Olympiad problems

2014 France Team Selection Test, 6

Tags: inequalities

Let $n$ be a positive integer and $x_1,x_2,\ldots,x_n$ be positive reals. Show that there are numbers $a_1,a_2,\ldots, a_n \in \{-1,1\}$ such that the following holds: \[a_1x_1^2+a_2x_2^2+\cdots+a_nx_n^2 \ge (a_1x_1+a_2x_2 +\cdots+a_nx_n)^2\]

1996 AMC 8, 7

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Brent has goldfish that quadruple (become four times as many) every month, and Gretel has goldfish that double every month. If Brent has $4$ goldfish at the same time that Gretel has $128$ goldfish, then in how many months from that time will they have the same number of goldfish? $\text{(A)}\ 4 \qquad \text{(B)}\ 5 \qquad \text{(C)}\ 6 \qquad \text{(D)}\ 7 \qquad \text{(E)}\ 8$

2022 Grosman Mathematical Olympiad, P3

Tags: combinatorics , combinatorial geometry

An ant crawled a total distance of $1$ in the plane and returned to its original position (so that its path is a closed loop of length $1$; the width is considered to be $0$). Prove that there is a circle of radius $\frac{1}{4}$ containing the path. Illustration of an example path:

2019 Bundeswettbewerb Mathematik, 2

Tags: digit , divisible , divisibility , number theory

The lettes $A,C,F,H,L$ and $S$ represent six not necessarily distinct decimal digits so that $S \ne 0$ and $F \ne 0$. We form the two six-digit numbers $SCHLAF$ and $FLACHS$. Show that the difference of these two numbers is divisible by $271$ if and only if $C=L$ and $H=A$. [i]Remark:[/i] The words "Schlaf" and "Flachs" are German for "sleep" and "flax".

1994 Taiwan National Olympiad, 4

Tags: arithmetic sequence , number theory

Prove that there are infinitely many positive integers $n$ with the following property: For any $n$ integers $a_{1},a_{2},...,a_{n}$ which form in arithmetic progression, both the mean and the standard deviation of the set $\{a_{1},a_{2},...,a_{n}\}$ are integers. [i]Remark[/i]. The mean and standard deviation of the set $\{x_{1},x_{2},...,x_{n}\}$ are defined by $\overline{x}=\frac{x_{1}+x_{2}+...+x_{n}}{n}$ and $\sqrt{\frac{\sum (x_{i}-\overline{x})^{2}}{n}}$, respectively.

2010 Bosnia Herzegovina Team Selection Test, 5

Tags: inequalities

Let $a$,$b$ and $c$ be sides of a triangle such that $a+b+c\le2$. Prove that $-3<{\frac{a^3}{b}+\frac{b^3}{c}+\frac{c^3}{a}-\frac{a^3}{c}-\frac{b^3}{a}-\frac{c^3}{b}}<3$

MBMT Team Rounds, 2020.9

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Consider a regular pentagon $ABCDE$, and let the intersection of diagonals $\overline{CA}$ and $\overline{EB}$ be $F$. Find $\angle AFB$. [i]Proposed by Justin Chen[/i]

2022 AMC 10, 2

Tags: rates

Mike cycled $15$ laps in $57$ minutes. Assume he cycled at a constant speed throughout. Approximately how many laps did he complete in the first $27$ minutes? $\textbf{(A) } 5 \qquad \textbf{(B) } 7 \qquad \textbf{(C) } 9 \qquad \textbf{(D) } 11 \qquad \textbf{(E) } 13$

1976 Swedish Mathematical Competition, 5

Tags: analysis , inequalities , function , algebra

$f(x)$ is defined for $x \geq 0$ and has a continuous derivative. It satisfies $f(0)=1$, $f'(0)=0$ and $(1+f(x))f''(x)=1+x$. Show that $f$ is increasing and that $f(1) \leq 4/3$.

1990 All Soviet Union Mathematical Olympiad, 524

Tags: geometry , regular polygon , angle bisector , angle

$A, B, C$ are adjacent vertices of a regular $2n$-gon and $D$ is the vertex opposite to $B$ (so that $BD$ passes through the center of the $2n$-gon). $X$ is a point on the side $AB$ and $Y$ is a point on the side $BC$ so that $XDY = \frac{\pi}{2n}$. Show that $DY$ bisects $\angle XYC$.

2023 Math Prize for Girls Problems, 17

Tags:

Let $C$ be a unit cube. Let $D$ be a translate of $C$ such that one corner of $D$ is located at the center of $C$ and one corner of $C$ is located at the center of $D$. Let $D^\prime$ be the image of $D$ under a $60^\circ$ clockwise rotation about the line that passes through both cube centers when looking from the center of $D$ to the center of $C$. What is the volume of the intersection of $C$ with $D^\prime$?

2021 Ukraine National Mathematical Olympiad, 5

Tags: diophantine equation , diophantine , number theory

Are there natural numbers $(m,n,k)$ that satisfy the equation $m^m+ n^n=k^k$ ?

