Found problems: 85335
1993 Bulgaria National Olympiad, 4
Find all natural numbers $n > 1$ for which there exists such natural numbers $a_1,a_2,...,a_n$ for which the numbers
$\{a_i +a_j | 1 \le i \le j \le n \}$ form a full system modulo $\frac{n(n+1)}{2}$.
2023 VIASM Summer Challenge, Problem 4
Let $ABCD$ be a parallelogram and $P$ be an arbitrary point in the plane. Let $O$ be the intersection of two diagonals $AC$ and $BD.$ The circumcircles of triangles $POB$ and $POC$ intersect the circumcircles of triangle $OAD$ at $Q$ and $R,$ respectively $(Q,R \ne O).$ Construct the parallelograms $PQAM$ and $PRDN.$
Prove that: the circumcircle of triangle $MNP$ passes through $O.$
[i]Proposed by Tran Quang Hung ([url=https://artofproblemsolving.com/community/user/68918]buratinogigle[/url])[/i]
1967 IMO Shortlist, 3
Suppose $\tan \alpha = \dfrac{p}{q}$, where $p$ and $q$ are integers and $q \neq 0$. Prove that the number $\tan \beta$ for which $\tan {2 \beta} = \tan {3 \alpha}$ is rational only when $p^2 + q^2$ is the square of an integer.
MOAA Accuracy Rounds, 2023.3
Ms. Raina's math class has 6 students, including the troublemakers Andy and Harry. For a group project, Ms. Raina randomly divides the students into three groups containing 1, 2, and 3 people. The probability that Andy and Harry unfortunately end up in the same group can be expressed in the form $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Find $m+n$.
[i]Proposed by Andy Xu[/i]
2004 Switzerland - Final Round, 1
Let $\Gamma$ be a circle and $P$ a point outside of $\Gamma$ . A tangent from $P$ to the circle intersects it in $A$. Another line through $P$ intersects $\Gamma$ at the points $B$ and $C$. The bisector of $\angle APB$ intersects $AB$ at $D$ and $AC$ at $E$. Prove that the triangle $ADE$ is isosceles.
1953 Putnam, A4
From the identity
$$ \int_{0}^{\pi \slash 2} \log \sin 2x \, dx = \int_{0}^{\pi \slash 2} \log \sin x \, dx + \int_{0}^{\pi \slash 2} \log \cos x \, dx +\int_{0}^{\pi \slash 2} \log 2 \, dx, $$
deduce the value of $\int_{0}^{\pi \slash 2} \log \sin x \, dx.$
2022 USAJMO, 4
Let $ABCD$ be a rhombus, and let $K$ and $L$ be points such that $K$ lies inside the rhombus, $L$ lies outside the rhombus, and $KA = KB = LC = LD$. Prove that there exist points $X$ and $Y$ on lines $AC$ and $BD$ such that $KXLY$ is also a rhombus.
[i]Proposed by Ankan Bhattacharya[/i]
1968 Bulgaria National Olympiad, Problem 5
The point $M$ is inside the tetrahedron $ABCD$ and the intersection points of the lines $AM,BM,CM$ and $DM$ with the opposite walls are denoted with $A_1,B_1,C_1,D_1$ respectively. It is given also that the ratios $\frac{MA}{MA_1}$, $\frac{MB}{MB_1}$, $\frac{MC}{MC_1}$, and $\frac{MD}{MD_1}$ are equal to the same number $k$. Find all possible values of $k$.
[i]K. Petrov[/i]
1990 IMO Longlists, 85
Let $A_1, A_2, \ldots, A_n (n \geq 4)$ be $n$ convex sets in plane. Knowing that every three convex sets have a common point. Prove that there exists a point belonging to all the sets.
2018 Romania National Olympiad, 3
Let $f:[a,b] \to \mathbb{R}$ be an integrable function and $(a_n) \subset \mathbb{R}$ such that $a_n \to 0.$
$\textbf{a) }$ If $A= \{m \cdot a_n \mid m,n \in \mathbb{N}^* \},$ prove that every open interval of strictly positive real numbers contains elements from $A.$
$\textbf{b) }$ If, for any $n \in \mathbb{N}^*$ and for any $x,y \in [a,b]$ with $|x-y|=a_n,$ the inequality $\left| \int_x^yf(t)dt \right| \leq |x-y|$ is true, prove that $$\left| \int_x^y f(t)dt \right| \leq |x-y|, \: \forall x,y \in [a,b]$$
[i]Nicolae Bourbacut[/i]
2019 IFYM, Sozopol, 2
$\Delta ABC$ is a triangle with center $I$ of its inscribed circle and $B_1$ and $C_1$ are feet of its angle bisectors through $B$ and $C$. Let $S$ be the middle point on the arc $\widehat{BAC}$ of the circumscribed circle of $\Delta ABC$ (denoted with $\Omega$) and let $\omega_a$ be the excircle of $\Delta ABC$ opposite to $A$. Let $\omega_a (I_a)$ be tangent to $AB$ and $AC$ in points $D$ and $E$ respectively and $SI\cap \Omega=\{S,P\}$. Let $M$ be the middle point of $DE$ and $N$ be the middle point of $SI$. If $MN\cap AP=K$, prove that $KI_a\perp B_1 C_1$.
