Olympiad problems

1983 IMO Longlists, 13

Let $p$ be a prime number and $a_1, a_2, \ldots, a_{(p+1)/2}$ different natural numbers less than or equal to $p.$ Prove that for each natural number $r$ less than or equal to $p$, there exist two numbers (perhaps equal) $a_i$ and $a_j$ such that \[p \equiv a_i a_j \pmod r.\]

2016 Canadian Mathematical Olympiad Qualification, 4

Tags: algebra , functional equation , function

Determine all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ such that $$f(x + f(y)) + f(x - f(y)) = x.$$

2024 Dutch BxMO/EGMO TST, IMO TSTST, 2

Tags: algebra

We define a sequence with $a_1=850$ and $$a_{n+1}=\frac{a_n^2}{a_n-1}$$ for $n\geq 1$. Find all values of $n$ for which $\lfloor a_n\rfloor =2024$.

MBMT Team Rounds, 2015 F8 E5

Tags:

Victor has $3$ piles of $3$ cards each. He draws all of the cards, but cannot draw a card until all the cards above it have been drawn. (For example, for his first card, Victor must draw the top card from one of the $3$ piles.) In how many orders can Victor draw the cards?

2019 USMCA, 4

Tags:

Find all functions $f: \mathbb R \rightarrow \mathbb R$ such that for all $x, y \in \mathbb R$, $$f(f(x) + y)^2 = (x-y)(f(x) - f(y)) + 4f(x) f(y).$$

2012 Today's Calculation Of Integral, 785

Tags: calculus , integration , derivative , calculus computations , logarithm

For a positive real number $x$, find the minimum value of $f(x)=\int_x^{2x} (t\ln t-t)dt.$

2000 Iran MO (2nd round), 1

Tags: combinatorics

$21$ distinct numbers are chosen from the set $\{1,2,3,\ldots,2046\}.$ Prove that we can choose three distinct numbers $a,b,c$ among those $21$ numbers such that \[bc<2a^2<4bc\]

2002 BAMO, 3

Tags: game theory , combinatorics

A game is played with two players and an initial stack of $n$ pennies $(n \geq 3)$. The players take turns choosing one of the stacks of pennies on the table and splitting it into two stacks. The winner is the player who makes a move that causes all stacks to be of height $1$ or $2.$ For which starting values of n does the player who goes first win, assuming best play by both players?

1994 Czech And Slovak Olympiad IIIA, 6

Tags: inequalities , algebra

Show that from any four distinct numbers lying in the interval $(0,1)$ one can choose two distinct numbers $a$ and $b$ such that $$\sqrt{(1-a^2)(1-b^2)} > \frac{a}{2b}+\frac{b}{2a}-ab-\frac{1}{8ab} $$

2024 Australian Mathematical Olympiad, P6

Tags: combinatorics

In a school, there are $1000$ students in each year level, from Year $1$ to Year $12$. The school has $12 000$ lockers, numbered from $1$ to $12 000$. The school principal requests that each student is assigned their own locker, so that the following condition is satisfied: For every pair of students in the same year level, the difference between their locker numbers must be divisible by their year-level number. Can the principal’s request be satisfied?

2007 IMS, 7

Tags: linear algebra , search , number theory

$x_{1},x_{2},\dots,x_{n}$ are real number such that for each $i$, the set $\{x_{1},x_{2},\dots,x_{n}\}\backslash \{x_{i}\}$ could be partitioned into two sets that sum of elements of first set is equal to the sum of the elements of the other. Prove that all of $x_{i}$'s are zero. [hide="Hint"]It is a number theory problem.[/hide]

PEN F Problems, 13

Tags: rational number , factorial

Prove that numbers of the form \[\frac{a_{1}}{1!}+\frac{a_{2}}{2!}+\frac{a_{3}}{3!}+\cdots,\] where $0 \le a_{i}\le i-1 \;(i=2, 3, 4, \cdots)$ are rational if and only if starting from some $i$ on all the $a_{i}$'s are either equal to $0$ ( in which case the sum is finite) or all are equal to $i-1$.

2020 Princeton University Math Competition, B3

Tags: geometry , geometric inequality , tangent circle

Let $ABC$ be a triangle and let the points $D, E$ be on the rays $AB$, $AC$ such that $BCED$ is cyclic. Prove that the following two statements are equivalent: $\bullet$ There is a point $X$ on the circumcircle of $ABC$ such that $BDX$, $CEX$ are tangent to each other. $\bullet$ $AB \cdot AD \le 4R^2$, where $R$ is the radius of the circumcircle of $ABC$.

