Olympiad problems

2012 Bogdan Stan, 1

Find the functions $ f:\mathbb{Z}\longrightarrow\mathbb{Z}_{\ge 0} $ that satisfy the following two conditions: $ \text{(a)} f(m+n)=f(n)+f(m)+2mn,\quad\forall m,n\in\mathbb{Z} $ $ \text{(b)} f(f(1))-f(1) $ is a perfect square [i]Marin Ionescu[/i]

2022 Harvard-MIT Mathematics Tournament, 10

Tags: number theory

Compute the smallest positive integer $n$ for which there are at least two odd primes $p$ such that $\sum_{k=1}^{n} (-1)^{v_p(k!)} < 0$. Note: for a prime $p$ and a positive integer $m$, $v_p(m)$ is the exponent of the largest power of $p$ that divides $m$; for example, $v_3(18) = 2$.

2023 LMT Fall, 8

Tags: combinatorics

To celebrate the $20$th LMT, the LHSMath Team bakes a cake. Each of the $n$ bakers places $20$ candles on the cake. When they count, they realize that there are $(n -1)!$ total candles on the cake. Find $n$. [i]Proposed by Christopher Cheng[/i]

2007 JBMO Shortlist, 3

Tags: perfect square , number theory

Let $n > 1$ be a positive integer and $p$ a prime number such that $n | (p - 1) $and $p | (n^6 - 1)$. Prove that at least one of the numbers $p- n$ and $p + n$ is a perfect square.

2012 AMC 8, 2

Tags: algebra

In the country of East Westmore, statisticians estimate there is a baby born every 8 hours and a death every day. To the nearest hundred, how many people are added to the population of East Westmore each year? $\textbf{(A)}\hspace{.05in}600 \qquad \textbf{(B)}\hspace{.05in}700 \qquad \textbf{(C)}\hspace{.05in}800 \qquad \textbf{(D)}\hspace{.05in}900 \qquad \textbf{(E)}\hspace{.05in}1000 $

2023 AMC 10, 7

Tags: probability

Janet rolls a standard 6-sided die 4 times and keeps a running total of the numbers she rolls. What is the probability that at some point, her running total will equal 3? $\textbf{(A) }\frac{2}{9}\qquad\textbf{(B) }\frac{49}{216}\qquad\textbf{(C) }\frac{25}{108}\qquad\textbf{(D) }\frac{17}{72}\qquad\textbf{(E) }\frac{13}{54}$

2015 Sharygin Geometry Olympiad, P17

Tags: simson line , geometry , equal angles

Let $O$ be the circumcenter of a triangle $ABC$. The projections of points $D$ and $X$ to the sidelines of the triangle lie on lines $\ell $ and $L $ such that $\ell // XO$. Prove that the angles formed by $L$ and by the diagonals of quadrilateral $ABCD$ are equal.

2023 VN Math Olympiad For High School Students, Problem 3

Tags: geometry

Given a triangle $ABC$ isosceles at $A.$ A point $P$ lying inside the triangle such that $\angle PBC=\angle PCA$ and let $M$ be the midpoint of $BC.$ Prove that: $\angle APB+ \angle MPC =180^{\circ}.$

2021 CHMMC Winter (2021-22), 2

Tags: number theory

For any positive integer $n$, let $p(n)$ be the product of its digits in base-$10$ representation. Find the maximum possible value of $\frac{p(n)}{n}$ over all integers $n \ge 10$.

2018 Malaysia National Olympiad, A6

Tags: power , number theory , prime number

Determine the smallest prime $p$ such that $2018!$ is divisible by $p^{3}$ , but not divisible by $p^{4}$.

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]