Olympiad problems

2021 AMC 10 Spring, 6

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Chantal and Jean start hiking from a trailhead toward a fire tower. Jean is wearing a heavy backpack and walks slower. Chantal starts walking at $4$ miles per hour. Halfway to the tower, the trail becomes really steep, and Chantal slows down to $2$ miles per hour. After reaching the tower, she immediately turns around and descends the steep part of the trail at $3$ miles per hour. She meets Jean at the halfway point. What was Jean's average speed, in miles per hour, until they meet? $\textbf{(A)}~\frac{12}{13}\qquad \textbf{(B)}~1\qquad \textbf{(C)}~\frac{13}{12}\qquad \textbf{(D)}~\frac{24}{13}\qquad \textbf{(E)}~2$

2015 Turkey EGMO TST, 2

Tags: geometry

Let $D$ be the midpoint of the side $BC$ of a triangle $ABC$ and $P$ be a point inside the $ABD$ satisfying $\angle PAD=90^\circ - \angle PBD=\angle CAD$. Prove that $\angle PQB=\angle BAC$, where $Q$ is the intersection point of the lines $PC$ and $AD$.

2002 Estonia National Olympiad, 5

Tags: combinatorics

There is a lottery at Juku’s birthday party with a number of identical prizes, where each guest can win at most one prize. It is known that if there was one prize less, then the number of possible distributions of the prizes among the guests would be $50\%$ less than it actually is, while if there was one prize more, then the number of possible distributions of the prizes would be $50\%$ more than it actually is. Find the number of possible distributions of the prizes.

2001 India IMO Training Camp, 3

Tags: polynomial , inequalities , induction , algebra

Let $P(x)$ be a polynomial of degree $n$ with real coefficients and let $a\geq 3$. Prove that \[\max_{0\leq j \leq n+1}\left | a^j-P(j) \right |\geq 1\]

2019 Ecuador NMO (OMEC), 5

Tags: number theory , diophantine

Let $a, b, c$ be integers not all the same with $a, b, c\ge 4$ that satisfy $$4abc = (a + 3) (b + 3) (c + 3).$$ Find the numerical value of $a + b + c$.

2017 Purple Comet Problems, 8

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The positive integer $m$ is a multiple of 111, and the positive integer $n$ is a multiple of 31. Their sum is 2017. Find $n - m$.

2007 Today's Calculation Of Integral, 252

Tags: calculus , integration , trigonometry , derivative , calculus computations

Compare $ \displaystyle f(\theta) \equal{} \int_0^1 (x \plus{} \sin \theta)^2\ dx$ and $ \ g(\theta) \equal{} \int_0^1 (x \plus{} \cos \theta)^2\ dx$ for $ 0\leqq \theta \leqq 2\pi .$

2014 Romania National Olympiad, 4

Tags: complex number , inequalities

Let $n \in \mathbb{N} , n \ge 2$ and $ a_0,a_1,a_2,\cdots,a_n \in \mathbb{C} ; a_n \not = 0 $. Then: [b][size=100][i]P.[/i][/size][/b] $|a_nz^n + a_{n-1}z^z{n-1} + \cdots + a_1z + a_0 | \le |a_n+a_0|$ for any $z \in \mathbb{C}, |z|=1$ [b][size=100][i]Q[/i][/size][/b]. $a_1=a_2=\cdots=a_{n-1}=0$ and $a_0/a_n \in [0,\infty)$ Prove that $ P \Longleftrightarrow Q$

2025 Harvard-MIT Mathematics Tournament, 3

Tags: combinatorics

Ben has $16$ balls labeled $1, 2, 3, \ldots, 16,$ as well as $4$ indistinguishable boxes. Two balls are [i]neighbors[/i] if their labels differ by $1.$ Compute the number of ways for him to put $4$ balls in each box such that each ball is in the same box as at least one of its neighbors. (The order in which the balls are placed does not matter.)

1982 IMO Longlists, 17

Tags: sequence , maximization , minimization , rearrangement , algebra

[b](a)[/b] Find the rearrangement $\{a_1, \dots , a_n\}$ of $\{1, 2, \dots, n\}$ that maximizes \[a_1a_2 + a_2a_3 + \cdots + a_na_1 = Q.\] [b](b)[/b] Find the rearrangement that minimizes $Q.$

2011 IberoAmerican, 2

Tags: number theory

Find all positive integers $n$ for which exist three nonzero integers $x, y, z$ such that $x+y+z=0$ and: \[\frac{1}{x}+\frac{1}{y}+\frac{1}{z}=\frac{1}{n}\]

2017 Harvard-MIT Mathematics Tournament, 2

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How many ways are there to insert $+$'s between the digits of $111111111111111$ (fifteen $1$'s) so that the result will be a multiple of $30$?

Oliforum Contest III 2012, 4

Tags: algebra , inequalities

Show that if $a \ge b \ge c \ge 0$ then $$a^2b(a - b) + b^2c(b - c) + c^2a(c - a) \ge 0.$$

2007 Indonesia TST, 1

Tags: circumcircle , cyclic quadrilateral , angle bisector , geometry

Let $ ABCD$ be a cyclic quadrilateral and $ O$ be the intersection of diagonal $ AC$ and $ BD$. The circumcircles of triangle $ ABO$ and the triangle $ CDO$ intersect at $ K$. Let $ L$ be a point such that the triangle $ BLC$ is similar to $ AKD$ (in that order). Prove that if $ BLCK$ is a convex quadrilateral, then it has an incircle.

