Olympiad problems

2009 Postal Coaching, 5

Let $ABCD$ be a quadrilateral that has an incircle with centre $O$ and radius $r$. Let $P = AB \cap CD$, $Q = AD \cap BC$, $E = AC \cap BD$. Show that $OE \cdot d = r^2$, where $d$ is the distance of $O$ from $PQ$.

2005 Mediterranean Mathematics Olympiad, 4

Tags: algebra , polynomial , function , geometric transformation , modular arithmetic , number theory , geometry

Let $A$ be the set of all polynomials $f(x)$ of order $3$ with integer coefficients and cubic coefficient $1$, so that for every $f(x)$ there exists a prime number $p$ which does not divide $2004$ and a number $q$ which is coprime to $p$ and $2004$, so that $f(p)=2004$ and $f(q)=0$. Prove that there exists a infinite subset $B\subset A$, so that the function graphs of the members of $B$ are identical except of translations

1963 German National Olympiad, 3

Tags: algebra , inequalities

It has to be proven: If at least two of the real numbers $a, b, c$ are different from zero, then the inequality holds $$\frac{a^2}{b^2 + c^2} + \frac{b^2}{c^2 + a^2} + \frac{c^2}{a^2 + b^2} \ge \frac32$$ Under what conditions does equality occur?

2022 MMATHS, 12

Tags: geometry

Let triangle $ABC$ with incenter $I$ satisfy $AB = 10$, $BC = 21$, and $CA = 17$. Points $D$ and E lie on side $BC$ such that $BD = 4$, $DE = 6$, and $EC = 11$. The circumcircles of triangles $BIE$ and $CID$ meet again at point $P$, and line $IP$ meets the altitude from $A$ to $BC$ at $X$. Find $(DX \cdot EX)^2$.

1988 Mexico National Olympiad, 8

Tags: geometry , 3-dimensional geometry , octahedron , sphere

Compute the volume of a regular octahedron circumscribed about a sphere of radius $1$.

2012 USA Team Selection Test, 3

Tags: induction , arithmetic sequence , strong induction , combinatorics

Determine all positive integers $n$, $n\ge2$, such that the following statement is true: If $(a_1,a_2,...,a_n)$ is a sequence of positive integers with $a_1+a_2+\cdots+a_n=2n-1$, then there is block of (at least two) consecutive terms in the sequence with their (arithmetic) mean being an integer.

KoMaL A Problems 2020/2021, A. 794

Tags: combinatorics

A polyomino $P$ occupies $n$ cells of an infinite grid of unit squares. In each move, we lift $P$ off the grid and then we place it back into a new position, possibly rotated and reflected, so that the preceding and the new position have $n-1$ cells in common. We say that $P$ is a caterpillar of area $n$ if, by means of a series of moves, we can free up all cells initially occupied by $P$. How many caterpillars of area $n=10^{6}+1$ are there? Proposed by Nikolai Beluhov, Bulgaria

2020 Taiwan TST Round 2, 5

Tags: number theory

A finite set $K$ consists of at least 3 distinct positive integers. Suppose that $K$ can be partitioned into two nonempty subsets $A,B\in K$ such that $ab+1$ is always a perfect square whenever $a\in A$ and $b\in B$. Prove that \[\max_{k\in K}k\geq \left\lfloor (2+\sqrt{3})^{\min\{|A|,|B|\}-1}\right\rfloor+1,\]where $|X|$ stands for the cartinality of the set $X$, and for $x\in \mathbb{R}$, $\lfloor x\rfloor$ is the greatest integer that does not exceed $x$.

IV Soros Olympiad 1997 - 98 (Russia), 9.1

Tags: geometry

Through vertices $A$ and $B$ of the unit square $ABCD$ , passes a circle intersecting lines $AD$ and $AC$ at points $K$ and $M$, other than $A$. Find the length of the projection $KM$ onto $AC$.

2014 Contests, 2

Tags: symmetry , circumcircle , geometry

Let $ABC$ be a triangle. Let $H$ be the foot of the altitude from $C$ on $AB$. Suppose that $AH = 3HB$. Suppose in addition we are given that (a) $M$ is the midpoint of $AB$; (b) $N$ is the midpoint of $AC$; (c) $P$ is a point on the opposite side of $B$ with respect to the line $AC$ such that $NP = NC$ and $PC = CB$. Prove that $\angle APM = \angle PBA$.

2025 District Olympiad, P3

Tags: function , limit , real analysis

Let $f:[0,\infty)\rightarrow [0,\infty)$ be a continuous and bijective function, such that $$\lim_{x\rightarrow\infty}\frac{f^{-1}(f(x)/x)}{x}=1.$$ [list=a] [*] Show that $\lim_{x\rightarrow\infty}\frac{f(x)}{x}=\infty$ and $\lim_{x\rightarrow\infty}\frac{f^{-1}(ax)}{f^{-1}(x)}=1$ for any $a>0$. [*] Give an example of function which satisfies the hypothesis.

2005 Purple Comet Problems, 4

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A palindrome is a number that reads the same forwards and backwards such as $3773$ or $42924$. What is the smallest $9$ digit palindrome which is a multiple of $3$ and has at least two digits which are $5$'s and two digits which are $7$'s?

