Olympiad problems

2018 Romanian Masters in Mathematics, 1

Tags: geometry

Let $ABCD$ be a cyclic quadrilateral an let $P$ be a point on the side $AB.$ The diagonals $AC$ meets the segments $DP$ at $Q.$ The line through $P$ parallel to $CD$ mmets the extension of the side $CB$ beyond $B$ at $K.$ The line through $Q$ parallel to $BD$ meets the extension of the side $CB$ beyond $B$ at $L.$ Prove that the circumcircles of the triangles $BKP$ and $CLQ$ are tangent .

2019 Philippine TST, 3

Tags: quadratic , inequalities , functional inequality , function , absolute value , constant , algebra

Determine all ordered triples $(a, b, c)$ of real numbers such that whenever a function $f : \mathbb{R} \to \mathbb{R}$ satisfies $$|f(x) - f(y)| \le a(x - y)^2 + b(x - y) + c$$ for all real numbers $x$ and $y$, then $f$ must be a constant function.

2022 MIG, 15

Tags:

There exists a fraction $x$ that satisfies $ \sqrt{x^2+5} - x = \tfrac{1}{3}$. What is the sum of the numerator and denominator of this fraction? $\textbf{(A) }8\qquad\textbf{(B) }21\qquad\textbf{(C) }25\qquad\textbf{(D) }32\qquad\textbf{(E) }34$

2019-IMOC, G2

Tags: midpoint , geometry , symmetry , orthocenter , concyclic

Given a scalene triangle $\vartriangle ABC$ with orthocenter $H$. The midpoint of $BC$ is denoted by $M$. $AH$ intersects the circumcircle at $D \ne A$ and $DM$ intersects circumcircle of $\vartriangle ABC$ at $T\ne D$. Now, assume the reflection points of $M$ with respect to $AB,AC,AH$ are $F,E,S$. Show that the midpoints of $BE,CF,AM,TS$ are concyclic. [img]https://3.bp.blogspot.com/-v7D_A66nlD0/XnYNJussW9I/AAAAAAAALeQ/q6DMQ7w6QtI5vLwBcKqp4010c3XTCj3BgCK4BGAYYCw/s1600/imoc2019g2.png[/img]

2021 Serbia Team Selection Test, P3

Tags: diophantine equation , system of equations , number theory , prime number

Given is a prime number $p$. Find the number of positive integer solutions $(a, b, c, d)$ of the system of equations $ac+bd = p(a+c)$ and $bc-ad = p(b-d)$.

2012 Miklós Schweitzer, 7

Tags: probability , real analysis

Let $\Gamma$ be a simple curve, lying inside a circle of radius $r$, rectifiable and of length $\ell$. Prove that if $\ell > kr\pi$, then there exists a circle of radius $r$ which intersects $\Gamma$ in at least $k+1$ distinct points.

2011 South East Mathematical Olympiad, 2

Tags: greatest common divisor , number theory

If positive integers, $a,b,c$ are pair-wise co-prime, and, \[\ a^2|(b^3+c^3), b^2|(a^3+c^3), c^2|(a^3+b^3) \] find $a,b,$ and $c$

1970 Polish MO Finals, 4

Tags: geometry , rectangle , combinatorial geometry

In the plane are given two mutually perpendicular lines and $n$ rectangles with sides parallel to the two lines. Show that if every two rectangles have a common point, then all the rectangles have a common point.

2008 IMAR Test, 1

Tags: induction , combinatorics

An array $ n\times n$ is given, consisting of $ n^2$ unit squares. A [i]pawn[/i] is placed arbitrarily on a unit square. The pawn can move from a square of the $ k$-th column to any square of the $ k$-th row. Show that there exists a sequence of $ n^2$ moves of the pawn so that all the unit squares of the array are visited once, the pawn returning to its original position. [b]Dinu Serbanescu[/b]

2008 Romania National Olympiad, 3

Tags: function , algebra , domain , real analysis , calculus , integration , analytic geometry

Let $ f: \mathbb R \to \mathbb R$ be a function, two times derivable on $ \mathbb R$ for which there exist $ c\in\mathbb R$ such that \[ \frac { f(b)\minus{}f(a) }{b\minus{}a} \neq f'(c) ,\] for all $ a\neq b \in \mathbb R$. Prove that $ f''(c)\equal{}0$.

