Olympiad problems

2010 Mathcenter Contest, 2

A positive rational number $x$ is called [i]banzai [/i] if the following conditions are met: $\bullet$ $x=\frac{p}{q}>1$ where $p,q$ are comprime natural numbers $\bullet$ exist constants $\alpha,N$ such that for all integers $n\geq N$,$$\mid \left\{\,x^n\right\} -\alpha\mid \leq \dfrac{1}{2(p+q)}.$$ Find the total number of banzai numbers. Note:$\left\{\,x\right\}$ means fractional part of $x$ [i](tatari/nightmare)[/i]

2009 National Olympiad First Round, 28

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We divide entire $ Z$ into $ n$ subsets such that difference of any two elements in a subset will not be a prime number. $ n$ is at least ? $\textbf{(A)}\ 6 \qquad\textbf{(B)}\ 5 \qquad\textbf{(C)}\ 4 \qquad\textbf{(D)}\ 3 \qquad\textbf{(E)}\ \text{None}$

1998 Baltic Way, 3

Tags: number theory

Find all positive integer solutions to $2x^2+5y^2=11(xy-11)$.

1991 AMC 12/AHSME, 30

Tags: percent

For any set $S$, let $|S|$ denote the number of elements in $S$, and let $n(S)$ be the number of subsets of $S$, including the empty set and the set $S$ itself. If $A$, $B$ and $C$ are sets for which \[n(A) + n(B) + n(C) = n(A \cup B \cup C)\quad\text{and}\quad |A| = |B| = 100,\] then what is the minimum possible value of $|A \cap B \cap C|$? $ \textbf{(A)}\ 96\qquad\textbf{(B)}\ 97\qquad\textbf{(C)}\ 98\qquad\textbf{(D)}\ 99\qquad\textbf{(E)}\ 100 $

2018 Portugal MO, 2

Tags: geometry , incircle , square

In the figure, $[ABCD]$ is a square of side $1$. The points $E, F, G$ and $H$ are such that $[AFB], [BGC], [CHD]$ and $[DEA]$ are right-angled triangles. Knowing that the circles inscribed in each of these triangles and the circle inscribed in the square $[EFGH]$ has all the same radius, what is the measure of the radius of the circles? [img]https://1.bp.blogspot.com/-l37AEXa7_-c/X4KaJwe6HQI/AAAAAAAAMk4/14wvIipf26cRge_GqKSRwH32bp291vX4QCLcBGAsYHQ/s0/2018%2Bportugal%2Bp2.png[/img]

1990 Swedish Mathematical Competition, 1

Tags: sum , number theory , divisor

Let $d_1, d_2, ... , d_k$ be the positive divisors of $n = 1990!$. Show that $\sum \frac{d_i}{\sqrt{n}} = \sum \frac{\sqrt{n}}{d_i}$.

2005 All-Russian Olympiad, 1

Tags: symmetry , number theory

Ten mutually distinct non-zero reals are given such that for any two, either their sum or their product is rational. Prove that squares of all these numbers are rational.

1979 Putnam, A4

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Let $A$ be a set of $2n$ points in the plane, no three of which are collinear. Suppose that $n$ of them are colored red and the remaining $n$ blue. Prove or disprove: there are $n$ closed straight line segments, no two with a point in common, such that the endpoints of each segment are points of $A$ having different colors.

2010 Kazakhstan National Olympiad, 3

Tags: inequalities

Positive real $A$ is given. Find maximum value of $M$ for which inequality $ \frac{1}{x}+\frac{1}{y}+\frac{A}{x+y} \geq \frac{M}{\sqrt{xy}} $ holds for all $x, y>0$

2021 AMC 10 Spring, 6

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Chantal and Jean start hiking from a trailhead toward a fire tower. Jean is wearing a heavy backpack and walks slower. Chantal starts walking at $4$ miles per hour. Halfway to the tower, the trail becomes really steep, and Chantal slows down to $2$ miles per hour. After reaching the tower, she immediately turns around and descends the steep part of the trail at $3$ miles per hour. She meets Jean at the halfway point. What was Jean's average speed, in miles per hour, until they meet? $\textbf{(A)}~\frac{12}{13}\qquad \textbf{(B)}~1\qquad \textbf{(C)}~\frac{13}{12}\qquad \textbf{(D)}~\frac{24}{13}\qquad \textbf{(E)}~2$

2015 Turkey EGMO TST, 2

Tags: geometry

Let $D$ be the midpoint of the side $BC$ of a triangle $ABC$ and $P$ be a point inside the $ABD$ satisfying $\angle PAD=90^\circ - \angle PBD=\angle CAD$. Prove that $\angle PQB=\angle BAC$, where $Q$ is the intersection point of the lines $PC$ and $AD$.

2002 Estonia National Olympiad, 5

Tags: combinatorics

There is a lottery at Juku’s birthday party with a number of identical prizes, where each guest can win at most one prize. It is known that if there was one prize less, then the number of possible distributions of the prizes among the guests would be $50\%$ less than it actually is, while if there was one prize more, then the number of possible distributions of the prizes would be $50\%$ more than it actually is. Find the number of possible distributions of the prizes.

