Olympiad problems

2018 Germany Team Selection Test, 1

A rectangle $\mathcal{R}$ with odd integer side lengths is divided into small rectangles with integer side lengths. Prove that there is at least one among the small rectangles whose distances from the four sides of $\mathcal{R}$ are either all odd or all even. [i]Proposed by Jeck Lim, Singapore[/i]

2009 India IMO Training Camp, 1

Tags: geometry , trigonometry , geometric transformation , homothety , ratio , circumcircle , incenter

Let $ ABC$ be a triangle with $ \angle A = 60^{\circ}$.Prove that if $ T$ is point of contact of Incircle And Nine-Point Circle, Then $ AT = r$, $ r$ being inradius.

2016 CCA Math Bonanza, L3.1

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How many 3-digit positive integers have the property that the sum of their digits is greater than the product of their digits? [i]2016 CCA Math Bonanza Lightning #3.1[/i]

2025 PErA, P2

Tags: number theory

Let $m$ be a positive integer. We say that a positive integer $x$ is $m$-good if $a^m$ divides $x$ for some integer $a > 1$. We say a positive integer $x$ is $m$-bad if it is not $m$-good. (a) Is it true that for every positive integer $n$ there exist $n$ consecutive $m$-bad positive integers? (b) Is it true that for every positive integer $n$ there exist $n$ consecutive $m$-good positive integers?

2019 USAMTS Problems, 4

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Let $FIG$ be a triangle and let $D$ be a point on $\overline{FG}$. The line perpendicular to $\overline{FI}$ passing through the midpoint of $\overline{FD}$ and the line perpendicular to $\overline{IG}$ passing through the midpoint of $\overline{DG}$ intersect at $T$. Prove that $FT = GT$ if and only if $\overline{ID}$ is perpendicular to $\overline{FG}$.

1982 Tournament Of Towns, (020) 1

Tags: inequalities , algebra

(a) Prove that for any positive numbers $x_1,x_2,...,x_k$ ($k > 3$), $$\frac{x_1}{x_k+x_2}+ \frac{x_2}{x_1+x_3}+...+\frac{x_k}{x_{k-1}+x_1}\ge 2$$ (b) Prove that for every $k$ this inequality cannot be sharpened, i.e. prove that for every given $k$ it is not possible to change the number $2$ in the right hand side to a greater number in such a way that the inequality remains true for every choice of positive numbers $x_1,x_2,...,x_k$. (A Prokopiev)

2021 AMC 12/AHSME Spring, 20

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Suppose that on a parabola with vertex $V$ and a focus $F$ there exists a point $A$ such that $AF=20$ and $AV=21$. What is the sum of all possible values of the length $FV?$ $\textbf{(A) }13 \qquad \textbf{(B) }\frac{40}3 \qquad \textbf{(C) }\frac{41}3 \qquad \textbf{(D) }14\qquad \textbf{(E) }\frac{43}3$ Proposed by [b]djmathman[/b]

1992 Denmark MO - Mohr Contest, 3

Tags: algebra , inequalities

Let $x$ and $y$ be positive numbers with $x +y=1$. Show that $$\left(1+\frac{1}{x}\right)\left(1+\frac{1}{y}\right) \ge 9.$$

2025 Romania EGMO TST, P2

Tags: combinatorics

Prove that any finite set $H$ of lattice points on the plane has a subset $K$ with the following properties: [list] [*]any vertical or horizontal line in the plane cuts $K$ in at most $2$ points, [*]any point of $H\setminus K$ is contained by a segment with endpoints from $K$.[/list]

1975 Putnam, B4

Tags: topology , general topology

Does a circle have a subset which is topologically closed and which contains exactly one point of each pair of diametrically opposite points?

