Olympiad problems

2020 USAMTS Problems, 2:

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Find distinct points $A, B, C,$ and $D$ in the plane such that the length of the segment $AB$ is an even integer, and the lengths of the segments $AC, AD, BC, BD,$ and $CD$ are all odd integers. In addition to stating the coordinates of the points and distances between points, please include a brief explanation of how you found the configuration of points and computed the distances.

1954 Moscow Mathematical Olympiad, 274

Solve the system $\begin{cases} 10x_1 + 3x_2 + 4x_3 + x_4 + x_5 = 0 \\ 11x_2 + 2x_3 + 2x_4 + 3x_5 + x_6 = 0 \\ 15x_3 + 4x_4 + 5x_5 + 4x_6 + x_7 = 0 \\ 2x_1 + x_2 - 3x_3 + 12x_4 - 3x_5 + x_6 + x_7 = 0 \\ 6x_1 - 5x_2 + 3x_3 - x_4 + 17x_5 + x_6 = 0 \\ 3x_1 + 2x_2 - 3x_3 + 4x_4 + x_5 - 16x_6 + 2x_7 = 0\\ 4x_1 - 8x_2 + x_3 + x_4 + 3x_5 + 19x_7 = 0 \end{cases}$

2006 Tournament of Towns, 1

Tags: geometry , incircle , circumcircle , area , heptagon , 17-gon , regular polygon

Two regular polygons, a $7$-gon and a $17$-gon are given. For each of them two circles are drawn, an inscribed circle and a circumscribed circle. It happened that rings containing the polygons have equal areas. Prove that sides of the polygons are equal. (3)

2007 Estonia National Olympiad, 4

Tags: limit , number theory

Find all pairs $ (m, n)$ of positive integers such that $ m^n \minus{} n^m \equal{} 3$.

2004 District Olympiad, 4

Tags: algebra , complex number

If $x,y \in (0, \frac{\pi}{2})$ such as $ (cosx+isiny)^n=cos(nx)+isin(ny)$ for two consecutive positive integers, then the relation is true for all positive integers.

2021 Alibaba Global Math Competition, 18

Tags: modular arithmetic , linear algebra , vector

Let $p$ be an odd prime number, and let $m \ge 0$ and $N \ge 1$ be integers. Let $\Lambda$ be a free $\mathbb{Z}/p^N\mathbb{Z}$-module of rank $2m+1$, equipped with a perfect symmetric $\mathbb{Z}/p^N\mathbb{Z}$-bilinear form \[(\, ,\,): \Lambda \times \Lambda \to \mathbb{Z}/p^N\mathbb{Z}.\] Here ``perfect'' means that the induced map \[\Lambda \to \text{Hom}_{\mathbb{Z}/p^N\mathbb{Z}}(\Lambda, \mathbb{Z}/p^N\mathbb{Z}), \quad x \mapsto (x,\cdot)\] is an isomorphism. Find the cardinality of the set \[\{x \in \Lambda: (x,x)=0\},\] expressed in terms of $p,m,N$.

2012 Today's Calculation Of Integral, 838

Tags: calculus , integration , calculus computations , inequalities

Prove that : $\frac{e-1}{e}<\int_0^1 e^{-x^2}dx<\frac{\pi}{4}.$

2009 Irish Math Olympiad, 4

Tags: combinatorics

At a strange party, each person knew exactly $22$ others. For any pair of people $X$ and $Y$ who knew each other, there was no other person at the party that they both knew. For any pair of people $X$ and $Y$ who did not know one another, there were exactly $6$ other people that they both knew. How many people were at the party?

2004 Thailand Mathematical Olympiad, 19

Tags: inequalities , algebra , sum , max

Find positive reals $a, b, c$ which maximizes the value of $a+ 2b+ 3c$ subject to the constraint that $9a^2 + 4b^2 + c^2 = 91$

2002 Portugal MO, 1

Tags: combinatorics

The keyword that Ana Viso chose for her computer has the $7$ characters of her name: A, N, A, V, I, S, O. Sorting all the different words alphabetically formed by all these $7$ characters, Ana's keyword appears in the $881$st position. What it is Ana's keyword?

2018 Iran Team Selection Test, 4

Tags: number theory

We say distinct positive integers $a_1,a_2,\ldots ,a_n $ are "good" if their sum is equal to the sum of all pairwise $\gcd $'s among them. Prove that there are infinitely many $n$ s such that $n$ good numbers exist. [i]Proposed by Morteza Saghafian[/i]

2009 VJIMC, Problem 3

Tags: matrix , linear algebra

Let $A$ be an $n\times n$ square matrix with integer entries. Suppose that $p^2A^{p^2}=q^2A^{q^2}+r^2I_n$ for some positive integers $p,q,r$ where $r$ is odd and $p^2=q^2+r^2$. Prove that $|\det A|=1$. (Here $I_n$ means the $n\times n$ identity matrix.)

