Olympiad problems

2015 May Olympiad, 2

Tags: weight , combinatorics

$6$ indistinguishable coins are given, $4$ are authentic, all of the same weight, and $2$ are false, one is more light than the real ones and the other one, heavier than the real ones. The two false ones together weigh same as two authentic coins. Find two authentic coins using a balance scale twice only by two plates, no weights. Clarification: A two-pan scale only reports if the left pan weighs more, equal or less that right.

2012 CentroAmerican, 1

Tags: floor function , combinatorics

Trilandia is a very unusual city. The city has the shape of an equilateral triangle of side lenght 2012. The streets divide the city into several blocks that are shaped like equilateral triangles of side lenght 1. There are streets at the border of Trilandia too. There are 6036 streets in total. The mayor wants to put sentinel sites at some intersections of the city to monitor the streets. A sentinel site can monitor every street on which it is located. What is the smallest number of sentinel sites that are required to monitor every street of Trilandia?

2024 Malaysian IMO Training Camp, 2

Tags: combinatorics

The sequence $1, 2, \dots, 2023, 2024$ is written on a whiteboard. Every second, Megavan chooses two integers $a$ and $b$, and four consecutive numbers on the whiteboard. Then counting from the left, he adds $a$ to the 1st and 3rd of those numbers, and adds $b$ to the 2nd and 4th of those numbers. Can he achieve the sequence $2024, 2023, \dots, 2, 1$ in a finite number of moves? [i](Proposed by Avan Lim Zenn Ee)[/i]

2013 F = Ma, 15

Tags: geometric transformation , quadratic , geometry , rotation

A uniform rod is partially in water with one end suspended, as shown in figure. The density of the rod is $5/9$ that of water. At equilibrium, what portion of the rod is above water? $\textbf{(A) } 0.25\\ \textbf{(B) } 0.33\\ \textbf{(C) } 0.5\\ \textbf{(D) } 0.67\\ \textbf{(E) } 0.75$

2018 Dutch BxMO TST, 2

Tags: integer , sides of a triangle , coprime , geometry

Let $\vartriangle ABC$ be a triangle of which the side lengths are positive integers which are pairwise coprime. The tangent in $A$ to the circumcircle intersects line $BC$ in $D$. Prove that $BD$ is not an integer.

2022 Bosnia and Herzegovina Junior BMO TST, 2

Tags: number theory

Let $a,b,c$ be positive integers greater than $1$ such that $$p=ab+bc+ac$$ is prime. A) Prove that $a^2, b^2, c^2$ all have different reminder $mod\ p$. B) Prove that $a^3, b^3, c^3$ all have different reminder $mod\ p$.

2008 District Olympiad, 3

Tags: euler , function , parameterization , algebra

For any real $ a$ define $ f_a : \mathbb{R} \rightarrow \mathbb{R}^2$ by the law $ f_a(t) \equal{} \left( \sin(t), \cos(at) \right)$. a) Prove that $ f_{\pi}$ is not periodic. b) Determine the values of the parameter $ a$ for which $ f_a$ is periodic. [b]Remark[/b]. L. Euler proved in $ 1737$ that $ \pi$ is irrational.

1972 IMO Shortlist, 3

Tags: inequality , optimization , maximization

The least number is $m$ and the greatest number is $M$ among $ a_1 ,a_2 ,\ldots,a_n$ satisfying $ a_1 \plus{}a_2 \plus{}...\plus{}a_n \equal{}0$. Prove that \[ a_1^2 \plus{}\cdots \plus{}a_n^2 \le\minus{}nmM\]

2009 National Olympiad First Round, 21

Tags:

$ AB \equal{} AC$, $ \angle BAC \equal{} 80^\circ$. Let $ E$ be a point inside $ \triangle ABC$ such that $ AE \equal{} EC$ and $ \angle EAC \equal{} 10^\circ$. What is the measure of $ \angle EBC$? $\textbf{(A)}\ 10^\circ \qquad\textbf{(B)}\ 15^\circ \qquad\textbf{(C)}\ 20^\circ \qquad\textbf{(D)}\ 25^\circ \qquad\textbf{(E)}\ 30^\circ$

2009 Today's Calculation Of Integral, 426

Tags: integration , calculus , polynomial , calculus computations , algebra

Consider the polynomial $ f(x) \equal{} ax^2 \plus{} bx \plus{} c$, with degree less than or equal to 2. When $ f$ varies with subject to the constrain $ f(0) \equal{} 0,\ f(2) \equal{} 2$, find the minimum value of $ S\equal{}\int_0^2 |f'(x)|\ dx$.

2022 VTRMC, 1

Tags: algebra

Give all possible representations of $2022$ as a sum of at least two consecutive positive integers and prove that these are the only representations.

