Olympiad problems

1987 Tournament Of Towns, (155) 6

Tags: combinatorics

There are $2000$ apples , contained in several baskets. One can remove baskets and /or remove apples from the baskets. Prove that it is possible to then have an equal number of apples in each of the remaining baskets, with the total number of apples being not less than $100$ . (A. Razborov)

1999 IMC, 4

Tags: function , real analysis

Find all strictly monotonic functions $f: \mathbb{R}^+\rightarrow\mathbb{R}^+$ for which $f\left(\frac{x^2}{f(x)}\right)=x$ for all $x$.

1998 Dutch Mathematical Olympiad, 4

Tags: geometry , rhombus , vector

Let $ABCD$ be a convex quadrilateral such that $AC \perp BD$. (a) Prove that $AB^2 + CD^2 = BC^2 + DA^2$. (b) Let $PQRS$ be a convex quadrilateral such that $PQ = AB$, $QR = BC$, $RS = CD$ and $SP = DA$. Prove that $PR \perp QS$.

2000 Miklós Schweitzer, 8

Tags: function , real analysis

Let $f\colon \mathbb{R}^n \rightarrow \mathbb{R}^m$ be a map such that the image of every compact set is compact, and the image of every connected set is connected. Prove that $f$ is continuous.

1969 AMC 12/AHSME, 12

Tags:

Let $F=\dfrac{6x^2+16x+3m}6$ be the square of an expression which is linear in $x$. Then $m$ has a particular value between: $\textbf{(A) }3\text{ and }4\qquad \textbf{(B) }4\text{ and }5\qquad \textbf{(C) }5\text{ and }6\qquad$ $\textbf{(D) }-4\text{ and }-3\qquad \textbf{(E) }-6\text{ and }-5$

2012 HMNT, 8

Tags: combinatorics

In the game of rock-paper-scissors-lizard-Spock, rock defeats scissors and lizard, paper defeats rock and Spock, scissors defeats paper and lizard, lizard defeats paper and Spock, and Spock defeats rock and scissors, as shown in the below diagram. As before, if two players choose the same move, then there is a draw. If three people each play a game of rock-paper-scissors-lizard-Spock at the same time by choosing one of the five moves at random, what is the probability that one player beats the other two? [img]https://cdn.artofproblemsolving.com/attachments/6/0/3129da5998a2e872673e34351f786ffd47e1a1.png[/img]

1999 BAMO, 1

Tags: number theory

Prove that among any $12$ consecutive positive integers there is at least one which is smaller than the sum of its proper divisors. (The proper divisors of a positive integer n are all positive integers other than $1$ and $n$ which divide $n$. For example, the proper divisors of $14$ are $2$ and $7$.)

1998 Brazil National Olympiad, 2

Tags: function , functional equation , algebra

Find all functions $f : \mathbb N \to \mathbb N$ satisfying, for all $x \in \mathbb N$, \[ f(2f(x)) = x + 1998 . \]

2006 Austrian-Polish Competition, 3

Tags: 3d geometry , tetrahedron , geometry

$ABCD$ is a tetrahedron. Let $K$ be the center of the incircle of $CBD$. Let $M$ be the center of the incircle of $ABD$. Let $L$ be the gravycenter of $DAC$. Let $N$ be the gravycenter of $BAC$. Suppose $AK$, $BL$, $CM$, $DN$ have one common point. Is $ABCD$ necessarily regular?

2019 Oral Moscow Geometry Olympiad, 1

Tags: geometry , square , incircle , angle

Circle inscribed in square $ABCD$ , is tangent to sides $AB$ and $CD$ at points $M$ and $K$ respectively. Line $BK$ intersects this circle at the point $L, X$ is the midpoint of $KL$. Find the angle $\angle MXK $.

MOAA Accuracy Rounds, 2023.9

Tags:

Let $\triangle{ABC}$ be a triangle with $AB = 10$ and $AC = 11$. Let $I$ be the center of the inscribed circle of $\triangle{ABC}$. If $M$ is the midpoint of $AI$ such that $BM = BC$ and $CM = 7$, then $BC$ can be expressed in the form $\frac{\sqrt{a}-b}{c}$ where $a$, $b$, and $c$ are positive integers. Find $a+b+c$. [color=#00f]Note that this problem is null because a diagram is impossible.[/color] [i]Proposed by Andy Xu[/i]

2009 AMC 12/AHSME, 2

Tags:

Paula the painter had just enough paint for $ 30$ identically sized rooms. Unfortunately, on the way to work, three cans of paint fell of her truck, so she had only enough paint for $ 25$ rooms. How many cans of paint did she use for the $ 25$ rooms? $ \textbf{(A)}\ 10 \qquad \textbf{(B)}\ 12 \qquad \textbf{(C)}\ 15 \qquad \textbf{(D)}\ 18 \qquad \textbf{(E)}\ 25$

2018 CMIMC Algebra, 8

Tags: algebra , polynomial , inequalities

Suppose $P$ is a cubic polynomial satisfying $P(0) = 3$ and \[(x^3 - 2x + 1 - P(x))(2x^3 - 5x^2 + 4 - P(x))\leq 0\] for all $x\in\mathbb R$. Determine all possible values of $P(-1)$.

