Olympiad problems

1985 Spain Mathematical Olympiad, 3

Solve the equation $tan^2 2x+2 tan2x tan3x = 1$

KoMaL A Problems 2019/2020, A.756

Tags: algebra , functional equation

Find all functions $f:\mathbb{R}\to\mathbb{R}$ which satisfy the following conditions: $f(x+1)=f(x)+1$ and $f(x^2)=f(x)^2.$ [i]Based on a problem of Romanian Masters of Mathematics[/i]

2024 Ukraine National Mathematical Olympiad, Problem 2

Tags: number theory , algebra , power of 2

You are given a positive integer $n$. Find the smallest positive integer $k$, for which there exist integers $a_1, a_2, \ldots, a_k$, for which the following equality holds: $$2^{a_1} + 2^{a_2} + \ldots + 2^{a_k} = 2^n - n + k$$ [i]Proposed by Mykhailo Shtandenko[/i]

2015 Serbia National Math Olympiad, 6

Tags: exponential equation , congruence , number theory

In nonnegative set of integers solve the equation: $$(2^{2015}+1)^x + 2^{2015}=2^y+1$$

1997 AMC 12/AHSME, 30

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For positive integers $ n$, denote by $ D(n)$ the number of pairs of different adjacent digits in the binary (base two) representation of $ n$. For example, $ D(3) \equal{} D(11_2) \equal{} 0$, $ D(21) \equal{} D(10101_2) \equal{} 4$, and $ D(97) \equal{} D(110001_2) \equal{} 2$. For how many positive integers $ n$ less than or equal to $ 97$ does $ D(n) \equal{} 2$? $ \textbf{(A)}\ 16\qquad \textbf{(B)}\ 20\qquad \textbf{(C)}\ 26\qquad \textbf{(D)}\ 30\qquad \textbf{(E)}\ 35$

2003 AMC 12-AHSME, 9

Tags: function , analytic geometry , graphing lines , slope

Let $ f$ be a linear function for which $ f(6)\minus{}f(2)\equal{}12$. What is $ f(12)\minus{}f(2)$? $ \textbf{(A)}\ 12 \qquad \textbf{(B)}\ 18 \qquad \textbf{(C)}\ 24 \qquad \textbf{(D)}\ 30 \qquad \textbf{(E)}\ 36$

2023 Moldova Team Selection Test, 10

Tags: geometry

Let $ABC$ be a triangle with $\angle ACB=90$ and $AC>BC.$ Let $\Omega$ be the circumcircle of $ABC.$ Point $ D $ is the midpoint of small arc $AC$ of $\Omega.$ Point $ M $ is symmetric with $ A$ with respect to $D.$ Point $ N$ is the midpoint of $MC.$ Line $AN$ intersects $\Omega$ in point $ P $ and line $BP$ intersects line $DN$ in point $Q.$ Prove that line $QM$ passes through the midpoint of $AC.$

2005 Today's Calculation Of Integral, 78

Tags: calculus , integration , trigonometry , calculus computations

Let $\alpha,\beta$ be the distinct positive roots of the equation of $2x=\tan x$. Evaluate \[\int_0^1 \sin \alpha x\sin \beta x\ dx\]

2012 AMC 10, 8

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The sums of three whole numbers taken in pairs are $12$, $17$, and $19$. What is the middle number? $ \textbf{(A)}\ 4\qquad\textbf{(B)}\ 5\qquad\textbf{(C)}\ 6\qquad\textbf{(D)}\ 7\qquad\textbf{(E)}\ 8 $

2023 Brazil Team Selection Test, 3

Tags: algebra , geometric sequence

Find all positive integers $n \geqslant 2$ for which there exist $n$ real numbers $a_1<\cdots<a_n$ and a real number $r>0$ such that the $\tfrac{1}{2}n(n-1)$ differences $a_j-a_i$ for $1 \leqslant i<j \leqslant n$ are equal, in some order, to the numbers $r^1,r^2,\ldots,r^{\frac{1}{2}n(n-1)}$.

2008 IMO Shortlist, 7

Tags: geometry , incenter , convex quadrilateral

Let $ ABCD$ be a convex quadrilateral with $ BA\neq BC$. Denote the incircles of triangles $ ABC$ and $ ADC$ by $ \omega_{1}$ and $ \omega_{2}$ respectively. Suppose that there exists a circle $ \omega$ tangent to ray $ BA$ beyond $ A$ and to the ray $ BC$ beyond $ C$, which is also tangent to the lines $ AD$ and $ CD$. Prove that the common external tangents to $ \omega_{1}$ and $\omega_{2}$ intersect on $ \omega$. [i]Author: Vladimir Shmarov, Russia[/i]

2015 IMO Shortlist, C6

Tags: combinatorics , additive combinatorics

Let $S$ be a nonempty set of positive integers. We say that a positive integer $n$ is [i]clean[/i] if it has a unique representation as a sum of an odd number of distinct elements from $S$. Prove that there exist infinitely many positive integers that are not clean.

2007 Peru Iberoamerican Team Selection Test, P4

Tags: table , combinatorics , board

Each of the squares on a $15$×$15$ board has a zero. At every step you choose a row or a column, we delete all the numbers from it and then we write the numbers from $1$ to $15$ in the empty cells, in an arbitrary order. find the sum possible maximum of the numbers on the board that can be achieved after a number finite number of steps.