2010 Belarus Team Selection Test, 3.1

Tags: circumcircle , incenter , geometry , inequalities , geometric inequality

Let $I$ be an incenter of a triangle $ABC, A_1,B_1,C_1$ be intersection points of the circumcircle of the triangle $ABC$ and the lines $AI, BI, Cl$ respectively. Prove that a) $\frac{AI}{IA_1}+ \frac{BI}{IB_1}+ \frac{CI}{IC_1}\ge 3$ b) $AI \cdot BI \cdot CI \le I_1A_1\cdot I_2B_1 \cdot I_1C_1$ (D. Pirshtuk)

1995 Chile National Olympiad, 2

Tags: geometry , arc , area , circles

In a circle of radius $1$, six arcs of radius $1$ are drawn, which cut the circle as in the figure. Determine the black area. [img]https://cdn.artofproblemsolving.com/attachments/8/9/0323935be8406ea0c452b3c8417a8148c977e3.jpg[/img]

2002 HKIMO Preliminary Selection Contest, 20

Tags: geometry

A rectangular piece of paper has integer side lengths. The paper is folded so that a pair of diagonally opposite vertices coincide, and it is found that the crease is of length 65. Find a possible value of the perimeter of the paper.

2013 Stanford Mathematics Tournament, 20

Tags: ceiling function , probability , expected value

Ben is throwing darts at a circular target with diameter 10. Ben never misses the target when he throws a dart, but he is equally likely to hit any point on the target. Ben gets $\lceil 5-x \rceil$ points for having the dart land $x$ units away from the center of the target. What is the expected number of points that Ben can earn from throwing a single dart? (Note that $\lceil y \rceil$ denotes the smallest integer greater than or equal to $y$.)

2021 AIME Problems, 15

Tags:

Let $f(n)$ and $g(n)$ be functions satisfying $$f(n) = \begin{cases}\sqrt{n} & \text{ if } \sqrt{n} \text{ is an integer}\\ 1 + f(n+1) & \text{ otherwise} \end{cases}$$and $$g(n) = \begin{cases}\sqrt{n} & \text{ if } \sqrt{n} \text{ is an integer}\\ 2 + g(n+2) & \text{ otherwise} \end{cases}$$for positive integers $n$. Find the least positive integer $n$ such that $\tfrac{f(n)}{g(n)} = \tfrac{4}{7}$.

2022 Canada National Olympiad, 5

Tags: geometry

A pentagon is inscribed in a circle, such that the pentagon has an incircle. All $10$ sets of $3$ vertices from the pentagon are chosen, and the incenters of each of the $10$ resulting triangles are drawn in. Prove these $10$ incenters lie on $2$ concentric circles. Note: I spent nearly no time on this, so if anyone took CMO and I misremembered just let me know.

2009 Moldova Team Selection Test, 3

Tags: algebra

[color=darkblue]The sequence $ (a_n)_{n \in \mathbb{N}}$ is defined as follows: \[ a_n \equal{} \dfrac{2}{3 \plus{} 1} \plus{} \dfrac{2^2}{3^2 \plus{} 1} \plus{} \dfrac{2^3}{3^4 \plus{} 1} \plus{} \ldots \plus{} \dfrac{2^{n \plus{} 1}}{3^{2^n} \plus{} 1} \] Prove that $ a_n < 1$ for any $ n \in \mathbb{N}$[/color]

1984 IMO Longlists, 62

Tags: circumcircle , geometry

From a point $P$ exterior to a circle $K$, two rays are drawn intersecting $K$ in the respective pairs of points $A, A'$ and $B,B' $. For any other pair of points $C, C'$ on $K$, let $D$ be the point of intersection of the circumcircles of triangles $PAC$ and $PB'C'$ other than point $P$. Similarly, let $D'$ be the point of intersection of the circumcircles of triangles $PA'C'$ and $PBC$ other than point $P$. Prove that the points $P, D$, and $D'$ are collinear.

1958 AMC 12/AHSME, 42

Tags: geometry , cyclic quadrilateral

In a circle with center $ O$, chord $ \overline{AB}$ equals chord $ \overline{AC}$. Chord $ \overline{AD}$ cuts $ \overline{BC}$ in $ E$. If $ AC \equal{} 12$ and $ AE \equal{} 8$, then $ AD$ equals: $ \textbf{(A)}\ 27\qquad \textbf{(B)}\ 24\qquad \textbf{(C)}\ 21\qquad \textbf{(D)}\ 20\qquad \textbf{(E)}\ 18$

2015 HMNT, 1

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Find the number of triples $(a, b, c)$ of positive integers such that $a+ab+abc = 11.$

2016 AMC 10, 12

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Two different numbers are selected at random from $( 1, 2, 3, 4, 5)$ and multiplied together. What is the probability that the product is even? $\textbf{(A)}\ 0.2\qquad\textbf{(B)}\ 0.4\qquad\textbf{(C)}\ 0.5\qquad\textbf{(D)}\ 0.7\qquad\textbf{(E)}\ 0.8$

2021 Indonesia TST, N

Tags: number theory

For a three-digit prime number $p$, the equation $x^3+y^3=p^2$ has an integer solution. Calculate $p$.

2021 BMT, 22

Tags: geometry

In $\vartriangle ABC$, let $D$ and $E$ be points on the angle bisector of $\angle BAC$ such that $\angle ABD = \angle ACE =90^o$ . Furthermore, let $F$ be the intersection of $AE$ and $BC$, and let $O$ be the circumcenter of $\vartriangle AF C$. If $\frac{AB}{AC} =\frac{3}{4}$, $AE = 40$, and $BD$ bisects $EF$, compute the perpendicular distance from $A$ to $OF$.