Olympiad problems

2009 Today's Calculation Of Integral, 505

In the $ xyz$ space with the origin $ O$, given a cuboid $ K: |x|\leq \sqrt {3},\ |y|\leq \sqrt {3},\ 0\leq z\leq 2$ and the plane $ \alpha : z \equal{} 2$. Draw the perpendicular $ PH$ from $ P$ to the plane. Find the volume of the solid formed by all points of $ P$ which are included in $ K$ such that $ \overline{OP}\leq \overline{PH}$.

1990 Mexico National Olympiad, 6

Tags: similar triangles , right triangle , geometry

$ABC$ is a triangle with $\angle C = 90^o$. $E$ is a point on $AC$, and $F$ is the midpoint of $EC$. $CH$ is an altitude. $I$ is the circumcenter of $AHE$, and $G$ is the midpoint of $BC$. Show that $ABC$ and $IGF$ are similar.

2010 F = Ma, 4

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Two teams of movers are lowering a piano from the window of a $10$ floor apartment building. The rope breaks when the piano is $30$ meters above the ground. The movers on the ground, alerted by the shouts of the movers above, first notice the piano when it is $14$ meters above the ground. How long do they have to get out of the way before the piano hits the ground? (A) $\text{0.66 sec}$ (B) $\text{0.78 sec}$ (C) $\text{1.67 sec}$ (D) $\text{1.79 sec}$ (E) $\text{2.45 sec}$

2002 Czech-Polish-Slovak Match, 1

Tags: inequalities , ceiling function , floor function , algebra

Let $a, b$ be distinct real numbers and $k,m$ be positive integers $k + m = n \ge 3, k \le 2m, m \le 2k$. Consider sequences $x_1,\dots , x_n$ with the following properties: (i) $k$ terms $x_i$, including $x_1$, are equal to $a$; (ii) $m$ terms $x_i$, including $x_n$, are equal to $b$; (iii) no three consecutive terms are equal. Find all possible values of $x_nx_1x_2 + x_1x_2x_3 + \cdots + x_{n-1}x_nx_1$.

2019 CCA Math Bonanza, T2

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A triangle has side lengths of $x,75,100$ where $x<75$ and altitudes of lengths $y,28,60$ where $y<28$. What is the value of $x+y$? [i]2019 CCA Math Bonanza Team Round #2[/i]

1955 AMC 12/AHSME, 21

Tags: geometry , area of a triangle

Represent the hypotenuse of a right triangle by $ c$ and the area by $ A$. The atltidue on the hypotenuse is: $ \textbf{(A)}\ \frac{A}{c} \qquad \textbf{(B)}\ \frac{2A}{c} \qquad \textbf{(C)}\ \frac{A}{2c} \qquad \textbf{(D)}\ \frac{A^2}{c} \qquad \textbf{(E)}\ \frac{A}{c^2}$

2003 IMAR Test, 1

Tags: pentagon , covering , disc , regular , geometry

Prove that the interior of a convex pentagon whose sides are all equal, is not covered by the open disks having the sides of the pentagon as diameter.

2011 N.N. Mihăileanu Individual, 4

Tags: geometry , incenter , inradius , pure geometry , vector geometry

Consider a triangle $ ABC $ having incenter $ I $ and inradius $ r. $ Let $ D $ be the tangency of $ ABC $ 's incircle with $ BC, $ and $ E $ on the line $ BC $ such that $ AE $ is perpendicular to $ BC, $ and $ M\neq E $ on the segment $ AE $ such that $ AM=r. $ [b]a)[/b] Give an idenity for $ \frac{BD}{DC} $ involving only the lengths of the sides of the triangle. [b]b)[/b] Prove that $ AB \cdot \overrightarrow{IC} +BC\cdot \overrightarrow{IA} +CA\cdot \overrightarrow{IB} =0. $ [b]c)[/b] Show that $ MI $ passes through the middle of the side $ BC. $ [i]Cătălin Zârnă[/i]

2015 Dutch BxMO/EGMO TST, 4

Tags: geometry , circumcircle , equal angles , angle chasing , angle bisector

In a triangle $ABC$ the point $D$ is the intersection of the interior angle bisector of $\angle BAC$ and side $BC$. Let $P$ be the second intersection point of the exterior angle bisector of $\angle BAC$ with the circumcircle of $\angle ABC$. A circle through $A$ and $P$ intersects line segment $BP$ internally in $E$ and line segment $CP$ internally in $F$. Prove that $\angle DEP = \angle DFP$.

