Olympiad problems

Novosibirsk Oral Geo Oly VIII, 2023.2

The rectangle is cut into $10$ squares as shown in the figure on the right. Find its sides if the side of the smallest square is $3$.[img]https://cdn.artofproblemsolving.com/attachments/e/5/1fe3a0e41b2d3182338a557d3d44ff5ef9385d.png[/img]

2015 Dutch IMO TST, 2

Tags: Polynomials , algebra , polynomial

Determine all polynomials P(x) with real coefficients such that [(x + 1)P(x − 1) − (x − 1)P(x)] is a constant polynomial.

2021 China Girls Math Olympiad, 1

Tags: inequalities , algebra , China , CGMO

Let $n \in \mathbb{N}^+,$ $x_1,x_2,...,x_{n+1},p,q\in \mathbb{R}^+ $ , $p<q$ and $x^p_{n+1}>\sum_{i=1}^{n}x^p_{i}.$ Prove that $(1)x^q_{n+1}>\sum_{i=1}^{n}x^q_{i};$ $(2)\left(x^p_{n+1}-\sum_{i=1}^{n}x^p_{i}\right)^{\frac{1}{p}}<\left(x^q_{n+1}-\sum_{i=1}^{n}x^q_{i}\right)^{\frac{1}{q}}.$

1977 Spain Mathematical Olympiad, 1

Tags: determinant , linear algebra

Given the determinant of order $n$ $$\begin{vmatrix} 8 & 3 & 3 & \dots & 3 \\ 3 & 8 & 3 & \dots & 3 \\ 3 & 3 & 8 & \dots & 3 \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ 3 & 3 & 3 & \dots & 8 \end{vmatrix}$$ Calculate its value and determine for which values of $n$ this value is a multiple of $10$.

2011 Today's Calculation Of Integral, 691

Tags: calculus , integration , logarithms , geometry , geometric transformation , rotation , calculus computations

Let $a$ be a constant. In the $xy$ palne, the curve $C_1:y=\frac{\ln x}{x}$ touches $C_2:y=ax^2$. Find the volume of the solid generated by a rotation of the part enclosed by $C_1,\ C_2$ and the $x$ axis about the $x$ axis. [i]2011 Yokohama National Universty entrance exam/Engineering[/i]

1997 IMO, 5

Tags: number theory , equation , IMO , IMO 1997 , prime factorization

Find all pairs $ (a,b)$ of positive integers that satisfy the equation: $ a^{b^2} \equal{} b^a$.

2009 Kosovo National Mathematical Olympiad, 3

Tags: algebra

Let $n\geq2$ be an integer. $n$ is a prime if it is only divisible by $1$ and $n$. Prove that there are infinitely many prime numbers.

1955 Moscow Mathematical Olympiad, 292

Tags: number theory , Divide

Let $a, b, n$ be positive integers, $b < 10$ and $2^n = 10a + b$. Prove that if $n > 3$, then $6$ divides $ab$.

2022 Assam Mathematical Olympiad, 18

Tags:

Let $f : \mathbb{N} \longrightarrow \mathbb{N}$ be a function such that (a) $ f(m) < f(n)$ whenever $m < n$. (b) $f(2n) = f(n) + n$ for all $n \in \mathbb{N}$. (c) $n$ is prime whenever $f(n)$ is prime. Find $$\sum_{n=1}^{2022} f(n).$$

2014 Romania National Olympiad, 1

Tags: number theory

Find x, y, z $\in Z$\\$x^2+y^2+z^2=2^n(x+y+z)$\\$n\in N$

2007 Princeton University Math Competition, 7

Tags:

In a $7 \times 7$ square table, some of the squares are colored black and the others white, such that each white square is adjacent (along an edge) to an edge of the table or to a black square. Find the minimum number of black squares on the table.

