Olympiad problems

2000 Junior Balkan Team Selection Tests - Moldova, 4

Find the smallest natural number nonzero n so that it exists in real numbers $x_1, x_2,..., x_n$ which simultaneously check the conditions: 1) $x_i \in [1/2 , 2]$ , $i = 1, 2,... , n$ 2) $x_1+x_2+...+x_n \ge \frac{7n}{6}$ 3) $\frac{1}{x_1}+\frac{1}{x_2}+...+\frac{1}{x_n}\ge \frac{4n}{3}$

2024 Belarusian National Olympiad, 9.3

Tags: geometry

On the side $AC$ of triangle $ABC$ point $D$ is chosen. The perpendicular bisector of segment $BD$ intersects the circumcircle $\Omega$ of triangle $ABC$ at $P$, $Q$. Point $E$ lies on the arc $AC$ of circle $\Omega$, that doesn't contain point $B$, such that $\angle ABD=\angle CBE$. Prove that the orthocenter of the triangle $PQE$ lies on the line $AC$ [i]M. Zorka[/i]

2024 USA IMO Team Selection Test, 3

Tags: number theory , USA TST , USA TST 2024 , combinatorics , Hi , bye

Let $n>k \geq 1$ be integers and let $p$ be a prime dividing $\tbinom{n}{k}$. Prove that the $k$-element subsets of $\{1,\ldots,n\}$ can be split into $p$ classes of equal size, such that any two subsets with the same sum of elements belong to the same class. [i]Ankan Bhattacharya[/i]

1980 IMO, 15

Tags: ratio , function , geometry , geometry unsolved

Three points $A,B,C$ are such that $B\in AC$. On one side of $AC$, draw the three semicircles with diameters $AB,BC,CA$. The common interior tangent at $B$ to the first two semicircles meets the third circle $E$. Let $U,V$ be the points of contact of the common exterior tangent to the first two semicircles. Evaluate the ratio $R=\frac{[EUV]}{[EAC]}$ as a function of $r_{1} = \frac{AB}{2}$ and $r_2 = \frac{BC}{2}$, where $[X]$ denotes the area of polygon $X$.

2011 Today's Calculation Of Integral, 762

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

Define a function $f_n(x)\ (n=0,\ 1,\ 2,\ \cdots)$ by \[f_0(x)=\sin x,\ f_{n+1}(x)=\int_0^{\frac{\pi}{2}} f_n\prime (t)\sin (x+t)dt.\] (1) Let $f_n(x)=a_n\sin x+b_n\cos x.$ Express $a_{n+1},\ b_{n+1}$ in terms of $a_n,\ b_n.$ (2) Find $\sum_{n=0}^{\infty} f_n\left(\frac{\pi}{4}\right).$

2011 Today's Calculation Of Integral, 720

Tags: calculus , integration , trigonometry , calculus computations

Evaluate $\int_0^{2\pi} |x^2-\pi ^ 2 -\sin ^ 2 x|\ dx$.

2012 Greece JBMO TST, 4

Tags: combinatorics , positive integers , integer solutions

Numbers $x,y,z$ are positive integers and satisfy the equation $x+y+z=2013$. (E) a) Find the number of the triplets $(x,y,z)$ that are solutions of the equation (E). b) Find the number of the solutions of the equation (E) for which $x=y$. c) Find the solution $(x,y,z)$ of the equation (E) for which the product $xyz$ becomes maximum.

2015 Dutch BxMO/EGMO TST, 4

Tags: geometry , circumcircle , equal angles , Angle Chasing , angle bisector

In a triangle $ABC$ the point $D$ is the intersection of the interior angle bisector of $\angle BAC$ and side $BC$. Let $P$ be the second intersection point of the exterior angle bisector of $\angle BAC$ with the circumcircle of $\angle ABC$. A circle through $A$ and $P$ intersects line segment $BP$ internally in $E$ and line segment $CP$ internally in $F$. Prove that $\angle DEP = \angle DFP$.

1998 Belarus Team Selection Test, 2

Tags: geometry , perimeter , geometric inequality , incircle , inradius

The incircle of the triangle $ABC$ touches its sides $AB,BC,CA$ at points $C_1,A_1,B_1$ respectively. If $r$ is the inradius of $\vartriangle ABC, P,P_1$ are the perimeters of $\vartriangle ABC, \vartriangle A_1B_1C_1$ respectively, prove that $P+P_1 \ge 9 \sqrt3 r$. I. Voronovich

1998 Junior Balkan Team Selection Tests - Romania, 3

Tags: number theory

Find the smallest natural number for which there exist that many natural numbers such that the sum of the squares of their squares is equal to $ 1998. $ [i]Gheorghe Iurea[/i]

2002 National Olympiad First Round, 16

Tags:

Which of the following cannot be equal to $x^2 + \dfrac 1{4x}$ where $x$ is a positive real number? $ \textbf{a)}\ \sqrt 3 -1 \qquad\textbf{b)}\ 2\sqrt 2 - 2 \qquad\textbf{c)}\ \sqrt 5 - 1 \qquad\textbf{d)}\ 1 \qquad\textbf{e)}\ \text{None of above} $

2013 Saint Petersburg Mathematical Olympiad, 7

Tags: combinatorics proposed , combinatorics

In the language of wolves has two letters $F$ and $P$, any finite sequence which forms a word. А word $Y$ is called 'subpart' of word $X$ if Y is obtained from X by deleting some letters (for example, the word $FFPF$ has 8 'subpart's: F, P, FF, FP, PF, FFP, FPF, FFF). Determine $n$ such that the $n$ is the greatest number of 'subpart's can have n-letter word language of wolves. F. Petrov, V. Volkov

1967 IMO Longlists, 51

Tags: combinatorics , Extremal combinatorics , counting , IMO Shortlist , IMO Longlist

A subset $S$ of the set of integers 0 - 99 is said to have property $A$ if it is impossible to fill a crossword-puzzle with 2 rows and 2 columns with numbers in $S$ (0 is written as 00, 1 as 01, and so on). Determine the maximal number of elements in the set $S$ with the property $A.$

1979 Bundeswettbewerb Mathematik, 1

Tags: combinatorics , football

There are $n$ teams in a football league. During a championship, every two teams play exactly one match, but no team can play more than one match in a week. At least, how many weeks are necessary for the championship to be held? Give an schedule for such a championship.

