Olympiad problems

1982 All Soviet Union Mathematical Olympiad, 328

Every member, starting from the third one, of two sequences $\{a_n\}$ and $\{b_n\}$ equals to the sum of two preceding ones. First members are: $a_1 = 1, a_2 = 2, b_1 = 2, b_2 = 1$. How many natural numbers are encountered in both sequences (may be on the different places)?

2015 Sharygin Geometry Olympiad, P16

Tags: geometry , circumcircle , incenter

The diagonals of a convex quadrilateral divide it into four triangles. Restore the quadrilateral by the circumcenters of two adjacent triangles and the incenters of two mutually opposite triangles

1989 IMO Shortlist, 6

Tags: geometry , circumcircle , geometric inequality , area of a triangle

For a triangle $ ABC,$ let $ k$ be its circumcircle with radius $ r.$ The bisectors of the inner angles $ A, B,$ and $ C$ of the triangle intersect respectively the circle $ k$ again at points $ A', B',$ and $ C'.$ Prove the inequality \[ 16Q^3 \geq 27 r^4 P,\] where $ Q$ and $ P$ are the areas of the triangles $ A'B'C'$ and $ABC$ respectively.

2013 IFYM, Sozopol, 1

Tags: concurrency , moving points , geometry

The points $P$ and $Q$ on the side $AC$ of the non-isosceles $\Delta ABC$ are such that $\angle ABP=\angle QBC<\frac{1}{2}\angle ABC$. The angle bisectors of $\angle A$ and $\angle C$ intersect the segment $BP$ in points $K$ and $L$ and the segment $BQ$ in points $M$ and $N$, respectively. Prove that $AC$,$KN$, and $LM$ are concurrent.

1984 IMO Shortlist, 5

Tags: inequality , three variable inequality , calculus , maximization

Prove that $0\le yz+zx+xy-2xyz\le{7\over27}$, where $x,y$ and $z$ are non-negative real numbers satisfying $x+y+z=1$.

1967 AMC 12/AHSME, 21

Tags:

In right triangle $ABC$ the hypotenuse $\overline{AB}=5$ and leg $\overline{AC}=3$. The bisector of angle $A$ meets the opposite side in $A_1$. A second right triangle $PQR$ is then constructed with hypotenuse $\overline{PQ}=A_1B$ and leg $\overline{PR}=A_1C$. If the bisector of angle $P$ meets the opposite side in $P_1$, the length of $PP_1$ is: $\textbf{(A)}\ \frac{3\sqrt{6}}{4}\qquad \textbf{(B)}\ \frac{3\sqrt{5}}{4}\qquad \textbf{(C)}\ \frac{3\sqrt{3}}{4}\qquad \textbf{(D)}\ \frac{3\sqrt{2}}{2}\qquad \textbf{(E)}\ \frac{15\sqrt{2}}{16}$

2003 IMC, 5

Tags: function , real analysis

Let $g:[0,1]\rightarrow \mathbb{R}$ be a continuous function and let $f_{n}:[0,1]\rightarrow \mathbb{R}$ be a sequence of functions defined by $f_{0}(x)=g(x)$ and $$f_{n+1}(x)=\frac{1}{x}\int_{0}^{x}f_{n}(t)dt.$$ Determine $\lim_{n\to \infty}f_{n}(x)$ for every $x\in (0,1]$.

2020 AIME Problems, 3

Tags: logarithm

The value of $x$ that satisfies $\log_{2^x} 3^{20} = \log_{2^{x+3}} 3^{2020}$ can be written as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

2020 Thailand TST, 4

Tags: algebra , polynomial

Let $n$ be a positive integer and let $P$ be the set of monic polynomials of degree $n$ with complex coefficients. Find the value of \[ \min_{p \in P} \left \{ \max_{|z| = 1} |p(z)| \right \} \]

1993 Miklós Schweitzer, 6

Tags: geometry , inequalities

Let $P_1 , P_2 , ...$ be arbitrary points and A be a connected compact set in the plane with a diameter greater than 4. Show that for some point P in A , $\overline {PP_1} \cdot \overline {PP_2} \cdots \overline {PP_n}>1$. Furthermore, prove that this is no longer necessarily true for compact sets of diameter 4.

2006 Iran MO (2nd round), 3

Tags: induction , least common multiple , combinatorics

Some books are placed on each other. Someone first, reverses the upper book. Then he reverses the $2$ upper books. Then he reverses the $3$ upper books and continues like this. After he reversed all the books, he starts this operation from the first. Prove that after finite number of movements, the books become exactly like their initial configuration.

