Olympiad problems

1999 Harvard-MIT Mathematics Tournament, 8

Tags: geometry

Squares $ABKL$, $BCMN$, $CAOP$ are drawn externally on the sides of a triangle $ABC$. The line segments $KL$, $MN$, $OP$, when extended, form a triangle $A'B'C'$. Find the area of $A'B'C'$ if $ABC$ is an equilateral triangle of side length $2$.

2007 AMC 12/AHSME, 15

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The set $ \{3,6,9,10\}$ is augmented by a fifth element $ n$, not equal to any of the other four. The median of the resulting set is equal to its mean. What is the sum of all possible values of $ n$? $ \textbf{(A)}\ 7 \qquad \textbf{(B)}\ 9 \qquad \textbf{(C)}\ 19 \qquad \textbf{(D)}\ 24 \qquad \textbf{(E)}\ 26$

2024 Israel National Olympiad (Gillis), P1

Tags: system of equations , algebra

Solve the following system (over the real numbers): \[\begin{cases}5x+5y+5xy-2xy^2-2x^2y=20 &\\ 3x+3y+3xy+xy^2+x^2y=23&\end{cases}\]

1968 AMC 12/AHSME, 30

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Convex polygons $P_1$ and $P_2$ are drawn in the same plane with $n_1$ and $n_2$ sides, respectively, $n_1 \le n_2$. If $P_1$ and $P_2$ do not have any line segment in common, then the maximum number of intersections of $P_1$ and $P_2$ is: $\textbf{(A)}\ 2n_1 \qquad\textbf{(B)}\ 2n_2 \qquad\textbf{(C)}\ n_1n_2 \qquad\textbf{(D)}\ n_1+n_2 \qquad\textbf{(E)}\ \text{none of these} $

2019 BMT Spring, 8

Tags: function

For a positive integer $ n $, define $ \phi(n) $ as the number of positive integers less than or equal to $ n $ that are relatively prime to $ n $. Find the sum of all positive integers $ n $ such that $ \phi(n) = 20 $.

2010 IFYM, Sozopol, 7

Tags: algebra , functional equation

Does there exist a function $f: \mathbb{R}\rightarrow \mathbb{R}$ such that: $f(f(x))=-x$, for all $x\in \mathbb{R}$?

Kvant 2024, M2784

Tags: geometry , locus

The bisectors $AD{}$ and $BE{}$ were drawn in the triangle $ABC{}$ and they intersected at point $I{}.$ Then everything was erased, leaving only the points $D{}$ and $E{}.$ Find the set of possible positions of the point $I{}.$ [i]Proposed by M. Didin[/i]

1991 Baltic Way, 9

Tags: function

Find the number of real solutions of the equation $a e^x = x^3$, where $a$ is a real parameter.

2022 Estonia Team Selection Test, 6

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Consider a $100\times 100$ square unit lattice $\textbf{L}$ (hence $\textbf{L}$ has $10000$ points). Suppose $\mathcal{F}$ is a set of polygons such that all vertices of polygons in $\mathcal{F}$ lie in $\textbf{L}$ and every point in $\textbf{L}$ is the vertex of exactly one polygon in $\mathcal{F}.$ Find the maximum possible sum of the areas of the polygons in $\mathcal{F}.$ [i]Michael Ren and Ankan Bhattacharya, USA[/i]

2016 Japan Mathematical Olympiad Preliminary, 3

Tags: angle , geometry

A hexagon $ABCDEF$ is inscribed in a circle. Let $P, Q, R, S$ be intersections of $AB$ and $DC$, $BC$ and $ED$, $CD$ and $FE$, $DE$ and $AF$, then $\angle BPC=50^{\circ}$, $\angle CQD=45^{\circ}$, $\angle DRE=40^{\circ}$, $\angle ESF=35^{\circ}$. Let $T$ be an intersection of $BE$ and $CF$. Find $\angle BTC$.

2000 Federal Competition For Advanced Students, Part 2, 1

Tags: binomial theorem , algebra

The sequence an is defined by $a_0 = 4, a_1 = 1$ and the recurrence formula $a_{n+1} = a_n + 6a_{n-1}$. The sequence $b_n$ is given by \[b_n=\sum_{k=0}^n \binom nk a_k.\] Find the coefficients $\alpha,\beta$ so that $b_n$ satisfies the recurrence formula $b_{n+1} = \alpha b_n + \beta b_{n-1}$. Find the explicit form of $b_n$.

