Olympiad problems

2002 National High School Mathematics League, 4

Line $\frac{x}{4}+\frac{y}{3}=1$ and ellipse $\frac{x^2}{16}+\frac{y^2}{9}=1$ intersect at $A$ and $B$. A point on the ellipse $P$ satisties that the area of $\triangle PAB$ is $3$. The number of such points is $\text{(A)}1\qquad\text{(B)}2\qquad\text{(C)}3\qquad\text{(D)}4$

2016 USAMTS Problems, 2:

Tags: USAMTS

A tower of height $h$ is a stack of contiguous rows of squares of height $h$ such that [list] [*] the bottom row of the tower has $h$ squares, [*] each row above the bottom row has one fewer square than the row below it, and within each row the squares are contiguous, [*] the squares in any given row all lie directly above a square in the row below. [/list] A tower is called balanced if when the squares of the tower are colored black and white in a checkerboard fashion, the number of black squares is equal to the number of white squares. For example, the figure above shows a tower of height 5 that is not balanced, since there are 7 white squares and 8 black squares. How many balanced towers are there of height 2016?

2022 Korea Junior Math Olympiad, 2

Tags: combinatorics , counting

For positive integer $n \ge 3$, find the number of ordered pairs $(a_1, a_2, ... , a_n)$ of integers that satisfy the following two conditions [list=disc] [*]For positive integer $i$ such that $1\le i \le n$, $1 \le a_i \le i$ [*]For positive integers $i,j,k$ such that $1\le i < j < k \le n$, if $a_i = a_j$ then $a_j \ge a_k$ [/list]

2023 Princeton University Math Competition, B2

Tags: geometry

The area of the largest square that can be inscribed in a regular hexagon with sidelength $1$ can be expressed as $a-b\sqrt{c}$ where $c$ is not divisible by the square of any prime. Find $a+b+c$.

1997 AIME Problems, 9

Tags: floor function

Given a nonnegative real number $x,$ let $\langle x\rangle$ denote the fractional part of $x;$ that is, $\langle x\rangle=x-\lfloor x\rfloor,$ where $\lfloor x\rfloor$ denotes the greatest integer less than or equal to $x.$ Suppose that $a$ is positive, $\langle a^{-1}\rangle=\langle a^2\rangle,$ and $2<a^2<3.$ Find the value of $a^{12}-144a^{-1}.$

2010 Harvard-MIT Mathematics Tournament, 5

Tags: calculus , function , algebra , polynomial

Let the functions $f(\alpha,x)$ and $g(\alpha)$ be defined as \[f(\alpha,x)=\dfrac{(\frac{x}{2})^\alpha}{x-1}\qquad\qquad\qquad g(\alpha)=\,\dfrac{d^4f}{dx^4}|_{x=2}\] Then $g(\alpha)$ is a polynomial is $\alpha$. Find the leading coefficient of $g(\alpha)$.

2022 Thailand Online MO, 9

Tags: number theory

The number $1$ is written on the blackboard. At any point, Kornny may pick two (not necessary distinct) of the numbers $a$ and $b$ written on the board and write either $ab$ or $\frac{1}{a}+\frac{1}{b}+\frac{1}{ab}$ on the board as well. Determine all possible numbers that Kornny can write on the board in finitely many steps.

LMT Team Rounds 2021+, 3

Tags: number theory

Beter Pai wants to tell you his fastest $40$-line clear time in Tetris, but since he does not want Qep to realize she is better at Tetris than he is, he does not tell you the time directly. Instead, he gives you the following requirements, given that the correct time is t seconds: $\bullet$ $t < 100$. $\bullet$ $t$ is prime. $\bullet$ $t -1$ has 5 proper factors. $\bullet$ all prime factors of $t +1$ are single digits. $\bullet$ $t -2$ is a multiple of $3$. $\bullet$ $t +2$ has $2$ factors. Find t.

