Olympiad problems

2008 Indonesia TST, 1

Tags: subset , combinatorics

Let $A$ be the subset of $\{1, 2, ..., 16\}$ that has $6$ elements. Prove that there exist $2$ subsets of $A$ that are disjoint, and the sum of their elements are the same.

2003 Flanders Junior Olympiad, 4

Tags: analytic geometry

The points in the plane with integer coordinates are numbered as below. [img]https://cdn.artofproblemsolving.com/attachments/0/2/122cb559c6fb4cb8401ffa215528a035346a3d.png[/img] What are the coordinates of the number $2003$?

2011 Today's Calculation Of Integral, 718

Tags: calculus computations , calculus , integration

Find $\sum_{n=1}^{\infty} \frac{1}{2^n}\int_{-1}^1 (1-x)^2(1+x)^n dx\ (n\geq 1).$

2010 Saudi Arabia BMO TST, 2

Tags: equal area , cevian , geometry , ratio

Consider a triangle $ABC$ and a point $P$ in its interior. Lines $PA$, $PB$, $PC$ intersect $BC$, $CA$, $AB$ at $A', B', C'$ , respectively. Prove that $$\frac{BA'}{BC}+ \frac{CB'}{CA}+ \frac{AC'}{AB}= \frac32$$ if and only if at least two of the triangles $PAB$, $PBC$, $PCA$ have the same area.

1998 Romania National Olympiad, 3

Tags: derivative , arithmetic sequence , real analysis , function , inequalities , calculus

Suppose $f:\mathbb{R}\to\mathbb{R}$ is a differentiable function for which the inequality $f'(x) \leq f'(x+\frac{1}{n})$ holds for every $x\in\mathbb{R}$ and every $n\in\mathbb{N}$.Prove that f is continiously differentiable

2017 ASDAN Math Tournament, 27

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How many primes between $2$ and $2^{30}$ are $1$ more than a multiple of $2017$? If $C$ is the correct answer and $A$ is your answer, then your score will be rounded up from $\max(0,25-15|\ln\tfrac{A}{C}|)$.

2001 Argentina National Olympiad, 4

Tags: number theory

Find all positive integers $k$ that can be expressed as the sum of $50$ fractions such that the numerators are the $50$ natural numbers from $1$ to $50$ and the denominators are positive integers, that is, $k = \dfrac{1}{a_1} + \dfrac{2}{a_2} + \ldots + \dfrac{50}{a_{50}}$ with a$_1 , a_2 , \ldots , a_n$ positive integers.

2001 Putnam, 6

Tags: parabola , conic

Can an arc of a parabola inside a circle of radius $1$ have a length greater than $4$?

2011 Bosnia And Herzegovina - Regional Olympiad, 2

Tags: round table , combinatorics

At the round table there are $10$ students. Every of the students thinks of a number and says that number to its immediate neighbors (left and right) such that others do not hear him. So every student knows three numbers. After that every student publicly says arithmetic mean of two numbers he found out from his neghbors. If those arithmetic means were $1$, $2$, $3$, $4$, $5$, $6$, $7$, $8$, $9$ and $10$, respectively, which number thought student who told publicly number $6$

1951 Polish MO Finals, 4

Tags: polynomial , algebra

Determine the coefficients of the equation $$ x^3 - ax^2 + bx - c = 0$$ in such a way that the roots of this equation are the numbers $ a $, $ b $, $ c $.

2008 USA Team Selection Test, 2

Tags: circumcircle , spiral similarity , pedal triangle , geometry

Let $ P$, $ Q$, and $ R$ be the points on sides $ BC$, $ CA$, and $ AB$ of an acute triangle $ ABC$ such that triangle $ PQR$ is equilateral and has minimal area among all such equilateral triangles. Prove that the perpendiculars from $ A$ to line $ QR$, from $ B$ to line $ RP$, and from $ C$ to line $ PQ$ are concurrent.

2017 Costa Rica - Final Round, 6

Tags: functional equation , functional , algebra

Let $f:] 0. \infty [ \to R$ be a strictly increasing function, such that $$f (x) f\left(f (x) +\frac{1}{x} \right)= 1.$$ Find $f (1)$.

