Olympiad problems

1984 AMC 12/AHSME, 5

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The largest integer $n$ for which $n^{200} < 5^{300}$ is $\textbf{(A) }8\qquad \textbf{(B) }9\qquad \textbf{(C) }10\qquad \textbf{(D) }11\qquad \textbf{(E) }12$

2014 Greece Team Selection Test, 3

Let $ABC$ be an acute,non-isosceles triangle with $AB<AC<BC$.Let $D,E,Z$ be the midpoints of $BC,AC,AB$ respectively and segments $BK,CL$ are altitudes.In the extension of $DZ$ we take a point $M$ such that the parallel from $M$ to $KL$ crosses the extensions of $CA,BA,DE$ at $S,T,N$ respectively (we extend $CA$ to $A$-side and $BA$ to $A$-side and $DE$ to $E$-side).If the circumcirle $(c_{1})$ of $\triangle{MBD}$ crosses the line $DN$ at $R$ and the circumcirle $(c_{2})$ of $\triangle{NCD}$ crosses the line $DM$ at $P$ prove that $ST\parallel PR$.

2023 Iran MO (3rd Round), 2

Tags: algebra

Does there exist bijections $f,g$ from positive integers to themselves st: $$g(n)=\frac{f(1)+f(2)+ \cdot \cdot \cdot +f(n)}{n}$$ holds for any $n$?

2006 Italy TST, 2

Tags: modular arithmetic , euler , number theory

Let $n$ be a positive integer, and let $A_{n}$ be the the set of all positive integers $a\le n$ such that $n|a^{n}+1$. a) Find all $n$ such that $A_{n}\neq \emptyset$ b) Find all $n$ such that $|{A_{n}}|$ is even and non-zero. c) Is there $n$ such that $|{A_{n}}| = 130$?

2025 Kosovo National Mathematical Olympiad`, P1

Tags: combinatorics

Anna wants to form a four-digit number with four different digits from the digits $1, 2, 3, 4, 5, 6, 7, 8, 9$. She wants the first digit of that number to be bigger than the sum of the other three digits. How many such numbers can she form?

2012 Princeton University Math Competition, A7

Tags: number theory

Let $a, b$, and $c$ be positive integers satisfying $a^4 + a^2b^2 + b^4 = 9633$ $2a^2 + a^2b^2 + 2b^2 + c^5 = 3605$. What is the sum of all distinct values of $a + b + c$?

2010 Purple Comet Problems, 24

Tags: sfft , special factorizations

Find the number of ordered pairs of integers $(m, n)$ that satisfy $20m-10n = mn$.

1973 Canada National Olympiad, 1

Tags: inequalities , calculus , integration , logarithm

(i) Solve the simultaneous inequalities, $x<\frac{1}{4x}$ and $x<0$; i.e. find a single inequality equivalent to the two simultaneous inequalities. (ii) What is the greatest integer that satisfies both inequalities $4x+13 < 0$ and $x^{2}+3x > 16$. (iii) Give a rational number between $11/24$ and $6/13$. (iv) Express 100000 as a product of two integers neither of which is an integral multiple of 10. (v) Without the use of logarithm tables evaluate \[\frac{1}{\log_{2}36}+\frac{1}{\log_{3}36}.\]

1989 AMC 8, 3

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Which of the following numbers is the largest? $\text{(A)}\ .99 \qquad \text{(B)}\ .9099 \qquad \text{(C)}\ .9 \qquad \text{(D)}\ .909 \qquad \text{(E)}\ .9009$

2023 HMNT, 6

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There are five people in a room. They each simultaneously pick two of the other people in the room independently and uniformly at random and point at them. Compute the probability that there exists a group of three people such that each of them is pointing at the other two in the group.

2019 IFYM, Sozopol, 2

Tags: combinatorics , game strategy , game theory , board , coloring

Let $n$ be a natural number. At first the cells of a table $2n$ x $2n$ are colored in white. Two players $A$ and $B$ play the following game. First is $A$ who has to color $m$ arbitrary cells in red and after that $B$ chooses $n$ rows and $n$ columns and color their cells in black. Player $A$ wins, if there is at least one red cell on the board. Find the least value of $m$ for which $A$ wins no matter how $B$ plays.

2008 ITest, 19

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Let $A$ be the set of positive integers that are the product of two consecutive integers. Let $B$ the set of positive integers that are the product of three consecutive integers. Find the sum of the two smallest elements of $A\cap B$.

