Olympiad problems

2002 Romania Team Selection Test, 1

Let $ABCD$ be a unit square. For any interior points $M,N$ such that the line $MN$ does not contain a vertex of the square, we denote by $s(M,N)$ the least area of the triangles having their vertices in the set of points $\{ A,B,C,D,M,N\}$. Find the least number $k$ such that $s(M,N)\le k$, for all points $M,N$. [i]Dinu Șerbănescu[/i]

1998 Harvard-MIT Mathematics Tournament, 7

Tags: polynomial , vieta , algebra

Given that three roots of $f(x)=x^4+ax^2+bx+c$ are $2$, $-3$, and $5$, what is the value of $a+b+c$?

2012 India Regional Mathematical Olympiad, 2

Tags: power , number theory , divide , divisible

Let $a,b,c$ be positive integers such that $a|b^4, b|c^4$ and $c|a^4$. Prove that $abc|(a+b+c)^{21}$

2017 Kyiv Mathematical Festival, 2

Tags: geometry , triangle

A triangle $ABC$ is given. Let $D$ be a point on the extension of the segment $AB$ beyond $A$ such that $AD=BC,$ and $E$ be a point on the extension of the segment $BC$ beyond $B$ such that $BE=AC.$ Prove that the circumcircle of the triangle $DEB$ passes through the incenter of the triangle $ABC.$

2012 Poland - Second Round, 1

Tags: function , algebra , functional equation

$f,g:\mathbb{R}\rightarrow\mathbb{R}$ find all $f,g$ satisfying $\forall x,y\in \mathbb{R}$: \[g(f(x)-y)=f(g(y))+x.\]

2021 Iran MO (3rd Round), 3

Tags: pigeonhole principle , c2

Let $n\ge 3$ be a fixed integer. There are $m\ge n+1$ beads on a circular necklace. You wish to paint the beads using $n$ colors, such that among any $n+1$ consecutive beads every color appears at least once. Find the largest value of $m$ for which this task is $\emph{not}$ possible. [i]Carl Schildkraut, USA[/i]

2010 IMAC Arhimede, 2

Tags: algebra , functional equation

Find all functions $ f: \mathbb{R}\to\mathbb{R}$ such that we have $f(x + y) = f(x) + f(y) + f(xy)$ for all $ x,y\in \mathbb{R}$

2024 AMC 8 -, 25

Tags: probability

A small airplane has $4$ rows of seats with $3$ seats in each row. Eight passengers have boarded the plane and are distributed randomly among the seats. A married couple is next to board. What is the probability there will be 2 adjacent seats in the same row for the couple?

1996 ITAMO, 6

Tags: combinatorics

What is the minimum number of squares that is necessary to draw on a white sheet to obtain a square grid of side $n$?

1989 IMO, 4

Tags: geometry , geometric inequality , convex quadrilateral

Let $ ABCD$ be a convex quadrilateral such that the sides $ AB, AD, BC$ satisfy $ AB \equal{} AD \plus{} BC.$ There exists a point $ P$ inside the quadrilateral at a distance $ h$ from the line $ CD$ such that $ AP \equal{} h \plus{} AD$ and $ BP \equal{} h \plus{} BC.$ Show that: \[ \frac {1}{\sqrt {h}} \geq \frac {1}{\sqrt {AD}} \plus{} \frac {1}{\sqrt {BC}} \]

I Soros Olympiad 1994-95 (Rus + Ukr), 11.3

Tags: algebra , polynomial

For each non-negative $a$, consider the equation $$x^3 + ax - a^3 - 29 = 0.$$ Let $x_o$ be the positive root of this equation. Prove that for all $a > 0$ such a root exists. What is the smallest value of $x_o$?

Kvant 2023, M2740

Tags: number theory

Let $a, b, c$ be positive integers such that no number divides some other number. If $ab-b+1 \mid abc+1$, prove that $c \geq b$.

2010 Harvard-MIT Mathematics Tournament, 8

Tags: algebra , polynomial

How many polynomials of degree exactly $5$ with real coefficients send the set $\{1, 2, 3, 4, 5, 6\}$ to a permutation of itself?

2018 District Olympiad, 1

Tags: function , continuity , supremum

Let $\mathcal{F}$ be the set of continuous functions $f : [0, 1]\to\mathbb{R}$ satisfying $\max_{0\le x\le 1} |f(x)| = 1$ and let $I : \mathcal{F} \to \mathbb{R}$, \[I(f) = \int_0^1 f(x)\, \text{d}x - f(0) + f(1).\] a) Show that $I(f) < 3$, for any $f \in \mathcal{F}$. b) Determine $\sup\{I(f) \mid f \in \mathcal{F}\}$.

