Olympiad problems

1960 Putnam, A5

Find all polynomials $f(x)$ with real coefficients having the property $f(g(x))=g(f(x))$ for every polynomial $g(x)$ with real coefficients.

2005 Croatia National Olympiad, 2

Tags: polynomial , limit , inequalities , algebra

Let $P(x)$ be a monic polynomial of degree $n$ with nonnegative coefficients and the free term equal to $1$. Prove that if all the roots of $P(x)$ are real, then $P(x) \geq (x+1)^{n}$ holds for every $x \geq 0$.

2019 Saint Petersburg Mathematical Olympiad, 5

Tags: combinatorics , number theory

Baron Munchhausen has a collection of stones, such that they are of $1000$ distinct whole weights, $2^{1000}$ stones of every weight. Baron states that if one takes exactly one stone of every weight, then the weight of all these $1000$ stones chosen will be less than $2^{1010}$, and there is no other way to obtain this weight by picking another set of stones of the collection. Can this statement happen to be true? [i](М. Антипов)[/i] [hide=Thanks]Thanks to the user Vlados021 for translating the problem.[/hide]

2021 Macedonian Mathematical Olympiad, Problem 4

Tags: combinatorics

For a fixed positive integer $n \geq 3$ we are given a $n$ $\times$ $n$ board with all unit squares initially white. We define a [i]floating plus [/i]as a $5$-tuple $(M,L,R,A,B)$ of unit squares such that $L$ is in the same row and left of $M$, $R$ is in the same row and right of $M$, $A$ is in the same column and above $M$ and $B$ is in the same column and below $M$. It is possible for $M$ to form a floating plus with unit squares that are not next to it. Find the largest positive integer $k$ (depending on $n$) such that we can color some $k$ squares black in such a way that there is no black colored floating plus. [i]Authored by Nikola Velov[/i]

2010 Contests, 2

Tags: integration , real analysis , infinite series , logarithm

Compute the sum of the series $\sum_{k=0}^{\infty} \frac{1}{(4k+1)(4k+2)(4k+3)(4k+4)} = \frac{1}{1\cdot2\cdot3\cdot4} + \frac{1}{5\cdot6\cdot7\cdot8} + ...$

2011 Today's Calculation Of Integral, 695

Tags: calculus , integration , limit , calculus computations , logarithm

For a positive integer $n$, let \[S_n=\int_0^1 \frac{1-(-x)^n}{1+x}dx,\ \ T_n=\sum_{k=1}^n \frac{(-1)^{k-1}}{k(k+1)}\] Answer the following questions: (1) Show the following inequality. \[\left|S_n-\int_0^1 \frac{1}{1+x}dx\right|\leq \frac{1}{n+1}\] (2) Express $T_n-2S_n$ in terms of $n$. (3) Find the limit $\lim_{n\to\infty} T_n.$

2018 PUMaC Team Round, 8

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Jackson has a $5\times 5$ grid of squares. He places coins in the grid squares $-$ at most one per square $-$ so that no row, column, or diagonal has five coins. What is the maximum number of coins that he can place?

2012-2013 SDML (High School), 14

Tags: number theory , greatest common divisor , arithmetic sequence

A finite arithmetic progression of positive integers $a_1,a_2,\ldots,a_n$ satisfies the condition that for all $1\leq i<j\leq n$, the number of positive divisors of $\gcd\left(a_i,a_j\right)$ is equal to $j-i$. Find the maximum possible value of $n$. $\text{(A) }2\qquad\text{(B) }3\qquad\text{(C) }4\qquad\text{(D) }5\qquad\text{(E) }6$

2022 Vietnam TST, 4

Tags: geometry , circumcircle , geometric transformation , reflection

An acute, non-isosceles triangle $ABC$ is inscribed in a circle with centre $O$. A line go through $O$ and midpoint $I$ of $BC$ intersects $AB, AC$ at $E, F$ respectively. Let $D, G$ be reflections to $A$ over $O$ and circumcentre of $(AEF)$, respectively. Let $K$ be the reflection of $O$ over circumcentre of $(OBC)$. $a)$ Prove that $D, G, K$ are collinear. $b)$ Let $M, N$ are points on $KB, KC$ that $IM\perp AC$, $IN\perp AB$. The midperpendiculars of $IK$ intersects $MN$ at $H$. Assume that $IH$ intersects $AB, AC$ at $P, Q$ respectively. Prove that the circumcircle of $\triangle APQ$ intersects $(O)$ the second time at a point on $AI$.

2004 AMC 12/AHSME, 11

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The average value of all the pennies, nickels, dimes, and quarters in Paula's purse is $ 20$ cents. If she had one more quarter, the average value would be $ 21$ cents. How many dimes does she have in her purse? $ \textbf{(A)}\ 0 \qquad \textbf{(B)}\ 1 \qquad \textbf{(C)}\ 2 \qquad \textbf{(D)}\ 3\qquad \textbf{(E)}\ 4$

2002 Iran MO (3rd Round), 5

Tags: circumcircle , power of a point , radical axis , perpendicular bisector , geometry

$\omega$ is circumcirlce of triangle $ABC$. We draw a line parallel to $BC$ that intersects $AB,AC$ at $E,F$ and intersects $\omega$ at $U,V$. Assume that $M$ is midpoint of $BC$. Let $\omega'$ be circumcircle of $UMV$. We know that $R(ABC)=R(UMV)$. $ME$ and $\omega'$ intersect at $T$, and $FT$ intersects $\omega'$ at $S$. Prove that $EF$ is tangent to circumcircle of $MCS$.

