Olympiad problems

2010 District Olympiad, 3

Let $ a < c < b$ be three real numbers and let $ f: [a,b]\rightarrow \mathbb{R}$ be a continuos function in $ c$. If $ f$ has primitives on each of the intervals $ [a,c)$ and $ (c,b]$, then prove that it has primitives on the interval $ [a,b]$.

2002 Junior Balkan Team Selection Tests - Romania, 1

Tags: number theory , divide , divisible , divisor , sum of powers

Let $n$ be an even positive integer and let $a, b$ be two relatively prime positive integers. Find $a$ and $b$ such that $a + b$ is a divisor of $a^n + b^n$.

2021 Middle European Mathematical Olympiad, 6

Tags: geometry , power of a point , circumcircle

Let $ABC$ be a triangle and let $M$ be the midpoint of the segment $BC$. Let $X$ be a point on the ray $AB$ such that $2 \angle CXA=\angle CMA$. Let $Y$ be a point on the ray $AC$ such that $2 \angle AYB=\angle AMB$. The line $BC$ intersects the circumcircle of the triangle $AXY$ at $P$ and $Q$, such that the points $P, B, C$, and $Q$ lie in this order on the line $BC$. Prove that $PB=QC$. [i]Proposed by Dominik Burek, Poland[/i]

2023 EGMO, 2

Tags: geometry , easy geometry

We are given an acute triangle $ABC$. Let $D$ be the point on its circumcircle such that $AD$ is a diameter. Suppose that points $K$ and $L$ lie on segments $AB$ and $AC$, respectively, and that $DK$ and $DL$ are tangent to circle $AKL$. Show that line $KL$ passes through the orthocenter of triangle $ABC$.

2019 Philippine MO, 3

Tags: system of equations , number theory , least common multiple

Find all triples $(a, b, c)$ of positive integers such that $a^2 + b^2 = n\cdot lcm(a, b) + n^2$ $b^2 + c^2 = n \cdot lcm(b, c) + n^2$ $c^2 + a^2 = n \cdot lcm(c, a) + n^2$ for some positive integer $n$.

1991 Greece National Olympiad, 3

Tags: number theory , digit

Find all 2-digit numbers$ n$ having the property: 'Number $n^2$ is 4-digit number of form $\overline{xxyy}$.

2019 Swedish Mathematical Competition, 4

Tags: combinatorial geometry , combinatorics , circles

Let $\Omega$ be a circle disk with radius $1$. Determine the minimum $r$ that has the following property: You can select three points on $\Omega$ so that each circle disk located in $\Omega$ and has a radius greater than $r$ contains at least one of the three points.

2010 Tournament Of Towns, 6

Tags: geometry , trapezoid , ratio , geometric transformation , reflection , incenter , analytic geometry

Quadrilateral $ABCD$ is circumscribed around the circle with centre $I$. Let points $M$ and $N$ be the midpoints of sides $AB$ and $CD$ respectively and let $\frac{IM}{AB} = \frac{IN}{CD}$. Prove that $ABCD$ is either a trapezoid or a parallelogram.

2012 Purple Comet Problems, 13

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Find the smallest positive integer $n$ such that the decimal representation of $n!(n+1)!(2n+1)! - 1$ has its last $30$ digits all equal to $9$.

1956 Czech and Slovak Olympiad III A, 2

Tags: geometry , 3d geometry , constructive geometry , construction , parallelogram

In a given plane $\varrho$ consider a convex quadrilateral $ABCD$ and denote $E=AC\cap BD.$ Moreover, consider a point $V\notin\varrho$. On rays $VA,VB,VC,VD$ find points $A',B',C',D'$ respectively such that $E,A',B',C',D'$ are coplanar and $A'B'C'D'$ is a parallelogram. Discuss conditions of solvability.

2017 Tournament Of Towns, 7

Tags: combinatorics , geometry , chessboard

$1\times 2$ dominoes are placed on an $8 \times 8$ chessboard without overlapping. They may partially stick out from the chessboard but the center of each domino must be strictly inside the chessboard (not on its border). Place on the chessboard in such a way: a) at least $40$ dominoes, (3 points) b) at least $41$ dominoes, (3 points) c) more than $41$ dominoes. (6 points) [i](Mikhail Evdokimov)[/i]

2015 All-Russian Olympiad, 6

Tags: inequalities

Let a,b,c,d be real numbers satisfying $|a|,|b|,|c|,|d|>1$ and $abc+abd+acd+bcd+a+b+c+d=0$. Prove that $\frac {1} {a-1}+\frac {1} {b-1}+ \frac {1} {c-1}+ \frac {1} {d-1} >0$