2025 Romania National Olympiad, 2

Tags: geometry , circumcircle , vector geometry

Let $\triangle ABC$ be an acute-angled triangle, with circumcenter $O$, circumradius $R$ and orthocenter $H$. Let $A_1$ be a point on $BC$ such that $A_1H+A_1O=R$. Define $B_1$ and $C_1$ similarly. If $\overrightarrow{AA_1} + \overrightarrow{BB_1} + \overrightarrow{CC_1} = \overrightarrow{0}$, prove that $\triangle ABC$ is equilateral.

2010 Mexico National Olympiad, 3

Tags: geometry

Let $\mathcal{C}_1$ and $\mathcal{C}_2$ be externally tangent at a point $A$. A line tangent to $\mathcal{C}_1$ at $B$ intersects $\mathcal{C}_2$ at $C$ and $D$; then the segment $AB$ is extended to intersect $\mathcal{C}_2$ at a point $E$. Let $F$ be the midpoint of $\overarc{CD}$ that does not contain $E$, and let $H$ be the intersection of $BF$ with $\mathcal{C}_2$. Show that $CD$, $AF$, and $EH$ are concurrent.

2021 Thailand TST, 3

Tags: geometry , circumcircle

Let $P$ be a point on the circumcircle of acute triangle $ABC$. Let $D,E,F$ be the reflections of $P$ in the $A$-midline, $B$-midline, and $C$-midline. Let $\omega$ be the circumcircle of the triangle formed by the perpendicular bisectors of $AD, BE, CF$. Show that the circumcircles of $\triangle ADP, \triangle BEP, \triangle CFP,$ and $\omega$ share a common point.

2006 Federal Competition For Advanced Students, Part 2, 3

Tags: system of equations , algebra

Let $ A$ be an integer not equal to $ 0$. Solve the following system of equations in $ \mathbb{Z}^3$. $ x \plus{} y^2 \plus{} z^3 \equal{} A$ $ \frac {1}{x} \plus{} \frac {1}{y^2} \plus{} \frac {1}{z^3} \equal{} \frac {1}{A}$ $ xy^2z^3 \equal{} A^2$

1985 IMO Longlists, 83

Tags: trigonometry , ratio , similar triangles , geometric sequence , geometry

Let $\Gamma_i, i = 0, 1, 2, \dots$ , be a circle of radius $r_i$ inscribed in an angle of measure $2\alpha$ such that each $\Gamma_i$ is externally tangent to $\Gamma_{i+1}$ and $r_{i+1} < r_i$. Show that the sum of the areas of the circles $\Gamma_i$ is equal to the area of a circle of radius $r =\frac 12 r_0 (\sqrt{ \sin \alpha} + \sqrt{\text{csc} \alpha}).$

2007 Moldova National Olympiad, 12.4

Tags: calculus , integration , inequalities , function , calculus computations

If the function $f\colon [1,2]\to R$ is such that $\int_{1}^{2}f(x) dx=\frac{73}{24}$, then show that there exists a $x_{0}\in (1;2)$ such that \[x_{0}^{2}<f(x_{0})<x_{0}^{3}\] [Edit: $f$ is continuous]

Swiss NMO - geometry, 2005.8

Tags: geometry , orthocenter , collinear , circles

Let $ABC$ be an acute-angled triangle. $M ,N$ are any two points on the sides $AB , AC$ respectively. The circles with the diameters $BN$ and $CM$ intersect at points $P$ and $Q$. Show that the points $P, Q$ and the orthocenter of the triangle $ABC$ lie on a straight line.