2020 German National Olympiad, 3

Tags: polynomial , root , algebra , inequalities

Show that the equation \[x(x+1)(x+2)\dots (x+2020)-1=0\] has exactly one positive solution $x_0$, and prove that this solution $x_0$ satisfies \[\frac{1}{2020!+10}<x_0<\frac{1}{2020!+6}.\]

2019 USEMO, 2

Tags: function , polynomial , algebra

Let $\mathbb{Z}[x]$ denote the set of single-variable polynomials in $x$ with integer coefficients. Find all functions $\theta : \mathbb{Z}[x] \to \mathbb{Z}[x]$ (i.e. functions taking polynomials to polynomials) such that [list] [*] for any polynomials $p, q \in \mathbb{Z}[x]$, $\theta(p + q) = \theta(p) + \theta(q)$; [*] for any polynomial $p \in \mathbb{Z}[x]$, $p$ has an integer root if and only if $\theta(p)$ does. [/list] [i]Carl Schildkraut[/i]

1971 Bulgaria National Olympiad, Problem 6

Tags: geometry , 3d geometry , pyramid , inequalities , geometric inequalities

In a triangular pyramid $SABC$ one of the plane angles with vertex $S$ is a right angle and the orthogonal projection of $S$ on the base plane $ABC$ coincides with the orthocenter of the triangle $ABC$. Let $SA=m$, $SB=n$, $SC=p$, $r$ is the inradius of $ABC$. $H$ is the height of the pyramid and $r_1,r_2,r_3$ are radii of the incircles of the intersections of the pyramid with the plane passing through $SA,SB,SC$ and the height of the pyramid. Prove that (a) $m^2+n^2+p^2\ge18r^2$; (b) $\frac{r_1}H,\frac{r_2}H,\frac{r_3}H$ are in the range $(0.4,0.5)$.

2010 AIME Problems, 13

Tags: geometry , rectangle , trapezoid , trigonometry , analytic geometry , graphing lines , ratio

Rectangle $ ABCD$ and a semicircle with diameter $ AB$ are coplanar and have nonoverlapping interiors. Let $ \mathcal{R}$ denote the region enclosed by the semicircle and the rectangle. Line $ \ell$ meets the semicircle, segment $ AB$, and segment $ CD$ at distinct points $ N$, $ U$, and $ T$, respectively. Line $ \ell$ divides region $ \mathcal{R}$ into two regions with areas in the ratio $ 1: 2$. Suppose that $ AU \equal{} 84$, $ AN \equal{} 126$, and $ UB \equal{} 168$. Then $ DA$ can be represented as $ m\sqrt {n}$, where $ m$ and $ n$ are positive integers and $ n$ is not divisible by the square of any prime. Find $ m \plus{} n$.

2024 IRN-SGP-TWN Friendly Math Competition, 2

Tags: number theory , divisor function

Let $d(n)$ denote the number of positive divisors of $n$. For any given integer $a \geq 3$, define a sequence $\{a_i\}_{i=0}^\infty$ satisfying [list] [*] $a_{0}=a$, and [*] $a_{n+1}=a_{n}+(-1)^{n} d(a_{n})$ for each integer $n \geq 0$. [/list] For example, if $a=275$, the sequence would be \[275, \overline{281,279,285,277,279,273}.\] Prove that for each positive integer $k$ there exists a positive integer $N$ such that if such a sequence has period $2k$ and all terms of the sequence are greater than $N$ then all terms of the sequence have the same parity. [i]Proposed by Navid[/i]

1992 Swedish Mathematical Competition, 5

Tags: right triangle , right angle , geometry , circumradius

A triangle has sides $a, b, c$ with longest side $c$, and circumradius $R$. Show that if $a^2 + b^2 = 2cR$, then the triangle is right-angled.