JOM 2025, 5

Tags: combinatorics

There are $n>1$ cities in Jansonland, with two-way roads joining certain pairs of cities. Janson will send a few robots one-by-one to build more roads. The robots operate as such: 1. Janson first selects an integer $k$ and a list of cities $a_0, a_1, \dots, a_k$ (cities can repeat). 2. The robot begins at $a_0$ and goes to $a_1$, then $a_2$, and so on until $a_k$. 3. When the robot goes from $a_i$ to $a_{i+1}$, if there is no road then the robot builds a road, but if there is a road then the robot destroys the road. In terms of $n$, determine the smallest constant $k$ such that Janson can always achieve a configuration such that every pair of cities has a road connecting them using no more than $k$ robots. [i](Proposed by Ho Janson)[/i]

2010 Spain Mathematical Olympiad, 2

Tags: function , floor function , induction , algebra proposed , algebra

Let $\mathbb{N}_0$ and $\mathbb{Z}$ be the set of all non-negative integers and the set of all integers, respectively. Let $f:\mathbb{N}_0\rightarrow\mathbb{Z}$ be a function defined as \[f(n)=-f\left(\left\lfloor\frac{n}{3}\right\rfloor \right)-3\left\{\frac{n}{3}\right\} \] where $\lfloor x \rfloor$ is the greatest integer smaller than or equal to $x$ and $\{ x\}=x-\lfloor x \rfloor$. Find the smallest integer $n$ such that $f(n)=2010$.

1960 AMC 12/AHSME, 33

Tags: number theory , prime numbers , AMC

You are given a sequence of $58$ terms; each term has the form $P+n$ where $P$ stands for the product $2 \times 3 \times 5 \times... \times 61$ of all prime numbers less than or equal to $61$, and $n$ takes, successively, the values $2, 3, 4, ...., 59$. let $N$ be the number of primes appearing in this sequence. Then $N$ is: $ \textbf{(A)}\ 0\qquad\textbf{(B)}\ 16\qquad\textbf{(C)}\ 17\qquad\textbf{(D)}\ 57\qquad\textbf{(E)}\ 58 $

1958 November Putnam, B5

Tags: Putnam , broken line

The lengths of successive segments of a broken line are represented by the successive terms of the harmonic progression $1, 1\slash 2, 1\slash 3, \ldots.$ Each segment makes with the preceding a given angle $\theta.$ What is the distance and what is the direction of the limiting points (if there is one) from the initial point of the first segment?

2023 Sharygin Geometry Olympiad, 8.3

Tags: geometry , Sharygin Geometry Olympiad , Sharygin 2023 , parallelogram

The altitudes of a parallelogram are greater than $1$. Does this yield that the unit square may be covered by this parallelogram?

2004 Baltic Way, 3

Tags: inequalities , inequalities proposed

Let $p, q, r$ be positive real numbers and $n$ a natural number. Show that if $pqr = 1$, then \[ \frac{1}{p^n+q^n+1} + \frac{1}{q^n+r^n+1} + \frac{1}{r^n+p^n+1} \leq 1. \]

2011 Today's Calculation Of Integral, 739

Tags: calculus , integration , function , trigonometry , derivative , algebra , functional equation

Find the function $f(x)$ such that : \[f(x)=\cos x+\int_0^{2\pi} f(y)\sin (x-y)\ dy\]

1978 IMO Shortlist, 2

Tags: symmetry , geometry , similar triangles , Equilateral Triangle , IMO Shortlist

Two identically oriented equilateral triangles, $ABC$ with center $S$ and $A'B'C$, are given in the plane. We also have $A' \neq S$ and $B' \neq S$. If $M$ is the midpoint of $A'B$ and $N$ the midpoint of $AB'$, prove that the triangles $SB'M$ and $SA'N$ are similar.