2018 Swedish Mathematical Competition, 4

Tags: consecutive , divisible , number theory , sum of digits

Find the least positive integer $n$ with the property: Among arbitrarily $n$ selected consecutive positive integers, all smaller than $2018$, there is at least one that is divisible by its sum of digits .

PEN I Problems, 6

Tags: floor function , inequalities

Prove that for all positive integers $n$, \[\lfloor \sqrt{n}+\sqrt{n+1}+\sqrt{n+2}\rfloor =\lfloor \sqrt{9n+8}\rfloor.\]

2023 Quang Nam Province Math Contest (Grade 11), Problem 4

Tags: number theory

a) Find all integer pairs $(x,y)$ satisfying $x^4+(y+2)^3=(x+2)^4.$ b) Prove that: if $p$ is a prime of the form $p=4k+3$ $(k$ is a non-negative number$),$ then there doesn's exist $p-1$ consecutive non-negative integers such that we can divide the set of these numbers into $2$ distinct subsets so that the product of all the numbers in one subset is equal to that in the remained subset.

LMT Team Rounds 2021+, 1

Tags: number theory

Let $x$ be the positive integer satisfying $5^2 +28^2 +39^2 = 24^2 +35^2 + x^2$. Find $x$.

2015 IFYM, Sozopol, 8

Tags: functional equation , function , algebra

Let $\mathbb{N} = \{1, 2, 3, \ldots\}$ be the set of positive integers. Find all functions $f$, defined on $\mathbb{N}$ and taking values in $\mathbb{N}$, such that $(n-1)^2< f(n)f(f(n)) < n^2+n$ for every positive integer $n$.

2022 Purple Comet Problems, 14

Tags:

Starting at $12:00:00$ AM on January $1,$ $2022,$ after $13!$ seconds it will be $y$ years (including leap years) and $d$ days later, where $d < 365.$ Find $y + d.$

2004 Austrian-Polish Competition, 4

Tags: number theory , polynomial , algebra

Determine all $n \in \mathbb{N}$ for which $n^{10} + n^5 + 1$ is prime.

2023 UMD Math Competition Part I, #10

Tags: algebra

There are $100$ people in a room. Some are [i]wise[/i] and some are [i]optimists[/i]. $\quad \bullet~$ A [i]wise[/i] person can look at someone and know if they are wise or if they are an optimist. $\quad \bullet~$ An [i]optimist[/i] thinks everyone is wise (including themselves). Everyone in the room writes down what they think is the number of wise people in the room. What is the smallest possible value for the average? $$ \mathrm a. ~ 10\qquad \mathrm b.~25\qquad \mathrm c. ~50 \qquad \mathrm d. ~75 \qquad \mathrm e. ~100 $$

2017 NZMOC Camp Selection Problems, 1

Tags: algebra

Alice has five real numbers $a < b < c < d < e$. She takes the sum of each pair of numbers and writes down the ten sums. The three smallest sums are $32$, $36$ and $37$, while the two largest sums are $48$ and $51$. Determine $e$.

2025 Belarusian National Olympiad, 11.4

Tags: number theory

A finite set $S$ consists of primes, and $3$ is not in $S$. Prove that there exists a positive integer $M$ such that for every $p \in S$ one can shuffle the digits of $M$ to get a number divisible by $p$ and not divisible by all other numbers in $S$. (Note: the first digit of a positive integer can not be zero). [i]A. Voidelevich[/i]

1979 Kurschak Competition, 1

Tags: 3d geometry , pyramid , geometry

The base of a convex pyramid has an odd number of edges. The lateral edges of the pyramid are all equal, and the angles between neighbouring faces are all equal. Show that the base must be a regular polygon.

2019 China Team Selection Test, 1

Tags: geometry , angle bisector , circumcircle

Cyclic quadrilateral $ABCD$ has circumcircle $(O)$. Points $M$ and $N$ are the midpoints of $BC$ and $CD$, and $E$ and $F$ lie on $AB$ and $AD$ respectively such that $EF$ passes through $O$ and $EO=OF$. Let $EN$ meet $FM$ at $P$. Denote $S$ as the circumcenter of $\triangle PEF$. Line $PO$ intersects $AD$ and $BA$ at $Q$ and $R$ respectively. Suppose $OSPC$ is a parallelogram. Prove that $AQ=AR$.

2023 CMIMC Algebra/NT, 4

Tags: algebra , number theory

An arithmetic sequence of exactly $10$ positive integers has the property that any two elements are relatively prime. Compute the smallest possible sum of the $10$ numbers. [i]Proposed by Kyle Lee[/i]

1977 Czech and Slovak Olympiad III A, 5

Tags: point , coloring , combinatorial geometry

Let $A_1,\ldots,A_n$ be different collinear points. Every point is dyed by one of four colors and every of these colors is used at least once. Show that there is a line segment where two colors are used exactly once and the other two are used at least once.