Olympiad problems

2011 Princeton University Math Competition, B2

Tags: algebra

If $a$ and $b$ are the roots of $x^2 - 2x + 5$, what is $|a^8 + b^8|$?

2023 Ukraine National Mathematical Olympiad, 10.4

Tags: algebra , sequence

Let $(x_n)$ be an infinite sequence of real numbers from interval $(0, 1)$. An infinite sequence $(a_n)$ of positive integers is defined as follows: $a_1 = 1$, and for $i \ge 1$, $a_{i+1}$ is equal to the smallest positive integer $m$, for which $[x_1 + x_2 + \ldots + x_m] = a_i$. Show that for any indexes $i, j$ holds $a_{i+j} \ge a_i + a_j$. [i]Proposed by Nazar Serdyuk[/i]

2008 Spain Mathematical Olympiad, 1

Tags: number theory

Let $p$ and $q$ be two different prime numbers. Prove that there are two positive integers, $a$ and $b$, such that the arithmetic mean of the divisors of $n=p^aq^b$ is an integer.

1973 Yugoslav Team Selection Test, Problem 3

Tags: vector , geometry , geometric transformation

Several points are denoted on a white piece of paper. The distance between each two of the points is greater than $24$. A drop of ink was sprinkled over the paper covering an area smaller than $\pi$. Prove that there exists a vector $\overrightarrow v$ with $\overrightarrow v<1$, such that after translating all of the points by $v$ none of them is covered in ink.

2019 India PRMO, 20

Tags: combinatorics

How many $4-$digit numbers $\overline{abcd}$ are there such that $a<b<c<d$ and $b-a<c-b<d-c$ ?

1997 Romania National Olympiad, 2

Tags: fractional part , algebra

Let $n\geq 3$ be a natural number and $x\in \mathbb{R}$, for which $\{ x\} =\{ x^2\} =\{ x^n\} $ (with $\{ x\} $ we denote the fractional part of $x$). Prove that $x$ is an integer.

2000 France Team Selection Test, 3

Tags: inequalities

$a,b,c,d$ are positive reals with sum $1$. Show that $\frac{a^2}{a+b}+\frac{b^2}{b+c}+\frac{c^2}{c+d}+\frac{d^2}{d+a} \ge \frac{1}{2}$ with equality iff $a=b=c=d=\frac{1}{4}$.

2001 Putnam, 2

Tags: algebra , system of equations

Find all pairs of real numbers $(x,y)$ satisfying the system of equations: \begin{align*}\frac{1}{x} + \frac{1}{2y} &= (x^2+3y^2)(3x^2+y^2)\\ \frac{1}{x} - \frac{1}{2y} &= 2(y^4-x^4)\end{align*}

1988 Canada National Olympiad, 3

Tags: combinatorics

Suppose that $S$ is a finite set of at least five points in the plane; some are coloured red, the others are coloured blue. No subset of three or more similarly coloured points is collinear. Show that there is a triangle (i) whose vertices are all the same colour, and (ii) at least one side of the triangle does not contain a point of the opposite colour.

2022 Bundeswettbewerb Mathematik, 2

Tags: combinatorics

On a table lie 2022 matches and a regular dice that has the number $a$ on top. Now Max and Moritz play the following game: Alternately, they take away matches according to the following rule, where Max begins: The player to make a move rolls the dice over one of its edges and then takes a way as many matches as the top number shows. The player that cannot make legal move after some number of moves loses. For which $a$ can Moritz force Max to lose?

2009 ISI B.Stat Entrance Exam, 2

Tags: function , calculus , derivative , integration , trigonometry , analytic geometry , graphing lines

Let $f(x)$ be a continuous function, whose first and second derivatives are continuous on $[0,2\pi]$ and $f''(x) \geq 0$ for all $x$ in $[0,2\pi]$. Show that \[\int_{0}^{2\pi} f(x)\cos x dx \geq 0\]

1995 China Team Selection Test, 2

Tags: tangent , geometric transformation , locus problems , geometry

Given a fixed acute angle $\theta$ and a pair of internally tangent circles, let the line $l$ which passes through the point of tangency, $A$, cut the larger circle again at $B$ ($l$ does not pass through the centers of the circles). Let $M$ be a point on the major arc $AB$ of the larger circle, $N$ the point where $AM$ intersects the smaller circle, and $P$ the point on ray $MB$ such that $\angle MPN = \theta$. Find the locus of $P$ as $M$ moves on major arc $AB$ of the larger circle.

1991 India National Olympiad, 3

Tags: trigonometry

Given a triangle $ABC$ let \begin{eqnarray*} x &=& \tan\left(\dfrac{B-C}{2}\right) \tan \left(\dfrac{A}{2}\right) \\ y &=& \tan\left(\dfrac{C-A}{2}\right) \tan \left(\dfrac{B}{2}\right) \\ z &=& \tan\left(\dfrac{A-B}{2}\right) \tan \left(\dfrac{C}{2}\right). \end{eqnarray*} Prove that $x+ y + z + xyz = 0$.