2007 All-Russian Olympiad, 8

Tags: number theory

Dima has written number $ 1/80!,\,1/81!,\,\dots,1/99!$ on $ 20$ infinite pieces of papers as decimal fractions (the following is written on the last piece: $ \frac {1}{99!} \equal{} 0{,}{00\dots 00}10715\dots$, 155 0-s before 1). Sasha wants to cut a fragment of $ N$ consecutive digits from one of pieces without the comma. For which maximal $ N$ he may do it so that Dima may not guess, from which piece Sasha has cut his fragment? [i]A. Golovanov[/i]

2004 Iran MO (2nd round), 3

Tags: ceiling function , combinatorics

The road ministry has assigned $80$ informal companies to repair $2400$ roads. These roads connect $100$ cities to each other. Each road is between $2$ cities and there is at most $1$ road between every $2$ cities. We know that each company repairs $30$ roads that it has agencies in each $2$ ends of them. Prove that there exists a city in which $8$ companies have agencies.

1999 AIME Problems, 2

Tags: geometry , parallelogram , analytic geometry , graphing lines , slope

Consider the parallelogram with vertices $(10,45),$ $(10,114),$ $(28,153),$ and $(28,84).$ A line through the origin cuts this figure into two congruent polygons. The slope of the line is $m/n,$ where $m$ and $n$ are relatively prime positive integers. Find $m+n.$

2025 Sharygin Geometry Olympiad, 2

Tags: geometry

Four points on the plane are not concyclic, and any three of them are not collinear. Prove that there exists a point $Z$ such that the reflection of each of these four points about $Z$ lies on the circle passing through three remaining points. Proposed by:A Kuznetsov

2011 Dutch IMO TST, 3

Tags: circles , tangent , geometry

Let $\Gamma_1$ and $\Gamma_2$ be two intersecting circles with midpoints respectively $O_1$ and $O_2$, such that $\Gamma_2$ intersects the line segment $O_1O_2$ in a point $A$. The intersection points of $\Gamma_1$ and $\Gamma_2$ are $C$ and $D$. The line $AD$ intersects $\Gamma_1$ a second time in $S$. The line $CS$ intersects $O_1O_2$ in $F$. Let $\Gamma_3$ be the circumcircle of triangle $AD$. Let $E$ be the second intersection point of $\Gamma_1$ and $\Gamma_3$. Prove that $O_1E$ is tangent to $\Gamma_3$.

1991 APMO, 2

Tags: induction , geometry , parallelogram , analytic geometry , combinatorial geometry , combinatorics

Suppose there are $997$ points given in a plane. If every two points are joined by a line segment with its midpoint coloured in red, show that there are at least $1991$ red points in the plane. Can you find a special case with exactly $1991$ red points?

1996 IMC, 11

Tags: limit , real analysis

i) Prove that $$ \lim_{x\to \infty}\,\sum_{n=1}^{\infty} \frac{nx}{(n^{2}+x)^{2}}=\frac{1}{2}$$. ii) Prove that there is a positive constant $c$ such that for every $x\in [1,\infty)$ we have $$\left|\sum_{n=1}^{\infty} \frac{nx}{(n^{2}+x)^{2}}-\frac{1}{2} \right| \leq \frac{c}{x}$$

2004 Manhattan Mathematical Olympiad, 1

Tags: pigeonhole principle

Seven line segments, with lengths no greater than $10$ inches, and no shorter than $1$ inch, are given. Show that one can choose three of them to represent the sides of a triangle. Give an example which shows that if only six segments are used, then such a choice may be impossible.

2019 Vietnam National Olympiad, Day 1

Tags: integer sequence , perfect number , sum of divisors

Let $({{x}_{n}})$ be an integer sequence such that $0\le {{x}_{0}}<{{x}_{1}}\le 100$ and $${{x}_{n+2}}=7{{x}_{n+1}}-{{x}_{n}}+280,\text{ }\forall n\ge 0.$$ a) Prove that if ${{x}_{0}}=2,{{x}_{1}}=3$ then for each positive integer $n,$ the sum of divisors of the following number is divisible by $24$ $${{x}_{n}}{{x}_{n+1}}+{{x}_{n+1}}{{x}_{n+2}}+{{x}_{n+2}}{{x}_{n+3}}+2018.$$ b) Find all pairs of numbers $({{x}_{0}},{{x}_{1}})$ such that ${{x}_{n}}{{x}_{n+1}}+2019$ is a perfect square for infinitely many nonnegative integer numbers $n.$

LMT Speed Rounds, 2010.17

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Al wishes to label the faces of his cube with the integers $2009,2010,$ and $2011,$ with one integer per face, such that adjacent faces (faces that share an edge) have integers that differ by at most $1.$ Determine the number of distinct ways in which he can label the cube, given that two configurations that can be rotated on to each other are considered the same, and that we disregard the orientation in which each number is written on to the cube.

1979 Spain Mathematical Olympiad, 5

Tags: trigonometry , integration , integral , calculus

Calculate the definite integral $$\int_2^4 \sin ((x-3)^3) dx$$