Olympiad problems

2019 Oral Moscow Geometry Olympiad, 1

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

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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

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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

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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.

1975 Bundeswettbewerb Mathematik, 4

Tags: combinatorics

Two brothers inherited $n$ gold pieces of the total weight $2n$. The weights of the pieces are integers, and the heaviest piece is not heavier than all the other pieces together. Show that if $n$ is even, the brother can divide the inheritance into two parts of equal weight.

2012 Bulgaria National Olympiad, 2

Tags: quadratic , function , algebra

Let $Q(x)$ be a quadratic trinomial. Given that the function $P(x)=x^{2}Q(x)$ is increasing in the interval $(0,\infty )$, prove that: \[P(x) + P(y) + P(z) > 0\] for all real numbers $x,y,z$ such that $x+y+z>0$ and $xyz>0$.

2014 JBMO Shortlist, 4

Tags: number theory , prime number

Prove that there are not intgers $a$ and $b$ with conditions, i) $16a-9b$ is a prime number. ii) $ab$ is a perfect square. iii) $a+b$ is also perfect square.

2015 JBMO Shortlist, A4

Tags: inequalities

Let $a,b,c$ be positive real numbers such that $a+b+c = 3$. Find the minimum value of the expression \[A=\dfrac{2-a^3}a+\dfrac{2-b^3}b+\dfrac{2-c^3}c.\]

2019 USMCA, 22

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Find the largest real number $\lambda$ such that \[a_1^2 + \cdots + a_{2019}^2 \ge a_1a_2 + a_2a_3 + \cdots + a_{1008}a_{1009} + \lambda a_{1009}a_{1010} + \lambda a_{1010}a_{1011} + a_{1011}a_{1012} + \cdots + a_{2018}a_{2019}\] for all real numbers $a_1, \ldots, a_{2019}$. The coefficients on the right-hand side are $1$ for all terms except $a_{1009}a_{1010}$ and $a_{1010}a_{1011}$, which have coefficient $\lambda$.

2020 Harvard-MIT Mathematics Tournament, 1

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Let $DIAL$, $FOR$, and $FRIEND$ be regular polygons in the plane. If $ID=1$, find the product of all possible areas of $OLA$. [i]Proposed by Andrew Gu.[/i]

2011 District Olympiad, 3

Tags: function , integral , riemann sum , real analysis

Let $ f:[0,1]\longrightarrow\mathbb{R} $ be a continuous and nondecreasing function. [b]a)[/b] Show that the sequence $ \left( \frac{1}{2^n}\sum_{i=1}^{2^n} f\left(\frac{i}{2^n}\right) \right)_{n\ge 1} $ is nonincreasing. [b]b)[/b] Prove that, if there exists some natural index at which the sequence above is equal to $ \int_0^1 f(x)dx, $ then $ f $ is constant.

1993 India Regional Mathematical Olympiad, 3

Tags: congruent triangles , geometry

Suppose $A_1, A_2, A_3, \ldots, A_{20}$is a 20 sides regular polygon. How many non-isosceles (scalene) triangles can be formed whose vertices are among the vertices of the polygon but the sides are not the sides of the polygon?

1986 IMO Longlists, 41

Tags: circumcircle , trigonometry , symmetry , geometry

Let $M,N,P$ be the midpoints of the sides $BC, CA, AB$ of a triangle $ABC$. The lines $AM, BN, CP$ intersect the circumcircle of $ABC$ at points $A',B', C'$, respectively. Show that if $A'B'C'$ is an equilateral triangle, then so is $ABC.$