Olympiad problems

2002 District Olympiad, 4

Let be a continuous and periodic function $ f:\mathbb{R}\longrightarrow [0,\infty ) $ of period $ 1. $ Show: [b]a)[/b] $ a\in\mathbb{R}\implies\int_a^{a+1} f(x)dx =\int_0^1 f(x) dx . $ [b]b)[/b] $ \lim_{n\to\infty} \int_0^1 f(x)f(nx) dx=\left( \int_0^1 f(x) dx \right)^2 . $ [i]C. Mortici[/i]

Denmark (Mohr) - geometry, 1994.1

Tags: geometry , cylinder , 3d geometry , sphere

A wine glass with a cross section as shown has the property of an orange in shape as a sphere with a radius of $3$ cm just can be placed in the glass without protruding above glass. Determine the height $h$ of the glass. [img]https://1.bp.blogspot.com/-IuLm_IPTvTs/XzcH4FAjq5I/AAAAAAAAMYY/qMi4ng91us8XsFUtnwS-hb6PqLwAON_jwCLcBGAsYHQ/s0/1994%2BMohr%2Bp1.png[/img]

2002 Balkan MO, 3

Tags: geometry , rectangle , geometric transformation , homothety , reflection , circumcircle , parallelogram

Two circles with different radii intersect in two points $A$ and $B$. Let the common tangents of the two circles be $MN$ and $ST$ such that $M,S$ lie on the first circle, and $N,T$ on the second. Prove that the orthocenters of the triangles $AMN$, $AST$, $BMN$ and $BST$ are the four vertices of a rectangle.

1999 Romania National Olympiad, 1

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Source: Romania 1999 7.1 Determine the side lengths of a right trianlge if they are intgers and the product of the leg lengths is equal to three times the perimeter.

2014 AMC 12/AHSME, 16

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The product $(8)(888\ldots 8)$, where the second factor has $k$ digits, is an integer whose digits have a sum of $1000$. What is $k$? $\textbf{(A) }901\qquad \textbf{(B) }911\qquad \textbf{(C) }919\qquad \textbf{(D) }991\qquad \textbf{(E) }999\qquad$

2014 Iranian Geometry Olympiad (junior), P5

Tags: circumcircle , arc midpoint , geometric inequality , geometry

Two points $X, Y$ lie on the arc $BC$ of the circumcircle of $\triangle ABC$ (this arc does not contain $A$) such that $\angle BAX = \angle CAY$ . Let $M$ denotes the midpoint of the chord $AX$ . Show that $BM +CM > AY$ . by Mahan Tajrobekar

2013 AIME Problems, 9

Tags: modular arithmetic , recurrence , combinatorics

A $7 \times 1$ board is completely covered by $m \times 1$ tiles without overlap; each tile may cover any number of consecutive squares, and each tile lies completely on the board. Each tile is either red, blue, or green. Let $N$ be the number of tilings of the $7 \times 1$ board in which all three colors are used at least once. For example, a $1 \times 1$ red tile followed by a $2 \times 1$ green tile, a $1 \times 1$ green tile, a $2 \times 1$ blue tile, and a $1 \times 1$ green tile is a valid tiling. Note that if the $2 \times 1$ blue tile is replaced by two $1 \times 1$ blue tiles, this results in a different tiling. Find the remainder when $N$ is divided by $1000$.

1990 National High School Mathematics League, 15

Tags: geometry , 3d geometry , pyramid , sphere

In pyramid $M-ABCD$, bottom surface $ABCD$ is a square. $MA=MC,MA\perp AB$. If the area of $\triangle AMD$ is $1$, find the maximum value of radius of sphere that can be put inside the pyramid.

1973 All Soviet Union Mathematical Olympiad, 179

Tags: combinatorics , tournament

The tennis federation has assigned numbers to $1024$ sportsmen, participating in the tournament, according to their skill. (The tennis federation uses the olympic system of tournaments. The looser in the pair leaves, the winner meets with the winner of another pair. Thus, in the second tour remains $512$ participants, in the third -- $256$, et.c. The winner is determined after the tenth tour.) It comes out, that in the play between the sportsmen whose numbers differ more than on $2$ always win that whose number is less. What is the greatest possible number of the winner?

1985 IMO, 6

Tags: limit , polynomial , calculus , fixed point , sequence

For every real number $x_1$, construct the sequence $x_1,x_2,\ldots$ by setting: \[ x_{n+1}=x_n(x_n+{1\over n}). \] Prove that there exists exactly one value of $x_1$ which gives $0<x_n<x_{n+1}<1$ for all $n$.

2001 Moldova National Olympiad, Problem 7

Tags: geometry

Let $ABCD$ and $AB’C’D’$ be equally oriented squares. Prove that the lines $BB_1,CC_1,DD_1$ are concurrent.

Today's calculation of integrals, 857

Tags: calculus , integration , limit , trigonometry , geometry , geometric transformation , rotation

Let $f(x)=\lim_{n\to\infty} (\cos ^ n x+\sin ^ n x)^{\frac{1}{n}}$ for $0\leq x\leq \frac{\pi}{2}.$ (1) Find $f(x).$ (2) Find the volume of the solid generated by a rotation of the figure bounded by the curve $y=f(x)$ and the line $y=1$ around the $y$-axis.

