Found problems: 85335
1992 All Soviet Union Mathematical Olympiad, 569
Circles $C$ and $C'$ intersect at $O$ and $X$. A circle center $O$ meets $C$ at $Q$ and $R$ and meets $C'$ at $P$ and $S$. $PR$ and $QS$ meet at $Y$ distinct from $X$. Show that $\angle YXO = 90^o$.
2018 Canadian Open Math Challenge, C3
Source: 2018 Canadian Open Math Challenge Part C Problem 3
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Consider a convex quadrilateral $ABCD$. Let rays $BA$ and $CD$ intersect at $E$, rays $DA$ and $CB$ intersect at $F$, and the diagonals $AC$ and $BD$ intersect at $G$. It is given that the triangles $DBF$ and $DBE$ have the same area.
$\text{(a)}$ Prove that $EF$ and $BD$ are parallel.
$\text{(b)}$ Prove that $G$ is the midpoint of $BD$.
$\text{(c)}$ Given that the area of triangle $ABD$ is 4 and the area of triangle $CBD$ is 6,
[color=transparent](C.)[/color]compute the area of triangle $EFG$.
2005 India IMO Training Camp, 1
Consider a $n$-sided polygon inscribed in a circle ($n \geq 4$). Partition the polygon into $n-2$ triangles using [b]non-intersecting[/b] diagnols. Prove that, irrespective of the triangulation, the sum of the in-radii of the triangles is a constant.
2022 ELMO Revenge, 4
Find all ordered pairs of integers $(a,b)$ such that there exists a function $f\colon \mathbb{N} \to \mathbb{N}$ satisfying
$$f^{f(n)}(n)=an+b$$
For all $n\in \mathbb{N}$.
2021 JHMT HS, 2
Compute the smallest positive integer $n$ such that $\int_{0}^{n} \lfloor x\rfloor\,dx$ is at least $2021.$
1999 Mongolian Mathematical Olympiad, Problem 2
Let $a,b,c$ be the real numbers with $a\ge\frac85b>0$ and $a\ge c>0$. Prove the inequality
$$\frac45\left(\frac1a+\frac1b\right)+\frac2c\ge\frac{27}2\cdot\frac1{a+b+c}.$$
2007 Indonesia MO, 7
Points $ A,B,C,D$ are on circle $ S$, such that $ AB$ is the diameter of $ S$, but $ CD$ is not the diameter. Given also that $ C$ and $ D$ are on different sides of $ AB$. The tangents of $ S$ at $ C$ and $ D$ intersect at $ P$. Points $ Q$ and $ R$ are the intersections of line $ AC$ with line $ BD$ and line $ AD$ with line $ BC$, respectively.
(a) Prove that $ P$, $ Q$, and $ R$ are collinear.
(b) Prove that $ QR$ is perpendicular to line $ AB$.
2002 IMO Shortlist, 4
Circles $S_1$ and $S_2$ intersect at points $P$ and $Q$. Distinct points $A_1$ and $B_1$ (not at $P$ or $Q$) are selected on $S_1$. The lines $A_1P$ and $B_1P$ meet $S_2$ again at $A_2$ and $B_2$ respectively, and the lines $A_1B_1$ and $A_2B_2$ meet at $C$. Prove that, as $A_1$ and $B_1$ vary, the circumcentres of triangles $A_1A_2C$ all lie on one fixed circle.
2021 Kyiv City MO Round 1, 11.2
Chess piece called [i]skew knight[/i], if placed on the black square, attacks all the gray squares.
[img]https://i.ibb.co/HdTDNjN/Kyiv-MO-2021-Round-1-11-2.png[/img]
What is the largest number of such knights that can be placed on the $8\times 8$ chessboard without them attacking each other?
[i]Proposed by Arsenii Nikolaiev[/i]
2004 Germany Team Selection Test, 2
Find all pairs of positive integers $\left(n;\;k\right)$ such that $n!=\left( n+1\right)^{k}-1$.
2010 Contests, 3
All sides and diagonals of a convex $n$-gon, $n\ge 3$, are coloured one of two colours. Show that there exist $\left[\frac{n+1}{3}\right]$ pairwise disjoint monochromatic segments.
[i](Two segments are disjoint if they do not share an endpoint or an interior point).[/i]
2007 Korea - Final Round, 5
For the vertex $ A$ of a triangle $ ABC$, let $ l_a$ be the distance between the projections on $ AB$ and $ AC$ of the intersection of the angle bisector of ∠$ A$ with side $ BC$. Define $ l_b$ and $ l_c$ analogously. If $ l$ is the perimeter of triangle $ ABC$, prove that $ \frac{l_a l_b l_c}{l^3}\le\frac{1}{64}$.
1983 Bulgaria National Olympiad, Problem 5
Can the polynomials $x^{5}-x-1$ and $x^{2}+ax+b$ , where $a,b\in Q$, have common complex roots?