1979 Poland - Second Round, 6

Tags: geometry , semicircle , rectangle

On the side $ \overline{DC} $ of the rectangle $ ABCD $ in which $ \frac{AB}{AD} = \sqrt{2} $ a semicircle is built externally. Any point $ M $ of the semicircle is connected by segments with $ A $ and $ B $ to obtain points $ K $ and $ L $ on $ \overline{DC} $, respectively. Prove that $ DL^2 + KC^2 = AB^2 $.

2021 Vietnam TST, 1

Tags: number theory , inequalities

Define the sequence $(a_n)$ as $a_1 = 1$, $a_{2n} = a_n$ and $a_{2n+1} = a_n + 1$ for all $n\geq 1$. a) Find all positive integers $n$ such that $a_{kn} = a_n$ for all integers $1 \leq k \leq n$. b) Prove that there exist infinitely many positive integers $m$ such that $a_{km} \geq a_m$ for all positive integers $k$.

2003 AMC 12-AHSME, 1

Tags: arithmetic sequence , AMC 10

What is the difference between the sum of the first $ 2003$ even counting numbers and the sum of the first $ 2003$ odd counting numbers? $ \textbf{(A)}\ 0 \qquad \textbf{(B)}\ 1 \qquad \textbf{(C)}\ 2 \qquad \textbf{(D)}\ 2003 \qquad \textbf{(E)}\ 4006$

2024 Nigerian MO Round 2, Problem 4

Tags: algebra , ap

Let the AP of the form $4$, $9$, $\ldots$ be $\mathbf{A}$, and the AP of the form $16$, $25$, $\ldots$ be $\mathbf{B}$. Find the number of integers from $1$ to $2024$ inclusive, that appear in only one of the AP's. For clarification, the AP's $\mathbf{A}$ and $\mathbf{B}$ start from 4 and 16 respectively. [hide=Answer]584[/hide]

1994 Turkey Team Selection Test, 1

Tags: ratio , geometry , geometry proposed

Let $P,Q,R$ be points on the sides of $\triangle ABC$ such that $P \in [AB],Q\in[BC],R\in[CA]$ and $\frac{|AP|}{|AB|} = \frac {|BQ|}{|BC|} =\frac{|CR|}{|CA|} =k < \frac 12$ If $G$ is the centroid of $\triangle ABC$, find the ratio $\frac{Area(\triangle PQG)}{Area(\triangle PQR)}$ .

2007 ISI B.Math Entrance Exam, 4

Tags: geometry proposed , geometry

Let $ABC$ be an isosceles triangle with $AB=AC=20$ . Let $P$ be a point inside the triangle $ABC$ such that the sum of the distances of $P$ to $AB$ and $AC$ is $1$ . Describe the locus of all such points inside triangle $ABC$.

2023 Dutch IMO TST, 4

Tags: number theory

Find all positive integers $n$, such that $\sigma(n) =\tau(n) \lceil {\sqrt{n}} \rceil$.

2010 Miklós Schweitzer, 3

Tags: college contests , Miklos Schweitzer , combinatorics , set theory , Subsets

Let $ A_i,i=1,2,\dots,t$ be distinct subsets of the base set $\{1,2,\dots,n\}$ complying to the following condition $$ \displaystyle A_ {i} \cap A_ {k} \subseteq A_ {j}$$for any $1 \leq i <j <k \leq t.$ Find the maximum value of $t.$ Thanks @dgrozev

1978 AMC 12/AHSME, 1

Tags: LaTeX , quadratics , AMC

If $1-\frac{4}{x}+\frac{4}{x^2}=0$, then $\frac{2}{x}$ equals $\textbf{(A) }-1\qquad\textbf{(B) }1\qquad\textbf{(C) }2\qquad\textbf{(D) }-1\text{ or }2\qquad \textbf{(E) }-1\text{ or }-2$

2018 Bosnia and Herzegovina Team Selection Test, 6

Tags: geometry , IMO Shortlist , Angle Chasing , projective geometry , similar triangles , ISL 2017 , G3

Let $O$ be the circumcenter of an acute triangle $ABC$. Line $OA$ intersects the altitudes of $ABC$ through $B$ and $C$ at $P$ and $Q$, respectively. The altitudes meet at $H$. Prove that the circumcenter of triangle $PQH$ lies on a median of triangle $ABC$.

Ukrainian TYM Qualifying - geometry, I.7

Tags: ratio , geometry , polygon , inscribed , areas , Ukrainian TYM

Given a natural number $n> 3$. On the plane are considered convex $n$ - gons $F_1$ and $F_2$ such that on each side of $F_1$ lies one vertex of $F_2$ and no two vertices $F_1$ and $F_2$ coincide. For each $n$, determine the limits of the ratio of the areas of the polygons $F_1$ and $F_2$. For each $n$, determine the range of the areas of the polygons $F_1$ and $F_2$, if $F_1$ is a regular $n$-gon. Determine the set of values of this value for other partial cases of the polygon $F_1$.