2012 Mediterranean Mathematics Olympiad, 4

Tags: geometry , circumcircle , geometry proposed

Let $O$ be the circumcenter,$R$ be the circumradius, and $k$ be the circumcircle of a triangle $ABC$ . Let $k_1$ be a circle tangent to the rays $AB$ and $AC$, and also internally tangent to $k$. Let $k_2$ be a circle tangent to the rays $AB$ and $AC$ , and also externally tangent to $k$. Let $A_1$ and $A_2$ denote the respective centers of $k_1$ and $k_2$. Prove that: $(OA_1+OA_2)^2-A_1A_2^2 = 4R^2.$

2014 AMC 12/AHSME, 25

Tags: conics , parabola , calculus , derivative , vector , analytic geometry , parameterization

The parabola $P$ has focus $(0,0)$ and goes through the points $(4,3)$ and $(-4,-3)$. For how many points $(x,y)\in P$ with integer coefficients is it true that $|4x+3y|\leq 1000$? $\textbf{(A) }38\qquad \textbf{(B) }40\qquad \textbf{(C) }42\qquad \textbf{(D) }44\qquad \textbf{(E) }46\qquad$

1998 Gauss, 8

Tags:

Tuesday’s high temperature was 4°C warmer than that of Monday’s. Wednesday’s high temperature was 6°C cooler than that of Monday’s. If Tuesday’s high temperature was 22°C, what was Wednesday’s high temperature? $\textbf{(A)}\ 20 \qquad \textbf{(B)}\ 24 \qquad \textbf{(C)}\ 12 \qquad \textbf{(D)}\ 32 \qquad \textbf{(E)}\ 16$

2014 NIMO Summer Contest, 15

Tags: Diophantine equation

Let $A = (0,0)$, $B=(-1,-1)$, $C=(x,y)$, and $D=(x+1,y)$, where $x > y$ are positive integers. Suppose points $A$, $B$, $C$, $D$ lie on a circle with radius $r$. Denote by $r_1$ and $r_2$ the smallest and second smallest possible values of $r$. Compute $r_1^2 + r_2^2$. [i]Proposed by Lewis Chen[/i]

2020 AMC 12/AHSME, 4

Tags: AMC , AMC 10 , prime numbers , 2020 AMC , 2020 AMC 10B , 2020 AMC 12B , AMC 12

The acute angles of a right triangle are $a^{\circ}$ and $b^{\circ}$, where $a>b$ and both $a$ and $b$ are prime numbers. What is the least possible value of $b$? $\textbf{(A) }2\qquad\textbf{(B) }3\qquad\textbf{(C) }5\qquad\textbf{(D) }7\qquad\textbf{(E) }11$

2022 JHMT HS, 10

Tags: geometry , incenter , circumcircle , 2022

In $\triangle JMT$, $JM=410$, $JT=49$, and $\angle{MJT}>90^\circ$. Let $I$ and $H$ be the incenter and orthocenter of $\triangle JMT$, respectively. The circumcircle of $\triangle JIH$ intersects $\overleftrightarrow{JT}$ at a point $P\neq J$, and $IH=HP$. Find $MT$.

2012-2013 SDML (Middle School), 13

Tags:

Let $a+\frac{1}{b}=8$ and $b+\frac{1}{a}=3$. Given that there are two possible real values for $a$, find their sum. $\text{(A) }\frac{3}{8}\qquad\text{(B) }\frac{8}{3}\qquad\text{(C) }3\qquad\text{(D) }5\qquad\text{(E) }8$

2005 Alexandru Myller, 1

Tags: function , integration , real analysis , real analysis solved

Let $f:[a,b]\to\mathbb R$ be a continous function with the property that there exists a constant $\lambda\in\mathbb R$ so that for every $x\in[a,b]$ there exists a $y\in[a,b]-\{x\}$ s.t. $\int_x^yf(x)dx=\lambda$. Prove that the function $f$ has at least two zeros in $(a,b)$. [i]Eugen Paltanea[/i]

2014 Singapore Senior Math Olympiad, 35

Tags: geometry , power of a point , radical axis

Two circles intersect at the points $C$ and $D$. The straight lines $CD$ and $BYXA$ intersect at the point $Z$. Moreever, the straight line $WB$ is tangent to both of the circles. Suppose $ZX=ZY$ and $AB\cdot AX=100$. Find the value of $BW$.

2003 CHKMO, 4

Tags: modular arithmetic , quadratics , number theory unsolved , number theory

Let $p$ be a prime number such that $p\equiv 1\pmod{4}$. Determine $\sum_{k=1}^{\frac{p-1}{2}}\left \lbrace \frac{k^2}{p} \right \rbrace$, where $\{x\}=x-[x]$.

