Olympiad problems

1995 IMO, 5

Let $ ABCDEF$ be a convex hexagon with $ AB \equal{} BC \equal{} CD$ and $ DE \equal{} EF \equal{} FA$, such that $ \angle BCD \equal{} \angle EFA \equal{} \frac {\pi}{3}$. Suppose $ G$ and $ H$ are points in the interior of the hexagon such that $ \angle AGB \equal{} \angle DHE \equal{} \frac {2\pi}{3}$. Prove that $ AG \plus{} GB \plus{} GH \plus{} DH \plus{} HE \geq CF$.

2020 Cono Sur Olympiad, 2

Tags: combinatorics , pigeonhole principle , number theory

Given $2021$ distinct positive integers non divisible by $2^{1010}$, show that it's always possible to choose $3$ of them $a$, $b$ and $c$, such that $|b^2-4ac|$ is not a perfect square.

2006 Flanders Math Olympiad, 3

Tags: logic , puzzle

Elfs and trolls are seated at a round table, 60 creatures in total. Trolls always lie, and all elfs always speak the truth, except when they make a little mistake. Everybody claims to sit between an elf and a troll, but exactly two elfs made a mistake! How many trolls are there at this table?

2014 Contests, 3

Tags: function , induction , algebra

Prove that: there exists only one function $f:\mathbb{N^*}\to\mathbb{N^*}$ satisfying: i) $f(1)=f(2)=1$; ii)$f(n)=f(f(n-1))+f(n-f(n-1))$ for $n\ge 3$. For each integer $m\ge 2$, find the value of $f(2^m)$.

2013 Stanford Mathematics Tournament, 12

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Suppose Robin and Eddy walk along a circular path with radius $r$ in the same direction. Robin makes a revolution around the circular path every $3$ minutes and Eddy makes a revolution every minute. Jack stands still at a distance $R>r$ from the center of the circular path. At time $t=0$, Robin and Eddy are at the same point on the path, and Jack, Robin, and Eddy, and the center of the path are collinear. When is the next time the three people (but not necessarily the center of the path) are collinear?

2024 IFYM, Sozopol, 2

Tags: geometry

Given an acute-angled triangle $ABC$ ($AB \neq AC$) with orthocenter $H$, circumcenter $O$, and midpoint $M$ of side $BC$. The line $AM$ intersects the circumcircle of triangle $BHC$ at point $K$, with $M$ between $A$ and $K$. The segments $HK$ and $BC$ intersect at point $N$. If $\angle BAM = \angle CAN$, prove that the lines $AN$ and $OH$ are perpendicular.

1999 National Olympiad First Round, 17

Tags: pyramid , geometry , 3d geometry , rectangle

In a regular pyramid with top point $ T$ and equilateral base $ ABC$, let $ P$, $ Q$, $ R$, $ S$ be the midpoints of $ \left[AB\right]$, $ \left[BC\right]$, $ \left[CT\right]$ and $ \left[TA\right]$, respectively. If $ \left|AB\right| \equal{} 6$ and the altitude of pyramid is equal to $ 2\sqrt {15}$, then area of $ PQRS$ will be $\textbf{(A)}\ 4\sqrt {15} \qquad\textbf{(B)}\ 8\sqrt {2} \qquad\textbf{(C)}\ 8\sqrt {3} \qquad\textbf{(D)}\ 6\sqrt {5} \qquad\textbf{(E)}\ 9\sqrt {2}$

2014 China National Olympiad, 1

Tags: circumcircle , symmetry , incenter , geometry , ratio , trigonometry , angle bisector

Let $ABC$ be a triangle with $AB>AC$. Let $D$ be the foot of the internal angle bisector of $A$. Points $F$ and $E$ are on $AC,AB$ respectively such that $B,C,F,E$ are concyclic. Prove that the circumcentre of $DEF$ is the incentre of $ABC$ if and only if $BE+CF=BC$.

2013 Hanoi Open Mathematics Competitions, 10

Tags: perimeter , rectangle , geometry

Consider the set of all rectangles with a given area $S$. Find the largest value o $ M = \frac{S}{2S+p + 2}$ where $p$ is the perimeter of the rectangle.

1985 National High School Mathematics League, 4

Tags: geometry , inequalities

Given 5 points on a plane. Let $\lambda$ be the ratio of maximum value between the points to minimum value between the points. Prove that $\lambda\geq2\sin\frac{3}{10}\pi$.

2015 Sharygin Geometry Olympiad, 1

Tags: geometry

Let $K$ be an arbitrary point on side $BC$ of triangle $ABC$, and $KN$ be a bisector of triangle $AKC$. Lines $BN$ and $AK$ meet at point $F$, and lines $CF$ and $AB$ meet at point $D$. Prove that $KD$ is a bisector of triangle $AKB$.

2022 CMIMC Integration Bee, 15

Tags: integration , calculus

\[\int_0^\infty 1+\frac{2}{\sqrt[x]{8}}-\frac{3}{\sqrt[x]{4}}\,\mathrm dx\] [i]Proposed by Connor Gordon[/i]

2012 Irish Math Olympiad, 3

Tags: algebra , combi , polynomial

Find, with proof, all polynomials $f$ such that $f$ has nonnegative integer coefficients, $f$($1$) = $8$ and $f$($2$) = $2012$.