2022 AMC 10, 22

Tags: geometry

Let $S$ be the set of circles in the coordinate plane that are tangent to each of the three circles with equations $x^{2}+y^{2}=4$, $x^{2}+y^{2}=64$, and $(x-5)^{2}+y^{2}=3$. What is the sum of the areas of all circles in $S$? $\textbf{(A)}~48\pi\qquad\textbf{(B)}~68\pi\qquad\textbf{(C)}~96\pi\qquad\textbf{(D)}~102\pi\qquad\textbf{(E)}~136\pi\qquad$

1991 Tournament Of Towns, (311) 1

Tags: geometry , tangent circle

Two circles with centres $A$ and $B$ lie inside an angle. They touch each other and both sides of the angle. Prove that the circle with the diameter $AB$ touches both sides of the angle. (V. Prasolov)

2012 Irish Math Olympiad, 4

Tags: number theory

There exists an infinite set of triangles with the following properties: (a) the lengths of the sides are integers with no common factors, and (b) one and only one angle is $60^\circ$. One such triangle has side lengths $5,7,8$. Find two more.

PEN A Problems, 22

Tags: polynomial , binomial theorem , divisibility theory , algebra

Prove that the number \[\sum_{k=0}^{n}\binom{2n+1}{2k+1}2^{3k}\] is not divisible by $5$ for any integer $n\geq 0$.

2014 ASDAN Math Tournament, 11

Tags:

Mr. Ambulando is at the intersection of $5^{\text{th}}$ and $\text{A St}$, and needs to walk to the intersection of $1^{\text{st}}$ and $\text{F St}$. There's an accident at the intersection of $4^{\text{th}}$ and $\text{B St}$, which he'd like to avoid. [center]<see attached>[/center] Given that Mr. Ambulando wants to walk the shortest distance possible, how many different routes through downtown can he take?

2013 Princeton University Math Competition, 1

Tags:

If $p,q,$ and $r$ are primes with $pqr=7(p+q+r)$, find $p+q+r$.

2003 Estonia Team Selection Test, 5

Tags: inequalities , algebra

Let $a, b, c$ be positive real numbers satisfying the condition $\frac{1}{ab}+\frac{1}{ac}+\frac{1}{bc}=1$ . Prove the inequality $$\frac{a}{\sqrt{1+a^2}}+\frac{b}{\sqrt{1+b^2}}+\frac{c}{\sqrt{1+c^2}} \le \frac{3\sqrt3}{2}$$ When does the equality hold? (L. Parts)

2022 India National Olympiad, 1

Tags: geometry , incenter , circumcircle

Let $D$ be an interior point on the side $BC$ of an acute-angled triangle $ABC$. Let the circumcircle of triangle $ADB$ intersect $AC$ again at $E(\ne A)$ and the circumcircle of triangle $ADC$ intersect $AB$ again at $F(\ne A)$. Let $AD$, $BE$, and $CF$ intersect the circumcircle of triangle $ABC$ again at $D_1(\ne A)$, $E_1(\ne B)$ and $F_1(\ne C)$, respectively. Let $I$ and $I_1$ be the incentres of triangles $DEF$ and $D_1E_1F_1$, respectively. Prove that $E,F, I, I_1$ are concyclic.

1995 Tuymaada Olympiad, 3

Tags: diophantine equation , algebra , number theory

Prove that the equation $(\sqrt5 +1)^{2x}+ (\sqrt5 -1)^{2x}=2^x(y^2+2)$ has an infinite number of solutions in natural numbers.

2019 Kosovo National Mathematical Olympiad, 4

Tags: circumcircle , geometry

Let $ABC$ be an acute triagnle with its circumcircle $\omega$. Let point $D$ be the foot of triangle $ABC$ from point $A$. Let points $E,F$ be midpoints of sides $AB,AC$, respectively. Let points $P$ and $Q$ be the second intersections of of circle $\omega$ with circumcircle of triangles $BDE$ and $CDF$, respectively. Suppose that $A,P,B,Q$ and $C$ be on a circle in this order. Show that the lines $EF,BQ$ and $CP$ are concurrent.

2013 ISI Entrance Examination, 6

Tags: polynomial , limit , algebra

Let $p(x)$ and $q(x)$ be two polynomials, both of which have their sum of coefficients equal to $s.$ Let $p,q$ satisfy $p(x)^3-q(x)^3=p(x^3)-q(x^3).$ Show that (i) There exists an integer $a\geq1$ and a polynomial $r(x)$ with $r(1)\neq0$ such that \[p(x)-q(x)=(x-1)^ar(x).\] (ii) Show that $s^2=3^{a-1},$ where $a$ is described as above.

2008 Oral Moscow Geometry Olympiad, 4

Tags: geometry , orthocenter , circumcenter , distance

Angle $A$ in triangle $ABC$ is equal to $120^o$. Prove that the distance from the center of the circumscribed circle to the orthocenter is equal to $AB + AC$. (V. Protasov)

2007 All-Russian Olympiad, 4

Tags: floor function , algebra

An infinite sequence $(x_{n})$ is defined by its first term $x_{1}>1$, which is a rational number, and the relation $x_{n+1}=x_{n}+\frac{1}{\lfloor x_{n}\rfloor}$ for all positive integers $n$. Prove that this sequence contains an integer. [i]A. Golovanov[/i]

PEN J Problems, 22

Tags: inequalities , divisor function

Let $n$ be an odd positive integer. Prove that $\sigma(n)^3 <n^4$.