2024 Argentina National Math Olympiad Level 3, 4

Tags: combinatorics

On a table, there are $10\,000$ matches, two of which are inside a box. Ana and Beto take turns playing the following game. On each turn, a player adds to the box a number of matches equal to a proper divisor of the current number of matches in the box. The game ends when, for the first time, there are more than $2024$ matches in the box and the person who played the last turn is the winner. If Ana starts the game, determine who has a winning strategy.

2015 Purple Comet Problems, 3

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The repeating decimal $2.0151515\ldots$ can be expressed as $\tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.

Kvant 2020, M2590

Tags: geometry , area

In an acute triangle $ABC$ the point $O{}$ is the circumcenter, $H_1$ is the foot of the perpendicular from $A{}$ onto $BC$, and $M_H$ and $N_H$ are the projections of $H_1$ on $AC$ and $AB{}$, respectively. Prove that the polyline $M_HON_H$ divides the triangle $ABC$ in two figures of equal area. [i]Proposed by I. A. Kushner[/i]

2022 Indonesia MO, 2

Tags: number theory , algebra , polynomial

Let $P(x)$ be a polynomial with integer coefficient such that $P(1) = 10$ and $P(-1) = 22$. (a) Give an example of $P(x)$ such that $P(x) = 0$ has an integer root. (b) Suppose that $P(0) = 4$, prove that $P(x) = 0$ does not have an integer root.

2011 Irish Math Olympiad, 5

Tags: combinatorics

In the mathematical talent show called “The $X^2$-factor” contestants are scored by a a panel of $8$ judges. Each judge awards a score of $0$ (‘fail’), $X$ (‘pass’), or $X^2$ (‘pass with distinction’). Three of the contestants were Ann, Barbara and David. Ann was awarded the same score as Barbara by exactly $4$ of the judges. David declares that he obtained different scores to Ann from at least $4$ of the judges, and also that he obtained different scores to Barbara from at least $4$ judges. In how many ways could scores have been allocated to David, assuming he is telling the truth?

2024 Austrian MO National Competition, 4

Tags: powerful numbers , number theory

A positive integer is called [i]powerful [/i]if all exponents in its prime factorization are $\ge 2$. Prove that there are infinitely many pairs of powerful consecutive positive integers. [i](Walther Janous)[/i]

2019 Durer Math Competition Finals, 2

Tags: number theory , right triangle , geometry , sidelengths , prime , integer

Prove that if a triangle has integral side lengths and its circumradius is a prime number then the triangle is right-angled.

2016 Online Math Open Problems, 4

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Given that $x$ is a real number, find the minimum value of $f(x)=|x+1|+3|x+3|+6|x+6|+10|x+10|.$ [i]Proposed by Yannick Yao[/i]

2017 Iran MO (3rd round), 3

Tags: number theory

Let $n$ be a positive integer. Prove that there exists a poisitve integer $m$ such that $$7^n \mid 3^m+5^m-1$$

1972 IMO Shortlist, 9

Tags: inequality system , algebra

Find all positive real solutions to: \begin{eqnarray*} (x_1^2-x_3x_5)(x_2^2-x_3x_5) &\le& 0 \\ (x_2^2-x_4x_1)(x_3^2-x_4x_1) &\le& 0 \\ (x_3^2-x_5x_2)(x_4^2-x_5x_2) &\le& 0 \\ (x_4^2-x_1x_3)(x_5^2-x_1x_3) &\le & 0 \\ (x_5^2-x_2x_4)(x_1^2-x_2x_4) &\le& 0 \\ \end{eqnarray*}

2017 Irish Math Olympiad, 5

Tags: algebra , sequence , sum

Given a positive integer $m$, a sequence of real numbers $a= (a_1,a_2,a_3,...)$ is called $m$-powerful if it satisfies $$(\sum_{k=1}^{n} a_k )^{m} = \sum_{k=1}^{n} a_k^{m}$$for all positive integers $n$. (a) Show that a sequence is $30$-powerful if and only if at most one of its terms is non-zero. (b) Find a sequence none of whose terms are zero but which is $2017$-powerful.

1972 All Soviet Union Mathematical Olympiad, 160

Tags: combinatorial geometry

Given $50$ segments on the line. Prove that one of the following statements is valid: 1. Some $8$ segments have the common point. 2. Some $8$ segments do not intersect each other.

1991 All Soviet Union Mathematical Olympiad, 551

Tags: number theory , recurrence relation , sequence , integer sequence , game , game strategy , algebra

A sequence of positive integers is constructed as follows. If the last digit of $a_n$ is greater than $5$, then $a_{n+1}$ is $9a_n$. If the last digit of $a_n$ is $5$ or less and an has more than one digit, then $a_{n+1}$ is obtained from $a_n$ by deleting the last digit. If $a_n$ has only one digit, which is $5$ or less, then the sequence terminates. Can we choose the first member of the sequence so that it does not terminate?