Olympiad problems

2007 APMO, 2

Tags: geometry , incenter

Let $ABC$ be an acute angled triangle with $\angle{BAC}=60^\circ$ and $AB > AC$. Let $I$ be the incenter, and $H$ the orthocenter of the triangle $ABC$ . Prove that $2\angle{AHI}= 3\angle{ABC}$.

An ice ballerina rotates at a constant angular velocity at one particular point. That is, she does not translationally move. Her arms are fully extended as she rotates. Her moment of inertia is $I$. Now, she pulls her arms in and her moment of inertia is now $\frac{7}{10}I$. What is the ratio of the new kinetic energy (arms in) to the initial kinetic energy (arms out)? $ \textbf {(A) } \dfrac {7}{10} \qquad \textbf {(B) } \dfrac {49}{100} \qquad \textbf {(C) } 1 \qquad \textbf {(C) } \dfrac {100}{49} \qquad \textbf {(E) } \dfrac {10}{7} $ [i]Problem proposed by Ahaan Rungta[/i]

2000 Harvard-MIT Mathematics Tournament, 2

Tags:

Simplify $\left(\dfrac{-1+i\sqrt{3}}{2}\right)^6+\left(\dfrac{-1-i\sqrt{3}}{2}\right)^6$ to the form $a+bi$.

2018 Kyiv Mathematical Festival, 4

Tags: Kyiv mathematical festival , number theory , Divisibility

Do there exist positive integers $a$ and $b$ such that each of the numbers $2^a+3^b,$ $3^a+5^b$ and $5^a+2^b$ is divisible by 29?

Kettering MO, 2015

Tags: algebra , geometry , combinatorics , number theory , Kettering MO

[b]p1.[/b] Solve the equation $\log_x (x + 2) = 2$. [b]p2.[/b] Solve the inequality: $0.5^{|x|} > 0.5^{x^2}$. [b]p3.[/b] The integers from 1 to 2015 are written on the blackboard. Two randomly chosen numbers are erased and replaced by their difference giving a sequence with one less number. This process is repeated until there is only one number remaining. Is the remaining number even or odd? Justify your answer. [b]p4.[/b] Four circles are constructed with the sides of a convex quadrilateral as the diameters. Does there exist a point inside the quadrilateral that is not inside the circles? Justify your answer. [b]p5.[/b] Prove that for any finite sequence of digits there exists an integer the square of which begins with that sequence. [b]p6.[/b] The distance from the point $P$ to two vertices $A$ and $B$ of an equilateral triangle are $|P A| = 2$ and $|P B| = 3$. Find the greatest possible value of $|P C|$. PS. You should use hide for answers.

2013 F = Ma, 9

Tags:

A truck is initially moving at velocity $v$. The driver presses the brake in order to slow the truck to a stop. The brake applies a constant force $F$ to the truck. The truck rolls a distance $x$ before coming to a stop, and the time it takes to stop is $t$. Which of the following expressions is equal the initial momentum of the truck (i.e. the momentum before the driver starts braking)? $\textbf{(A) } Fx\\ \textbf{(B) } Ft/2\\ \textbf{(C) } Fxt\\ \textbf{(D) } 2Ft\\ \textbf{(E) } 2Fx/v$

2007 Today's Calculation Of Integral, 249

Tags: calculus , integration , logarithms , calculus computations

Determine the sign of $ \int_{\frac{1}{2}}^2 \frac{\ln t}{1\plus{}t^n}\ dt\ (n\equal{}1, 2, \cdots)$.

2024 Turkey Junior National Olympiad, 3

Tags: combinatorics

Let $n\geq 2$ be an integer and $a_1,a_2,\cdots,a_n$ be distinct positive real numbers. For any $(i,j)$ in a country consisting of cities $C_1,C_2,\cdots,C_n$, there is a two-way flight between $C_i$ and $C_j$ that costs $a_i+a_j$.A traveler travels between cities of this country such that every time they pay a strictly higher cost than their previous flight. Find the maximum number of flight this traveler could take.

2005 China Team Selection Test, 2

Tags: combinatorics unsolved , combinatorics

Given positive integer $n (n \geq 2)$, find the largest positive integer $\lambda$ satisfying : For $n$ bags, if every bag contains some balls whose weights are all integer powers of $2$ (the weights of balls in a bag may not be distinct), and the total weights of balls in every bag are equal, then there exists a weight among these balls such that the total number of balls with this weight is at least $\lambda$.

2001 Brazil National Olympiad, 4

Tags: trigonometry , function , inequalities , algebra proposed , algebra

A calculator treats angles as radians. It initially displays 1. What is the largest value that can be achieved by pressing the buttons cos or sin a total of 2001 times? (So you might press cos five times, then sin six times and so on with a total of 2001 presses.)

1999 All-Russian Olympiad Regional Round, 10.7

Tags: combinatorics

Each voter in an election puts $n$ names of candidates on the ballot. There are $n + 1$ at the polling station urn. After the elections it turned out that each ballot box contained at least at least one ballot, for every choice of the $(n + 1)$-th ballot, one from each ballot box, there is a candidate whose surname appears in each of the selected ballots. Prove that in at least one ballot box all ballots contain the name of the same candidate.

2018 AIME Problems, 1

Tags: AMC , AIME , AIME I , 2018 AIME I

Let $S$ be the number of ordered pairs of integers $(a,b)$ with $1 \leq a \leq 100$ and $b \geq 0$ such that the polynomial $x^2+ax+b$ can be factored into the product of two (not necessarily distinct) linear factors with integer coefficients. Find the remainder when $S$ is divided by $1000$.

