Olympiad problems

2002 Rioplatense Mathematical Olympiad, Level 3, 1

Determine all pairs $(a, b)$ of positive integers for which $\frac{a^2b+b}{ab^2+9}$ is an integer number.

2016 Purple Comet Problems, 11

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One evening a theater sold 300 tickets for a concert. Each ticket sold for \$40, and all tickets were purchased using \$5, \$10, and \$20 bills. At the end of the evening the theater had received twice as many \$10 bills as \$20 bills, and 20 more \$5 bills than \$10 bills. How many bills did the theater receive altogether?

2022 India National Olympiad, 3

Tags: arrangement

For a positive integer $N$, let $T(N)$ denote the number of arrangements of the integers $1, 2, \cdots N$ into a sequence $a_1, a_2, \cdots a_N$ such that $a_i > a_{2i}$ for all $i$, $1 \le i < 2i \le N$ and $a_i > a_{2i+1}$ for all $i$, $1 \le i < 2i+1 \le N$. For example, $T(3)$ is $2$, since the possible arrangements are $321$ and $312$ (a) Find $T(7)$ (b) If $K$ is the largest non-negative integer so that $2^K$ divides $T(2^n - 1)$, show that $K = 2^n - n - 1$. (c) Find the largest non-negative integer $K$ so that $2^K$ divides $T(2^n + 1)$

2016 Balkan MO Shortlist, A5

Tags: algebra , system of equations , inequalities , minimum , maximum

Let $a, b,c$ and $d$ be real numbers such that $a + b + c + d = 2$ and $ab + bc + cd + da + ac + bd = 0$. Find the minimum value and the maximum value of the product $abcd$.

2009 Abels Math Contest (Norwegian MO) Final, 3b

Tags: point , combinatorics , combinatorial geometry , lattice points , geometry , circles

Show for any positive integer $n$ that there exists a circle in the plane such that there are exactly $n$ grid points within the circle. (A grid point is a point having integer coordinates.)

2007 Sharygin Geometry Olympiad, 4

Tags: geometry , reflection , angle bisector , cyclic quadrilateral , symmetry

A quadrilateral A$BCD$ is inscribed into a circle with center $O$. Points $C', D'$ are the reflections of the orthocenters of triangles $ABD$ and $ABC$ at point $O$. Lines $BD$ and $BD'$ are symmetric with respect to the bisector of angle $ABC$. Prove that lines $AC$ and $AC'$ are symmetric with respect to the bisector of angle $DAB$.

2009 ISI B.Math Entrance Exam, 4

Tags: trigonometry , algebra

Find the values of $x,y$ for which $x^2+y^2$ takes the minimum value where $(x+5)^2+(y-12)^2=14$.

2016 IFYM, Sozopol, 6

Tags: polynomial , algebra

Find all polynomials $P\in \mathbb{Q}[x]$, which satisfy the following equation: $P^2 (n)+\frac{1}{4}=P(n^2+\frac{1}{4})$ for $\forall$ $n\in \mathbb{N}$.

1993 IMO, 2

Tags: geometry , circumcircle , quadrilateral , area

Let $A$, $B$, $C$, $D$ be four points in the plane, with $C$ and $D$ on the same side of the line $AB$, such that $AC \cdot BD = AD \cdot BC$ and $\angle ADB = 90^{\circ}+\angle ACB$. Find the ratio \[\frac{AB \cdot CD}{AC \cdot BD}, \] and prove that the circumcircles of the triangles $ACD$ and $BCD$ are orthogonal. (Intersecting circles are said to be orthogonal if at either common point their tangents are perpendicuar. Thus, proving that the circumcircles of the triangles $ACD$ and $BCD$ are orthogonal is equivalent to proving that the tangents to the circumcircles of the triangles $ACD$ and $BCD$ at the point $C$ are perpendicular.)

2013 Tuymaada Olympiad, 4

Tags: inequalities , three variable inequality

Prove that if $x$, $y$, $z$ are positive real numbers and $xyz = 1$ then \[\frac{x^3}{x^2+y}+\frac{y^3}{y^2+z}+\frac{z^3}{z^2+x}\geq \dfrac {3} {2}.\] [i]A. Golovanov[/i]

2024 Indonesia TST, 2

Tags: geometry

Let $ABC$ be a triangle with $AC > BC,$ let $\omega$ be the circumcircle of $\triangle ABC,$ and let $r$ be its radius. Point $P$ is chosen on $\overline{AC}$ such taht $BC=CP,$ and point $S$ is the foot of the perpendicular from $P$ to $\overline{AB}$. Ray $BP$ mets $\omega$ again at $D$. Point $Q$ is chosen on line $SP$ such that $PQ = r$ and $S,P,Q$ lie on a line in that order. Finally, let $E$ be a point satisfying $\overline{AE} \perp \overline{CQ}$ and $\overline{BE} \perp \overline{DQ}$. Prove that $E$ lies on $\omega$.

2010 Contests, 2

Tags: pigeonhole principle , arithmetic sequence , induction

Let $n > 1$ be an integer. Find, with proof, all sequences $x_1 , x_2 , \ldots , x_{n-1}$ of positive integers with the following three properties: (a). $x_1 < x_2 < \cdots < x_{n-1}$ ; (b). $x_i + x_{n-i} = 2n$ for all $i = 1, 2, \ldots , n - 1$; (c). given any two indices $i$ and $j$ (not necessarily distinct) for which $x_i + x_j < 2n$, there is an index $k$ such that $x_i + x_j = x_k$.