CNCM Online Round 1, 5

Tags:

Positive reals $a,b,c \leq 1$ satisfy $\frac{a+b+c-abc}{1-ab-bc-ca} = 1$. Find the minimum value of $$\bigg(\frac{a+b}{1-ab} + \frac{b+c}{1-bc} + \frac{c+a}{1-ca}\bigg)^{2}$$ Proposed by Harry Chen (Extile)

2010 Iran MO (3rd Round), 4

Tags: analytic geometry , rotation , geometry , combinatorics , covering

[b]carpeting[/b] suppose that $S$ is a figure in the plane such that it's border doesn't contain any lattice points. suppose that $x,y$ are two lattice points with the distance $1$ (we call a point lattice point if it's coordinates are integers). suppose that we can cover the plane with copies of $S$ such that $x,y$ always go on lattice points ( you can rotate or reverse copies of $S$). prove that the area of $S$ is equal to lattice points inside it. time allowed for this question was 1 hour.

1960 AMC 12/AHSME, 14

Tags:

If $a$ and $b$ are real numbers, the equation $3x-5+a=bx+1$ has a unique solution $x$ [The symbol $a \neq 0$ means that $a$ is different from zero]: $ \textbf{(A) }\text{for all a and b} \qquad\textbf{(B) }\text{if a }\neq\text{2b}\qquad\textbf{(C) }\text{if a }\neq 6\qquad$ $\textbf{(D) }\text{if b }\neq 0\qquad\textbf{(E) }\text{if b }\neq 3 $

1995 National High School Mathematics League, 5

Tags:

The order of $\log_{\sin1}\cos1,\log_{\sin1}\tan1,\log_{\cos1}\sin1,\log_{\cos1}\tan1$ is (form small to large) $\text{(A)}\log_{\sin1}\cos1<\log_{\cos1}\sin1<\log_{\sin1}\tan1<\log_{\cos1}\tan1$ $\text{(B)}\log_{\cos1}\sin1<\log_{\cos1}\tan1<\log_{\sin1}\cos1<\log_{\sin1}\tan1$ $\text{(C)}\log_{\sin1}\tan1<\log_{\cos1}\tan1<\log_{\cos1}\sin1<\log_{\sin1}\cos1$ $\text{(D)}\log_{\cos1}\tan1<\log_{\sin1}\tan1<\log_{\sin1}\cos1<\log_{\cos1}\sin1$

2018 Azerbaijan BMO TST, 3

Tags: geometry , pentagon , inscribed circle

Let $ABCDE$ be a convex pentagon such that $AB=BC=CD$, $\angle{EAB}=\angle{BCD}$, and $\angle{EDC}=\angle{CBA}$. Prove that the perpendicular line from $E$ to $BC$ and the line segments $AC$ and $BD$ are concurrent.

2018 AIME Problems, 1

Tags:

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 $1000$.

2010 Switzerland - Final Round, 8

Tags: floor function , combinatorics

In a village with at least one inhabitant, there are several associations. Each inhabitant is a member of at least $ k$ associations, and any two associations have at most one common member. Prove that at least $ k$ associations have the same number of members.

2001 VJIMC, Problem 4

Tags: set theory

Let $A,B,C$ be nonempty sets in $\mathbb R^n$. Suppose that $A$ is bounded, $C$ is closed and convex, and $A+B\subseteq A+C$. Prove that $B\subseteq C$. Recall that $E+F=\{e+f:e\in E,f\in F\}$ and $D\subseteq\mathbb R^n$ is convex iff $tx+(1-t)y\in D$ for all $x,y\in D$ and any $t\in[0,1]$.

1994 Cono Sur Olympiad, 1

Tags: algebra

The positive integrer number $n$ has $1994$ digits. $14$ of its digits are $0$'s and the number of times that the other digits: $1, 2, 3, 4, 5, 6, 7, 8, 9$ appear are in proportion $1: 2: 3: 4: 5: 6: 7: 8: 9$, respectively. Prove that $n$ is not a perfect square.

2001 Slovenia National Olympiad, Problem 3

Tags: geometry

Let $E$ and $F$ be points on the side $AB$ of a rectangle $ABCD$ such that $AE = EF$. The line through $E$ perpendicular to $AB$ intersects the diagonal $AC$ at $G$, and the segments $FD$ and $BG$ intersect at $H$. Prove that the areas of the triangles $FBH$ and $GHD$ are equal.

2024 Indonesia TST, 2

Tags: number theory

For positive integers $n$ and $k \geq 2$, define $E_k(n)$ as the greatest exponent $r$ such that $k^r$ divides $n!$. Prove that there are infinitely many $n$ such that $E_{10}(n) > E_9(n)$ and infinitely many $m$ such that $E_{10}(m) < E_9(m)$.

2013 Romania Team Selection Test, 1

Tags: floor function , ratio , limit , number theory , greatest common divisor , inequalities , algebra

Suppose that $a$ and $b$ are two distinct positive real numbers such that $\lfloor na\rfloor$ divides $\lfloor nb\rfloor$ for any positive integer $n$. Prove that $a$ and $b$ are positive integers.