2003 All-Russian Olympiad, 3

Tags: combinatorics unsolved , combinatorics

There are $100$ cities in a country, some of them being joined by roads. Any four cities are connected to each other by at least two roads. Assume that there is no path passing through every city exactly once. Prove that there are two cities such that every other city is connected to at least one of them.

1952 AMC 12/AHSME, 9

Tags:

If $ m \equal{} \frac {cab}{a \minus{} b}$, then $ b$ equals: $ \textbf{(A)}\ \frac {m(a \minus{} b)}{ca} \qquad\textbf{(B)}\ \frac {cab \minus{} ma}{ \minus{} m} \qquad\textbf{(C)}\ \frac {1}{1 \plus{} c} \qquad\textbf{(D)}\ \frac {ma}{m \plus{} ca}$ $ \textbf{(E)}\ \frac {m \plus{} ca}{ma}$

2009 F = Ma, 25

Tags: 2009 , problem 25

Two discs are mounted on thin, lightweight rods oriented through their centers and normal to the discs. These axles are constrained to be vertical at all times, and the discs can pivot frictionlessly on the rods. The discs have identical thickness and are made of the same material, but have differing radii $r_\text{1}$ and $r_\text{2}$. The discs are given angular velocities of magnitudes $\omega_\text{1}$ and $\omega_\text{2}$, respectively, and brought into contact at their edges. After the discs interact via friction it is found that both discs come exactly to a halt. Which of the following must hold? Ignore effects associated with the vertical rods. [asy] //Code by riben, Improved by CalTech_2023 // Solids import solids; //bigger cylinder draw(shift(0,0,-1)*scale(0.1,0.1,0.59)*unitcylinder,surfacepen=white,black); draw(shift(0,0,-0.1)*unitdisk, surfacepen=black); draw(unitdisk, surfacepen=white,black); draw(scale(0.1,0.1,1)*unitcylinder,surfacepen=white,black); //smaller cylinder draw(rotate(5,X)*shift(-2,3.2,-1)*scale(0.1,0.1,0.6)*unitcylinder,surfacepen=white,black); draw(rotate(4,X)*scale(0.5,0.5,1)*shift(1,8,0.55)*unitdisk, surfacepen=black); draw(rotate(4,X)*scale(0.5,0.5,1)*shift(1,8,0.6)*unitdisk, surfacepen=white,black); draw(rotate(5,X)*shift(-2,3.2,-0.2)*scale(0.1,0.1,1)*unitcylinder,surfacepen=white,black); // Lines draw((0,-2)--(1,-2),Arrows(size=5)); draw((4,-2)--(4.7,-2),Arrows(size=5)); // Labels label("r1",(0.5,-2),S); label("r2",(4.35,-2),S); // Curved Lines path A=(-0.694, 0.897)-- (-0.711, 0.890)-- (-0.742, 0.886)-- (-0.764, 0.882)-- (-0.790, 0.873)-- (-0.815, 0.869)-- (-0.849, 0.867)-- (-0.852, 0.851)-- (-0.884, 0.844)-- (-0.895, 0.837)-- (-0.904, 0.824)-- (-0.879, 0.800)-- (-0.841, 0.784)-- (-0.805, 0.772)-- (-0.762, 0.762)-- (-0.720, 0.747)-- (-0.671, 0.737)-- (-0.626, 0.728)-- (-0.591, 0.720)-- (-0.556, 0.715)-- (-0.504, 0.705)-- (-0.464, 0.700)-- (-0.433, 0.688)-- (-0.407, 0.683)-- (-0.371, 0.685)-- (-0.316, 0.673)-- (-0.271, 0.672)-- (-0.234, 