2001 AMC 10, 2

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A number $ x$ is $ 2$ more than the product of its reciprocal and its additive inverse. In which interval does the number lie? $ \textbf{(A) }\minus{}4\le x\le \minus{}2\qquad\textbf{(B) }\minus{}2 < x\le 0\qquad\textbf{(C) }0 < x\le 2$ $ \textbf{(D) }2 < x\le 4\qquad\textbf{(E) }4 < x\le 6$

2011 Purple Comet Problems, 14

Tags: trigonometry , pythagorean theorem , geometry

The lengths of the three sides of a right triangle form a geometric sequence. The sine of the smallest of the angles in the triangle is $\tfrac{m+\sqrt{n}}{k}$ where $m$, $n$, and $k$ are integers, and $k$ is not divisible by the square of any prime. Find $m + n + k$.

2007 Bulgarian Autumn Math Competition, Problem 12.3

Tags: symmetric inequality , three variable inequality , algebra , inequalities

Find all real numbers $r$, such that the inequality \[r(ab+bc+ca)+(3-r)\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right)\geq 9\] holds for any real $a,b,c>0$.

2018 AMC 10, 11

Tags: prime number

Which of the following expressions is never a prime number when $p$ is a prime number? $\textbf{(A) } p^2+16 \qquad \textbf{(B) } p^2+24 \qquad \textbf{(C) } p^2+26 \qquad \textbf{(D) } p^2+46 \qquad \textbf{(E) } p^2+96$

2007 All-Russian Olympiad Regional Round, 8.2

Tags: combinatorics

Pete chose a positive integer $ n$. For each (unordered) pair of its decimal digits, he wrote their difference on the blackboard. After that, he erased some of these differences, and the remaining ones are $ 2,0,0,7$. Find the smallest number $ n$ for which this situation is possible.

2011 Today's Calculation Of Integral, 704

Tags: calculus , integration , function , induction , calculus computations

A function $f_n(x)\ (n=0,\ 1,\ 2,\ 3,\ \cdots)$ satisfies the following conditions: (i) $f_0(x)=e^{2x}+1$. (ii) $f_n(x)=\int_0^x (n+2t)f_{n-1}(t)dt-\frac{2x^{n+1}}{n+1}\ (n=1,\ 2,\ 3,\ \cdots).$ Find $\sum_{n=1}^{\infty} f_n'\left(\frac 12\right).$

2010 Korea Junior Math Olympiad, 7

Tags: collinear , cyclic quadrilateral , perpendicular , geometry

Let $ABCD$ be a cyclic convex quadrilateral. Let $E$ be the intersection of lines $AB,CD$. $P$ is the intersection of line passing $B$ and perpendicular to $AC$, and line passing $C$ and perpendicular to $BD$. $Q$ is the intersection of line passing $D$ and perpendicular to $AC$, and line passing $A$ and perpendicular to $BD$. Prove that three points $E, P,Q$ are collinear.

2019 Brazil EGMO TST, 4

Tags: combinatorics

Twenty players participated in a chess tournament. Each player faced every other player exactly once and each match ended with either player winning or a draw. In this tournament, it was noticed that for every match that ended in a draw, each of the other $18$ players won at least one of the two players involved in it. We also know that at least two games ended in a draw. Show that it is possible to name the players as $P_1, P_2, ...., P_{20}$ so that player $P_k$ beat player $P_{k+1}$, to each $k \in \{ 1, 2, 3,... , 19\}$.

2011 APMO, 2

Tags: geometry , geometric transformation , combinatorics

Five points $A_1,A_2,A_3,A_4,A_5$ lie on a plane in such a way that no three among them lie on a same straight line. Determine the maximum possible value that the minimum value for the angles $\angle A_iA_jA_k$ can take where $i, j, k$ are distinct integers between $1$ and $5$.

2004 Rioplatense Mathematical Olympiad, Level 3, 3

Tags: combinatorics

Consider a partition of $\{1,2,\ldots,900\}$ into $30$ subsets $S_1,S_2,\ldots,S_{30}$ each with $30$ elements. In each $S_k$, we paint the fifth largest number blue. Is it possible that, for $k=1,2,\ldots,30$, the sum of the elements of $S_k$ exceeds the sum of the blue numbers?

2025 239 Open Mathematical Olympiad, 7

Tags: combinatorics

Given a $2025 \times 2025$ board and $k$ chips lying on the table. Initially, the board is empty. It is allowed to place a chip from the table on any free square $A$ if two conditions are met simultaneously: – the cell next to $A$ from below has a chip (or $A$ is on the bottom edge of the board); – the cell next to $A$ on the left has a chip (or $A$ is on the left edge of the board). In addition, it is allowed to remove any chip from the board and put it on the table. At what minimum $k$ can a chip appear in the upper-right corner of the board?

LMT Team Rounds 2021+, 1

Tags: algebra

Kevin writes the multiples of three from $1$ to $100$ on the whiteboard. How many digits does he write?

2011 HMNT, 8

Tags: geometry

Points $D$, $E$, $F$ lie on circle $O$ such that the line tangent to $O$ at $D$ intersects ray $\overrightarrow{EF}$ at $P$. Given that $PD = 4$, $PF = 2$, and $\angle FPD = 60^o$, determine the area of circle $O$.