2018 AMC 10, 8

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Joe has a collection of 23 coins, consisting of 5-cent coins, 10-cent coins, and 25-cent coins. He has 3 more 10-cent coins than 5-cent coins, and the total value of his collection is 320 cents. How many more 25-cent coins does Joe have than 5-cent coins? $\textbf{(A) } 0 \qquad \textbf{(B) } 1 \qquad \textbf{(C) } 2 \qquad \textbf{(D) } 3 \qquad \textbf{(E) } 4 $

2013 National Olympiad First Round, 35

Tags: floor function , algebra

What is the least positive integer $n$ such that $\overbrace{f(f(\dots f}^{21 \text{ times}}(n)))=2013$ where $f(x)=x+1+\lfloor \sqrt x \rfloor$? ($\lfloor a \rfloor$ denotes the greatest integer not exceeding the real number $a$.) $ \textbf{(A)}\ 1214 \qquad\textbf{(B)}\ 1202 \qquad\textbf{(C)}\ 1186 \qquad\textbf{(D)}\ 1178 \qquad\textbf{(E)}\ \text{None of above} $

2001 Cuba MO, 9

Tags: geometry , area

In triangle $ABC$, right at $C$, let $F$ be the intersection point of the altitude $CD$ with the angle bisector $AE$ and $G$ be the intersection point of $ED$ with $BF$. Prove that the area of the quadrilateral $CEGF$ is equal to the area of the triangle $BDG$ .

2012 India Regional Mathematical Olympiad, 2

Tags: algebra , polynomial , vieta , inequalities , binomial theorem , sum of roots

Let $P(x)=x^n+a_{n-1}x^{n-1}+\cdots+a_0$ be a polynomial of degree $n\geq 3.$ Knowing that $a_{n-1}=-\binom{n}{1}$ and $a_{n-2}=\binom{n}{2},$ and that all the roots of $P$ are real, find the remaining coefficients. Note that $\binom{n}{r}=\frac{n!}{(n-r)!r!}.$

2010 Indonesia TST, 2

Tags: geometric transformation , projective geometry , power of a point , radical axis , geometry

Circles $ \Gamma_1$ and $ \Gamma_2$ are internally tangent to circle $ \Gamma$ at $ P$ and $ Q$, respectively. Let $ P_1$ and $ Q_1$ are on $ \Gamma_1$ and $ \Gamma_2$ respectively such that $ P_1Q_1$ is the common tangent of $ P_1$ and $ Q_1$. Assume that $ \Gamma_1$ and $ \Gamma_2$ intersect at $ R$ and $ R_1$. Define $ O_1,O_2,O_3$ as the intersection of $ PQ$ and $ P_1Q_1$, the intersection of $ PR$ and $ P_1R_1$, and the intersection $ QR$ and $ Q_1R_1$. Prove that the points $ O_1,O_2,O_3$ are collinear. [i]Rudi Adha Prihandoko, Bandung[/i]

2020 USOJMO, 5

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Suppose that $(a_1,b_1),$ $(a_2,b_2),$ $\dots,$ $(a_{100},b_{100})$ are distinct ordered pairs of nonnegative integers. Let $N$ denote the number of pairs of integers $(i,j)$ satisfying $1\leq i<j\leq 100$ and $|a_ib_j-a_jb_i|=1$. Determine the largest possible value of $N$ over all possible choices of the $100$ ordered pairs. [i]Proposed by Ankan Bhattacharya[/i]

2017 CMIMC Geometry, 7

Tags: geometry

Two non-intersecting circles, $\omega$ and $\Omega$, have centers $C_\omega$ and $C_\Omega$ respectively. It is given that the radius of $\Omega$ is strictly larger than the radius of $\omega$. The two common external tangents of $\Omega$ and $\omega$ intersect at a point $P$, and an internal tangent of the two circles intersects the common external tangents at $X$ and $Y$. Suppose that the radius of $\omega$ is $4$, the circumradius of $\triangle PXY$ is $9$, and $XY$ bisects $\overline{PC_\Omega}$. Compute $XY$.

1986 Austrian-Polish Competition, 8

Tags: combinatorics , table

Pairwise distinct real numbers are arranged into an $m \times n$ rectangular array. In each row the entries are arranged increasingly from left to right. Each column is then rearranged in decreasing order from top to bottom. Prove that in the reorganized array, the rows remain arranged increasingly.