Olympiad problems

2015 India IMO Training Camp, 2

Tags: algebra

Let $A$ be a finite set of pairs of real numbers such that for any pairs $(a,b)$ in $A$ we have $a>0$. Let $X_0=(x_0, y_0)$ be a pair of real numbers(not necessarily from $A$). We define $X_{j+1}=(x_{j+1}, y_{j+1})$ for all $j\ge 0$ as follows: for all $(a,b)\in A$, if $ax_j+by_j>0$ we let $X_{j+1}=X_j$; otherwise we choose a pair $(a,b)$ in $A$ for which $ax_j+by_j\le 0$ and set $X_{j+1}=(x_j+a, y_j+b)$. Show that there exists an integer $N\ge 0$ such that $X_{N+1}=X_N$.

2003 Mediterranean Mathematics Olympiad, 2

Tags: trigonometry , calculus , derivative , hyperbola , conic , geometry

In a triangle $ABC$ with $BC = CA + \frac 12 AB$, point $P$ is given on side $AB$ such that $BP : PA = 1 : 3$. Prove that $\angle CAP = 2 \angle CPA.$

1999 May Olympiad, 1

Tags: number theory , digit

A three-digit natural number is called [i]tricubic [/i] if it is equal to the sum of the cubes of its digits. Find all pairs of consecutive numbers such that both are tricubic.

2023 Turkey Team Selection Test, 5

Tags: geometry , circumcircle , incenter

Let $ABC$ be a scalene triangle with circumcentre $O$, incentre $I$ and orthocentre $H$. Let the second intersection point of circle which passes through $O$ and tangent to $IH$ at point $I$, and the circle which passes through $H$ and tangent to $IO$ at point $I$ be $M$. Prove that $M$ lies on circumcircle of $ABC$.

1977 USAMO, 3

Tags: algebra , polynomial

If $ a$ and $ b$ are two of the roots of $ x^4\plus{}x^3\minus{}1\equal{}0$, prove that $ ab$ is a root of $ x^6\plus{}x^4\plus{}x^3\minus{}x^2\minus{}1\equal{}0$.

2011 ISI B.Math Entrance Exam, 1

Tags: trigonometry , algebra

Given $a,x\in\mathbb{R}$ and $x\geq 0$,$a\geq 0$ . Also $\sin(\sqrt{x+a})=\sin(\sqrt{x})$ . What can you say about $a$??? Justify your answer.

2007 Today's Calculation Of Integral, 236

Tags: calculus , integration , trigonometry , calculus computations

Let $a$ be a positive constant. Evaluate the following definite integrals $A,\ B$. \[A=\int_0^{\pi} e^{-ax}\sin ^ 2 x\ dx,\ B=\int_0^{\pi} e^{-ax}\cos ^ 2 x\ dx\]. [i]1998 Shinsyu University entrance exam/Textile Science[/i]

1992 Chile National Olympiad, 1

Tags: perfect square , number theory

Determine all naturals $n$ such that $2^n + 5$ is a perfect square.

2025 USAMO, 4

Tags: geometry

Let $H$ be the orthocenter of acute triangle $ABC$, let $F$ be the foot of the altitude from $C$ to $AB$, and let $P$ be the reflection of $H$ across $BC$. Suppose that the circumcircle of triangle $AFP$ intersects line $BC$ at two distinct points $X$ and $Y$. Prove that $C$ is the midpoint of $XY$.

2006 Petru Moroșan-Trident, 3

Tags: linear algebra , matrix

Let be a $ 2\times 2 $ real matrix such that $ \det \left( A^6+64I \right) =0. $ Show that $ \det A=4. $ [i]Viorel Botea[/i]

2023 IRN-SGP-TWN Friendly Math Competition, 1

Tags: number theory

Prove that there are infinitely many positive integers $a$ such that \[a!+(a+2)!\mid (a+2\left\lfloor\sqrt{a}\right\rfloor)!.\] [i]Proposed by Navid and the4seasons.[/i]

2013 Online Math Open Problems, 11

Tags: analytic geometry , graphing lines , slope , trigonometry

Let $A$, $B$, and $C$ be distinct points on a line with $AB=AC=1$. Square $ABDE$ and equilateral triangle $ACF$ are drawn on the same side of line $BC$. What is the degree measure of the acute angle formed by lines $EC$ and $BF$? [i]Ray Li[/i]

2007 Macedonia National Olympiad, 2

Tags: parallelogram , trapezoid , circumcircle , geometry

In a trapezoid $ABCD$ with a base $AD$, point $L$ is the orthogonal projection of $C$ on $AB$, and $K$ is the point on $BC$ such that $AK$ is perpendicular to $AD$. Let $O$ be the circumcenter of triangle $ACD$. Suppose that the lines $AK , CL$ and $DO$ have a common point. Prove that $ABCD$ is a parallelogram.

