This website contains problems from math contests. Problems and corresponding tags were obtained from the Art of Problem Solving website.

Tags were heavily modified to better represent problems.

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Found problems: 85335

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 $\max\{\lfloor18.5-\tfrac{|A-C|^{1.8}}{40}\rfloor,0\}.$

2022 New Zealand MO, 4
Tags: geometry , incircle , right triangle
Triangle $ABC$ is right-angled at $B$ and has incentre $I$. Points $D$, $E$ and $F$ are the points where the incircle of the triangle touches the sides $BC$, $AC$ and AB respectively. Lines $CI$ and $EF$ intersect at point $P$. Lines $DP$ and $AB$ intersect at point $Q$. Prove that $AQ = BF$.

1985 IMO Longlists, 27
Tags: trigonometry , geometry
Let $O$ be a point on the oriented Euclidean plane and $(\mathbf i, \mathbf j)$ a directly oriented orthonormal basis. Let $C$ be the circle of radius $1$, centered at $O$. For every real number $t$ and non-negative integer$ n$ let $M_n$ be the point on $C$ for which $\langle \mathbf i , \overrightarrow{OM_n} \rangle = \cos 2^n t.$ (or $\overrightarrow{OM_n} =\cos 2^n t \mathbf i +\sin 2^n t \mathbf j$). Let $k \geq 2$ be an integer. Find all real numbers $t \in [0, 2\pi)$ that satisfy [b](i)[/b] $M_0 = M_k$, and [b](ii)[/b] if one starts from $M0$ and goes once around $C$ in the positive direction, one meets successively the points $M_0,M_1, \dots,M_{k-2},M_{k-1}$, in this order.

2011 Olympic Revenge, 3
Tags: ellipse , analytic geometry , conic , combinatorics
Let $E$ to be an infinite set of congruent ellipses in the plane, and $r$ a fixed line. It is known that each line parallel to $r$ intersects at least one ellipse belonging to $E$. Prove that there exist infinitely many triples of ellipses belonging to $E$, such that there exists a line that intersect the triple of ellipses.

1973 All Soviet Union Mathematical Olympiad, 184
Tags: combinatorics , chessboard
The king have revised the chess-board $8\times 8$ having visited all the fields once only and returned to the starting point. When his trajectory was drawn (the centres of the squares were connected with the straight lines), a closed broken line without self-intersections appeared. a) Give an example that the king could make $28$ steps parallel the sides of the board only. b) Prove that he could not make less than $28$ such a steps. c) What is the maximal and minimal length of the broken line if the side of a field is $1$?

1986 IMO Longlists, 73
Tags: limit , algebra
Let $(a_i)_{i\in \mathbb N}$ be a strictly increasing sequence of positive real numbers such that $\lim_{i \to \infty} a_i = +\infty$ and $a_{i+1}/a_i \leq 10$ for each $i$. Prove that for every positive integer $k$ there are infinitely many pairs $(i, j)$ with $10^k \leq a_i/a_j \leq 10^{k+1}.$

2012 May Olympiad, 1
Tags:
A four digit number is called [i]stutterer[/i] if its first two digits are the same and its last two digits are also the same, e.g. $3311$ and $2222$ are stutterer numbers. Find all stutterer numbers that are square numbers.