Olympiad problems

1991 AMC 8, 11

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There are several sets of three different numbers whose sum is $15$ which can be chosen from $\{ 1,2,3,4,5,6,7,8,9 \} $. How many of these sets contain a $5$? $\text{(A)}\ 3 \qquad \text{(B)}\ 4 \qquad \text{(C)}\ 5 \qquad \text{(D)}\ 6 \qquad \text{(E)}\ 7$

2023/2024 Tournament of Towns, 6

Tags: combinatorics

A table $2 \times 2024$ is filled with positive integers. Specifically, the first row is filled with numbers from the set $\{1, \ldots, 2023\}$. It turned out that for any two columns the difference of numbers from the first row is divisible by the difference of numbers from the second row, while all numbers in the second row are pairwise different. Is it true for sure that the numbers in the first row are equal? Ivan Kukharchuk

2024 CCA Math Bonanza, TB4

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Equilateral triangle $A_1A_2A_3$ has side length $15$ and circumcenter $M$. Let $N$ be a point such that $\angle A_3MN = 72^{\circ}$ and $MN = 7$. The circle with diameter $MN$ intersects lines $MA_1$, $MA_2$, and $MA_3$ again at $B_1$, $B_2$, and $B_3$, respectively. The value of $NB_1^2+NB_2^2+NB_3^2$ can be expressed as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$. [i]Tiebreaker #4[/i]

2022 Stanford Mathematics Tournament, 8

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Given that \[A=\sum_{n=1}^\infty\frac{\sin(n)}{n},\] determine $\lfloor100A\rfloor$.

2016 ASMT, 5

Tags: geometry , analytic geometry , distance

Plane $A$ passes through the points $(1, 0, 0)$, $(0, 1, 0)$, and $(0, 0, 1)$. Plane $B$ is parallel to plane $A$, but passes through the point $(1, 0, 1)$. Find the distance between planes $A$ and $B$.

2022 USAMTS Problems, 5

Tags: analytic geometry , graphing lines , slope , lattice points

A lattice point is a point on the coordinate plane with integer coefficients. Prove or disprove : there exists a finite set $S$ of lattice points such that for every line $l$ in the plane with slope $0,1,-1$, or undefined, either $l$ and $S$ intersect at exactly $2022$ points, or they do not intersect.

1966 German National Olympiad, 3

Tags: geometry , trigonometry , area of a triangle , counting , combinatorics

Consider all segments dividing the area of a triangle $ABC$ in two equal parts. Find the length of the shortest segment among them, if the side lengths $a,$ $b,$ $c$ of triangle $ABC$ are given. How many of these shortest segments exist ?

MathLinks Contest 1st, 2

Tags: complex number , inequalities , algebra

Let $m$ be the greatest number such that for any set of complex numbers having the sum of all modulus of all the elements $1$, there exists a subset having the modulus of the sum of the elements in the subset greater than $m$. Prove that $$\frac14 \le m \le \frac12.$$ (Optional Task for 3p) Find a smaller value for the RHS.

2011 All-Russian Olympiad, 3

Tags: modular arithmetic , graph theory , extremal principle , combinatorics

There are 999 scientists. Every 2 scientists are both interested in exactly 1 topic and for each topic there are exactly 3 scientists that are interested in that topic. Prove that it is possible to choose 250 topics such that every scientist is interested in at most 1 theme. [i]A. Magazinov[/i]

1998 All-Russian Olympiad Regional Round, 11.1

Tags: algebra

Two identical decks have 36 cards each. One deck is shuffled and put on top of the second. For each card of the top deck, we count the number of cards between it and the corresponding card of the bottom deck. What is the sum of these numbers? Sorry if this has been posted before but I would like to know if I solved it correctly. Thanks!

2021 Indonesia TST, N

Tags: number theory

A positive integer $n$ is said to be $interesting$ if there exist some coprime positive integers $a$ and $b$ such that $n = a^2 - ab + b^2$. Show that if $n^2$ is $interesting$, then $n$ or $3n$ is $interesting$.

2015 Taiwan TST Round 3, 2

Tags: algebra , functional inequality , polynomial

Consider all polynomials $P(x)$ with real coefficients that have the following property: for any two real numbers $x$ and $y$ one has \[|y^2-P(x)|\le 2|x|\quad\text{if and only if}\quad |x^2-P(y)|\le 2|y|.\] Determine all possible values of $P(0)$. [i]Proposed by Belgium[/i]

2013 Purple Comet Problems, 9

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Find the sum of all four-digit integers whose digits are a rearrangement of the digits $1$, $2$, $3$, $4$, such as $1234$, $1432$, or $3124$.

