Olympiad problems

2002 Austria Beginners' Competition, 4

In a trapezoid $ABCD$ with base $AB$ let $E$ be the midpoint of side $AD$. Suppose further that $2CD=EC=BC=b$. Let $\angle ECB=120^{\circ}$. Construct the trapezoid and determine its area based on $b$.

2007 Junior Macedonian Mathematical Olympiad, 4

Tags: algebra , inequality

The numbers $a_{1}, a_{2}, ..., a_{20}$ satisfy the following conditions: $a_{1} \ge a_{2} \ge ... \ge a_{20} \ge 0$ $a_{1} + a_{2} = 20$ $a_{3} + a_{4} + ... + a_{20} \le 20$ . What is maximum value of the expression: $a_{1}^2 + a_{2}^2 + ... + a_{20}^2$ ? For which values of $a_{1}, a_{2}, ..., a_{20}$ is the maximum value achieved?

1999 Bosnia and Herzegovina Team Selection Test, 4

Tags: angle bisector , parallel , geometry

Let angle bisectors of angles $\angle BAC$ and $\angle ABC$ of triangle $ABC$ intersect sides $BC$ and $AC$ in points $D$ and $E$, respectively. Let points $F$ and $G$ be foots of perpendiculars from point $C$ on lines $AD$ and $BE$, respectively. Prove that $FG \mid \mid AB$

2017 Taiwan TST Round 2, 1

Tags: geometry

Given a circle and four points $B,C,X,Y$ on it. Assume $A$ is the midpoint of $BC$, and $Z$ is the midpoint of $XY$. Let $L_1,L_2$ be lines perpendicular to $BC$ and pass through $B,C$ respectively. Let the line pass through $X$ and perpendicular to $AX$ intersects $L_1,L_2$ at $X_1,X_2$ respectively. Similarly, let the line pass through $Y$ and perpendicular to $AY$ intersects $L_1,L_2$ at $Y_1,Y_2$ respectively. Assume $X_1Y_2$ intersects $X_2Y_1$ at $P$. Prove that $\angle AZP=90^o.$ [i]Proposed by William Chao[/i]

1999 Singapore Team Selection Test, 2

Tags: floor function , algebra

Find all possible values of $$ \lfloor \frac{x - p}{p} \rfloor + \lfloor \frac{-x-1}{p} \rfloor $$ where $x$ is a real number and $p$ is a nonzero integer. Here $\lfloor z \rfloor$ denotes the greatest integer less than or equal to $z$.

1974 IMO Longlists, 18

Tags: geometry , triangle , congruent triangles , circles

Let $A_r,B_r, C_r$ be points on the circumference of a given circle $S$. From the triangle $A_rB_rC_r$, called $\Delta_r$, the triangle $\Delta_{r+1}$ is obtained by constructing the points $A_{r+1},B_{r+1}, C_{r+1} $on $S$ such that $A_{r+1}A_r$ is parallel to $B_rC_r$, $B_{r+1}B_r$ is parallel to $C_rA_r$, and $C_{r+1}C_r$ is parallel to $A_rB_r$. Each angle of $\Delta_1$ is an integer number of degrees and those integers are not multiples of $45$. Prove that at least two of the triangles $\Delta_1,\Delta_2, \ldots ,\Delta_{15}$ are congruent.

2017 AMC 12/AHSME, 19

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Let $N = 123456789101112\dots4344$ be the $79$-digit number obtained that is formed by writing the integers from $1$ to $44$ in order, one after the other. What is the remainder when $N$ is divided by $45$? $\textbf{(A)}\ 1 \qquad\textbf{(B)}\ 4 \qquad\textbf{(C)}\ 9 \qquad\textbf{(D)}\ 18 \qquad\textbf{(E)}\ 44$

1936 Moscow Mathematical Olympiad, 022

Tags: perfect square , 4-digit , number theory

Find a four-digit perfect square whose first digit is the same as the second, and the third is the same as the fourth.

2004 Regional Olympiad - Republic of Srpska, 1

Tags: number theory

Prove that the cube of any positive integer greater than 1 can be represented as a difference of the squares of two positive integers.

2006 Indonesia MO, 6

Tags: geometry , probability , combinatorics

Every phone number in an area consists of eight digits and starts with digit $ 8$. Mr Edy, who has just moved to the area, apply for a new phone number. What is the chance that Mr Edy gets a phone number which consists of at most five different digits?

2014 PUMaC Combinatorics A, 6

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Let $f(n)$ be the number of points of intersection of diagonals of a $n$-dimensional hypercube that is not the vertex of the cube. For example, $f(3) = 7$ because the intersection points of a cube’s diagonals are at the centers of each face and the center of the cube. Find $f(5)$.