2017-2018 SDPC, 6
Let $ABC$ be an acute triangle with circumcenter $O$. Let the parallel to $BC$ through $A$ intersect line $BO$ at $B_A$ and $CO$ at $C_A$. Lines $B_AC$ and $BC_A$ intersect at $A'$. Define $B'$ and $C'$ similarly.
(a) Prove that the the perpendicular from $A'$ to $BC$, the perpendicular from $B'$ to $AC$, and $C'$ to $AB$ are concurrent.
(b) Prove that likes $AA'$, $BB'$, and $CC'$ are concurrent.
2024 India Iran Friendly Math Competition, 2
Let $ABCD$ be a cyclic quadrilateral with circumcentre $O_1$. The diagonals $AC$ and $BD$ meet at point $P$. Suppose the four incentres of triangles $PAB, PBC, PCD, PDA$ lie on a circle with centre $O_2$. Prove that $P, O_1, O_2$ are collinear.
[i]Proposed by Shantanu Nene[/i]
2016 Indonesia TST, 2
Let $a,b$ be two positive integers, such that $ab\neq 1$. Find all the integer values that $f(a,b)$ can take, where \[ f(a,b) = \frac { a^2+ab+b^2} { ab- 1} . \]
Kyiv City MO Juniors 2003+ geometry, 2003.9.4
The diagonals of a convex quadrilateral divide it into four triangles. The radii of the circles circumscribed around these triangles are equal. Can such a property have a quadrilateral other than:
a) parallelogram,
b) rhombus?
(Sharygin Igor)
2006 South africa National Olympiad, 6
Consider the function $f$ defined by
\[f(n)=\frac{1}{n}\left (\left \lfloor\frac{n}{1}\right \rfloor+\left \lfloor\frac{n}{2}\right \rfloor+\cdots+\left \lfloor\frac{n}{n}\right \rfloor \right )\]
for all positive integers $n$. (Here $\lfloor x\rfloor$ denotes the greatest integer less than or equal to $x$.) Prove that
(a) $f(n+1)>f(n)$ for infinitely many $n$.
(b) $f(n+1)<f(n)$ for infinitely many $n$.
2020 Australian Maths Olympiad, 3
Let $ABC$ be a triangle with $\angle ACB=90^{\circ}$. Suppose that the tangent line at $C$ to the circle passing through $A,B,C$ intersects the line $AB$ at $D$. Let $E$ be the midpoint of $CD$ and let $F$ be a point on $EB$ such that $AF$ is parallel to $CD$.
Prove that the lines $AB$ and $CF$ are perpendicular.
2023 4th Memorial "Aleksandar Blazhevski-Cane", P4
Let $ABCD$ be a cyclic quadrilateral such that $AB = AD + BC$ and $CD < AB$. The diagonals $AC$ and $BD$ intersect at $P$, while the lines $AD$ and $BC$ intersect at $Q$. The angle bisector of $\angle APB$ meets $AB$ at $T$. Show that the circumcenter of the triangle $CTD$ lies on the circumcircle of the triangle $CQD$.
[i]Proposed by Nikola Velov[/i]
2000 Junior Balkan Team Selection Tests - Romania, 4
Let be a triangle $ ABC, $ and three points $ A',B',C' $ on the segments $ BC,CA, $ respectively, $ AB, $ such that the lines $ AA',BB',CC' $ are concurent at $ M. $ Name $ a,b,c,x,y,z $ the areas of the triangles $ AB'M,BC'M,CA'M,AC'M,BA'M, $ respectively, $ CB'M. $ Show that:
[b]a)[/b] $ abc=xyz $
[b]b)[/b] $ ab+bc+ca=xy+yz+zx $
[i]Bogdan Enescu[/i] and [i]Marcel Chiriță[/i]
2015 IFYM, Sozopol, 8
A cross with length $p$ (or [i]p-cross[/i] for short) will be called the figure formed by a unit square and 4 rectangles $p-1$ x $1$ on its sides. What’s the least amount of colors one has to use to color the cells of an infinite table, so that each [i]p-cross[/i] on it covers cells, no two of which are in the same color?
2013 Purple Comet Problems, 1
Two years ago Tom was $25\%$ shorter than Mary. Since then Tom has grown $20\%$ taller, and Mary has grown $4$ inches taller. Now Mary is $20\%$ taller than Tom. How many inches tall is Tom now?
Kyiv City MO 1984-93 - geometry, 1988.10.2
Given an arbitrary tetrahedron. Prove that its six edges can be divided into two triplets so that from each triple it was possible to form a triangle.
PEN A Problems, 35
Let $p \ge 5$ be a prime number. Prove that there exists an integer $a$ with $1 \le a \le p-2$ such that neither $a^{p-1} -1$ nor $(a+1)^{p-1} -1$ is divisible by $p^2$.
2021 Cyprus JBMO TST, 2
Let $x,y$ be real numbers with $x \geqslant \sqrt{2021}$ such that
\[ \sqrt[3]{x+\sqrt{2021}}+\sqrt[3]{x-\sqrt{2021}} = \sqrt[3]{y}\]
Determine the set of all possible values of $y/x$.
1969 All Soviet Union Mathematical Olympiad, 118
Given positive numbers $a,b,c,d$. Prove that the set of inequalities
$$a+b<c+d$$
$$(a+b)(c+d)<ab+cd$$
$$(a+b)cd<ab(c+d)$$
contain at least one wrong.