1955 AMC 12/AHSME, 17
If $ \log x\minus{}5 \log 3\equal{}\minus{}2$, then $ x$ equals:
$ \textbf{(A)}\ 1.25 \qquad
\textbf{(B)}\ 0.81 \qquad
\textbf{(C)}\ 2.43 \qquad
\textbf{(D)}\ 0.8 \qquad
\textbf{(E)}\ \text{either 0.8 or 1.25}$
2024 AMC 12/AHSME, 24
A $\textit{disphenoid}$ is a tetrahedron whose triangular faces are congruent to one another. What is the least total surface area of a disphenoid whose faces are scalene triangles with integer side lengths?
$\textbf{(A) }\sqrt{3}\qquad\textbf{(B) }3\sqrt{15}\qquad\textbf{(C) }15\qquad\textbf{(D) }15\sqrt{7}\qquad\textbf{(E) }24\sqrt{6}$
May Olympiad L1 - geometry, 2022.5
Vero had an isosceles triangle made of paper. Using scissors, he divided it into three smaller triangles and painted them blue, red and green. Having done so, he observed that:
$\bullet$ with the blue triangle and the red triangle an isosceles triangle can be formed,
$\bullet$ with the blue triangle and the green triangle an isosceles triangle can be formed,
$\bullet$ with the red triangle and the green triangle an isosceles triangle can be formed.
Show what Vero's triangle looked like and how he might have made the cuts to make this situation be possible.
2002 AMC 10, 5
Let $(a_n)_{n\geq 1}$ be a sequence such that $a_1=1$ and $3a_{n+1}-3a_n=1$ for all $n\geq 1$. Find $a_{2002}$.
$\textbf{(A) }666\qquad\textbf{(B) }667\qquad\textbf{(C) }668\qquad\textbf{(D) }669\qquad\textbf{(E) }670$
2018 Bundeswettbewerb Mathematik, 3
Let $H$ be the orthocenter of the acute triangle $ABC$. Let $H_a$ be the foot of the perpendicular from $A$ to $BC$ and let the line through $H$ parallel to $BC$ intersect the circle with diameter $AH_a$ in the points $P_a$ and $Q_a$. Similarly, we define the points $P_b, Q_b$ and $P_c,Q_c$.
Show that the six points $P_a,Q_a,P_b,Q_b,P_c,Q_c$ lie on a common circle.
2007 VJIMC, Problem 3
Let $f:[0,1]\to\mathbb R$ be a continuous function such that $f(0)=f(1)=0$. Prove that the set
$$A:=\{h\in[0,1]:f(x+h)=f(x)\text{ for some }x\in[0,1]\}$$is Lebesgue measureable and has Lebesgue measure at least $\frac12$.
2003 IMC, 4
Determine the set of all pairs (a,b) of positive integers for which the set of positive integers can be decomposed into 2 sets A and B so that $a\cdot A=b\cdot B$.
2020 Israel National Olympiad, 4
At the start of the day, the four numbers $(a_0,b_0,c_0,d_0)$ were written on the board. Every minute, Danny replaces the four numbers written on the board with new ones according to the following rule: if the numbers written on the board are $(a,b,c,d)$, then Danny first calculates the numbers
\begin{align*}
a'&=a+4b+16c+64d\\
b'&=b+4c+16d+64a\\
c'&=c+4d+16a+64b\\
d'&=d+4a+16b+64c
\end{align*}
and replaces the numbers $(a,b,c,d)$ with the numbers $(a'd',c'd',c'b',b'a')$.
For which initial quadruples $(a_0,b_0,c_0,d_0)$, will Danny write at some point a quadruple of numbers all of which are divisible by $5780^{5780}$?
2009 AIME Problems, 7
Define $ n!!$ to be $ n(n\minus{}2)(n\minus{}4)\ldots3\cdot1$ for $ n$ odd and $ n(n\minus{}2)(n\minus{}4)\ldots4\cdot2$ for $ n$ even. When $ \displaystyle \sum_{i\equal{}1}^{2009} \frac{(2i\minus{}1)!!}{(2i)!!}$ is expressed as a fraction in lowest terms, its denominator is $ 2^ab$ with $ b$ odd. Find $ \displaystyle \frac{ab}{10}$.
1997 Israel National Olympiad, 3
Let $n?$ denote the product of all primes smaller than $n$.
Prove that $n? > n$ holds for any natural number $n > 3$.
1994 IMO Shortlist, 4
Let $ \mathbb{R}$ denote the set of all real numbers and $ \mathbb{R}^\plus{}$ the subset of all positive ones. Let $ \alpha$ and $ \beta$ be given elements in $ \mathbb{R},$ not necessarily distinct. Find all functions $ f: \mathbb{R}^\plus{} \mapsto \mathbb{R}$ such that
\[ f(x)f(y) \equal{} y^{\alpha} f \left( \frac{x}{2} \right) \plus{} x^{\beta} f \left( \frac{y}{2} \right) \forall x,y \in \mathbb{R}^\plus{}.\]
2014 IMC, 1
Determine all pairs $(a, b)$ of real numbers for which there exists a unique symmetric $2\times 2$ matrix $M$ with real entries satisfying $\mathrm{trace}(M)=a$ and $\mathrm{det}(M)=b$.
(Proposed by Stephan Wagner, Stellenbosch University)