1965 AMC 12/AHSME, 4

Tags: geometry , geometric transformation , reflection

Line $ l_2$ intersects line $ l_1$ and line $ l_3$ is parallel to $ l_1$. The three lines are distinct and lie in a plane. The number of points equidistant from all three lines is: $ \textbf{(A)}\ 0 \qquad \textbf{(B)}\ 1 \qquad \textbf{(C)}\ 2 \qquad \textbf{(D)}\ 4 \qquad \textbf{(E)}\ 8$

2017 Germany, Landesrunde - Grade 11/12, 3

Tags: number theory , prime number , exponent , power , catalan

Find the smallest prime number that can not be written in the form $\left| 2^a-3^b \right|$ with non-negative integers $a,b$.

2008 BAMO, 5

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A positive integer $N$ is called stable if it is possible to split the set of all positive divisors of $N$ (including $1$ and $N$) into two subsets that have no elements in common, which have the same sum. For example, 6 is stable, because $1+2+3=6$, but 10 is not stable. Is $2^{2008}\cdot2008$ stable?

2022 New Zealand MO, 2

Tags: algebra , inequalities

Find all triples $(a, b, c) $ of real numbers such that $a^2 + b^2 + c^2 = 1$ and $a(2b - 2a - c) \ge \frac12$.

2016 NIMO Problems, 5

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The equation $x^3 - 3x^2 - 7x - 1 = 0$ has three distinct real roots $a$, $b$, and $c$. If \[\left( \dfrac{1}{\sqrt[3]{a}-\sqrt[3]{b}} + \dfrac{1}{\sqrt[3]{b}-\sqrt[3]{c}} + \dfrac{1}{\sqrt[3]{c}-\sqrt[3]{a}} \right)^2 = \dfrac{p\sqrt[3]{q}}{r}\] where $p$, $q$, $r$ are positive integers such that $\gcd(p, r) = 1$ and $q$ is not divisible by the cube of a prime, find $100p + 10q + r$. [i]Proposed by Michael Tang and David Altizio[/i]

1958 AMC 12/AHSME, 1

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The value of $ [2 \minus{} 3(2 \minus{} 3)^{\minus{}1}]^{\minus{}1}$ is: $ \textbf{(A)}\ 5\qquad \textbf{(B)}\ \minus{}5\qquad \textbf{(C)}\ \frac{1}{5}\qquad \textbf{(D)}\ \minus{}\frac{1}{5}\qquad \textbf{(E)}\ \frac{5}{3}$

2010 Contests, 2

Tags: circumcircle , power of a point , radical axis , geometry

Let $ABC$ be a triangle with $ \widehat{BAC}\neq 90^\circ $. Let $M$ be the midpoint of $BC$. We choose a variable point $D$ on $AM$. Let $(O_1)$ and $(O_2)$ be two circle pass through $ D$ and tangent to $BC$ at $B$ and $C$. The line $BA$ and $CA$ intersect $(O_1),(O_2)$ at $ P,Q$ respectively. [b]a)[/b] Prove that tangent line at $P$ on $(O_1)$ and $Q$ on $(O_2)$ must intersect at $S$. [b]b)[/b] Prove that $S$ lies on a fix line.

1969 IMO Longlists, 51

Tags: geometry , calculus , counting , function

$(NET 6)$ A curve determined by $y =\sqrt{x^2 - 10x+ 52}, 0\le x \le 100,$ is constructed in a rectangular grid. Determine the number of squares cut by the curve.

2014 Harvard-MIT Mathematics Tournament, 8

Tags: geometry , circumcircle , trigonometry , analytic geometry , graphing lines

Let $ABC$ be a triangle with sides $AB = 6$, $BC = 10$, and $CA = 8$. Let $M$ and $N$ be the midpoints of $BA$ and $BC$, respectively. Choose the point $Y$ on ray $CM$ so that the circumcircle of triangle $AMY$ is tangent to $AN$. Find the area of triangle $NAY$.

2016 Japan MO Preliminary, 4

Tags: combinatorics , rectangle , geometry

There is a $11\times 11$ square grid. We divided this in $5$ rectangles along unit squares. How many ways that one of the rectangles doesn’t have a edge on basic circumference. Note that we count as different ways that one way coincides with another way by rotating or reversing.

LMT Guts Rounds, 2020 F35

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Estimate the number of ordered pairs $(p,q)$ of positive integers at most $2020$ such that the cubic equation $x^3-px-q=0$ has three distinct real roots. If your estimate is $E$ and the answer is $A$, your score for this problem will be \[\Big\lfloor15\min\Big(\frac{A}{E},\frac{E}{A}\Big)\Big\rfloor.\] [i]Proposed by Alex Li[/i]

1955 AMC 12/AHSME, 32

Tags: quadratic , arithmetic sequence , geometric sequence

If the discriminant of $ ax^2\plus{}2bx\plus{}c\equal{}0$ is zero, then another true statement about $ a$, $ b$, and $ c$ is that: $ \textbf{(A)}\ \text{they form an arithmetic progression} \\ \textbf{(B)}\ \text{they form a geometric progression} \\ \textbf{(C)}\ \text{they are unequal} \\ \textbf{(D)}\ \text{they are all negative numbers} \\ \textbf{(E)}\ \text{only b is negative and a and c are positive}$