1988 All Soviet Union Mathematical Olympiad, 480

Tags: algebra , inequalities , minimum

Find the minimum value of $\frac{xy}{z} + \frac{yz}{x} +\frac{ zx}{y}$ for positive reals $x, y, z$ with $x^2 + y^2 + z^2 = 1$.

1989 China Team Selection Test, 3

Tags: gauss , calculus , derivative , algebra , polynomial , combinatorial geometry

Find the greatest $n$ such that $(z+1)^n = z^n + 1$ has all its non-zero roots in the unitary circumference, e.g. $(\alpha+1)^n = \alpha^n + 1, \alpha \neq 0$ implies $|\alpha| = 1.$

2007 JBMO Shortlist, 5

Tags: quadratic , modular arithmetic , number theory

Prove that if $ p$ is a prime number, then $ 7p+3^{p}-4$ is not a perfect square.

2013 Chile TST Ibero, 2

Tags: number theory

Let $a \in \mathbb{N}$ such that $a + n^2$ can be expressed as the sum of two squares for all $n \in \mathbb{N}$. Prove that $a$ is the square of a natural number.

PEN I Problems, 15

Tags: function , floor function

Find the total number of different integer values the function \[f(x) = \lfloor x\rfloor+\lfloor 2x\rfloor+\left\lfloor \frac{5x}{3}\right\rfloor+\lfloor 3x\rfloor+\lfloor 4x\rfloor\] takes for real numbers $x$ with $0 \leq x \leq 100$.

1996 Hungary-Israel Binational, 4

Tags: algebra

$ a_1, a_2, \cdots, a_n$ is a sequence of real numbers, and $ b_1, b_2, \cdots, b_n$ are real numbers that satisfy the condition $ 1 \ge b_1 \ge b_2 \ge \cdots \ge b_n \ge 0$. Prove that there exists a natural number $ k \le n$ that satisifes $ |a_1b_1 \plus{} a_2b_2 \plus{} \cdots \plus{} a_nb_n| \le |a_1 \plus{} a_2 \plus{} \cdots \plus{} a_k|$

2020 Jozsef Wildt International Math Competition, W21

Tags: limit , real analysis

Evaluate $$\lim_{n\to\infty}\left(\frac{1+\frac13+\ldots+\frac1{2n+1}}{\ln\sqrt n}\right)^{\ln\sqrt n}$$ [i]Proposed by Ángel Plaza[/i]

2014 China Team Selection Test, 6

Tags: induction , number theory

For positive integer $k>1$, let $f(k)$ be the number of ways of factoring $k$ into product of positive integers greater than $1$ (The order of factors are not countered, for example $f(12)=4$, as $12$ can be factored in these $4$ ways: $12,2\cdot 6,3\cdot 4, 2\cdot 2\cdot 3$. Prove: If $n$ is a positive integer greater than $1$, $p$ is a prime factor of $n$, then $f(n)\leq \frac{n}{p}$

2021 Romanian Master of Mathematics, 3

Tags: combinatorics

A number of $17$ workers stand in a row. Every contiguous group of at least $2$ workers is a $\textit{brigade}$. The chief wants to assign each brigade a leader (which is a member of the brigade) so that each worker’s number of assignments is divisible by $4$. Prove that the number of such ways to assign the leaders is divisible by $17$. [i]Mikhail Antipov, Russia[/i]

2002 Mediterranean Mathematics Olympiad, 3

Tags: circumcircle , perpendicular bisector , geometry

In an acute-angled triangle $ABC$, $M$ and $N$ are points on the sides $AC$ and $BC$ respectively, and $K$ the midpoint of $MN$. The circumcircles of triangles $ACN$ and $BCM$ meet again at a point $D$. Prove that the line $CD$ contains the circumcenter $O$ of $\triangle ABC$ if and only if $K$ is on the perpendicular bisector of $AB.$