2021 JHMT HS, 9

Tags: general , geometry

Squares of side lengths $1,$ $2,$ $3,$ and $4,$ are placed on a line segment $\ell$ from left to right, respectively, and these squares lie on the same side of $\ell,$ forming a polygon $P.$ An equilateral triangle whose base is $\ell$ is drawn around the squares such that its other two sides intersect $P$ at its leftmost and rightmost vertices (that are not on $\ell$). The area of the triangle can be written in the form $\tfrac{a + b\sqrt{3}}{c},$ where $a,$ $b,$ and $c$ are positive integers, and $b$ and $c$ are relatively prime. Find $a + b + c.$

2021 Malaysia IMONST 1, 18

Tags: algebra , equation

How many real numbers $x$ are solutions to the equation $|x - 2| - 4 =\frac{1}{|x - 3|}$ ?

2012 Princeton University Math Competition, B3

Tags: number theory

Find, with proof, all pairs $(x, y)$ of integers satisfying the equation $3x^2+ 4 = 2y^3$.

2020 SJMO, 6

Tags: number theory , combinatorics

We say a positive integer $n$ is [i]$k$-tasty[/i] for some positive integer $k$ if there exists a permutation $(a_0, a_1, a_2, \ldots , a_n)$ of $(0,1,2, \ldots, n)$ such that $|a_{i+1} - a_i| \in \{k, k+1\}$ for all $0 \le i \le n-1$. Prove that for all positive integers $k$, there exists a constant $N$ such that all integers $n \geq N$ are $k$-tasty. [i]Proposed by Anthony Wang[/i]

2003 Korea Junior Math Olympiad, 5

Tags: integer , number theory

Four odd positive intgers $a, b, c, d (a\leq b \leq c\leq d)$ are given. Choose any three numbers among them and divide their sum by the un-chosen number, and you will always get the remainder as $1$. Find all $(a, b, c, d)$ that satisfies this.

2019 CCA Math Bonanza, L1.3

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Points $P$ and $Q$ are chosen on diagonal $AC$ of square $ABCD$ such that $AB=AP=CQ=1$. What is the measure of $\angle{PBQ}$ in degrees? [i]2019 CCA Math Bonanza Lightning Round #1.3[/i]

2016 Postal Coaching, 2

Tags: number theory , totient function , inequalities

Find all $n \in \mathbb N$ such that $n = \varphi (n) + 402$, where $\varphi$ denotes the Euler phi function.

Russian TST 2015, P1

Tags: number theory , prime number

Prove that there exist two natural numbers $a,b$ such that $|a-m|+|b-n|>1000$ for any relatively prime natural numbers $m,n$.

2008 Flanders Math Olympiad, 3

Tags: geometry , 3d geometry , tetrahedron , pyramid

A quadrilateral pyramid and a regular tetrahedron have edges that are all equal in length. They are glued together so that they have in common $1$ equilateral triangle . Prove that the resulting body has exactly $5$ sides.

2025 Bulgarian Spring Mathematical Competition, 9.3

Tags: algorithm , graph theory , dominating sets , spanning tree , connected graph , combinatorics

In a country, there are towns, some of which are connected by roads. There is a route (not necessarily direct) between every two towns. The Minister of Education has ensured that every town without a school is connected via a direct road to a town that has a school. The Minister of State Optimization wants to ensure that there is a unique path between any two towns (without repeating traveled segments), which may require removing some roads. Is it always possible to achieve this without constructing additional schools while preserving what the Minister of Education has accomplished?

MIPT Undergraduate Contest 2019, 1.3

Tags: linear algebra , matrix

Given a natural number $n$, for what maximal value $k$ it is possible to construct a matrix of size $k \times n$ consisting only of elements $\pm 1$ in such a way that for any interchange of a $+1$ with a $-1$ or vice versa, its rank is equal to $k$?

1994 Romania TST for IMO, 2:

Tags: geometry

Let $S_1, S_2,S_3$ be spheres of radii $a, b, c$ respectively whose centers lie on a line $l$. Sphere $S_2$ is externally tangent to $S_1$ and $S_3$, whereas $S_1$ and $S_3$ have no common points. A straight line t touches each of the spheres, Find the sine of the angle between $l$ and $t$

2016 Kyiv Mathematical Festival, P5

Tags: combinatorics , digit , divisibility

On the board a 20-digit number which have 10 ones and 10 twos in its decimal form is written. It is allowed to choose two different digits and to reverse the order of digits in the interval between them. Is it always possible to get a number divisible by 11 using such operations?