Olympiad problems

2001 National Olympiad First Round, 7

Tags:

How many ordered triples of positive integers $(a,b,c)$ are there such that $(2a+b)(2b+a)=2^c$? $ \textbf{(A)}\ 0 \qquad\textbf{(B)}\ 1 \qquad\textbf{(C)}\ 2 \qquad\textbf{(D)}\ 3 \qquad\textbf{(E)}\ \text{None of the preceding} $

1938 Eotvos Mathematical Competition, 1

Prove that an integer $n$ can be expressed as the sum of two squares if and only if $2n$ can be expressed as the sum of two squares.

2017 Harvard-MIT Mathematics Tournament, 9

Tags: roots of unity , number theory

Let $n$ be an odd positive integer greater than $2$, and consider a regular $n$-gon $\mathcal{G}$ in the plane centered at the origin. Let a [i]subpolygon[/i] $\mathcal{G}'$ be a polygon with at least $3$ vertices whose vertex set is a subset of that of $\mathcal{G}$. Say $\mathcal{G}'$ is [i]well-centered[/i] if its centroid is the origin. Also, say $\mathcal{G}'$ is [i]decomposable[/i] if its vertex set can be written as the disjoint union of regular polygons with at least $3$ vertices. Show that all well-centered subpolygons are decomposable if and only if $n$ has at most two distinct prime divisors.

2012 Tournament of Towns, 2

Tags: number theory , divisor

The number $4$ has an odd number of odd positive divisors, namely $1$, and an even number of even positive divisors, namely $2$ and $4$. Is there a number with an odd number of even positive divisors and an even number of odd positive divisors?

2010 CHMMC Winter, 4

Tags: number theory

Compute the number of integer solutions $(x, y)$ to $xy - 18x - 35y = 1890$.

2020 Tournament Of Towns, 4

Tags: algebra , polynomial

We say that a nonconstant polynomial $p(x)$ with real coefficients is split into two squares if it is represented as $a(x) +b(x)$ where $a(x)$ and $b(x)$ are squares of polynomials with real coefficients. Is there such a polynomial $p(x)$ that it may be split into two squares: a) in exactly one way; b) in exactly two ways? Note: two splittings that differ only in the order of summands are considered to be the same. Sergey Markelov

Kyiv City MO 1984-93 - geometry, 1992.7.2

Tags: geometry , equilateral , right angle

Inside a right angle is given a point $A$. Construct an equilateral triangle, one of the vertices of which is point $A$, and two others lie on the sides of the angle (one on each side).

2018 Malaysia National Olympiad, B2

Tags: number theory , digit

Let $a$ and $b$ be positive integers such that (i) both $a$ and $b$ have at least two digits; (ii) $a + b$ is divisible by $10$; (iii) $a$ can be changed into $b$ by changing its last digit. Prove that the hundreds digit of the product $ab$ is even.

2015 BMT Spring, 16

Tags: combinatorics

A binary decision tree is a list of $n$ yes/no questions, together with instructions for the order in which they should be asked (without repetition). For instance, if $n = 3$, there are $12$ possible binary decision trees, one of which asks question $2$ first, then question $3$ (followed by question $ 1$) if the answer was yes or question $1$ (followed by question $3$) if the answer was no. Determine the greatest possible $k$ such that $2^k$ divides the number of binary decision trees on $n = 13$ questions.

1958 AMC 12/AHSME, 27

Tags: analytic geometry , graphing lines , slope

The points $ (2,\minus{}3)$, $ (4,3)$, and $ (5, k/2)$ are on the same straight line. The value(s) of $ k$ is (are): $ \textbf{(A)}\ 12\qquad \textbf{(B)}\ \minus{}12\qquad \textbf{(C)}\ \pm 12\qquad \textbf{(D)}\ {12}\text{ or }{6}\qquad \textbf{(E)}\ {6}\text{ or }{6\frac{2}{3}}$

2021 Indonesia TST, G

Tags: geometric transformation , geometry

Let $P$ be a point in the plane of $\triangle ABC$, and $\gamma$ a line passing through $P$. Let $A', B', C'$ be the points where the reflections of lines $PA, PB, PC$ with respect to $\gamma$ intersect lines $BC, AC, AB$ respectively. Prove that $A', B', C'$ are collinear.

1996 Cono Sur Olympiad, 3

Tags: combinatorics , algebra

A shop sells bottles with this capacity: $1L, 2L, 3L,..., 1996L$, the prices of bottles satifies this $2$ conditions: $1$. Two bottles have the same price, if and only if, your capacities satifies $m - n = 1000$ $2$. The price of bottle $m$($1001>m>0$) is $1996 - m$ dollars. Find all pair(s) $m$ and $n$ such that: a) $m + n = 1000$ b) the cost is smallest possible!!! c) with the pair, the shop can measure $k$ liters, with $0<k<1996$(for all $k$ integer) Note: The operations to measure are: i) To fill or empty any one of two bottles ii)Pass water of a bottle for other bottle We can measure $k$ liters when the capacity of one bottle plus the capacity of other bottle is equal to $k$

VI Soros Olympiad 1999 - 2000 (Russia), 10.3

Tags: geometry , euler circle

he center of the circle passing through the midpoints of all sides of triangle $ABC$ lies on the bisector of its angle $C$. Find the side $AB$ if $BC = a$, $AC = b$ ($a$ is not equal to $b$).