2023 Ukraine National Mathematical Olympiad, 9.1

Tags: combinatorics , equality

$n \ge 4$ real numbers are arranged in a circle. It turned out that for any four consecutive numbers $a, b, c, d$, that lie on the circle in this order, holds $a+d = b+c$. For which $n$ does it follow that all numbers on the circle are equal? [i]Proposed by Oleksiy Masalitin[/i]

2008 Gheorghe Vranceanu, 2

Tags: inequalities , kronecker , number theory , fractional part

Show that there is a natural number $ n $ that satisfies the following inequalities: $$ \sqrt{3} -\frac{1}{10}<\{ n\sqrt 3\} +\{ (n+1)\sqrt 3 \} <\sqrt 3. $$

1995 Niels Henrik Abels Math Contest (Norwegian Math Olympiad) Round 2, 6

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How many 11-digit bank account numbers are there consisting of 1's and 2's only, and such that there are no two consecutive 1's? A. 64 B. 233 C. 1024 D. 1279 E. 1365

2004 India IMO Training Camp, 2

Tags: polynomial , algebra

Let $P(x) = x^4 + ax^3 + bx^2 + cx + d$ and $Q(x) = x^2 + px + q$be two real polynomials. Suppose that there exista an interval $(r,s)$ of length greater than $2$ SUCH THAT BOTH $P(x)$ AND $Q(x)$ ARE nEGATIVE FOR $X \in (r,s)$ and both are positive for $x > s$ and $x<r$. Show that there is a real $x_0$ such that $P(x_0) < Q(x_0)$

LMT Speed Rounds, 2010.7

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Let $ABCD$ be a square with $AB=6.$ A point $P$ in the interior is $2$ units away from side $BC$ and $3$ units away from side $CD.$ What is the distance from $P$ to $A?$

1987 AIME Problems, 2

Tags: geometry , 3d geometry , sphere , inequalities , triangle inequality

What is the largest possible distance between two points, one on the sphere of radius 19 with center $(-2, -10, 5)$ and the other on the sphere of radius 87 with center $(12, 8, -16)$?

2006 Germany Team Selection Test, 2

Tags: inequalities , combinatorics

In a room, there are $2005$ boxes, each of them containing one or several sorts of fruits, and of course an integer amount of each fruit. [b]a)[/b] Show that we can find $669$ boxes, which altogether contain at least a third of all apples and at least a third of all bananas. [b]b)[/b] Can we always find $669$ boxes, which altogether contain at least a third of all apples, at least a third of all bananas and at least a third of all pears?

2018 Sharygin Geometry Olympiad, 6

Tags: geometry , circumcircle

Let $\omega$ be the circumcircle of $ABC$, and $KL$ be the diameter of $\omega$ passing through $M$ midpoint of $AB$ ($K,C$ lies on different sides of $AB$). A circle passing through $L$ and $M$ meets $CK$ at points $P$ and $Q$ ($Q$ lies on $KP$). Let $LQ$ meet the circumcircle of $KMQ$ again at $R$. Prove that $APBR$ is cyclic.

2020-21 IOQM India, 23

Tags: geometry , inradius , tangent circle

The incircle $\Gamma$ of a scalene triangle $ABC$ touches $BC$ at $D, CA$ at $E$ and $AB$ at $F$. Let $r_A$ be the radius of the circle inside $ABC$ which is tangent to $\Gamma$ and the sides $AB$ and $AC$. Define $r_B$ and $r_C$ similarly. If $r_A = 16, r_B = 25$ and $r_C = 36$, determine the radius of $\Gamma$.

2010 District Olympiad, 3

2002 Junior Balkan Team Selection Tests - Romania, 1

2021 Middle European Mathematical Olympiad, 6

2023 EGMO, 2

2019 Philippine MO, 3

1991 Greece National Olympiad, 3

2019 Swedish Mathematical Competition, 4

2010 Tournament Of Towns, 6

2012 Purple Comet Problems, 13

1956 Czech and Slovak Olympiad III A, 2

2017 Tournament Of Towns, 7

2015 All-Russian Olympiad, 6

2023 Ukraine National Mathematical Olympiad, 9.1

2008 Gheorghe Vranceanu, 2

1995 Niels Henrik Abels Math Contest (Norwegian Math Olympiad) Round 2, 6

2004 India IMO Training Camp, 2

LMT Speed Rounds, 2010.7

1987 AIME Problems, 2

2006 Germany Team Selection Test, 2

2018 Sharygin Geometry Olympiad, 6

2020-21 IOQM India, 23

2013 IFYM, Sozopol, 1

2018 Saint Petersburg Mathematical Olympiad, 2

2012 May Olympiad, 5

2006 MOP Homework, 6