2001 India IMO Training Camp, 3

Tags: polynomial , inequalities , induction , algebra

Let $P(x)$ be a polynomial of degree $n$ with real coefficients and let $a\geq 3$. Prove that \[\max_{0\leq j \leq n+1}\left | a^j-P(j) \right |\geq 1\]

2019 Ecuador NMO (OMEC), 5

Tags: number theory , diophantine

Let $a, b, c$ be integers not all the same with $a, b, c\ge 4$ that satisfy $$4abc = (a + 3) (b + 3) (c + 3).$$ Find the numerical value of $a + b + c$.

2017 Purple Comet Problems, 8

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The positive integer $m$ is a multiple of 111, and the positive integer $n$ is a multiple of 31. Their sum is 2017. Find $n - m$.

2007 Today's Calculation Of Integral, 252

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

Compare $ \displaystyle f(\theta) \equal{} \int_0^1 (x \plus{} \sin \theta)^2\ dx$ and $ \ g(\theta) \equal{} \int_0^1 (x \plus{} \cos \theta)^2\ dx$ for $ 0\leqq \theta \leqq 2\pi .$

2014 Romania National Olympiad, 4

Tags: complex number , inequalities

Let $n \in \mathbb{N} , n \ge 2$ and $ a_0,a_1,a_2,\cdots,a_n \in \mathbb{C} ; a_n \not = 0 $. Then: [b][size=100][i]P.[/i][/size][/b] $|a_nz^n + a_{n-1}z^z{n-1} + \cdots + a_1z + a_0 | \le |a_n+a_0|$ for any $z \in \mathbb{C}, |z|=1$ [b][size=100][i]Q[/i][/size][/b]. $a_1=a_2=\cdots=a_{n-1}=0$ and $a_0/a_n \in [0,\infty)$ Prove that $ P \Longleftrightarrow Q$

2025 Harvard-MIT Mathematics Tournament, 3

Tags: combinatorics

Ben has $16$ balls labeled $1, 2, 3, \ldots, 16,$ as well as $4$ indistinguishable boxes. Two balls are [i]neighbors[/i] if their labels differ by $1.$ Compute the number of ways for him to put $4$ balls in each box such that each ball is in the same box as at least one of its neighbors. (The order in which the balls are placed does not matter.)

1982 IMO Longlists, 17

Tags: sequence , maximization , minimization , rearrangement , algebra

[b](a)[/b] Find the rearrangement $\{a_1, \dots , a_n\}$ of $\{1, 2, \dots, n\}$ that maximizes \[a_1a_2 + a_2a_3 + \cdots + a_na_1 = Q.\] [b](b)[/b] Find the rearrangement that minimizes $Q.$

2011 IberoAmerican, 2

Tags: number theory

Find all positive integers $n$ for which exist three nonzero integers $x, y, z$ such that $x+y+z=0$ and: \[\frac{1}{x}+\frac{1}{y}+\frac{1}{z}=\frac{1}{n}\]

2017 Harvard-MIT Mathematics Tournament, 2

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How many ways are there to insert $+$'s between the digits of $111111111111111$ (fifteen $1$'s) so that the result will be a multiple of $30$?

Oliforum Contest III 2012, 4

Tags: algebra , inequalities

Show that if $a \ge b \ge c \ge 0$ then $$a^2b(a - b) + b^2c(b - c) + c^2a(c - a) \ge 0.$$

2007 Indonesia TST, 1

Tags: circumcircle , cyclic quadrilateral , angle bisector , geometry

Let $ ABCD$ be a cyclic quadrilateral and $ O$ be the intersection of diagonal $ AC$ and $ BD$. The circumcircles of triangle $ ABO$ and the triangle $ CDO$ intersect at $ K$. Let $ L$ be a point such that the triangle $ BLC$ is similar to $ AKD$ (in that order). Prove that if $ BLCK$ is a convex quadrilateral, then it has an incircle.

2007 All-Russian Olympiad, 8

Tags: number theory

Dima has written number $ 1/80!,\,1/81!,\,\dots,1/99!$ on $ 20$ infinite pieces of papers as decimal fractions (the following is written on the last piece: $ \frac {1}{99!} \equal{} 0{,}{00\dots 00}10715\dots$, 155 0-s before 1). Sasha wants to cut a fragment of $ N$ consecutive digits from one of pieces without the comma. For which maximal $ N$ he may do it so that Dima may not guess, from which piece Sasha has cut his fragment? [i]A. Golovanov[/i]

2004 Iran MO (2nd round), 3

Tags: ceiling function , combinatorics

The road ministry has assigned $80$ informal companies to repair $2400$ roads. These roads connect $100$ cities to each other. Each road is between $2$ cities and there is at most $1$ road between every $2$ cities. We know that each company repairs $30$ roads that it has agencies in each $2$ ends of them. Prove that there exists a city in which $8$ companies have agencies.