Olympiad problems

2016 Balkan MO Shortlist, A5

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

Tags:

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 $O$.

2012 Purple Comet Problems, 19

Tags:

A teacher suggests four possible books for students to read. Each of six students selects one of the four books. How many ways can these selections be made if each of the books is read by at least one student?

2019 Purple Comet Problems, 17

Tags: factorial , number theory

Find the greatest integer $n$ such that $5^n$ divides $2019! - 2018! + 2017!$.

2008 Mongolia Team Selection Test, 2

Tags: gauss , combinatorics

Given positive integers$ m,n$ such that $ m < n$. Integers $ 1,2,...,n^2$ are arranged in $ n \times n$ board. In each row, $ m$ largest number colored red. In each column $ m$ largest number colored blue. Find the minimum number of cells such that colored both red and blue.