
VMEO III 2006, 10.4
Find the least real number $\alpha$ such that there is a real number $\beta$ so that for all triples of real numbers $(a, b,c)$ satisfying $2006a + 10b + c = 0$, the equation $ax^2 + bx + c = 0$ always has real root in the interval $[\beta, \beta + \alpha]$.
2015 Latvia Baltic Way TST, 7
Two circle $\Gamma_1$ and $\Gamma_2$ intersect at points $A$ and $B$, point $P$ is not on the line $AB$. Line $AP$ intersects again $\Gamma_1$ and $\Gamma_2$ at points $K$ and $L$ respectively, line $BP$ intersects again $\Gamma_1$ and $\Gamma_2$ at points $M$ and $N$ respectively and all the points mentioned so far are different. The centers of the circles circumscribed around the triangles $KMP$ and $LNP$ are $O_1$ and $O_2$ respectively. Prove that $O_1O_2$ is perpendicular to $AB$.
2013 Brazil Team Selection Test, 1
Let $ABC$ be an acute triangle with altitudes $AD$, $BE$, and $CF$, and let $O$ be the center of its circumcircle. Show that the segments $OA$, $OF$, $OB$, $OD$, $OC$, $OE$ dissect the triangle $ABC$ into three pairs of triangles that have equal areas.