2011 Purple Comet Problems, 4

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Five non-overlapping equilateral triangles meet at a common vertex so that the angles between adjacent triangles are all congruent. What is the degree measure of the angle between two adjacent triangles? [asy] size(100); defaultpen(linewidth(0.7)); path equi=dir(300)--dir(240)--origin--cycle; for(int i=0;i<=4;i=i+1) draw(rotate(72*i,origin)*equi); [/asy]

1980 All Soviet Union Mathematical Olympiad, 302

Tags: 3d geometry , orthogonal , trigonometry , tetrahedron , geometry

The edge $[AC]$ of the tetrahedron $ABCD$ is orthogonal to $[BC]$, and $[AD]$ is orthogonal to $[BD]$. Prove that the cosine of the angle between lines $(AC)$ and $(BD)$ is less than $|CD|/|AB|$.

2002 Nordic, 2

Tags: maximum , combinatorics

In two bowls there are in total ${N}$ balls, numbered from ${1}$ to ${N}$. One ball is moved from one of the bowls into the other. The average of the numbers in the bowls is increased in both of the bowls by the same amount, ${x}$. Determine the largest possible value of ${x}$.

2008 Finnish National High School Mathematics Competition, 3

Tags: modular arithmetic , number theory

Solve the diophantine equation \[x^{2008}- y^{2008} = 2^{2009}.\]

2023 ELMO Shortlist, A5

Tags: algebra

Find the least positive integer $M$ for which there exist a positive integer $n$ and polynomials $P_1(x)$, $P_2(x)$, $\ldots$, $P_n(x)$ with integer coefficients satisfying \[Mx=P_1(x)^3+P_2(x)^3+\cdots+P_n(x)^3.\] [i]Proposed by Karthik Vedula[/i]

2011 Argentina National Olympiad Level 2, 5

Tags: number theory

Let $a$ and $b$ be integers such that the remainder of dividing $a$ by $17$ is equal to the remainder of dividing $b$ by $19$, and the remainder of dividing $a$ by $19$ is equal to the remainder of dividing $b$ by $17$. Determine the possible values of the remainder of $a + b$ when divided by $323$.

2019 MIG, 2

Tags:

On Monday, Lyndon receives a $80$ on his daily math quiz. After being scolded by his parents, he works harder and gets an $83$ on Tuesday. From Tuesday onward, his score improves by $3$ points each day. What will be Lyndon's score that Friday? $\textbf{(A) }89\qquad\textbf{(B) }92\qquad\textbf{(C) }95\qquad\textbf{(D) }98\qquad\textbf{(E) }100$

LMT Team Rounds 2021+, 8

Tags: number theory

An odd positive integer $n$ can be expressed as the sum of two or more consecutive integers in exactly $2023$ ways. Find the greatest possible nonnegative integer $k$ such that $3^k$ is a factor of the least possible value of $n$.

2024 HMNT, 13

Tags: guts

Let $f$ and $g$ be two quadratic polynomials with real coefficients such that the equation $f(g(x)) = 0$ has four distinct real solutions: $112, 131, 146,$ and $a.$ Compute the sum of all possible values of $a.$

2021 Iran Team Selection Test, 4

Tags: number theory , polynomial , combi , prime number , function , algebra

Assume $\Omega(n),\omega(n)$ be the biggest and smallest prime factors of $n$ respectively . Alireza and Amin decided to play a game. First Alireza chooses $1400$ polynomials with integer coefficients. Now Amin chooses $700$ of them, the set of polynomials of Alireza and Amin are $B,A$ respectively . Amin wins if for all $n$ we have : $$\max_{P \in A}(\Omega(P(n))) \ge \min_{P \in B}(\omega(P(n)))$$ Who has the winning strategy. Proposed by [i]Alireza Haghi[/i]

2010 Argentina National Olympiad, 1

Tags: combinatorics

Given several integers, the allowed operation is to replace two of them by their non-negative difference. The operation is repeated until only one number remains. If the initial numbers are $1, 2, … , 2010$, what can be the last remaining number?

1958 AMC 12/AHSME, 4

Tags:

In the expression $ \frac{x \plus{} 1}{x \minus{} 1}$ each $ x$ is replaced by $ \frac{x \plus{} 1}{x \minus{} 1}$. The resulting expression, evaluated for $ x \equal{} \frac{1}{2}$, equals: $ \textbf{(A)}\ 3\qquad \textbf{(B)}\ \minus{}3\qquad \textbf{(C)}\ 1\qquad \textbf{(D)}\ \minus{}1\qquad \textbf{(E)}\ \text{none of these}$

1988 Bulgaria National Olympiad, Problem 4

Tags: geometry

Let $A,B,C$ be non-collinear points. For each point $D$ of the ray $AC$, we denote by $E$ and $F$ the points of tangency of the incircle of $\triangle ABD$ with $AB$ and $AD$, respectively. Prove that, as point $D$ moves along the ray $AC$, the line $EF$ passes through a fixed point.