1987 Romania Team Selection Test, 1

Tags: matrix , absolute value , linear algebra

Let $a,b,c$ be distinct real numbers such that $a+b+c>0$. Let $M$ be the set of $3\times 3$ matrices with the property that each line and each column contain all given numbers $a,b,c$. Find $\{\max \{ \det A \mid A \in M \}$ and the number of matrices which realise the maximum value. [i]Mircea Becheanu[/i]

1985 Yugoslav Team Selection Test, Problem 3

Tags: algebra , inequality , inequalities

1) proove for positive $a, b, c, d$ $ \frac{a}{b+c} + \frac{b}{c+d} + \frac{c}{d+a} + \frac{d}{a+b} \ge 2$

2005 Singapore MO Open, 1

Tags: floor function , number theory

An integer is square-free if it is not divisible by $a^2$ for any integer $a>1$. Let $S$ be the set of positive square-free integers. Determine, with justification, the value of\[\sum_{k\epsilon S}\left[\sqrt{\frac{10^{10}}{k}}\right]\]where $[x]$ denote the greatest integer less than or equal to $x$

1954 AMC 12/AHSME, 42

Tags: parabola , geometry , geometric transformation , conic

Consider the graphs of (1): $ y\equal{}x^2\minus{}\frac{1}{2}x\plus{}2$ and (2) $ y\equal{}x^2\plus{}\frac{1}{2}x\plus{}2$ on the same set of axis. These parabolas are exactly the same shape. Then: $ \textbf{(A)}\ \text{the graphs coincide.} \\ \textbf{(B)}\ \text{the graph of (1) is lower than the graph of (2).} \\ \textbf{(C)}\ \text{the graph of (1) is to the left of the graph of (2).} \\ \textbf{(D)}\ \text{the graph of (1) is to the right of the graph of (2).} \\ \textbf{(E)}\ \text{the graph of (1) is higher than the graph of (2).}$

KoMaL A Problems 2019/2020, A. 762

Tags: combinatorics

In a forest, there are $n$ different trees (considered as points), no three of which lie on the same line. John takes photographs of the forest such that all trees are visible (and no two trees are behind each other). What is the largest number of orders of in which the trees that can appear on the photos? [i]Proposed by Gábor Mészáros, Sunnyvale, Kalifornia[/i]

1970 AMC 12/AHSME, 27

Tags: geometry , inradius

In a triangle, the area is numerically equal to the perimeter. What is the radius of the inscribed circle? $\textbf{(A) }2\qquad\textbf{(B) }3\qquad\textbf{(C) }4\qquad\textbf{(D) }5\qquad \textbf{(E) }6$

MathLinks Contest 1st, 3

Tags: inequalities

Prove that if the positive reals $a, b, c$ have sum $1$ then the following inequality holds $$(ab)^{ \frac54} + (bc)^{\frac54} + (ca)^{\frac54} < \frac14 .$$

2017 IMC, 2

Tags: function , calculus

Let $f:\mathbb R\to(0,\infty)$ be a differentiabe function, and suppose that there exists a constant $L>0$ such that $$|f'(x)-f'(y)|\leq L|x-y|$$ for all $x,y$. Prove that $$(f'(x))^2<2Lf(x)$$ holds for all $x$.

2018 Moscow Mathematical Olympiad, 1

Tags: number theory

Is there a number in the decimal notation of the square which has a sequence of digits "$2018$"?

2016 CMIMC, 3

Tags: number theory

How many pairs of integers $(a,b)$ are there such that $0\leq a < b \leq 100$ and such that $\tfrac{2^b-2^a}{2016}$ is an integer?

2022 Greece Team Selection Test, 1

Tags: divisibility , cubefree , number theory

Find all positive integers $n\geq1$ such that there exists a pair $(a,b)$ of positive integers, such that $a^2+b+3$ is not divisible by the cube of any prime, and $$n=\frac{ab+3b+8}{a^2+b+3}.$$

2010 AMC 12/AHSME, 4

Tags: function

If $ x < 0$, then which of the following must be positive? $ \textbf{(A)}\ \frac{x}{|x|}\qquad \textbf{(B)}\ \minus{}x^2\qquad \textbf{(C)}\ \minus{}2^x\qquad \textbf{(D)}\ \minus{}x^{\minus{}1}\qquad \textbf{(E)}\ \sqrt[3]{x}$

2009 Today's Calculation Of Integral, 510

Tags: calculus , integration , trigonometry , function , derivative , calculus computations

(1) Evaluate $ \int_0^{\frac{\pi}{2}} (x\cos x\plus{}\sin ^ 2 x)\sin x\ dx$. (2) For $ f(x)\equal{}\int_0^x e^t\sin (x\minus{}t)\ dt$, find $ f''(x)\plus{}f(x)$.

Cono Sur Shortlist - geometry, 1993.13

Tags: obtuse , geometry

Determine the real values of $x$ such that the triangle with sides $5$, $8$, and $x$ is obtuse.

2013 Serbia National Math Olympiad, 5

Tags: geometry , trigonometry , ratio , circumcircle , geometric transformation , reflection , parallelogram

Let $A'$ and $B'$ be feet of altitudes from $A$ and $B$, respectively, in acute-angled triangle $ABC$ ($AC\not = BC$). Circle $k$ contains points $A'$ and $B'$ and touches segment $AB$ in $D$. If triangles $ADA'$ and $BDB'$ have the same area, prove that \[\angle A'DB'= \angle ACB.\]