2015 Online Math Open Problems, 25

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Define $\left\lVert A-B \right\rVert = (x_A-x_B)^2+(y_A-y_B)^2$ for every two points $A = (x_A, y_A)$ and $B = (x_B, y_B)$ in the plane. Let $S$ be the set of points $(x,y)$ in the plane for which $x,y \in \left\{ 0,1,\dots,100 \right\}$. Find the number of functions $f : S \to S$ such that $\left\lVert A-B \right\rVert \equiv \left\lVert f(A)-f(B) \right\rVert \pmod{101}$ for any $A, B \in S$. [i] Proposed by Victor Wang [/i]

2022 Macedonian Team Selection Test, Problem 5

Tags: number theory

Given is an arithmetic progression {$a_n$} of positive integers. Prove that there exist infinitely many $k$, such that $\omega (a_k)$ is even and $\omega (a_{k+1})$ is odd ($\omega (n)$ is the number of distinct prime factors of $n$). $\textit {Proposed by Viktor Simjanoski and Nikola Velov}$

2019 Kazakhstan National Olympiad, 6

Tags: geometry

The tangent line $l$ to the circumcircle of an acute triangle $ABC$ intersects the lines $AB, BC$, and $CA$ at points $C', A'$ and $B'$, respectively. Let $H$ be the orthocenter of a triangle $ABC$. On the straight lines A'H, B′H and C'H, respectively, points $A_1, B_1$ and $C_1$ (other than $H$) are marked such that $AH = AA_1, BH = BB_1$ and $CH = CC_1$. Prove that the circumcircles of triangles $ABC$ and $A_1B_1C_1$ are tangent.

2016 Bosnia and Herzegovina Junior BMO TST, 1

Tags: perfect square , number theory

Prove that it is not possible that numbers $(n+1)\cdot 2^n$ and $(n+3)\cdot 2^{n+2}$ are perfect squares, where $n$ is positive integer.

2015 Denmark MO - Mohr Contest, 3

Tags: equal segments , equilateral , geometry

Triangle $ABC$ is equilateral. The point $D$ lies on the extension of $AB$ beyond $B$, the point $E$ lies on the extension of $CB$ beyond $B$, and $|CD| = |DE|$. Prove that $|AD| = |BE|$. [img]https://1.bp.blogspot.com/-QnAXFw3ijn0/XzR0YjqBQ3I/AAAAAAAAMU0/0TvhMQtBNjolYHtgXsQo2OPGJzEYSfCwACLcBGAsYHQ/s0/2015%2BMohr%2Bp3.png[/img]

PEN A Problems, 104

Tags: divisibility theory

A wobbly number is a positive integer whose $digits$ in base $10$ are alternatively non-zero and zero the units digit being non-zero. Determine all positive integers which do not divide any wobbly number.

2013 AMC 8, 17

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The sum of six consecutive positive integers is 2013. What is the largest of these six integers? $\textbf{(A)}\ 335 \qquad \textbf{(B)}\ 338 \qquad \textbf{(C)}\ 340 \qquad \textbf{(D)}\ 345 \qquad \textbf{(E)}\ 350$

2008 Harvard-MIT Mathematics Tournament, 7

Tags: trigonometry

Given that $ x \plus{} \sin y \equal{} 2008$ and $ x \plus{} 2008 \cos y \equal{} 2007$, where $ 0 \leq y \leq \pi/2$, find the value of $ x \plus{} y$.

1986 AIME Problems, 14

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The shortest distances between an interior diagonal of a rectangular parallelepiped, P, and the edges it does not meet are $2\sqrt{5}$, $\frac{30}{\sqrt{13}}$, and $\frac{15}{\sqrt{10}}$. Determine the volume of $P$.

Ukrainian From Tasks to Tasks - geometry, 2011.8

Tags: isosceles , angle bisector , parallel , geometry

On the median $AD$ of the isosceles triangle $ABC$, point $E$ is marked. Point $F$ is the projection of point $E$ on the line $BC$, point $M$ lies on the segment $EF$, points $N$ and $P$ are projections of point $M$ on the lines $AC$ and $AB$, respectively. Prove that the bisectors of the angles $PMN$ and $PEN$ are parallel.

2005 IMO Shortlist, 5

Tags: trigonometry , circumcircle , spiral similarity , miquel point , geometry

Let $\triangle ABC$ be an acute-angled triangle with $AB \not= AC$. Let $H$ be the orthocenter of triangle $ABC$, and let $M$ be the midpoint of the side $BC$. Let $D$ be a point on the side $AB$ and $E$ a point on the side $AC$ such that $AE=AD$ and the points $D$, $H$, $E$ are on the same line. Prove that the line $HM$ is perpendicular to the common chord of the circumscribed circles of triangle $\triangle ABC$ and triangle $\triangle ADE$.

2024 ELMO Shortlist, N3

Tags: ratio , number theory

Given a positive integer $k$, find all polynomials $P$ of degree $k$ with integer coefficients such that for all positive integers $n$ where all of $P(n)$, $P(2024n)$, $P(2024^2n)$ are nonzero, we have $$\frac{\gcd(P(2024n), P(2024^2n))}{\gcd(P(n), P(2024n))}=2024^k.$$ [i]Allen Wang[/i]