Olympiad problems

2002 Moldova Team Selection Test, 3

A triangle $ABC$ is inscribed in a circle $G$. Points $M$ and $N$ are the midpoints of the arcs $BC$ and $AC$ respectively, and $D$ is an arbitrary point on the arc $AB$ (not containing $C$). Points $I_1$ and $I_2$ are the incenters of the triangles $ADC$ and $BDC$, respectively. If the circumcircle of triangle $DI_1I_2$ meets $G$ again at $P$, prove that triangles $PNI_1$ and $PMI_2$ are similar.

2010 Iran MO (3rd Round), 4

Tags: floor function , ceiling function , combinatorics

suppose that $\mathcal F\subseteq X^{(K)}$ and $|X|=n$. we know that for every three distinct elements of $\mathcal F$ like $A,B$ and $C$ we have $A\cap B \not\subset C$. a)(10 points) Prove that : \[|\mathcal F|\le \dbinom{k}{\lfloor\frac{k}{2}\rfloor}+1\] b)(15 points) if elements of $\mathcal F$ do not necessarily have $k$ elements, with the above conditions show that: \[|\mathcal F|\le \dbinom{n}{\lceil\frac{n-2}{3}\rceil}+2\]

2015 Hanoi Open Mathematics Competitions, 15

Tags: maximum , inequalities , algebra

Let the numbers $a, b,c$ satisfy the relation $a^2+b^2+c^2 \le 8$. Determine the maximum value of $M = 4(a^3 + b^3 + c^3) - (a^4 + b^4 + c^4)$

Novosibirsk Oral Geo Oly IX, 2021.6

Tags: geometry , rectangle

Two congruent rectangles are located as shown in the figure. Find the area of the shaded part. [img]https://cdn.artofproblemsolving.com/attachments/2/e/10b164535ab5b3a3b98ce1a0b84892cd11d76f.png[/img]

2021 BMT, 1

Tags: combinatorics

Towa has a hand of three different red cards and three different black cards. How many ways can Towa pick a set of three cards from her hand that uses at least one card of each color?

1997 Baltic Way, 5

Tags: number theory

In a sequence $u_0,u_1,\ldots $ of positive integers, $u_0$ is arbitrary, and for any non-negative integer $n$, \[ u_{n+1}=\begin{cases}\frac{1}{2}u_n & \text{for even }u_n \\ a+u_n & \text{for odd }u_n \end{cases} \] where $a$ is a fixed odd positive integer. Prove that the sequence is periodic from a certain step.

1980 IMO, 12

Tags: circumcircle , geometry

There is a triangle $ABC$. Its circumcircle and its circumcentre are given. Show how the orthocentre of $ABC$ may be constructed using only a straightedge (unmarked ruler). [The straightedge and paper may be assumed large enough for the construction to be completed]

2023 South East Mathematical Olympiad, 3

Tags: geometry , angle bisector

In $\triangle {ABC}$, ${D}$ is on the internal angle bisector of $\angle BAC$ and $\angle ADB=\angle ACD$. $E, F$ is on the external angle bisector of $\angle BAC$, such that $AE=BE$ and $AF=CF$. The circumcircles of $\triangle ACE$ and $\triangle ABF$ intersects at ${A}$ and ${K}$ and $A'$ is the reflection of ${A}$ with respect to $BC$. Prove that: if $AD=BC$, then the circumcenter of $\triangle AKA'$ is on line $AD$.

2017 Peru Iberoamerican Team Selection Test, P2

Tags: number theory

Determine if there exists a positive integer $n$ such that $n^2+11$ is a prime number and $n+4$ is a perfect cube.

2022 Caucasus Mathematical Olympiad, 8

Tags: combinatorics , graph theory

There are $n > 2022$ cities in the country. Some pairs of cities are connected with straight two-ways airlines. Call the set of the cities {\it unlucky}, if it is impossible to color the airlines between them in two colors without monochromatic triangle (i.e. three cities $A$, $B$, $C$ with the airlines $AB$, $AC$ and $BC$ of the same color). The set containing all the cities is unlucky. Is there always an unlucky set containing exactly 2022 cities?

2005 China Team Selection Test, 1

Tags: geometry

Convex quadrilateral $ABCD$ is cyclic in circle $(O)$, $P$ is the intersection of the diagonals $AC$ and $BD$. Circle $(O_{1})$ passes through $P$ and $B$, circle $(O_{2})$ passes through $P$ and $A$, Circles $(O_{1})$ and $(O_{2})$ intersect at $P$ and $Q$. $(O_{1})$, $(O_{2})$ intersect $(O)$ at another points $E$, $F$ (besides $B$, $A$), respectively. Prove that $PQ$, $CE$, $DF$ are concurrent or parallel.