2008 AMC 10, 24

Tags: geometry , rhombus , trigonometry , geometric transformation , parallelogram

Quadrilateral $ABCD$ has $AB=BC=CD$, $\angle ABC=70^\circ$, and $\angle BCD=170^\circ$. What is the degree measure of $\angle BAD$? $ \textbf{(A)}\ 75\qquad \textbf{(B)}\ 80\qquad \textbf{(C)}\ 85\qquad \textbf{(D)}\ 90\qquad \textbf{(E)}\ 95$

1985 Greece National Olympiad, 3

Tags: analytic geometry , geometry , distance , number theory , lattice points

Consider the line (E): $5x-10y+3=0$ . Prove that: a) Line $(E)$ doesn't pass through points with integer coordinates. b) There is no point $A(a_1,a_2)$ with $ a_1,a_2 \in \mathbb{Z}$ with distance from $(E)$ less then $\frac{\sqrt3}{20}$.

2023 JBMO TST - Turkey, 3

Tags: geometry , tangent

Let $ABC$ is triangle and $D \in AB$,$E \in AC$ such that $DE//BC$. Let $(ABC)$ meets with $(BDE)$ and $(CDE)$ at the second time $K,L$ respectively. $BK$ and $CL$ intersect at $T$. Prove that $TA$ is tangent to the $(ABC)$

2019 Flanders Math Olympiad, 2

Tags: algebra , sum

Calculate the sum of all unsimplified fractions whose numerator and denominator are positive divisors of $1000$.

1989 IMO Shortlist, 28

Tags: geometry , area of a triangle , geometric inequality , point set

Consider in a plane $ P$ the points $ O,A_1,A_2,A_3,A_4$ such that \[ \sigma(OA_iA_j) \geq 1 \quad \forall i, j \equal{} 1, 2, 3, 4, i \neq j.\] where $ \sigma(OA_iA_j)$ is the area of triangle $ OA_iA_j.$ Prove that there exists at least one pair $ i_0, j_0 \in \{1, 2, 3, 4\}$ such that \[ \sigma(OA_iA_j) \geq \sqrt{2}.\]

2022 Germany Team Selection Test, 1

Tags: algebra , real analysis , inequalities

Let $n\geq 2$ be an integer and let $a_1, a_2, \ldots, a_n$ be positive real numbers with sum $1$. Prove that $$\sum_{k=1}^n \frac{a_k}{1-a_k}(a_1+a_2+\cdots+a_{k-1})^2 < \frac{1}{3}.$$

2023 Regional Olympiad of Mexico Southeast, 3

Tags: grid , coloring

Let $n$ be a positive integer. A grid of $n\times n$ has some black-colored cells. Drini can color a cell if at least three cells that share a side with it are also colored black. Drini discovers that by repeating this process, all the cells in the grid can be colored. Prove that if there are initially $k$ colored cells, then $$k\geq \frac{n^2+2n}{3}.$$

2018 HMNT, 6

Tags: geometry

Triangle $\triangle PQR$, with $PQ=PR=5$ and $QR=6$, is inscribed in circle $\omega$. Compute the radius of the circle with center on $\overline{QR}$ which is tangent to both $\omega$ and $\overline{PQ}$.

2004 Estonia National Olympiad, 3

Tags: bisects area , convex , parallel , geometry

On the sides $AB , BC$ of the convex quadrilateral $ABCD$ lie points $M$ and $N$ such that $AN$ and $CM$ each divide the quadrilateral $ABCD$ into two equal area parts. Prove that the line $MN$ and $AC$ are parallel.

2006 Bulgaria Team Selection Test, 1

Tags: circumcircle , parallelogram , symmetry , angle bisector , power of a point , geometry

[b]Problem 1.[/b] Points $D$ and $E$ are chosen on the sides $AB$ and $AC$, respectively, of a triangle $\triangle ABC$ such that $DE\parallel BC$. The circumcircle $k$ of triangle $\triangle ADE$ intersects the lines $BE$ and $CD$ at the points $M$ and $N$ (different from $E$ and $D$). The lines $AM$ and $AN$ intersect the side $BC$ at points $P$ and $Q$ such that $BC=2\cdot PQ$ and the point $P$ lies between $B$ and $Q$. Prove that the circle $k$ passes through the point of intersection of the side $BC$ and the angle bisector of $\angle BAC$. [i]Nikolai Nikolov[/i]

2022 Costa Rica - Final Round, 5

Tags: number theory

The $1$st edition of OLCOMA was organized in $1989$, so in $2022$ the $34$th edition will be celebrated. Suppose that the Olympics will continue to be held annually without interruption. We say that a year $N$ is [i]good [/i] if the OLCOMA edition number of that year divides the product $N(N +1)$. For example, the year $2022$ is good because $34$ divides $2022 \cdot 2023$. Determine the last year $N$ in the $21$st century, $2000\le N \le 2099$, which is good.

1974 Poland - Second Round, 4

Tags: geometry , area

In a convex quadrilateral $ ABCD $ with area $ S $, each side was divided into 3 equal parts and segments were drawn connecting the appropriate points of division of the opposite sides in such a way that the quadrilateral was divided into 9 quadrilaterals. Prove that the sum of the areas of the following three quadrilaterals resulting from the division: the one containing the vertex $ A $, the middle one and the one containing the vertex $ C $ is equal to $ \frac{S}{3} $.