1978 Austrian-Polish Competition, 3

Tags: trigonometry , inequalities

Prove that $$\sqrt[44]{\tan 1^\circ\cdot \tan 2^\circ\cdot \dots\cdot \tan 44^\circ}<\sqrt 2-1<\frac{\tan 1^\circ+ \tan 2^\circ+\dots+\tan 44^\circ}{44}.$$

1965 Spain Mathematical Olympiad, 7

Tags: density , mass , geometry

A truncated cone has the bigger base of radius $r$ centimetres and the generatrix makes an angle, with that base, whose tangent equals $m$. The truncated cone is constructed of a material of density $d$ (g/cm$^3$) and the smaller base is covered by a special material of density $p$ (g/cm$^2$). Which is the height of the truncated cone that maximizes the total mass?

2012 Centers of Excellency of Suceava, 2

Tags: function , find all functions , algebra

Find all functions $ f:\mathbb{R}\longrightarrow\mathbb{R} $ that verify, for any nonzero real number $ x $ the relation $$ xf(x/a)-f(a/x)=b, $$ where $ a\neq 0,b $ are two real numbers. [i]Dan Popescu[/i]

2021 USMCA, 17

Tags:

Let $X_1X_2X_3X_4$ be a quadrilateral inscribed in circle $\Omega$ such that $\triangle{X_1X_2X_3}$ has side lengths $13,14,15$ in some order. For $1 \le i \le 4$, let $l_i$ denote the tangent to $\Omega$ at $X_i$, and let $Y_i$ denote the intersection of $l_i$ and $l_{i+1}$ (indices taken modulo $4$). Find the least possible area of $Y_1Y_2Y_3Y_4$.

2000 Moldova National Olympiad, Problem 8

Tags: geometry , geometric inequality

A circle with radius $r$ touches the sides $AB,BC,CD,DA$ of a convex quadrilateral $ABCD$ at $E,F,G,H$, respectively. The inradii of the triangles $EBF,FCG,GDH,HAE$ are equal to $r_1,r_2,r_3,r_4$. Prove that $$r_1+r_2+r_3+r_4\ge2\left(2-\sqrt2\right)r.$$

2014 India Regional Mathematical Olympiad, 3

Tags: trigonometry , geometry , angle bisector

Let $ABC$ be an acute-angled triangle in which $\angle ABC$ is the largest angle. Let $O$ be its circumcentre. The perpendicular bisectors of $BC$ and $AB$ meet $AC$ at $X$ and $Y$ respectively. The internal angle bisectors of $\angle AXB$ and $\angle BYC$ meet $AB$ and $BC$ at $D$ and $E$ respectively. Prove that $BO$ is perpendicular to $AC$ if $DE$ is parallel to $AC$.

2016 Tuymaada Olympiad, 4

Tags: inequalities

Non-negative numbers $a$, $b$, $c$ satisfy $a^2+b^2+c^2\geq 3$. Prove the inequality $$ (a+b+c)^3\geq 9(ab+bc+ca). $$

1993 Tournament Of Towns, (386) 4

Tags: combinatorics

Diagonals of a $1$ by $1$ square are arranged in an $8$ by $8$ table (one in each $1$ by $1 $ square). Consider the union $W$ of all $64$ diagonals drawn. The set $W$ consists of several connected pieces (two points belong to the same piece if and only if W contains a path between them). Can the number of the pieces be greater than (a) $15$, (b) $20$? (NB Vassiliev)

2008 Swedish Mathematical Competition, 4

Tags: geometry , combinatorial geometry , angle

A convex $n$-side polygon has angles $v_1,v_2,\dots,v_n$ (in degrees), where all $v_k$ ($k = 1,2,\dots,n$) are positive integers divisible by $36$. (a) Determine the largest $n$ for which this is possible. (b) Show that if $n>5$, two of the sides of the $n$-polygon must be parallel.

2011 National Olympiad First Round, 31

Tags: floor function

For the integer numbers $i,j,k$ satisfying the condtion $i^2+j^2+k^2=2011$, what is the largest value of $i+j+k$? $\textbf{(A)}\ 71 \qquad\textbf{(B)}\ 73 \qquad\textbf{(C)}\ 74 \qquad\textbf{(D)}\ 76 \qquad\textbf{(E)}\ 77$

2018 AIME Problems, 15

Tags: trigonometry , geometry , cyclic quadrilateral

David found four sticks of different lengths that can be used to form three non-congruent convex cyclic quadrilaterals, $A$, $B$, $C$, which can each be inscribed in a circle with radius $1$. Let $\varphi_A$ denote the measure of the acute angle made by the diagonals of quadrilateral $A$, and define $\varphi_B$ and $\varphi_C$ similarly. Suppose that $\sin\varphi_A=\frac{2}{3}$, $\sin\varphi_B=\frac{3}{5}$, and $\sin\varphi_C=\frac{6}{7}$. All three quadrilaterals have the same area $K$, which can be written in the form $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

2001 Austrian-Polish Competition, 1

Tags: modular arithmetic , number theory

Determine the number of positive integers $a$, so that there exist nonnegative integers $x_0,x_1,\ldots,x_{2001}$ which satisfy the equation \[ \displaystyle a^{x_0} = \sum_{i=1}^{2001} a^{x_i} \]