2000 National Olympiad First Round, 11

Tags:

In how many ways can $7$ red, $7$ white balls be distributed into $7$ boxes such that every box contains exactly $2$ balls? $ \textbf{(A)}\ 163 \qquad\textbf{(B)}\ 393 \qquad\textbf{(C)}\ 858 \qquad\textbf{(D)}\ 1716 \qquad\textbf{(E)}\ \text{None} $

1995 May Olympiad, 1

Tags: number theory , game , game strategy , divisor

Veronica, Ana and Gabriela are forming a round and have fun with the following game. One of them chooses a number and says out loud, the one to its left divides it by its largest prime divisor and says the result out loud and so on. The one who says the number out loud $1$ wins , at which point the game ends. Ana chose a number greater than $50$ and less than $100$ and won. Veronica chose the number following the one chosen by Ana and also won. Determine all the numbers that could have been chosen by Ana.

1985 Putnam, B2

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Define polynomials $f_{n}(x)$ for $n \geq 0$ by $f_{0}(x)=1, f_{n}(0)=0$ for $n \geq 1,$ and $$ \frac{d}{d x} f_{n+1}(x)=(n+1) f_{n}(x+1) $$ for $n \geq 0 .$ Find, with proof, the explicit factorization of $f_{100}(1)$ into powers of distinct primes.

2012 AMC 10, 22

Tags: modular arithmetic , number theory , diophantine equation , system of equations , algebra

The sum of the first $m$ positive odd integers is $212$ more than the sum of the first $n$ positive even integers. What is the sum of all possible values of $n$? $ \textbf{(A)}\ 255 \qquad\textbf{(B)}\ 256 \qquad\textbf{(C)}\ 257 \qquad\textbf{(D)}\ 258 \qquad\textbf{(E)}\ 259 $

2015 All-Russian Olympiad, 5

Tags: combinatorics , grid , tiling

It is known that a cells square can be cut into $n$ equal figures of $k$ cells. Prove that it is possible to cut it into $k$ equal figures of $n$ cells.

2012 Mathcenter Contest + Longlist, 2 sl9

Tags: algebra , inequalities

Let $a,b,c \in \mathbb{R}^+$ where $a^2+b^2+c^2=1$. Find the minimum value of . $$a+b+c+\frac{3}{ab+bc+ca}$$ [i](PP-nine)[/i]

1950 Moscow Mathematical Olympiad, 178

Tags: trigonometry , angle

Let $A$ be an arbitrary angle,let $B$ and $C$ be acute angles. Is there an angle $x$ such that $$\sin x =\frac{\sin B \cdot \sin C}{1 - \cos B \cdot \cos C \cdot \cos A} ?$$

2013 ELMO Shortlist, 1

Tags: incenter , circumcircle , inradius , geometry

Let $ABC$ be a triangle with incenter $I$. Let $U$, $V$ and $W$ be the intersections of the angle bisectors of angles $A$, $B$, and $C$ with the incircle, so that $V$ lies between $B$ and $I$, and similarly with $U$ and $W$. Let $X$, $Y$, and $Z$ be the points of tangency of the incircle of triangle $ABC$ with $BC$, $AC$, and $AB$, respectively. Let triangle $UVW$ be the [i]David Yang triangle[/i] of $ABC$ and let $XYZ$ be the [i]Scott Wu triangle[/i] of $ABC$. Prove that the David Yang and Scott Wu triangles of a triangle are congruent if and only if $ABC$ is equilateral. [i]Proposed by Owen Goff[/i]

1993 Poland - First Round, 6

Tags: function

The function $f: R \longrightarrow R$ is continuous. Prove that if for every real number $x$, there exists a positive integer $n$, such that $\underbrace{(f \circ f \circ ... \circ f)}_{n}(x) = 1$, then $f(1) = 1$.

2000 All-Russian Olympiad Regional Round, 9.7

Tags: geometry , perpendicular

On side $AB$ of triangle $ABC$, point $D$ is selected. Circle circumscribed around triangle $BCD$, intersects side $AC$ at point $M$, and the circumcircle of triangle $ACD$ intersects the side $BC$ at point $ N$ ($M,N \ne C$). Let $O$ be the circumcenter of the triangle $CMN$. Prove that line $OD$ is perpendicular to side $AB$.

2016 ASDAN Math Tournament, 12

Tags: team test

Let $$f(x)=\frac{2016^x}{2016^x+\sqrt{2016}}.$$ Evaluate $$\sum_{k=0}^{2016}f\left(\frac{k}{2016}\right).$$

2022 Thailand TSTST, 3

Tags: combinatorics

An odd positive integer $n$ is called pretty if there exists at least one permutation $a_1, a_2,..., a_n$, of $1,2,...,n$, such that all $n$ sums $a_1-a_2+a_3-...+a_n$, $a_2-a_3+a_4-...+a_1$,..., $a_n-a_1+a_2-...+a_{n-1}$ are positive. Find all pretty integers.