1998 AIME Problems, 6

Tags: geometry , parallelogram , similar triangles

Let $ABCD$ be a parallelogram. Extend $\overline{DA}$ through $A$ to a point $P,$ and let $\overline{PC}$ meet $\overline{AB}$ at $Q$ and $\overline{DB}$ at $R.$ Given that $PQ=735$ and $QR=112,$ find $RC.$

2023 AMC 12/AHSME, 25

Tags: trigonometry , sequence

There is a unique sequence of integers $a_1, a_2, \cdots a_{2023}$ such that $$ \tan2023x = \frac{a_1 \tan x + a_3 \tan^3 x + a_5 \tan^5 x + \cdots + a_{2023} \tan^{2023} x}{1 + a_2 \tan^2 x + a_4 \tan^4 x \cdots + a_{2022} \tan^{2022} x} $$ whenever $\tan 2023x$ is defined. What is $a_{2023}?$ $\textbf{(A) } -2023 \qquad\textbf{(B) } -2022 \qquad\textbf{(C) } -1 \qquad\textbf{(D) } 1 \qquad\textbf{(E) } 2023$

2009 International Zhautykov Olympiad, 3

Tags: inequalities , trigonometry , trig identities , law of cosines , geometry

For a convex hexagon $ ABCDEF$ with an area $ S$, prove that: \[ AC\cdot(BD\plus{}BF\minus{}DF)\plus{}CE\cdot(BD\plus{}DF\minus{}BF)\plus{}AE\cdot(BF\plus{}DF\minus{}BD)\geq 2\sqrt{3}S \]

1948 Moscow Mathematical Olympiad, 141

Tags: number theory , sum , reciprocal

The sum of the reciprocals of three positive integers is equal to $1$. What are all the possible such triples?

2014 Federal Competition For Advanced Students, P2, 2

Tags: functional equation , algebra

Let $S$ be the set of all real numbers greater than or equal to $1$. Determine all functions$ f: S \to S$, so that for all real numbers $x ,y \in S$ with $x^2 -y^2 \in S$ the condition $f (x^2 -y^2) = f (xy)$ is fulfilled.

2004 Miklós Schweitzer, 2

Tags: graph theory

Write $t(G)$ for the number of complete quadrilaterals in the graph $G$ and $e_G(S)$ for the number of edges spanned by a subset $S$ of vertices of $G$. Let $G_1, G_2$ be two (simple) graphs on a common underlying set $V$ of vertices, $|V|-n$, and assume that $|e_{G_1}(S)-e_{G_2}(S)|<\frac{n^2}{1000}$ holds for any subset $S\subseteq V$. Prove that $|t(G_1)-t(G_2)|\le \frac{n^4}{1000}$.

2014 AMC 12/AHSME, 13

Tags: binomial coefficient , counting , distinguishability , complementary counting

A fancy bed and breakfast inn has $5$ rooms, each with a distinctive color-coded decor. One day $5$ friends arrive to spend the night. There are no other guests that night. The friends can room in any combination they wish, but with no more than $2$ friends per room. In how many ways can the innkeeper assign the guests to the rooms? $\textbf{(A) }2100\qquad \textbf{(B) }2220\qquad \textbf{(C) }3000\qquad \textbf{(D) }3120\qquad \textbf{(E) }3125\qquad$

2006 Junior Balkan Team Selection Tests - Romania, 3

Tags: trigonometry , inequalities , algebra

Find all real numbers $ a$ and $ b$ such that \[ 2(a^2 \plus{} 1)(b^2 \plus{} 1) \equal{} (a \plus{} 1)(b \plus{} 1)(ab \plus{} 1). \] [i]Valentin Vornicu[/i]

Novosibirsk Oral Geo Oly IX, 2019.3

Tags: area , square , geometry

The circle touches the square and goes through its two vertices as shown in the figure. Find the area of the square. (Distance in the picture is measured horizontally from the midpoint of the side of the square.) [img]https://cdn.artofproblemsolving.com/attachments/7/5/ab4b5f3f4fb4b70013e6226ce5189f3dc2e5be.png[/img]

2021 Latvia Baltic Way TST, P8

Tags: combinatorics , process

Initially on the blackboard eight zeros are written. In one step, it is allowed to choose numbers $a,b,c,d$, erase them and replace them with the numbers $a+1$, $b+2$, $c+3$, $d+3$. Determine: a) the minimum number of steps required to achieve $8$ consecutive integers on the board b) whether it is possible to achieve that sum of the numbers is $2021$ c) whether it is possible to achieve that product of the numbers is $2145$

1982 IMO, 2

Tags: geometry , triangle , midpoint , incircle , concurrency

A non-isosceles triangle $A_{1}A_{2}A_{3}$ has sides $a_{1}$, $a_{2}$, $a_{3}$ with the side $a_{i}$ lying opposite to the vertex $A_{i}$. Let $M_{i}$ be the midpoint of the side $a_{i}$, and let $T_{i}$ be the point where the inscribed circle of triangle $A_{1}A_{2}A_{3}$ touches the side $a_{i}$. Denote by $S_{i}$ the reflection of the point $T_{i}$ in the interior angle bisector of the angle $A_{i}$. Prove that the lines $M_{1}S_{1}$, $M_{2}S_{2}$ and $M_{3}S_{3}$ are concurrent.

2018 Azerbaijan Junior NMO, 2

Tags: algebra

$x^{11}+x^7+x^3=1$. $$x^{\alpha}=x^4+x^3-1.\hspace{4mm} \alpha=?$$

2013 National Olympiad First Round, 33

Tags: geometry , angle bisector , perpendicular bisector

Let $D$ be a point on side $[BC]$ of triangle $ABC$ such that $[AD]$ is an angle bisector, $|BD|=4$, and $|DC|=3$. Let $E$ be a point on side $[AB]$ and different than $A$ such that $m(\widehat{BED})=m(\widehat{DEC})$. If the perpendicular bisector of segment $[AE]$ meets the line $BC$ at $M$, what is $|CM|$? $ \textbf{(A)}\ 12 \qquad\textbf{(B)}\ 9 \qquad\textbf{(C)}\ 7 \qquad\textbf{(D)}\ 5 \qquad\textbf{(E)}\ \text { None of above} $