PEN I Problems, 9

Tags: floor function

Show that for all positive integers $m$ and $n$, \[\gcd(m, n) = m+n-mn+2\sum^{m-1}_{k=0}\left \lfloor \frac{kn}{m}\right \rfloor.\]

Mathematical Minds 2023, P7

Tags: algebra , sequence

Does there exist an increasing sequence of positive integers for which any large enough integer can be expressed uniquely as the sum of two (possibly equal) terms of the sequence? [i]Proposed by Vlad Spătaru and David Anghel[/i]

Denmark (Mohr) - geometry, 2007.4

Tags: geometry , circles , angle , tangent circle

The figure shows a $60^o$ angle in which are placed $2007$ numbered circles (only the first three are shown in the figure). The circles are numbered according to size. The circles are tangent to the sides of the angle and to each other as shown. Circle number one has radius $1$. Determine the radius of circle number $2007$. [img]https://1.bp.blogspot.com/-1bsLIXZpol4/Xzb-Nk6ospI/AAAAAAAAMWk/jrx1zVYKbNELTWlDQ3zL9qc_22b2IJF6QCLcBGAsYHQ/s0/2007%2BMohr%2Bp4.png[/img]

1988 AIME Problems, 10

Tags:

A convex polyhedron has for its faces 12 squares, 8 regular hexagons, and 6 regular octagons. At each vertex of the polyhedron one square, one hexagon, and one octagon meet. How many segments joining vertices of the polyhedron lie in the interior of the polyhedron rather than along an edge or a face?

2016 ELMO Problems, 4

Tags: polynomial , number theory , algebra

Big Bird has a polynomial $P$ with integer coefficients such that $n$ divides $P(2^n)$ for every positive integer $n$. Prove that Big Bird's polynomial must be the zero polynomial. [i]Ashwin Sah[/i]

2010 Contests, 1

Tags: number theory , algebra

a) Factorize $xy - x - y + 1$. b) Prove that if integers $a$ and $b$ satisfy $ |a + b| > |1 + ab|$, then $ab = 0$.

2006 QEDMO 3rd, 9

Tags: angle bisector , geometry

Let $ABC$ be a triangle, and $C^{\prime}$ and $A^{\prime}$ the midpoints of its sides $AB$ and $BC$. Consider two lines $g$ and $g^{\prime}$ which both pass through the vertex $A$ and are symmetric to each other with respect to the angle bisector of the angle $CAB$. Further, let $Y$ and $Y^{\prime}$ be the orthogonal projections of the point $B$ on these lines $g$ and $g^{\prime}$. Show that the points $Y$ and $Y^{\prime}$ are symmetric to each other with respect to the line $C^{\prime}A^{\prime}$.

2021 Princeton University Math Competition, A4 / B6

Tags: algebra

The roots of a monic cubic polynomial $p$ are positive real numbers forming a geometric sequence. Suppose that the sum of the roots is equal to $10$. Under these conditions, the largest possible value of $|p(-1)|$ can be written as $\frac{m}{n}$, where $m$, $n$ are relatively prime integers. Find $m + n$.

2012 JBMO ShortLists, 2

Tags: geometry , circumcircle , geometric transformation , reflection , perpendicular bisector

Let $ABC$ be an isosceles triangle with $AB=AC$ . Let also $\omega$ be a circle of center $K$ tangent to the line $AC$ at $C$ which intersects the segment $BC$ again at $H$ . Prove that $HK \bot AB $.

2010 Today's Calculation Of Integral, 632

Tags: calculus , integration , limit , trigonometry , floor function , ceiling function , calculus computations

Find $\lim_{n\to\infty} \int_0^1 |\sin nx|^3dx\ (n=1,\ 2,\ \cdots).$ [i]2010 Kyoto Institute of Technology entrance exam/Textile, 2nd exam[/i]

2011 Dutch Mathematical Olympiad, 5

Tags: coloring , combinatorics

The number devil has coloured the integer numbers: every integer is coloured either black or white. The number $1$ is coloured white. For every two white numbers $a$ and $b$ ($a$ and $b$ are allowed to be equal) the numbers $a-b$ and $a + $b have different colours. Prove that $2011$ is coloured white.

2009 Junior Balkan Team Selection Tests - Moldova, 4

Tags: combinatorics

Petrică, Vasile and Tudor participated at a math contest. At the contest $ 5$ problems where proposed, each worth distinct integer numbers of points. Petrică solved $4$ problems completely and got $21$ points and Vasile solved $3 $ problems completely and got $22$ points. Tudor solved all problems completely. What are the lowest and highest possible scores of Tudor?

2015 HMNT, 9

Tags:

A graph consists of 6 vertices. For each pair of vertices, a coin is flipped, and an edge connecting the two vertices is drawn if and only if the coin shows heads. Such a graph is $\it{good}$ if, starting from any vertex $V$ connected to at least one other vertex, it is possible to draw a path starting and ending at $V$ that traverses each edge exactly once. What is the probability that the graph is good?

2019 Dutch Mathematical Olympiad, 3

Tags: geometry , angle bisector , reflection , circumcircle , circumcenter

Points $A, B$, and $C$ lie on a circle with centre $M$. The reflection of point $M$ in the line $AB$ lies inside triangle $ABC$ and is the intersection of the angle bisectors of angles $A$ and $B$. Line $AM$ intersects the circle again in point $D$. Show that $|CA| \cdot |CD| = |AB| \cdot |AM|$.