2020 Princeton University Math Competition, 13

Tags: algebra , combinatorics

Will and Lucas are playing a game. Will claims that he has a polynomial $f$ with integer coefficients in mind, but Lucas doesn’t believe him. To see if Will is lying, Lucas asks him on minute $i$ for the value of $f(i)$, starting from minute $ 1$. If Will is telling the truth, he will report $f(i)$. Otherwise, he will randomly and uniformly pick a positive integer from the range $[1,(i+1)!]$. Now, Lucas is able to tell whether or not the values that Will has given are possible immediately, and will call out Will if this occurs. If Will is lying, say the probability that Will makes it to round $20$ is $a/b$. If the prime factorization of $b$ is $p_1^{e_1}... p_k^{e_k}$ , determine the sum $\sum_{i=1}^{k} e_i$.

2017 Pan-African Shortlist, N2

Tags: number theory , prime numbers , sum of cubes , algebra , factorization

For which prime numbers $p$ can we find three positive integers $n$, $x$ and $y$ such that $p^n = x^3 + y^3$?

2001 All-Russian Olympiad, 1

Tags: induction , combinatorics unsolved , combinatorics

The total mass of $100$ given weights with positive masses equals $2S$. A natural number $k$ is called [i]middle[/i] if some $k$ of the given weights have the total mass $S$. Find the maximum possible number of middle numbers.

2007 South East Mathematical Olympiad, 2

Tags: geometry , circumcircle , cyclic quadrilateral , geometry unsolved

In right-angle triangle $ABC$, $\angle C=90$°, Point $D$ is the midpoint of side $AB$. Points $M$ and $C$ lie on the same side of $AB$ such that $MB\bot AB$, line $MD$ intersects side $AC$ at $N$, line $MC$ intersects side $AB$ at $E$. Show that $\angle DBN=\angle BCE$.

2014 Purple Comet Problems, 18

Tags: function

Let $f$ be a real-valued function such that $4f(x)+xf\left(\tfrac1x\right)=x+1$ for every positive real number $x$. Find $f(2014)$.

2022 CMIMC Integration Bee, 1

Tags: integration bee , calculus , integration

\[\int_0^{\pi/1011}\sin^2(2022x)+\cos^2(2022x)\mathrm dx\] [i]Proposed by Connor Gordon[/i]

2010 ELMO Shortlist, 6

Tags: combinatorics proposed , combinatorics

Hamster is playing a game on an $m \times n$ chessboard. He places a rook anywhere on the board and then moves it around with the restriction that every vertical move must be followed by a horizontal move and every horizontal move must be followed by a vertical move. For what values of $m,n$ is it possible for the rook to visit every square of the chessboard exactly once? A square is only considered visited if the rook was initially placed there or if it ended one of its moves on it. [i]Brian Hamrick.[/i]

1983 IMO Longlists, 3

Tags: geometry , 3D geometry , tetrahedron , geometry proposed

[b](a)[/b] Given a tetrahedron $ABCD$ and its four altitudes (i.e., lines through each vertex, perpendicular to the opposite face), assume that the altitude dropped from $D$ passes through the orthocenter $H_4$ of $\triangle ABC$. Prove that this altitude $DH_4$ intersects all the other three altitudes. [b](b)[/b] If we further know that a second altitude, say the one from vertex A to the face $BCD$, also passes through the orthocenter $H_1$ of $\triangle BCD$, then prove that all four altitudes are concurrent and each one passes through the orthocenter of the respective triangle.

2023 Sharygin Geometry Olympiad, 20

Tags: geometry , Isogonal conjugate , tangent , symmedian , Parallel Lines , Sharygin Geometry Olympiad , Sharygin 2023

Let a point $D$ lie on the median $AM$ of a triangle $ABC$. The tangents to the circumcircle of triangle $BDC$ at points $B$ and $C$ meet at point $K$. Prove that $DD'$ is parallel to $AK$, where $D'$ is isogonally conjugated to $D$ with respect to $ABC$.