1975 AMC 12/AHSME, 30

Tags: trigonometry , geometry

Let $x=\cos 36^{\circ} - \cos 72^{\circ}$. Then $x$ equals $ \textbf{(A)}\ \frac{1}{3} \qquad\textbf{(B)}\ \frac{1}{2} \qquad\textbf{(C)}\ 3-\sqrt{6} \qquad\textbf{(D)}\ 2\sqrt{3}-3 \qquad\textbf{(E)}\ \text{none of these} $

2009 Romania Team Selection Test, 3

Tags: quadratic , algebra , polynomial , search , number theory

Show that there are infinitely many pairs of prime numbers $(p,q)$ such that $p\mid 2^{q-1}-1$ and $q\mid 2^{p-1}-1$.

2018 Macedonia JBMO TST, 5

Tags: combinatorics

A regular $2018$-gon is inscribed in a circle. The numbers $1, 2, ..., 2018$ are arranged on the vertices of the $2018$-gon, with each vertex having one number on it, such that the sum of any $2$ neighboring numbers ($2$ numbers are neighboring if the vertices they are on lie on a side of the polygon) equals the sum of the $2$ numbers that are on the antipodes of those $2$ vertices (with respect to the given circle). Determine the number of different arrangements of the numbers. (Two arrangements are identical if you can get from one of them to the other by rotating around the center of the circle).

1898 Eotvos Mathematical Competition, 2

Tags: geometry

Prove the following theorem: If two triangles have a common angle, then the sum of the sines of the angles will be larger in that triangle where the difference of the remaining two angles is smaller. On the basis of this theorem, determine the shape of that triangle for which the sum of the sines of its angles is a maximum.

1993 Tournament Of Towns, (358) 1

Tags: geometry , orthocenter

Let $M$ be a point on the side $AB$ of triangle $ABC$. The length $AB = c$ and $\angle CMA=\phi$ are given. Find the distance between the orthocentres (intersection points of altitudes) of the triangles $AMC$ and $BMC$. (IF Sharygin)

1988 IMO, 2

Tags: geometry , incenter , ratio , right triangle , area of a triangle

In a right-angled triangle $ ABC$ let $ AD$ be the altitude drawn to the hypotenuse and let the straight line joining the incentres of the triangles $ ABD, ACD$ intersect the sides $ AB, AC$ at the points $ K,L$ respectively. If $ E$ and $ E_1$ dnote the areas of triangles $ ABC$ and $ AKL$ respectively, show that \[ \frac {E}{E_1} \geq 2. \]

1972 IMO Longlists, 4

Tags: trigonometry , search , princeton , college

You have a triangle, $ABC$. Draw in the internal angle trisectors. Let the two trisectors closest to $AB$ intersect at $D$, the two trisectors closest to $BC$ intersect at $E$, and the two closest to $AC$ at $F$. Prove that $DEF$ is equilateral.

2023 Malaysian APMO Camp Selection Test, 1

Tags: algebra

For which $n\ge 3$ does there exist positive integers $a_1<a_2<\cdots <a_n$, such that: $$a_n=a_1+...+a_{n-1}, \hspace{0.5cm} \frac{1}{a_1}=\frac{1}{a_2}+...+\frac{1}{a_n}$$ are both true? [i]Proposed by Ivan Chan Kai Chin[/i]

1969 Bulgaria National Olympiad, Problem 4

Tags: triangle , geometry

Find the sides of a triangle if it is known that the inscribed circle meets one of its medians in two points and these points divide the median into three equal segments and the area of the triangle is equal to $6\sqrt{14}\text{ cm}^2$.

2014 Purple Comet Problems, 14

Tags: probability , number theory , relatively prime

Steve needed to address a letter to $2743$ Becker Road. He remembered the digits of the address, but he forgot the correct order of the digits, so he wrote them down in random order. The probability that Steve got exactly two of the four digits in their correct positions is $\tfrac m n$ where $m$ and $n$ are relatively prime positive integers. Find $m + n$.

2011 National Chemistry Olympiad, 29

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Introduction of two drops of concentrated sulfuric acid, $\text{H}_2\text{SO}_4$, speeds up an esterification reaction. Introduction of a piece of platinum metal, $\text{Pt}$, speeds up the reaction of $\text{H}_2$ and $\text{O}_2$ gas. Which of the following statements is true? $ \textbf{(A)}\ \text{Pt is a homogeneous catalyst; sulfuric acid is a heterogeneous catalyst}\qquad$ $\textbf{(B)}\ \text{Pt is a heterogeneous catalyst; sulfuric acid is a homogeneous catalyst}\qquad$ $\textbf{(C)}\ \text{Pt and sulfuric acid are both heterogeneous catalysts}\qquad$ $\textbf{(D)}\ \text{Pt and sulfuric acid are both homogeneous catalysts}\qquad$

2004 Manhattan Mathematical Olympiad, 2

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Consider the sequence $1, \dfrac12 , \dfrac13 , \ldots$. Show that every positive rational number can be written as a finite sum of different terms in this sequence.

2010 Contests, 1

Tags: combinatorics , symmetry , subset , bounding

A finite set of integers is called [i]bad[/i] if its elements add up to $2010$. A finite set of integers is a [i]Benelux-set[/i] if none of its subsets is bad. Determine the smallest positive integer $n$ such that the set $\{502, 503, 504, . . . , 2009\}$ can be partitioned into $n$ Benelux-sets. (A partition of a set $S$ into $n$ subsets is a collection of $n$ pairwise disjoint subsets of $S$, the union of which equals $S$.) [i](2nd Benelux Mathematical Olympiad 2010, Problem 1)[/i]

2015 Switzerland Team Selection Test, 1

Tags: combinatorics , chessboard

What is the maximum number of 1 × 1 boxes that can be colored black in a n × n chessboard so that any 2 × 2 square contains a maximum of 2 black boxes?