1990 Tournament Of Towns, (246) 4

Tags: weighing , combinatorics

A set of $61$ coins that look alike is given. Two coins (whose weights are equal) are counterfeit. The other $59$ (genuine) coins also have the same weight, but a different weight from that of the counterfeit coins. However it is not known whether it is the genuine coins or the counterfeit coins which are heavier. How can this question be resolved by three weighings on the one balance? (It is not required to separate the counterfeit coins from the genuine ones.) (D. Fomin, Leningrad)

1968 IMO Shortlist, 4

Tags: algebra , system of equations , polynomial , discriminant

Let $a,b,c$ be real numbers with $a$ non-zero. It is known that the real numbers $x_1,x_2,\ldots,x_n$ satisfy the $n$ equations: \[ ax_1^2+bx_1+c = x_{2} \]\[ ax_2^2+bx_2 +c = x_3\]\[ \ldots \quad \ldots \quad \ldots \quad \ldots\]\[ ax_n^2+bx_n+c = x_1 \] Prove that the system has [b]zero[/b], [u]one[/u] or [i]more than one[/i] real solutions if $(b-1)^2-4ac$ is [b]negative[/b], equal to [u]zero[/u] or [i]positive[/i] respectively.

2017 USAMO, 1

Tags: number theory

Prove that there are infinitely many distinct pairs $(a, b)$ of relatively prime integers $a>1$ and $b>1$ such that $a^b+b^a$ is divisible by $a+b$.

1955 Miklós Schweitzer, 6

Tags: number theory

[b]6.[/b] For a prime factorisation of a positive integer $N$ let us call the exponent of a prime $p$ the integer $k$ for which $p^{k} \mid N$ but $p^{k+1} \nmid N$; let, further, the power $p^{k}$ be called the "contribution" of $p$ to $N$. Show that for any positive integer $n$ and for any primes $p$ and $q$ the contibution of $p$ to $n!$ is greater than the contribution of $q$ if and only if the exponent of $p$ is greater than that of $q$.

1995 USAMO, 5

Tags: pigeonhole principle , combinatorics

Suppose that in a certain society, each pair of persons can be classified as either [i]amicable [/i]or [i]hostile[/i]. We shall say that each member of an amicable pair is a [i]friend[/i] of the other, and each member of a hostile pair is a [i]foe[/i] of the other. Suppose that the society has $\, n \,$ persons and $\, q \,$ amicable pairs, and that for every set of three persons, at least one pair is hostile. Prove that there is at least one member of the society whose foes include $\, q(1 - 4q/n^2) \,$ or fewer amicable pairs.

2012 NIMO Summer Contest, 14

Tags: analytic geometry , graphing lines , slope

A set of lattice points is called [i]good[/i] if it does not contain two points that form a line with slope $-1$ or slope $1$. Let $S = \{(x, y)\ |\ x, y \in \mathbb{Z}, 1 \le x, y \le 4\}$. Compute the number of non-empty good subsets of $S$. [i]Proposed by Lewis Chen[/i]

1998 Akdeniz University MO, 1

Tags: number theory , equation , perfect square

Prove that, for $k \in {\mathbb Z^+}$ $$k(k+1)(k+2)(k+3)$$ is not a perfect square.

2019 Oral Moscow Geometry Olympiad, 4

Tags: excircle , geometry , angle , right triangle

Given a right triangle $ABC$ ($\angle C=90^o$). The $C$-excircle touches the hypotenuse $AB$ at point $C_1, A_1$ is the touchpoint of $B$-excircle with line $BC, B_1$ is the touchpoint of $A$-excircle with line $AC$. Find the angle $\angle A_1C_1B_1$.

2022 CCA Math Bonanza, I12

Tags:

Find the number of 8-tuples of binary inputs $\{A, B, C, D, E, F, G, H\}$ such that \[ \{ (A \text{ AND } B)\text{ OR } (C \text{ AND } D)\} \text{ AND } \{ (E \text{ AND } F)\text{ OR } (G \text{ AND } H)\}\]\[ = \{ (A \text{ OR } B)\text{ AND } (C \text{ OR } D)\} \text{ OR } \{ (E \text{ OR } F)\text{ AND } (G \text{ OR } H)\}\] The AND gates produce an output that is ON only if both the inputs are ON, and the OR gates produce an output that is OFF only if both inputs are OFF. [i]2022 CCA Math Bonanza Individual Round #12[/i]

2008 Baltic Way, 17

Tags: inequalities , geometry

Assume that $ a$, $ b$, $ c$ and $ d$ are the sides of a quadrilateral inscribed in a given circle. Prove that the product $ (ab \plus{} cd)(ac \plus{} bd)(ad \plus{} bc)$ acquires its maximum when the quadrilateral is a square.

2018 PUMaC Live Round, Estimation 3

Tags: probability , expected value

Andrew starts with the $2018$-tuple of binary digits $(0,0,\dots,0)$. On each turn, he randomly chooses one index (between $1$ and $2018$) and flips the digit at that index (makes it $1$ if it was a $0$ and vice versa). What is the smallest $k$ such that, after $k$ steps, the expected number of ones in the sequence is greater than $1008?$ You must give your answer as a nonnegative integer. If your answer is $A$ and the correct answer is $C$, then your score will be $\max\{\lfloor18.5-\tfrac{|A-C|^{1.8}}{40}\rfloor,0\}.$