CNCM Online Round 3, 6

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Triangle $ABC$ has side lengths $AB=13, BC=14,$ and $CA=15$. Let $\Gamma$ denote the circumcircle of $\triangle ABC$. Let $H$ be the orthocenter of $\triangle ABC$. Let $AH$ intersect $\Gamma$ at a point $D$ other than $A$. Let $BH$ intersect $AC$ at $F$ and $\Gamma$ at point $G$ other than $B$. Suppose $DG$ intersects $AC$ at $X$. Compute the greatest integer less than or equal to the area of quadrilateral $HDXF$. [i]Proposed by Kenan Hasanaliyev (claserken)[/i]

2014 PUMaC Team, 2

Tags: geometry , circumcircle

Given a Pacman of radius $1$, and mouth opening angle $90^\circ$, what is the largest (circular) pellet it can eat? The pellet must lie entirely outside the yellow portion and entirely inside the circumcircle of the Pacman. Let the radius be equal to $a\sqrt b+c$. where $b$ is square free. Find $a+b+c$.

2015 Iran Geometry Olympiad, 5

Tags: geometry

a) Do there exist 5 circles in the plane such that every circle passes through centers of exactly 3 circles? b) Do there exist 6 circles in the plane such that every circle passes through centers of exactly 3 circles?

2023 ISL, G7

Tags: geometry

Let $ABC$ be an acute, scalene triangle with orthocentre $H$. Let $\ell_a$ be the line through the reflection of $B$ with respect to $CH$ and the reflection of $C$ with respect to $BH$. Lines $\ell_b$ and $\ell_c$ are defined similarly. Suppose lines $\ell_a$, $\ell_b$, and $\ell_c$ determine a triangle $\mathcal T$. Prove that the orthocentre of $\mathcal T$, the circumcentre of $\mathcal T$, and $H$ are collinear. [i]Fedir Yudin, Ukraine[/i]

2017 Turkey EGMO TST, 4

Tags: geometry

2004 May Olympiad, 2

Tags: combinatorics

Pepito's mother wants to prepare $n$ packages of $3$ candies to give away at the birthday party, and for this she will buy assorted candies of $3$ different flavors. You can buy any number of candies but you can't choose how many of each taste. She wants to put one candy of each flavor in each package, and if this is not possible she will use only candy of one flavor and all the packages will have $3$ candies of that flavor. Determine the least number of candies that must be purchased in order to assemble the n packages. He explains why if he buys fewer candies, he is not sure that he will be able to assemble the packages the way he wants.

2016 Saudi Arabia BMO TST, 3

Tags: polynomial , integer polynomial , algebra

Does there exist a polynomial $P(x)$ with integral coefficients such that a) $P(\sqrt[3]{25 }+ \sqrt[3]{5}) = 220\sqrt[3]{25} + 284\sqrt[3]{5}$ ? b) $P(\sqrt[3]{25 }+ \sqrt[3]{5}) = 1184\sqrt[3]{25} + 1210\sqrt[3]{5}$ ?

2021 MOAA, 14

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Sinclair starts with the number $1$. Every minute, he either squares his number or adds $1$ to his number, both with equal probability. What is the expected number of minutes until his number is divisible by $3$? [i]Proposed by Nathan Xiong[/i]

2001 All-Russian Olympiad, 3

Tags: geometry , circumcircle , parallelogram , similar triangles

Let $N$ be a point on the longest side $AC$ of a triangle $ABC$. The perpendicular bisectors of $AN$ and $NC$ intersect $AB$ and $BC$ respectively in $K$ and $M$. Prove that the circumcenter $O$ of $\triangle ABC$ lies on the circumcircle of triangle $KBM$.

2011 Tokio University Entry Examination, 3

Tags: calculus , integration , analytic geometry , geometry , perimeter , limit , logarithm

Let $L$ be a positive constant. For a point $P(t,\ 0)$ on the positive part of the $x$ axis on the coordinate plane, denote $Q(u(t),\ v(t))$ the point at which the point reach starting from $P$ proceeds by　distance $L$ in counter-clockwise on the perimeter of a circle passing the point $P$ with center $O$. (1) Find $u(t),\ v(t)$. (2) For real number $a$ with $0<a<1$, find $f(a)=\int_a^1 \sqrt{\{u'(t)\}^2+\{v'(t)\}^2}\ dt$. (3) Find $\lim_{a\rightarrow +0} \frac{f(a)}{\ln a}$. [i]2011 Tokyo University entrance exam/Science, Problem 3[/i]

2014 Online Math Open Problems, 5

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A crazy physicist has discovered a new particle called an omon. He has a machine, which takes two omons of mass $a$ and $b$ and entangles them; this process destroys the omon with mass $a$, preserves the one with mass $b$, and creates a new omon whose mass is $\frac 12 (a+b)$. The physicist can then repeat the process with the two resulting omons, choosing which omon to destroy at every step. The physicist initially has two omons whose masses are distinct positive integers less than $1000$. What is the maximum possible number of times he can use his machine without producing an omon whose mass is not an integer? [i]Proposed by Michael Kural[/i]

2005 Postal Coaching, 2

Tags: geometry

Let $< \Gamma _j >$ be a sequnce of concentric circles such that the sequence $< R_j >$ , where $R_j$ denotes the radius of $\Gamma_j$, is increasing and $R_j \longrightarrow \infty$ as $j \longrightarrow \infty$. Let $A_1 B_1 C_1$ be a triangle inscribed in $\Gamma _1$. extend the rays $\vec{A_i B_1} , \vec{B_1 C_1 }, \vec{C_1 A_1}$ to meet $\Gamma_2$ in $B_2, C_2$and $A_2$ respectively and form the triangle $A_2 B_2 C_2$. Continue this process. Show that the sequence of triangles $< A_n B_n C_n >$ tends to an equilateral triangle as $n \longrightarrow \infty$

2010 Portugal MO, 2

Tags: geometry

On a circumference, points $A$ and $B$ are on opposite arcs of diameter $CD$. Line segments $CE$ and $DF$ are perpendicular to $AB$ such that $A-E-F-B$ (i.e., $A$, $E$, $F$ and $B$ are collinear on this order). Knowing $AE=1$, find the length of $BF$.