Olympiad problems

2020 Spain Mathematical Olympiad, 5

Tags: length , geometry

In an acute-angled triangle $ABC$, let $M$ be the midpoint of $AB$ and $P$ the foot of the altitude to $BC$. Prove that if $AC+BC = \sqrt{2}AB$, then the circumcircle of triangle $BMP$ is tangent to $AC$.

2021 Turkey Team Selection Test, 4

Tags: combinatorics

In a fish shop with 28 kinds of fish, there are 28 fish sellers. In every seller, there exists only one type of each fish kind, depending on where it comes, Mediterranean or Black Sea. Each of the $k$ people gets exactly one fish from each seller and exactly one fish of each kind. For any two people, there exists a fish kind which they have different types of it (one Mediterranean, one Black Sea). What is the maximum possible number of $k$?

1983 Canada National Olympiad, 3

Tags: 3d geometry , tetrahedron , area of a triangle , geometry

The area of a triangle is determined by the lengths of its sides. Is the volume of a tetrahedron determined by the areas of its faces?

2017 Iran Team Selection Test, 3

Tags: combinatorics

There are $27$ cards, each has some amount of ($1$ or $2$ or $3$) shapes (a circle, a square or a triangle) with some color (white, grey or black) on them. We call a triple of cards a [i]match[/i] such that all of them have the same amount of shapes or distinct amount of shapes, have the same shape or distinct shapes and have the same color or distinct colors. For instance, three cards shown in the figure are a [i]match[/i] be cause they have distinct amount of shapes, distinct shapes but the same color of shapes. What is the maximum number of cards that we can choose such that non of the triples make a [i]match[/i]? [i]Proposed by Amin Bahjati[/i]

2010 Regional Olympiad of Mexico Center Zone, 1

Tags: angle bisector , circumcircle , geometry

In the acute triangle $ABC$, $\angle BAC$ is less than $\angle ACB $. Let $AD$ be a diameter of $\omega$, the circle circumscribed to said triangle. Let $E$ be the point of intersection of the ray $AC$ and the tangent to $\omega$ passing through $B$. The perpendicular to $AD$ that passes through $E$ intersects the circle circumscribed to the triangle $BCE$, again, at the point $F$. Show that $CD$ is an angle bisector of $\angle BCF$.

2020 Purple Comet Problems, 9

Tags: algebra

Let $a, b$, and $c$ be real numbers such that $3^a = 125$, $5^b = 49$,and $7^c = 8$1. Find the product $abc$.

2016 CHMMC (Fall), 13

Tags: abstract algebra

A sequence of numbers $a_1, a_2 , \dots a_m$ is a [i]geometric sequence modulo n of length m[/i] for $n,m \in \mathbb{Z}^+$ if for every index $i$, $a_i \in \{ 0, 1, 2, \dots , m-1\}$ and there exists an integer $k$ such that $n | a_{j+1} - ka_{j}$ for $1 \leq j \leq m-1$. How many geometric sequences modulo $14$ of length $14$ are there?

2025 AMC 8, 10

Tags:

In the figure below, $ABCD$ is a rectangle with sides of length $AB = 5$ inches and $AD = 3$ inches. Rectangle $ABCD$ is rotated $90^{\circ}$ clockwise about the midpoint of side $\overline{DC}$ to give a second rectangle. What is the total area, in square inches, covered by the two overlapping rectangles? [img]https://i.imgur.com/NyhZpL6.png[/img] $\textbf{(A) }21 \qquad\textbf{(B) }22.25 \qquad\textbf{(C) }23\qquad\textbf{(D) }23.75 \qquad\textbf{(E) }25$

2014 Belarus Team Selection Test, 2

Tags: algebra , inequalities

Prove that for all even positive integers $n$ the following inequality holds a) $\{n\sqrt6\} > \frac{1}{n}$ b)$ \{n\sqrt6\}> \frac{1}{n-1/(5n)} $ (I. Voronovich)

1994 All-Russian Olympiad Regional Round, 10.5

Tags: number theory , prime , difference

Find all primes that can be written both as a sum and as a difference of two primes (note that $ 1$ is not a prime).

2018 Ramnicean Hope, 1

Tags: geometry , trigonometry , algebra , inequalities

Show that $ 2/3+\sin 2018^{\circ } >0. $ [i]Costică Ambrinoc[/i]

2015 Dutch BxMO/EGMO TST, 1

Tags: number theory , divisor , divide

Let $m$ and $n$ be positive integers such that $5m+ n$ is a divisor of $5n +m$. Prove that $m$ is a divisor of $n$.