2005 Taiwan TST Round 2, 3

Tags: ellipse , geometry , conic

In the interior of an ellipse with major axis 2 and minor axis 1, there are more than 6 segments with total length larger than 15. Prove that there exists a line passing through all of the segments.

1998 Romania Team Selection Test, 3

Tags: number theory

Let $m\ge 2$ be an integer. Find the smallest positive integer $n>m$ such that for any partition with two classes of the set $\{ m,m+1,\ldots ,n \}$ at least one of these classes contains three numbers $a,b,c$ (not necessarily different) such that $a^b=c$. [i]Ciprian Manolescu[/i]

2007 Singapore MO Open, 4

Tags: algebra , functional equation

find all functions $f:\mathbb{N}\rightarrow\mathbb{N}$ st $f(f(m)+f(n))=m+n \,\forall m,n\in\mathbb{N}$ related: https://artofproblemsolving.com/community/c6h381298

1989 Canada National Olympiad, 3

Tags: modular arithmetic , algebra , binomial theorem , number theory , divisibility tests , logarithm

Define $ \{ a_n \}_{n\equal{}1}$ as follows: $ a_1 \equal{} 1989^{1989}; \ a_n, n > 1,$ is the sum of the digits of $ a_{n\minus{}1}$. What is the value of $ a_5$?

2008 VJIMC, Problem 2

Tags: calculus , integration , inequalities

Find all continuously differentiable functions $f:[0,1]\to(0,\infty)$ such that $\frac{f(1)}{f(0)}=e$ and $$\int^1_0\frac{\text dx}{f(x)^2}+\int^1_0f'(x)^2\text dx\le2.$$

2021 Latvia Baltic Way TST, P2

Tags: inequalities

Determine all functions $f: \mathbb{R} \backslash \{0 \} \rightarrow \mathbb{R}$ such that, for all nonzero $x$: $$ f(\frac{1}{x}) \ge 1 -f(x) \ge x^2f(x) $$

2010 Princeton University Math Competition, 8

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Matt is asked to write the numbers from 1 to 10 in order, but he forgets how to count. He writes a permutation of the numbers $\{1, 2, 3\ldots , 10\}$ across his paper such that: [list] [*]The leftmost number is 1. [*]The rightmost number is 10. [*]Exactly one number (not including 1 or 10) is less than both the number to its immediate left and the number to its immediate right.[/list] How many such permutations are there?

2004 Mexico National Olympiad, 2

Tags: integer , number theory , inequality

Find the maximum number of positive integers such that any two of them $a, b$ (with $a \ne b$) satisfy that$ |a - b| \ge \frac{ab}{100} .$

2006 Alexandru Myller, 1

Tags: binomial coefficient , modular arithmetic , number theory

For an odd prime $ p, $ show that $ \sum_{k=1}^{p-1} \frac{k^p-k}{p}\equiv \frac{1+p}{2}\pmod p . $

2007 Singapore Team Selection Test, 2

Tags: geometry

Let $ABCD$ be a convex quadrilateral inscribed in a circle with $M$ and $N$ the midpoints of the diagonals $AC$ and $BD$ respectively. Suppose that $AC$ bisects $\angle BMD$. Prove that $BD$ bisects $\angle ANC$.

1991 AIME Problems, 15

Tags: ratio , rotation , inequalities , trigonometry , triangle inequality

For positive integer $n$, define $S_n$ to be the minimum value of the sum \[ \sum_{k=1}^n \sqrt{(2k-1)^2+a_k^2}, \] where $a_1,a_2,\ldots,a_n$ are positive real numbers whose sum is 17. There is a unique positive integer $n$ for which $S_n$ is also an integer. Find this $n$.

2024 Stars of Mathematics, P3

Tags: combinatorics

Let $\mathcal{P}$ be a partition of $\{1,2,\dots ,2024\}$ into sets of two elements, such that for any $\{a,b\}\in\mathcal{P}$, either $|a-b|=1$ or $|a-b|=506$. Suppose that $\{1518,1519\}\in\mathcal{P}$. Determine the pair of $505$ in the partition.