2023 Indonesia TST, N

Tags: primitive root , fermat's little theorem , number theory

Let $p,q,r$ be primes such that for all positive integer $n$, $$n^{pqr}\equiv n (\mod{pqr})$$ Prove that this happens if and only if $p,q,r$ are pairwise distinct and $LCM(p-1,q-1,r-1)|pqr-1$

1993 National High School Mathematics League, 2

Tags: function

$f(x)=a\sin x+b\sqrt[3]{x}+4$. If $f(\lg\log_{3}10)=5$, then the value of $f(\lg\lg 3)$ is $\text{(A)}-5\qquad\text{(B)}-3\qquad\text{(C)}3\qquad\text{(D)}$ not sure

1992 Baltic Way, 18

Tags: perimeter , inequalities , circumcircle , geometry

Show that in a non-obtuse triangle the perimeter of the triangle is always greater than two times the diameter of the circumcircle.

2004 Romania National Olympiad, 2

Tags: inequalities

The sidelengths of a triangle are $a,b,c$. (a) Prove that there is a triangle which has the sidelengths $\sqrt a,\sqrt b,\sqrt c$. (b) Prove that $\displaystyle \sqrt{ab}+\sqrt{bc}+\sqrt{ca} \leq a+b+c < 2 \sqrt{ab} + 2 \sqrt{bc} + 2 \sqrt{ca}$.

2021 USMCA, 13

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An ant is currently located in the center (vertex $S$) of the adjoined hexagonal configuration, as shown in the figure below. Each minute, it walks along $1$ of the $15$ possible edges, traveling from one vertex to another. How many ways are there for the ant to be back to its original position after $2020$ minutes?

2005 China Second Round Olympiad, 3

Tags: function , floor function , geometry , geometric transformation , algebra

For each positive integer, define a function \[ f(n)=\begin{cases}0, &\text{if n is the square of an integer}\\ \\ \left\lfloor\frac{1}{\{\sqrt{n}\}}\right\rfloor, &\text{if n is not the square of an integer}\end{cases}. \] Find the value of $\sum_{k=1}^{200} f(k)$.

2017 IMAR Test, 4

Tags: combinatorial geometry , combinatorics , geometry , polygon

Let $n$ be an integer greater than or equal to $3$, and let $P_n$ be the collection of all planar (simple) $n$-gons no two distinct sides of which are parallel or lie along some line. For each member $P$ of $P_n$, let $f_n(P)$ be the least cardinal a cover of $P$ by triangles formed by lines of support of sides of $P$ may have. Determine the largest value $f_n(P)$ may achieve, as $P$ runs through $P_n$.

1978 Romania Team Selection Test, 3

Tags: geometry , trapezoid , coloring

Let $ A_1,A_2,...,A_{3n} $ be $ 3n\ge 3 $ planar points such that $ A_1A_2A_3 $ is an equilateral triangle and $ A_{3k+1} ,A_{3k+2} ,A_{3k+3} $ are the midpoints of the sides of $ A_{3k-2}A_{3k-1}A_{3k} , $ for all $ 1\le k<n. $ Of two different colors, each one of these points are colored, either with one, either with another. [b]a)[/b] Prove that, if $ n\ge 7, $ then some of these points form a monochromatic (only one color) isosceles trapezoid. [b]b)[/b] What about $ n=6? $

2005 International Zhautykov Olympiad, 1

Tags: algebra , polynomial , quadratic , modular arithmetic , number theory

Prove that the equation $ x^{5} \plus{} 31 \equal{} y^{2}$ has no integer solution.

1993 IberoAmerican, 1

Tags: number theory

A number is called [i]capicua[/i] if when it is written in decimal notation, it can be read equal from left to right as from right to left; for example: $8, 23432, 6446$. Let $x_1<x_2<\cdots<x_i<x_{i+1},\cdots$ be the sequence of all capicua numbers. For each $i$ define $y_i=x_{i+1}-x_i$. How many distinct primes contains the set $\{y_1,y_2, \ldots\}$?

2004 Thailand Mathematical Olympiad, 18

Tags: algebra , inequalities , max , product

Find positive reals $a, b, c$ which maximizes the value of $abc$ subject to the constraint that $b(a^2 + 2) + c(a + 2) = 12$.

2005 Junior Balkan Team Selection Tests - Romania, 15

Tags:

Let $n>3$ be a positive integer. Consider $n$ sets, each having two elements, such that the intersection of any two of them is a set with one element. Prove that the intersection of all sets is non-empty. [i]Sever Moldoveanu[/i]

2020 Canadian Junior Mathematical Olympiad, 4

Tags: geometry , rhombus

$ABCD$ is a fixed rhombus. Segment $PQ$ is tangent to the inscribed circle of $ABCD$, where $P$ is on side $AB$, $Q$ is on side $AD$. Show that, when segment $PQ$ is moving, the area of $\Delta CPQ$ is a constant.

STEMS 2024 Math Cat A, P3

Tags: geometry

Let $ABC$ be a triangle. Let $I$ be the Incenter of $ABC$ and $S$ be the midpoint of arc $BAC$. Define $IA$ as the $A$-excenter wrt $ABC$. Define $\omega$ to be the circle centred at $S$ with radius $SB$. Let $AI_A \cap \omega = X$, $Y$. Show that $\angle BCX = \angle ACY$.