0.667)-- (-0.192, 0.664)-- (-0.156, 0.663)-- (-0.114, 0.663)-- (-0.070, 0.660)-- (-0.033, 0.662)-- (0.000, 0.663)-- (0.036, 0.663)-- (0.067, 0.665)-- (0.095, 0.667)-- (0.125, 0.666)-- (0.150, 0.673)-- (0.187, 0.675)-- (0.223, 0.676)-- (0.245, 0.681)-- (0.274, 0.687)-- (0.300, 0.696)-- (0.327, 0.707)-- (0.357, 0.709)-- (0.381, 0.718)-- (0.408, 0.731)-- (0.443, 0.740)-- (0.455, 0.754)-- (0.458, 0.765)-- (0.453, 0.781)-- (0.438, 0.795)-- (0.411, 0.809)-- (0.383, 0.817)-- (0.344, 0.829)-- (0.292, 0.839)-- (0.254, 0.846)-- (0.216, 0.851)-- (0.182, 0.857)-- (0.153, 0.862)-- (0.124, 0.867); draw(shift(0.2,0)*A,EndArrow(size=5)); path B=(2.804, 0.844)-- (2.790, 0.838)-- (2.775, 0.838)-- (2.758, 0.831)-- (2.740, 0.831)-- (2.709, 0.827)-- (2.688, 0.825)-- (2.680, 0.818)-- (2.660, 0.810)-- (2.639, 0.810)-- (2.628, 0.803)-- (2.618, 0.799)-- (2.604, 0.790)-- (2.598, 0.778)-- (2.596, 0.769)-- (2.606, 0.757)-- (2.630, 0.748)-- (2.666, 0.733)-- (2.696, 0.721)-- (2.744, 0.707)-- (2.773, 0.702)-- (2.808, 0.697)-- (2.841, 0.683)-- (2.867, 0.680)-- (2.912, 0.668)-- (2.945, 0.665)-- (2.973, 0.655)-- (3.010, 0.648)-- (3.040, 0.647)-- (3.069, 0.642)-- (3.102, 0.640)-- (3.136, 0.632)-- (3.168, 0.629)-- (3.189, 0.627)-- (3.232, 0.619)-- (3.254, 0.624)-- (3.281, 0.621)-- (3.328, 0.618)-- (3.355, 0.618)-- (3.397, 0.617)-- (3.442, 0.616)-- (3.468, 0.611)-- (3.528, 0.611)-- (3.575, 0.617)-- (3.611, 0.619)-- (3.634, 0.625)-- (3.666, 0.622)-- (3.706, 0.626)-- (3.742, 0.635)-- (3.772, 0.635)-- (3.794, 0.641)-- (3.813, 0.646)-- (3.837, 0.654)-- (3.868, 0.659)-- (3.886, 0.672)-- (3.903, 0.681)-- (3.917, 0.688)-- (3.931, 0.697)-- (3.943, 0.711)-- (3.951, 0.720)-- (3.948, 0.731)-- (3.924, 0.745)-- (3.900, 0.757)-- (3.874, 0.774)-- (3.851, 0.779)-- (3.821, 0.779)-- (3.786, 0.786)-- (3.754, 0.792)-- (3.726, 0.797)-- (3.677, 0.806)-- (3.642, 0.812); draw(shift(0.7,0)*B,EndArrow(size=5)); [/asy] (A) $\omega_\text{1}^2r_\text{1}=\omega_\text{2}^2r_\text{2}$ (B) $\omega_\text{1}r_\text{1}=\omega_\text{2}r_\text{2}$ (C) $\omega_\text{1}r_\text{1}^2=\omega_\text{2}r_\text{2}^2$ (D) $\omega_\text{1}r_\text{1}^3=\omega_\text{2}r_\text{2}^3$ (E) $\omega_\text{1}r_\text{1}^4=\omega_\text{2}r_\text{2}^4$

2019 Iran Team Selection Test, 6

Tags: number theory

$\{a_{n}\}_{n\geq 0}$ and $\{b_{n}\}_{n\geq 0}$ are two sequences of positive integers that $a_{i},b_{i}\in \{0,1,2,\cdots,9\}$. There is an integer number $M$ such that $a_{n},b_{n}\neq 0$ for all $n\geq M$ and for each $n\geq 0$ $$(\overline{a_{n}\cdots a_{1}a_{0}})^{2}+999 \mid(\overline{b_{n}\cdots b_{1}b_{0}})^{2}+999 $$ prove that $a_{n}=b_{n}$ for $n\geq 0$.\\ (Note that $(\overline{x_nx_{n-1}\dots x_0}) = 10^n\times x_n + \dots + 10\times x_1 + x_0$.) [i]Proposed by Yahya Motevassel[/i]