2017 Iranian Geometry Olympiad, 2

Tags: geometry

Two circles $\omega_1,\omega_2$ intersect at $A,B$. An arbitrary line through $B$ meets $\omega_1,\omega_2$ at $C,D$ respectively. The points $E,F$ are chosen on $\omega_1,\omega_2$ respectively so that $CE=CB,\ BD=DF$. Suppose that $BF$ meets $\omega_1$ at $P$, and $BE$ meets $\omega_2$ at $Q$. Prove that $A,P,Q$ are collinear. [i]Proposed by Iman Maghsoudi[/i]

2013 Today's Calculation Of Integral, 886

Tags: calculus , integration , function , trigonometry , calculus computations

Find the functions $f(x),\ g(x)$ such that $f(x)=e^{x}\sin x+\int_0^{\pi} ug(u)\ du$ $g(x)=e^{x}\cos x+\int_0^{\pi} uf(u)\ du$

2024-IMOC, A2

Tags: algebra , minimum value , inequalities , absolute value

Given integer $n \geq 3$ and $x_1$, $x_2$, …, $x_n$ be $n$ real numbers satisfying $|x_1|+|x_2|+…+|x_n|=1$. Find the minimum of \[|x_1+x_2|+|x_2+x_3|+…+|x_{n-1}+x_n|+|x_n+x_1|.\] [i]Proposed by snap7822[/i]

2016 SDMO (High School), 3

Tags: inequalities , algebra

Let $ u, v, w$ be positive real numbers such that $ u\sqrt {vw} \plus{} v\sqrt {wu} \plus{} w\sqrt {uv} \geq 1$. Find the smallest value of $ u \plus{} v \plus{} w$.

2013 ELMO Shortlist, 7

Tags: ratio , parallelogram , circumcircle , trigonometry , collinear , complex , geometry

Let $ABC$ be a triangle inscribed in circle $\omega$, and let the medians from $B$ and $C$ intersect $\omega$ at $D$ and $E$ respectively. Let $O_1$ be the center of the circle through $D$ tangent to $AC$ at $C$, and let $O_2$ be the center of the circle through $E$ tangent to $AB$ at $B$. Prove that $O_1$, $O_2$, and the nine-point center of $ABC$ are collinear. [i]Proposed by Michael Kural[/i]

2014 Lusophon Mathematical Olympiad, 4

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From a point $K$ of a circle, a chord $KA$ (arc $AK$ is greather than $90^{o}$) and a tangent $l$ are drawn. The line that passes through the center of the circle and that is perpendicular to the radius $OA$, intersects $KA$ at $B$ and $l$ at $C$. Show that $KC = BC$.

2016 Purple Comet Problems, 2

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The figure below was formed by taking four squares, each with side length 5, and putting one on each side of a square with side length 20. Find the perimeter of the figure below. [center][img]https://snag.gy/LGimC8.jpg[/img][/center]

2010 Princeton University Math Competition, 7

Tags: function , induction , modular arithmetic

Let $f$ be a function such that $f(x)+f(x+1)=2^x$ and $f(0)=2010$. Find the last two digits of $f(2010)$.

1993 IMO Shortlist, 1

Tags: geometry , incenter , circumcircle , ratio

Let $ABC$ be a triangle, and $I$ its incenter. Consider a circle which lies inside the circumcircle of triangle $ABC$ and touches it, and which also touches the sides $CA$ and $BC$ of triangle $ABC$ at the points $D$ and $E$, respectively. Show that the point $I$ is the midpoint of the segment $DE$.

2016 AMC 12/AHSME, 24

Tags: algebra , polynomial

There is a smallest positive real number $a$ such that there exists a positive real number $b$ such that all the roots of the polynomial $x^3-ax^2+bx-a$ are real. In fact, for this value of $a$ the value of $b$ is unique. What is this value of $b$? $\textbf{(A)}\ 8\qquad\textbf{(B)}\ 9\qquad\textbf{(C)}\ 10\qquad\textbf{(D)}\ 11\qquad\textbf{(E)}\ 12$

1988 IMO Shortlist, 13

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 \[ \frac {E}{E_1} \geq 2. \]

2016 Thailand Mathematical Olympiad, 4

Tags: isosceles triangle , coloring , combinatorial geometry , isosceles , combinatorics

Each point on the plane is colored either red, green, or blue. Prove that there exists an isosceles triangle whose vertices all have the same color.

2016 Ukraine Team Selection Test, 3

Tags: geometry

Let $ABC$ be a triangle with $CA \neq CB$. Let $D$, $F$, and $G$ be the midpoints of the sides $AB$, $AC$, and $BC$ respectively. A circle $\Gamma$ passing through $C$ and tangent to $AB$ at $D$ meets the segments $AF$ and $BG$ at $H$ and $I$, respectively. The points $H'$ and $I'$ are symmetric to $H$ and $I$ about $F$ and $G$, respectively. The line $H'I'$ meets $CD$ and $FG$ at $Q$ and $M$, respectively. The line $CM$ meets $\Gamma$ again at $P$. Prove that $CQ = QP$. [i]Proposed by El Salvador[/i]

2017 Romania National Olympiad, 3

Tags: number theory , combinatorics

Let be two natural numbers $ n $ and $ a. $ [b]a)[/b] Prove that there exists an $ n\text{-tuplet} $ of natural numbers $ \left( a_1,a_2,\ldots ,a_n\right) $ that satisfy the following equality. $$ 1+\frac{1}{a} =\prod_{i=1}^n \left( 1+\frac{1}{a_i} \right) $$ [b]b)[/b] Show that there exist only finitely such $ n\text{-tuplets} . $