1998 IMO Shortlist, 1

Tags: inequalities , algebra , n-variable inequality , arithmetic mean-geometric mean

Let $a_{1},a_{2},\ldots ,a_{n}$ be positive real numbers such that $a_{1}+a_{2}+\cdots +a_{n}<1$. Prove that \[ \frac{a_{1} a_{2} \cdots a_{n} \left[ 1 - (a_{1} + a_{2} + \cdots + a_{n}) \right] }{(a_{1} + a_{2} + \cdots + a_{n})( 1 - a_{1})(1 - a_{2}) \cdots (1 - a_{n})} \leq \frac{1}{ n^{n+1}}. \]

1963 Miklós Schweitzer, 4

Tags: superior algebra , algebra , polynomial

Call a polynomial positive reducible if it can be written as a product of two nonconstant polynomials with positive real coefficients. Let $ f(x)$ be a polynomial with $ f(0)\not\equal{}0$ such that $ f(x^n)$ is positive reducible for some natural number $ n$. Prove that $ f(x)$ itself is positive reducible. [L. Redei]

2024 China Girls Math Olympiad, 3

Tags: number theory

Find the smallest real $\lambda$, such that for any positive integers $n, a, b$, such that $n \nmid a+b$, there exists a positive integer $1 \leq k \leq n-1$, satisfying $$\{\frac{ak} {n}\}+\{\frac{bk} {n}\} \leq \lambda.$$

2020 LIMIT Category 1, 3

Tags: geometry

The diagnols $\overline{AC}$ and $\overline{BD}$ of a quaderilateral $ABCD$ meet at $O$. Let $s_1$ be the area of $\triangle{AOB}$ and $s_2$ be the area of $\triangle{OCD}$. Then show that $$\sqrt{s_1}+\sqrt{s_2} \leq \sqrt{s}$$ Also find a geometrical condition for equality to hold (By geometrical condition we mean something like parallel lines, perpendicular lines,bisecting lines etc.)

2013 Hong kong National Olympiad, 4

Tags: geometric transformation , reflection , graph theory , geometry , combinatorics

In a chess tournament there are $n>2$ players. Every two players play against each other exactly once. It is known that exactly $n$ games end as a tie. For any set $S$ of players, including $A$ and $B$, we say that $A$ [i]admires[/i] $B$ [i]in that set [/i]if i) $A$ does not beat $B$; or ii) there exists a sequence of players $C_1,C_2,\ldots,C_k$ in $S$, such that $A$ does not beat $C_1$, $C_k$ does not beat $B$, and $C_i$ does not beat $C_{i+1}$ for $1\le i\le k-1$. A set of four players is said to be [i]harmonic[/i] if each of the four players admires everyone else in the set. Find, in terms of $n$, the largest possible number of harmonic sets.

2012-2013 SDML (Middle School), 14

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Sammy and Tammy run laps around a circular track that has a radius of $1$ kilometer. They begin and end at the same point and at the same time. Sammy runs $3$ laps clockwise while Tammy runs $4$ laps counterclockwise. How many times during their run is the straight-line distance between Sammy and Tammy exactly $1$ kilometer? $\text{(A) }7\qquad\text{(B) }8\qquad\text{(C) }13\qquad\text{(D) }14\qquad\text{(E) }21$

1982 USAMO, 1

Tags: combinatorics

In a party with $1982$ persons, among any group of four there is at least one person who knows each of the other three. What is the minimum number of people in the party who know everyone else?

2024 CMIMC Geometry, 1

Tags: geometry

Let $ABCD$ be a rectangle with $AB=5$. Let $E$ be on $\overline{AB}$ and $F$ be on $\overline {CD}$ such that $AE=CF=4$. Let $P$ and $Q$ lie inside $ABCD$ such that triangles $AEP$ and $CFQ$ are equilateral. If $E$, $P$, $Q$, and $F$ lie on a single line, find $\overline{BC}$. [i]Proposed by Connor Gordon[/i]

1974 IMO Longlists, 32

Tags: inequalities

Let $a_1,a_2,\ldots ,a_n$ be $n$ real numbers such that $0<a\le a_k\le b$ for $k=1,2,\ldots ,n$. If $m_1=\frac{1}{n}(a_1+a_2+\cdots+a_n)$ and $m_2=\frac{1}{n}(a_1^2+a_2^2+\cdots + a_n^2)$, prove that $m_2\le\frac{(a+b)^2}{4ab}m_1^2$ and find a necessary and sufficient condition for equality.

VMEO III 2006, 12.4

Tags: combinatorics , symmetry

Given a binary serie $A=a_1a_2...a_k$ is called "symmetry" if $a_i=a_{k+1-i}$ for all $i=1,2,3,...,k$, and $k$ is the length of that binary serie. If $A=11...1$ or $A=00...0$ then it is called "special". Find all positive integers $m$ and $n$ such that there exist non "special" binary series $A$ (length $m$) and $B$ (length $n$) satisfying when we place them next to each other, we receive a "symmetry" binary serie $AB$

2005 Today's Calculation Of Integral, 56

Tags: limit , integration , calculus , calculus computations

Evaluate \[\lim_{n\to\infty} \sum_{k=1}^n \frac{[\sqrt{2n^2-k^2}\ ]}{n^2}\] $[x]$ is the greatest integer $\leq x$.

2004 Silk Road, 4

Tags: group theory , abstract algebra , combinatorics

Natural $n \geq 2$ is given. Group of people calls $n-compact$, if for any men from group, we can found $n$ people (without he), each two of there are familiar. Find maximum $N$ such that for any $n-compact$ group, consisting $N$ people contains subgroup from $n+1$ people, each of two of there are familiar.