2018 Polish Junior MO First Round, 7

Tags: 3d geometry , square , volume , plane , geometry

Square $ABCD$ with sides of length $4$ is a base of a cuboid $ABCDA'B'C'D'$. Side edges $AA'$, $BB'$, $CC'$, $DD'$ of this cuboid have length $7$. Points $K, L, M$ lie respectively on line segments $AA'$, $BB'$, $CC'$, and $AK = 3$, $BL = 2$, $CM = 5$. Plane passing through points $K, L, M$ cuts cuboid on two blocks. Calculate volumes of these blocks.

2010 JBMO Shortlist, 5

Tags: inequalities

Let $ x, y, z > 0 $ with $ x \leq 2, \;y \leq 3 \;$ and $ x+y+z = 11 $ Prove that $xyz \leq 36$

2013 National Olympiad First Round, 6

Tags:

What is the $111^{\text{st}}$ smallest positive integer which does not have $3$ and $4$ in its base-$5$ representation? $ \textbf{(A)}\ 760 \qquad\textbf{(B)}\ 756 \qquad\textbf{(C)}\ 755 \qquad\textbf{(D)}\ 752 \qquad\textbf{(E)}\ 750 $

2004 Turkey Junior National Olympiad, 3

Tags: inequalities

On the evening, more than $\frac 13$ of the students of a school are going to the cinema. On the same evening, More than $\frac {3}{10}$ are going to the theatre, and more than $\frac {4}{11}$ are going to the concert. At least how many students are there in this school?

2022 VIASM Summer Challenge, Problem 1

Tags: number theory

Given prime numbers $p$ and $q.$ a) Assume that $2^xq=p^y+1,$ with $x,y$ are integers greater than $1$. Can $x$ be a composite number? b) Assume that $2^uq=p^v-1,$ with $u$ is a prime number and $v$ is an integer greater than $1$. Find all possible values of $p.$

1968 Yugoslav Team Selection Test, Problem 1

Tags: inequalities , combinatorics , geometry

Given $6$ points in a plane, assume that each two of them are connected by a segment. Let $D$ be the length of the longest, and $d$ the length of the shortest of these segments. Prove that $\frac Dd\ge\sqrt3$.

2009 Mexico National Olympiad, 2

Tags: geometry

Consider a triangle $ABC$ and a point $M$ on side $BC$. Let $P$ be the intersection of the perpendiculars from $M$ to $AB$ and from $B$ to $BC$, and let $Q$ be the intersection of the perpendiculars from $M$ to $AC$ and from $C$ to $BC$. Show that $PQ$ is perpendicular to $AM$ if and only if $M$ is the midpoint of $BC$.

2008 Princeton University Math Competition, A2

Tags: algebra

What is the polynomial of smallest degree that passes through $(-2, 2), (-1, 1), (0, 2),(1,-1)$, and $(2, 10)$?

2013 India Regional Mathematical Olympiad, 5

Tags: geometry , geometric transformation , circumcircle , incenter , homothety , reflection

In a triangle $ABC$, let $H$ denote its orthocentre. Let $P$ be the reflection of $A$ with respect to $BC$. The circumcircle of triangle $ABP$ intersects the line $BH$ again at $Q$, and the circumcircle of triangle $ACP$ intersects the line $CH$ again at $R$. Prove that $H$ is the incentre of triangle $PQR$.

Kyiv City MO Juniors Round2 2010+ geometry, 2017.7.4

Tags: geometry , perimeter , rectangle , geometric inequality

On the sides $AD$ and $BC$ of a rectangle $ABCD$ select points $M, N$ and $P, Q$ respectively such that $AM = MN = ND = BP = PQ = QC$. On segment $QC$ selected point $X$, different from the ends of the segment. Prove that the perimeter of $\vartriangle ANX$ is more than the perimeter of $\vartriangle MDX$.

2007 Today's Calculation Of Integral, 208

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

Find the values of real numbers $a,\ b$ for which the function $f(x)=a|\cos x|+b|\sin x|$ has local minimum at $x=-\frac{\pi}{3}$ and satisfies $\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}\{f(x)\}^{2}dx=2$.

PEN N Problems, 15

Tags: more sequences

In the sequence $00$, $01$, $02$, $03$, $\cdots$, $99$ the terms are rearranged so that each term is obtained from the previous one by increasing or decreasing one of its digits by $1$ (for example, $29$ can be followed by $19$, $39$, or $28$, but not by $30$ or $20$). What is the maximal number of terms that could remain on their places?

2018 Saint Petersburg Mathematical Olympiad, 7

Tags: combinatorics

In $10\times 10$ square we choose $n$ cells. In every chosen cell we draw one arrow from the angle to opposite angle. It is known, that for any two arrows, or the end of one of them coincides with the beginning of the other, or the distance between their ends is at least 2. What is the maximum possible value of $n$?