2013 Saudi Arabia BMO TST, 1

Tags: geometry , Cyclic

$ABCD$ is a cyclic quadrilateral such that $AB = BC = CA$. Diagonals $AC$ and $BD$ intersect at $E$. Given that $BE = 19$ and $ED = 6$, find the possible values of $AD$.

2015 AMC 12/AHSME, 3

Tags: AMC

Mr. Patrick teaches math to $15$ students. He was grading tests and found that when he graded everyone's test except Payton's, the average grade for the class was $80$. After he graded Payton's test, the class average became $81$. What was Payton's score on the test? $\textbf{(A) }81\qquad\textbf{(B) }85\qquad\textbf{(C) }91\qquad\textbf{(D) }94\qquad\textbf{(E) }95$

1997 South africa National Olympiad, 4

Tags: function , algebra , functional equation , algebra unsolved

Find all functions $f: \mathbb{Z} \rightarrow \mathbb{Z}$ which satisfy \[ f(m + f(n)) = f(m) + n \] for all $m,n \in \mathbb{Z}$.

2021 Taiwan TST Round 3, A

Tags: inequalities , n-variable inequality , IMO Shortlist , IMO Shortlist 2020

Let $n$ and $k$ be positive integers. Prove that for $a_1, \dots, a_n \in [1,2^k]$ one has \[ \sum_{i = 1}^n \frac{a_i}{\sqrt{a_1^2 + \dots + a_i^2}} \le 4 \sqrt{kn}. \]

1957 Polish MO Finals, 5

Tags: geometry , trisection , construction

Given a line $ m $ and a segment $ AB $ parallel to it. Divide the segment $ AB $ into three equal parts using only a ruler, i.e. drawing only the lines.

1991 Federal Competition For Advanced Students, 2

Tags: algebra proposed , algebra

Solve in real numbers the equation: $ \frac{1}{x}\plus{}\frac{1}{x\plus{}2}\minus{}\frac{1}{x\plus{}4}\minus{}\frac{1}{x\plus{}6}\minus{}\frac{1}{x\plus{}8}\minus{}\frac{1}{x\plus{}10}\plus{}\frac{1}{x\plus{}12}\plus{}\frac{1}{x\plus{}14}\equal{}0.$

1959 IMO Shortlist, 2

Tags: algebra , equation , square roots , IMO , IMO 1959

For what real values of $x$ is \[ \sqrt{x+\sqrt{2x-1}}+\sqrt{x-\sqrt{2x-1}}=A \] given a) $A=\sqrt{2}$; b) $A=1$; c) $A=2$, where only non-negative real numbers are admitted for square roots?

2003 Estonia National Olympiad, 5

Tags: winning strategy , game , game strategy , combinatorics

The game [i]Clobber [/i] is played by two on a strip of $2k$ squares. At the beginning there is a piece on each square, the pieces of both players stand alternatingly. At each move the player shifts one of his pieces to the neighbouring square that holds a piece of his opponent and removes his opponent’s piece from the table. The moves are made in turn, the player whose opponent cannot move anymore is the winner. Prove that if for some $k$ the player who does not start the game has the winning strategy, then for $k + 1$ and $k + 2$ the player who makes the first move has the winning strategy.

1994 Balkan MO, 1

Tags: geometry , parallelogram , analytic geometry , trigonometry , parameterization , conics , parabola

An acute angle $XAY$ and a point $P$ inside the angle are given. Construct (using a ruler and a compass) a line that passes through $P$ and intersects the rays $AX$ and $AY$ at $B$ and $C$ such that the area of the triangle $ABC$ equals $AP^2$. [i]Greece[/i]

2001 India IMO Training Camp, 3

Tags: combinatorics unsolved , combinatorics

Each vertex of an $m\times n$ grid is colored blue, green or red in such a way that all the boundary vertices are red. We say that a unit square of the grid is properly colored if: $(i)$ all the three colors occur at the vertices of the square, and $(ii)$ one side of the square has the endpoints of the same color. Show that the number of properly colored squares is even.

1992 IMO Longlists, 15

Tags: combinatorics , Intersection , counting , straight lines , IMO Shortlist , IMO Longlist

Prove that there exist $78$ lines in the plane such that they have exactly $1992$ points of intersection.

2024 Mexican University Math Olympiad, 5

Tags: Sequence , permutations , combinatorics

Consider two finite sequences of real numbers $ a_1, a_2, \dots, a_n $ and $ b_1, b_2, \dots, b_n $. Let $ \alpha(x) = \#\{i | a_i = x \} $ and $ \beta(x) = \#\{i | b_i = -x \} $. Prove that there exists a permutation $ \sigma \in S_n $ (the symmetric group of $ n $ elements) such that $ a_{\sigma(i)} + b_i \neq 0 $ for all $ i = 1, \dots, n $ if and only if $ \alpha(x) + \beta(x) \leq n $ for all $ x \in \mathbb{R} $.

1993 All-Russian Olympiad Regional Round, 9.7

Tags: geometry , rhombus , collinear , equal segments

On the diagonal $AC$ of the rhombus $ABCD$, a point $E$ is taken, which is different from points $A$ and $C$, and on the lines $AB$ and $BC$ are points $N$ and $M$, respectively, with $AE = NE$ and $CE = ME$. Let $K$ be the intersection point of lines $AM$ and $CN$. Prove that points $K, E$ and $D$ are collinear.